Advances in Mechanical Engineering: Selected Contributions from the Conference “Modern Engineering: Science and Education”, Saint Petersburg, Russia, June 2020 [1st ed.] 9783030620615, 9783030620622

Table of contents :
Front Matter ....Pages i-ix
Organization of Remote Education for Higher Mathematics. Challenges and Solutions (Marina V. Lagunova, Liubov A. Ivanova, Natalja V. Ezhova)....Pages 1-8
Industrial Cyber-Physical Systems Engineering: The Educational Aspect (Oman A. Abyshev, Eugeny I. Yablochnikov, Juho Mäkiö)....Pages 9-19
System Analysis of Cold Axial Rotary Forging of Thin-Walled Tube Blanks (L. B. Aksenov, S. N. Kunkin, N. M. Potapov)....Pages 20-29
Mechanical Challenges of Inspection Robot Moving Along the Electrical Line: Effect of Flexural Rigidity (Mohammad Reza Bahrami)....Pages 30-37
Development and Research of Mechatronic Spring Drives with Energy Recovery for Rod Depth Pumps (Valentina P. Belogur, Victor L. Zhavner, Milana V. Zhavner, Wen Zhao)....Pages 38-51
Solving the Problem of Decomposition of an Orthogonal Polyhedron of Arbitrary Dimension (Vladislav A. Chekanin, Alexander V. Chekanin)....Pages 52-59
Shock Spectra in Non-linear Interactions (Alexander N. Evgrafov, Vladimir I. Karazin, Valery A. Tereshin, Igor O. Khlebosolov)....Pages 60-70
Algorithm of Determining Reactions in Hinge Joints of Linkage Mechanisms with Non-ideal Kinematic Pairs (Alexander N. Evgrafov, Gennady N. Petrov, Sergey A. Evgrafov)....Pages 71-79
Axisymmetric Vibrations of the Cylindrical Shell Loaded with Pointed Masses (George V. Filippenko, Tatiana V. Zinovieva)....Pages 80-91
Gear Shift Analysis in Dual-Clutch Transmissions Using Impact Theory (Georgy K. Korendyasev, Konstantin B. Salamandra)....Pages 92-101
Technological Schemes for Elongated Foramen Internal Surface Finishing by Forced Electrolytic-Plasma Polishing (Mihail Mihailovich Radkevich, Ivan Sergeevich Kuzmichev)....Pages 102-111
Local Buckling of Curvilinear Plates in Axial Compression (Konstantin P. Manzhula, Anastasia A. Valiulina)....Pages 112-121
Interactive Synthesis of Technological Dimensional Schemes (Kirill P. Pompeev, Andrey A. Pleshkov, Viktoriya A. Borbotko)....Pages 122-135
To Calculation of Corones Lens Compensators Bimetallic Pipelines of Hydro Power and Nuclear Power (Vladimir A. Pukhliy, Efim A. Kogan, Sergey T. Miroshnichenko)....Pages 136-149
Thermoelasticity of Flexes Bimetallic Pipelines Atomic Energy (Vladimir A. Pukhliy, Efim A. Kogan, Sergey T. Miroshnichenko)....Pages 150-162
Vibration Active of Machines with Elastic Transmission Mechanism (Yuri A. Semenov, Nadezhda S. Semenova)....Pages 163-172
Method for Calculating and Selection the Optimal Profile of Clearance in Pairs of Cylinder-Piston Group of Piston Engine (Alexander Yu. Shabanov, Anatolii A. Sidorov, Pavel N. Brodnev, Oleg V. Abysov, Victor V. Rumyantsev)....Pages 173-184
Oscillations of Double Mathematical Pendulum with Noncollinear Joints (Alexey S. Smirnov, Boris A. Smolnikov)....Pages 185-193
The Method for Determining the Risks of the Transport of Dangerous Goods (Dmitry R. Stakhin, Dmitry G. Plotnikov)....Pages 194-203
The Method of Exponential Differential Inequality in the Estimation of Solutions of Nonlinear Systems in the Vicinity of the Zero of the State Space (G. I. Melnikov, V. G. Melnikov, N. A. Dudarenko, V. V. Talapov)....Pages 204-211
The Application of the Tubular Sorbing Elements Based on the Composition Powders of Zeolites in Adsorption Cryogeno-Vacuum Pumps (Aleksey V. Gotsiridze, Pavel A. Kuznetsov, Alexandra O. Prostorova, Valeriy P. Tretyakov)....Pages 212-218
Research of Air Suspension of Shock Machine (Mikhail N. Polishchuck, Arkadii N. Popov, Alexey K. Vasiliev, Dmitrii V. Reshetov)....Pages 219-230
Optimization of Parameters of Cyclic Machines When Crossing Resonance Zones (Iosif I. Vulfson)....Pages 231-246
Back Matter ....Pages 247-248

Citation preview

Lecture Notes in Mechanical Engineering

Alexander N. Evgrafov   Editor

Advances in Mechanical Engineering Selected Contributions from the Conference “Modern Engineering: Science and Education”, Saint Petersburg, Russia, June 2020

Lecture Notes in Mechanical Engineering Series Editors Francisco Cavas-Martínez, Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesco Gherardini, Dipartimento di Ingegneria, Università di Modena e Reggio Emilia, Modena, Italy Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Vitalii Ivanov, Department of Manufacturing Engineering Machine and Tools, Sumy State University, Sumy, Ukraine Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Justyna Trojanowska, Poznan University of Technology, Poznan, Poland

Lecture Notes in Mechanical Engineering (LNME) publishes the latest developments in Mechanical Engineering—quickly, informally and with high quality. Original research reported in proceedings and post-proceedings represents the core of LNME. Volumes published in LNME embrace all aspects, subfields and new challenges of mechanical engineering. Topics in the series include: • • • • • • • • • • • • • • • • •

Engineering Design Machinery and Machine Elements Mechanical Structures and Stress Analysis Automotive Engineering Engine Technology Aerospace Technology and Astronautics Nanotechnology and Microengineering Control, Robotics, Mechatronics MEMS Theoretical and Applied Mechanics Dynamical Systems, Control Fluid Mechanics Engineering Thermodynamics, Heat and Mass Transfer Manufacturing Precision Engineering, Instrumentation, Measurement Materials Engineering Tribology and Surface Technology

To submit a proposal or request further information, please contact the Springer Editor of your location: China: Dr. Mengchu Huang at [email protected] India: Priya Vyas at [email protected] Rest of Asia, Australia, New Zealand: Swati Meherishi at [email protected] All other countries: Dr. Leontina Di Cecco at [email protected] To submit a proposal for a monograph, please check our Springer Tracts in Mechanical Engineering at http://www.springer.com/series/11693 or contact [email protected] Indexed by SCOPUS. The books of the series are submitted for indexing to Web of Science.

More information about this series at http://www.springer.com/series/11236

Alexander N. Evgrafov Editor

Advances in Mechanical Engineering Selected Contributions from the Conference “Modern Engineering: Science and Education”, Saint Petersburg, Russia, June 2020

123

Editor Alexander N. Evgrafov Saint Petersburg Polytechnic University Saint Petersburg, Russia

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-3-030-62061-5 ISBN 978-3-030-62062-2 (eBook) https://doi.org/10.1007/978-3-030-62062-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The “Modern Mechanical Engineering: Science and Education” (MMESE) conference was initially organized by the Mechanical Engineering Department of Peter the Great St. Petersburg Polytechnic University in June 2011 in St. Petersburg (Russia). It was envisioned as a forum to bring together scientists, university professors, graduate students, and mechanical engineers, presenting new science, technology, and engineering ideas and achievements. The idea of holding such a forum proved to be highly relevant. Moreover, both location and timing of the conference were quite appealing. Late June is a wonderful and romantic season in St. Petersburg—one of the most beautiful cities, located on the Neva river banks and surrounded by charming greenbelts. The conference attracted many participants, working in various fields of engineering: design, mechanics, materials, etc. The success of the conference inspired the organizers to turn the conference into an annual event. More than 60 papers were presented at the 9th Conference MMESE 2020. They covered topics ranging from mechanics of machines, materials engineering, structural strength to transport technologies, machinery quality, and innovations, in addition to dynamics of machines, walking mechanisms, and computational methods. All presenters contributed greatly to the success of the conference. However, for the purposes of this book, only 23 papers, authored by research groups representing various universities and institutes, were selected for inclusion. I am particularly grateful to the authors for their contributions and all the participating experts for their valuable advice. Furthermore, I thank the staff and management of the university for their cooperation and support, and especially, all members of the program committee and the organizing committee for their work in preparing and organizing the conference. Last but not least, I thank Springer for its professional assistance and particularly Mr. Pierpaolo Riva who supported this publication. Alexander N. Evgrafov

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Contents

Organization of Remote Education for Higher Mathematics. Challenges and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marina V. Lagunova, Liubov A. Ivanova, and Natalja V. Ezhova

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Industrial Cyber-Physical Systems Engineering: The Educational Aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oman A. Abyshev, Eugeny I. Yablochnikov, and Juho Mäkiö

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System Analysis of Cold Axial Rotary Forging of Thin-Walled Tube Blanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. B. Aksenov, S. N. Kunkin, and N. M. Potapov

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Mechanical Challenges of Inspection Robot Moving Along the Electrical Line: Effect of Flexural Rigidity . . . . . . . . . . . . . . . . . . . . Mohammad Reza Bahrami

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Development and Research of Mechatronic Spring Drives with Energy Recovery for Rod Depth Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valentina P. Belogur, Victor L. Zhavner, Milana V. Zhavner, and Wen Zhao Solving the Problem of Decomposition of an Orthogonal Polyhedron of Arbitrary Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladislav A. Chekanin and Alexander V. Chekanin Shock Spectra in Non-linear Interactions . . . . . . . . . . . . . . . . . . . . . . . . Alexander N. Evgrafov, Vladimir I. Karazin, Valery A. Tereshin, and Igor O. Khlebosolov Algorithm of Determining Reactions in Hinge Joints of Linkage Mechanisms with Non-ideal Kinematic Pairs . . . . . . . . . . . . . . . . . . . . . Alexander N. Evgrafov, Gennady N. Petrov, and Sergey A. Evgrafov

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Axisymmetric Vibrations of the Cylindrical Shell Loaded with Pointed Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . George V. Filippenko and Tatiana V. Zinovieva

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Gear Shift Analysis in Dual-Clutch Transmissions Using Impact Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Georgy K. Korendyasev and Konstantin B. Salamandra

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Technological Schemes for Elongated Foramen Internal Surface Finishing by Forced Electrolytic-Plasma Polishing . . . . . . . . . . . . . . . . . 102 Mihail Mihailovich Radkevich and Ivan Sergeevich Kuzmichev Local Buckling of Curvilinear Plates in Axial Compression . . . . . . . . . . 112 Konstantin P. Manzhula and Anastasia A. Valiulina Interactive Synthesis of Technological Dimensional Schemes . . . . . . . . . 122 Kirill P. Pompeev, Andrey A. Pleshkov, and Viktoriya A. Borbotko To Calculation of Corones Lens Compensators Bimetallic Pipelines of Hydro Power and Nuclear Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Vladimir A. Pukhliy, Efim A. Kogan, and Sergey T. Miroshnichenko Thermoelasticity of Flexes Bimetallic Pipelines Atomic Energy . . . . . . . 150 Vladimir A. Pukhliy, Efim A. Kogan, and Sergey T. Miroshnichenko Vibration Active of Machines with Elastic Transmission Mechanism . . . 163 Yuri A. Semenov and Nadezhda S. Semenova Method for Calculating and Selection the Optimal Profile of Clearance in Pairs of Cylinder-Piston Group of Piston Engine . . . . . . . . . . . . . . . . 173 Alexander Yu. Shabanov, Anatolii A. Sidorov, Pavel N. Brodnev, Oleg V. Abysov, and Victor V. Rumyantsev Oscillations of Double Mathematical Pendulum with Noncollinear Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Alexey S. Smirnov and Boris A. Smolnikov The Method for Determining the Risks of the Transport of Dangerous Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Dmitry R. Stakhin and Dmitry G. Plotnikov The Method of Exponential Differential Inequality in the Estimation of Solutions of Nonlinear Systems in the Vicinity of the Zero of the State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 G. I. Melnikov, V. G. Melnikov, N. A. Dudarenko, and V. V. Talapov The Application of the Tubular Sorbing Elements Based on the Composition Powders of Zeolites in Adsorption Cryogeno-Vacuum Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Aleksey V. Gotsiridze, Pavel A. Kuznetsov, Alexandra O. Prostorova, and Valeriy P. Tretyakov

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Research of Air Suspension of Shock Machine . . . . . . . . . . . . . . . . . . . . 219 Mikhail N. Polishchuck, Arkadii N. Popov, Alexey K. Vasiliev, and Dmitrii V. Reshetov Optimization of Parameters of Cyclic Machines When Crossing Resonance Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Iosif I. Vulfson Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Organization of Remote Education for Higher Mathematics. Challenges and Solutions Marina V. Lagunova(B) , Liubov A. Ivanova, and Natalja V. Ezhova Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia [email protected]

Abstract. The paper considers the organization of distance learning in higher mathematics during the period of quarantine measures. It is told about the means already available and their further development. The issue of the possibilities of intermediate certification in the remote mode is discussed, as well as the experience of conducting a colloquium with proctoring elements is highlighted. Interim students’ results on studying higher mathematics of one of the streams are given. The results showed that remote learning, which, of course, cannot completely replace personal communication with lecturers, can and should be used not only during the period of self-isolation, but also in the normal educational process. Keywords: Distance learning · Remote learning · Test · Moodle · Proctoring · Point-rating system

1 Introduction During the period of distance learning, university lecturers faced many problems: • • • • • •

how to gather students; how to track attendance; how to organize the educational process; how to monitor the timely completion of tasks; how to evaluate the work of students; how to take into account the degree of student’s personal participation in mastering the material. This is perhaps the most difficult question - the question of proctoring.

In this paper, we share our experience and some of the results obtained. To begin with, in two streams of mechanical engineering students of the Institute of Metallurgy, Mechanical Engineering and Transport (IMMET) and in one stream of the Institute of Energy (IE), a distance course in higher mathematics has been widely used for a long time, covering all three semesters of the course, which mechanical engineering students study [1–3]. This course contains video lectures, text files of recommended textbooks, questions and tasks for self-testing, as well as control and final tests. A year © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 1–8, 2021. https://doi.org/10.1007/978-3-030-62062-2_1

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ago, two courses were released on the open portal “Open Education”: https://openedu.ru/ course/spbstu/HIMAT/ and https://openedu.ru/course/spbstu/HIMAT2/. They cover the materials of the first and second terms [4–6]. It should be noted that this is so far the only course on the Russian open portal that covers exactly the course of higher mathematics being studied in the general technical educational programs of universities. More than 3000 participants signed up for this course. We are constantly in touch with them - we answer questions that arise in the learning process. The third semester has not yet been released due to the great complexity of the work. But, given the current situation, we plan to speed up the process and place the third semester on the Open Portal in the near future. As for the university site https://lms.spb stu.ru/, which is being used by students of Peter the Great St. Petersburg Polytechnic University (SPbPU), it is based on the Moodle platform, which is more convenient and has a lot of features that are not available on the Open Portal. For this reason, we prefer to use this site when working with our students.

2 New Challenges At the very beginning of the quarantine measures, on March 16, 2020, an order was issued to start distance learning. Against the background of general confusion, and in some cases even panic, we invited many lecturers of our Department of Higher Mathematics to our course. For each lecturer his/her own forum was created, which was open to access only to groups studying with this lecturer. In this forum a task was given for each study day. In addition, each student had an opportunity to submit his/her work to this site for review. As for the material that they had to master, the site, as already mentioned, has a full set of video lectures (40 in total), assignments for independent solving and control tests. In addition to the distance course, social networks were involved. Each of the authors of this report has a personal page on Vkontakte (VK), as well as specially created groups on this social network, where we post materials for preparing for exams and tests, as well as the results of exams, colloquia and tests. In order to gather students, we used a general conversation with the group leaders. By the evening of Monday, March 16, almost all students, except for inveterate truants, went to the site lms.spbstu.ru and registered in the forum. The forum was set up so that the mark of participation was reflected in the table of results for both the student and the lecturer. Since March 25, the entire SPbPU has switched to remote learning using various sites on the same Moodle platform. All lecturers, having entered their electronic timetable, went to the LMS website (Learning Management System), where they had an opportunity to conduct webinars, answer questions and conduct other educational activities: surveys, questionnaires, testing, receiving and issuing calculation assignments [7, 8]. We use the following method to check attendance: we make a forum for each specific study day and students should go to the special topic “Attendance” and put +, and they can do this only in a limited period of time, during the lesson itself.

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3 New Approach to the Educational Process The organization of the educational process is mainly based on the materials that have already been posted on the site https://lms.spbstu.ru/, but, in fairness, we note that this was not enough. Some of the topics that are studied in the fourth semester of IE were missing the necessary material. Also, there were no practical lessons on the site that could adequately compensate for the lack of live communication between students and a lecturer in the atmosphere of a normal practical lesson. Therefore, we had to post a lot of additional materials. Topics such as field theory, curvilinear and surface integrals, and a short course on the theory of probability were added. In order to conduct more or less full-fledged practical exercises, practical assignments in the “Lecture” format were created (and are still being created). This format allows you to convey material with elements of interactivity, i.e. with the participation of the student himself/herself [9]. When a specific task is analyzed, it is described in detail how it can be solved, if possible, illustrations are used that would be rather difficult to depict with chalk on a blackboard. After analyzing typical examples, it is proposed to solve a similar problem on your own and enter an answer. If the answer is correct, then the student receives points, if incorrect, then possible errors are explained and another attempt is proposed, but without adding points. As a result, the lesson is considered passed if there are 60% correct answers. The number of tasks of various levels of complexity, intended for independent solution, from 6 to 8 in each practical lesson. The passage time is limited - from 45 min to 3 h. The test can be completed several times and the average score is counted. It should be noted that the students took an active part in the creation of these practical exercises. A group of about 10 volunteers, mostly excellent students, was organized, who were the first to test these assignments, providing valuable comments and correcting inevitable mistakes. These students actively communicate with each other and with the lecturer, discussing assignments, and identifying their mistakes in solving problems. We also used social network VK to communicate. Subsequently, they explain various subtle points to their classmates, taking on part of the teacher’s functions, for which we are very grateful. In total, about 20 practical lessons in the “Lecture” format have been created to date, and the work is still ongoing. A lesson looks something like one on Fig. 1. (Sorry for our Russian). Lectures for part of the study streams are presented on the website in video format. There is probably no particular point in conducting webinars on them. Any webinar simply announces which lecture is on the agenda. If some of the lectures have not yet been recorded in the studio, then a presentation with animation is created on this topic so that each new phrase or new formula appears in the desired sequence. At the webinar, the presentation is voiced, recorded and uploaded in PowerPoint format on two sites. Now about how it is proposed to monitor the timely completion of tasks. Of course, each student works in his/her own pace, and sometimes in his own time zone, which must also be taken into account. As you now, Russia is a vast country and sometimes the difference between time zones reached 4 h. There are deadlines - for not completing work before a certain point of time, the student receives penalty points in his/her rating. We must say that for several years now we have been using a point-rating system, which takes into account all the student’s work during the term. This includes home

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Fig. 1. Screen shot of the lecture about function level lines. Students see the behavior of level lines for some function of several variables. Next, they need to answer the question about the level lines of another function.

control tests, the colloquium and the final test, as well as bonuses in the form of extra points for participating in the Olympiads and completing additional tasks, which students themselves named «carrots», reflecting typical stimulation of domestic animals like donkeys. Every month, the group leaders are provided with an Excel files with intermediate results. The task of the group leader is to convey this information to all students in the group. Figure 2 shows the results of part of group 3331504/90001 as of May 6 of this year. The colored cells are debts that should be closed in the near future. As mentioned above, the student’s work is assessed throughout the term. Points can be earned for home tests (T6 - T8 in the table), which must be completed on time. These points are taken with a weight of 1/3. Practical exercises (the table shows that the majority is gaining 100%) are not included in the rating at all. But students know that if they score less than 66%, they will not get the final test. The rest of the points can be obtained for the colloquium, which we will talk about later, and, in fact, for the exam. If, for example, the maximum possible number of points received is 100, then the marks are distributed as follows: starting from 51 - satisfactory, from 71 - good, above 90 – excellent.

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Fig. 2. Distance learning results for one of the student groups /May 6, 2020/.

4 Methods of Proctoring The most difficult issue when organizing distance learning is the issue of proctoring. We need to be sure that we give a well-deserved mark to a student who did everything on his own. According to the rules approved by order of the Ministry of Education and Science of Russia No. 816 of 23.08.2017, SPbPU approved the Internal Regulations for Interim Certification using exclusively e-learning and distance learning technologies (the full text can be found on the university website). Interim certification can be carried out in the following forms: • computer testing, • oral interview; • a combination of the listed forms. Taking into account the large number of students in study streams passing higher mathematics, and the specifics of the discipline itself, we stopped at the first approach computer testing. Computer testing is carried out using the specialized free software Safe Exam Browser, which blocks the opening of windows on the student’s computer, except for the window with the task (test). In this case, before testing begins, the teacher in the MS Teams webinar room begins the meeting by turning on video recording, conducting identification of individuals and inspecting the premises of all students participating in the testing, fixing students who did not appear for interim certification. A student who has begun to perform the test before the identification of his personality, according to the results of the intermediate certification, receives an unsatisfactory grade. The advantage of this type of certification is the ability to conduct it for several groups at the same time (for a stream). The disadvantages include, firstly, that the Safe Exam Browser software must be installed on the student’s computer prior to the start of the intermediate certification. Secondly, the very process of performing computer testing by a student cannot be recorded, i.e. students take the test without proctor supervision. Thirdly, for a large stream, identification of individuals and inspection of the premises of all students is simply impossible. For our part, we tested the capabilities of the webinar in MS Teams and would like to share the following observations. At the beginning of May, at the numerous

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requests of students, a traditional colloquium in our course was held according to a very unconventional scheme. First, a survey was created on the desire to participate in the colloquium and on the availability of the necessary funds. This refers to uninterrupted Internet access and the availability of an acceptable web camera. 143 students out of 159 registered on the site indicated their desire to participate. The number 159 does not quite correspond to reality; unfortunately, it also includes already expelled students, there are about 10 of them. The colloquium was held in two days, with only 3 volunteers participating on the first day. This allowed us to test our capabilities in a sparing manner. On the second day, the number of participants was already 140. For the colloquium, a special test was compiled, as expected, of 30 tasks. For each task, 10–15 variants were created, so the test variants were equivalent, but not the same. A small part of the questions was borrowed from those tests that students already completed as homework, the rest were created a new. Test writing time is limited to two hours. Only one attempt planned. Login is carried out with a password, which was changed after the first 3 students passed the test. They learned their results only together with classmates. Since the colloquium includes not only practical questions, but also theoretical ones, the student had to prove something, derive or write down one of the formulas. For this purpose, questions of the “Essay” type were introduced into the test, in which the student had to either type the required text, including formulas, or write it on paper, photograph and insert the answer as a file. In order not to use homework, the theoretical tasks were concretized, that is, the derivation of the formula was required not in general form, but in a particular case. The big disadvantage of this type of questions is that they are tested separately, and the student does not see his result in points immediately after testing. We had 5 such questions in each test, it is probably worth reducing this number to one or two. Before the test, the students’ record-books were checked, not for all, but selectively, for two reasons - firstly, we already know our students by sight, and secondly, it was technically impossible, given the number of participants. A video was included that recorded the entire process of passing the colloquium. In total, there were three lecturers who supervised the work of students. The students did not know what the two of them saw, but the observation was carried out constantly. The lecturers’ cameras were turned off. The students’ microphones were also turned off, but at the right moment it was possible to select any student, display his image on the screen, ask a question or answer a question that arose. After completing the work, in question 31, any student had to take a photo of his draft and upload it to the test. This was the sixth Essay question that was not scored. This was where his participation ended. We must say that the ability to send large files to the LMS-site is not the same for everyone. It all strongly depends on the quality of the local Internet connection. Therefore, those who were unable to do this within two hours of testing could send drafts to the social network VK, but no later than within 20 min after the end of testing. The students were able to find out their results the next day because the lecturers had to check the essay. Analysis of errors was carried out at the webinar, during the practical sessions. Any student could see his/her answers, the lecturer explained mistakes in MS Teams.

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5 Results and Conclusion Now a few words about the results. The diagram below (Fig. 3) shows the results obtained by the students.

Fig. 3.

The highest score is 69, it was received by a single student, as is usually the case when conducting a colloquium in the classroom. Two students lost contact for various reasons - they will rewrite work with those who scored less than 50% of the points. Rewriting, by the way, also entails the accrual of penalty points, but this, of course, will not apply to those who have lost communication. In conclusion, a few words about how you can use the gained experience in the normal learning mode. It seems to us that practical exercises in the «Lecture» format may well replace the usual homework assignments, the completion of which is easy to check. Even if the student enters the answers known in advance, this is no different from the fact that he copied the answer from the end of a textbook. The fact is that whoever wants to learn will learn, and whoever wants to deceive someone will be deceived himself/herself. In the regular (not remote) mode, the student will have to solve similar problems on a full-time exam or on a test. You can successfully use the distance test while in the classroom. At the same time, there is no doubt that the student performs the test on his/her own, especially if we take into account the fact that he/she will certainly have to pass a draft. The great advantage of this approach is that the time for verification is significantly reduce and free up in order to further improving the distance course. Information about distance learning both in Russia and other countries and experience obtained during pandemic time can also be found in [10–16].

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References 1. Vasiliev, V.E., Lagunova, M.V., Ezhova, N.V.: Distance education of correspondence students of the mechanical engineering department of IMMiT. In: Modern Engineering. Science and Education: Materials Intern. Scientific-Practical Conferences, pp. 72–81. Polytechnic Publishing House. University, SPb. (2016) 2. Lagunova, M.V., Ezhova, N.V.: An interactive approach in the distance teaching of a mathematics course to students in the mechanical engineering field. In: Modern Engineering. Science and Education: Materials Intern. Scientific-Practical Conferences, pp. 67–71. Polytechnic Publishing House. University, SPb. (2014) 3. Lagunova, M.V., Ezhova, N.V., Ketov, D.V.: The use of Internet technologies in teaching a mathematics course for students of the mechanical engineering field of IMMiT. In: Modern Engineering. Science and Education: Materials Intern. Scientific-Practical Conferences, pp. 118–123. Polytechnic Publishing House. University, SPb. (2013) 4. Kozlikin, D.P., Radkevich, M.M.: Features of the preparation of bachelors according to the new federal state educational standard. In: Modern Engineering. Science and Education: Materials of the International Scientific-Practical Conference, 14–15 June 2011, 410 p. Polytechnic Publishing House. University, St. Petersburg (2011) 5. Lagunova, M.V., Ezhova, N.V.: The distribution of time in the course of mathematics and the organization of independent work of students. In: Modern Engineering. Science and Education: Materials of the International Scientific-Practical Conference, pp. 41–49. Publishing House of the Polytechnic. University, SPb. (2017) 6. Afanasyeva, I.B., Matveev, I.A., Merkulova, O.V.: Pedagogical conditions for managing the learning process. In: Modern Engineering. Science and Education: Materials Mezhdunar. Scientific-Practical Conferences, pp. 16–30. Publishing House of the Polytechnic. University, SPb. (2017) 7. Afanasyeva, I.B.: The formation of intellectual skills in learning. SPbSPU Scientific Technical Journal: Scientific Journal, Series “Science and Education”, no. 4 (110), pp. 349–356 (2010) 8. Skelcher, S., Yang, D., Trespalacios, J., Snelson, C.: Connecting online students to their higher learning institution. Distance Educ. 41(1), 128–147 (2020) 9. Gusev, V.A.: Psychological and pedagogical foundations of teaching mathematics, 267 p. Verbum Publishing House, Publ. Center “Academy” (2003) 10. Jordan, K.: Initial trends in enrolment and completion of massive open online courses. Int. Rev. Res. Open Distance Learn. 15(1), 133–160 (2014) 11. Eaton, P.W., Pasquini, L.A.: Networked practices in higher education: a netnography of the #AcAdv chat community. Internet High. Educ. 45 (2020) 12. Sun, Z., Xie, K.: How do students prepare in the pre-class setting of a flipped undergraduate math course? A latent profile analysis of learning behavior and the impact of achievement goals. Internet High. Educ. 46 (2020) 13. Ayupova, L.I.: Distance learning and russian realities. Educational Bulletin “Consciousness”, no 9 (2016) 14. Naidu, S.: The MOOC is dead—long live MOOC 2.0! Distance Educ. 4(1), 1–5 (2020) 15. Moorhouse, B.L.: Adaptations to a face-to-face initial teacher education course ‘forced’ online due to the COVID-19 pandemic. J. Educ. Teach. (2020) 16. Bu, W.: Teaching study of engineering graphics for expressions of innovative design. In: Majewski, M., Kacalak, W. (eds.) Innovations Induced by Research in Technical Systems. IIRTS 2019. Lecture Notes in Mechanical Engineering. Springer, Cham (2020)

Industrial Cyber-Physical Systems Engineering: The Educational Aspect Oman A. Abyshev1(B) , Eugeny I. Yablochnikov1 , and Juho Mäkiö2 1 ITMO University, Saint Petersburg, Russia [email protected], [email protected] 2 University of Applied Sciences Emden/Lehr, Emden, Germany [email protected]

Abstract. This work discusses issues and approaches to organizing training for specialists in the field of engineering of industrial cyber-physical systems (ICPS) at technical universities. An overview of the current state of research in the field of competence models of a future specialist, expectations of industrial enterprises is presented. This work proposes a novel educational approach T-CHAT and its application in education of engineers in the field of ICPS. Additionally, a comprehensive organization and provision of students are examined connecting the competency model, the architectural model of the “smart factory”, the formation of curricula, and individual educational trajectories of students. Furthermore, this work formulates recommendations for the use of the indicated practices and approaches in the training of specialists in the framework of consideration of the educational aspect of the engineering of industrial cyber-physical systems. Keywords: Engineering training · Competency model · T-CHAT · Smart manufacturing · Smart factory · Educational factory · Learning factory · RAMI · ICPS · Industrial cyber physical system

1 Introduction Modern industry is interested in the new generation of specialists who will design and create “smart” factories with a high degree of automation, flexibility, self-organization, and the ability to quickly respond to individual consumer requests. The training of such specialists is based on the study of new information and manufacturing technologies, such as digital design, modeling and simulation, digital manufacturing, additive and hybrid technologies, robotics, industrial sensing, industrial Internet of things (IIoT), service-oriented architecture (SOA), processing large data (Big Data), artificial intelligence, machine learning, information systems for production and enterprise management and others. A modern industrial enterprise that operates and organizes its production processes on the basis of aforementioned technologies can be considered as an ICPS. In general terms, cyber-physical systems are understood as a network technical system consisting © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 9–19, 2021. https://doi.org/10.1007/978-3-030-62062-2_2

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of digital (virtual) and physical systems (components) interacting with each other. In an ICPS physical components are machines and robots, devices and tools, transport systems and any other active (that is, identified, capable of interacting with other objects and control system) material objects used in the manufacturing processes of products. Virtual objects are digital representations (digital twins in perspective) of physical objects. The software is provided upon request from the ICPS components as cloud services when the system performs the corresponding tasks. The basic technology for organizing the communication of all components is the industrial Internet of things (IIoT). Thus, the main parameters of an ICPS allow us to consider it as one of the basic concepts of Industry 4.0 [22]. The new requirements put forward by the industry require constant improvement of master’s programs, within the framework of which it is possible to organize the study of the construction and operation of ICPS. This is an urgent task for the Russian Federation and is being actively discussed abroad [2, 3]. The content of the program is determined by the chosen direction of training and the need to form competencies among students in accordance with accepted standards. The purpose of the research is to select or develop educational approaches that could take into account the specifics of this topic, determine and justify the application of the basic methodology that logically connects the proposed academic disciplines, and propose an approach to organizing a real production and technological environment for experimental research and development of ICPS components.

2 Educational Approaches for ICPS Education – Literature Review According to [1], knowledge is a product of cognition, an attitude is an automated (i.e. carried out without the direct participation of consciousness) component of activity, formed during repeated repetitions (exercises, training). Attitudes and skills are related as part and whole: an attitude is specific (automated) component of skill. Skills (synonym – competences) – mastered by a person the ability to perform actions, provided by a set of acquired knowledge and skills. Skills is the highest human quality, the formation of which is the goal of the educational process, its completion. Core competencies are context independent and versatile. Disciplinary competencies can be understood as fundamental knowledge and professional knowledge in the subject area. Disciplinary competencies differ depending on the subject, since each area requires different competencies [25]. Training programs for specialists in the field of cyber physical systems include a list of mandatory disciplinary competencies, according to [2, 3]. The necessary list of disciplinary knowledge, abilities and skills form the basis in understanding the essence of cyber physical systems and their non-functional characteristics, systems engineering, business and humanities. Key competencies include cooperation and communication (especially in multidisciplinary multicultural teams), cooperation and communication with clients, problem identification and solving, analytical skills, creativity, critical thinking and critical attitude to technological developments, entrepreneurship, including lifelong learning. Knowledge in English is a key competency, especially in international projects, since in such projects the working language is usually English.

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ICPS is an interdisciplinary field of knowledge that requires the use of special teaching methods. The use of interdisciplinary teaching allows learners to integrate information, data, methods, tools, concepts and theories from two or more disciplines to create ideas and models, explain phenomena, or solve problems in ways that are difficult to describe and logically construct using the approaches of only one discipline [4]. Thus, teaching methods that support interdisciplinary learning are the preferred choices for ICPS teaching. Today, despite the use of modern educational approaches, graduates of engineering specialties do not meet industry expectations in hard-skills and soft-skills [8–10]. One of the options for solving the significant gap between the requirements of the industry and the system of training engineers is the use of a holistic adaptive educational approach T-CHAT, focused on solving problems in the process of training engineers in the field of industrial cyber physical systems [6, 7]. This approach is aimed at improving the methodological, social and personal competencies of students along with the development of disciplinary knowledge and skills. The main idea of T-CHAT, according to the authors, is to use five pedagogical teaching approaches: perceptual, design, problem, research and full-time, and combine them in order to increase the efficiency of this process by flexibly varying pedagogical methods in accordance with changing needs students and industry. This educational approach is being introduced and developed as part of a new curriculum for the preparation of undergraduates at ITMO University in the field of ICPS. Teaching students is carried out in accordance with the accepted model of modular training of students [5].

3 Educational Factories as a Tool for Training Specialists An educational, training factory is a learning environment, the processes and technologies of which are based on a real industrial site, which allows you to directly approach the process of creating products and products. Learning factories are based on a didactic concept that emphasizes experiential and problem-based learning [11]. The issue of creating, engineering and developing educational factories for training engineers in the field of ICPS involves considering the approaches and achievements of various research groups through a harmonious and detailed analysis of the proposed solutions. Of interest is the guide to the creation of ICPS proposed by the authors [12], compiled based on the results of their own research in the field of engineering of educational factories. The approach described in the work allows for the systematic, purposeful design of training systems for industrial engineering by integrating the configuration and design of the entire training factory infrastructure, training course, as well as individual training and training situations. This manual promotes the development of a practical approach to the effective development of competencies in the environment of the educational factory by solving the problems of intuitively designed learning systems. The experience of interaction between industrial partners and students of the University of Patras, Greece based on the two pilot projects under consideration within the framework of the Learning Factory concept is described in [13]. The proposed implementation option is based on the principle of remote interaction between industry representatives through video communications and remote access with students in specially

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equipped classrooms according to the “one-to-one” or “one-to-many” schemes. The experience of interaction between the parties through the transfer to production of the results of ongoing academic research based on university laboratories (robot complexes) - the “Academia-to-Industry” model is described. A similar approach to building a multi-purpose academic laboratory Learning Factory with a focus on the requirements of local manufacturers of mechanical components and mechatronic systems was considered in [14]. In this case, the homogeneity of the composition of components and information systems should be noted as a special factor, which contributes to the construction of an integral problem-oriented educational environment. The project of an educational factory with a key focus on small-node assembly and intra-shop logistics is of interest. Within the framework of this project, based on these technologies, students to a greater extent independently master the main areas of knowledge of the technologies and methods used. The role of the teacher changes from instructor to mentor of the educational process. Special attention is paid to considering the role of a person and the ergonomics of his work in a working environment with ICPS [15]. The CubeFactory [16] implements the original model of an autonomous, waste-free, environmentally friendly educational factory. The aim of the CubeFactory is to teach human-machine interaction with a high significance of zero waste of the applied processes. For this purpose, a special module for 3D prototyping from polymeric materials and their subsequent regeneration is applied. Among them, a number of works can be distinguished, in which great attention is paid to the consideration of the concept of digital educational modular methodology based on the use of a competency model formed by the results of the analysis of literature sources and the method of situational tasks (cases) [17, 18]. This technique is used within the educational factory - LEAD Factory when training specialists in lean management in the workplace. Students are invited to go through several modules: independent study of the case, passing an introductory theoretical course, developing their own project, working on a project in the perimeter of the LEAD Factory, final theoretical training. The presented examples of educational factories have their own unique specifics and strong differentiation of their purpose as a project, due to the significant dependence on the tasks and conceptual concepts underlying its design. The specific target orientation of the factories does not provide universality; however, it should be noted the importance of ensuring multitask project design training of students in accordance with the multidisciplinary nature of the subject under consideration. In this regard, an important aspect is the definition of general conceptual principles that ensure its effectiveness as an educational environment for the training of engineers in the field of ICPS. A wide variety of tasks and learning objectives must be considered in the further development of such factories. Next, we will consider some provisions and concepts, the use of which can be recommended in the training of modern engineering specialists of a wide profile - engineers of industrial cyber physical systems.

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4 Conceptual Provisions for the Development of Curricula and the Choice of Specialization One of the possible variants of the conceptual principle in the formation of a methodology that ensures the coherence of academic disciplines and factories can be the reference architecture of an enterprise, which determines the general principles of building a new type of production. The use of the reference architecture of the enterprise allows you to determine the stages in the study of technologies by a student, to build a logical connection between educational courses and research projects, to form a holistic view of modern technologies, and also to provide a connection between the theoretical and experimental-applied aspects of the educational process. The latter can be achieved through external collaboration with industrial partners of the university, as well as the creation of experienced educational factories and laboratories of industrial cyber physical systems. In the framework of this work, the use of the reference architecture model Reference Architecture Model for Industry 4.0 (RAMI), which defines the standards for the engineering of industrial cyber physical systems [22], is proposed. This architectural model was proposed in 2013 by BITCOM, VDMA and ZWEI as a development of the existing Smart Grid Architecture Model for the electric power industry for the needs of the industry, as well as developed by the Platform Industrie 4.0 association. This model is a service-oriented architecture that describes the structure and relationships of Smart Factory elements in three-dimensional coordinate space within the framework of the Industrie 4.0 concept. It combines the concepts of hardware and virtual components in a seven-level hierarchical automation model - IEC 62264, IEC 61512 and IEC 628990 life cycle. The model also allows you to decompose complex processes into simple modules, making the notation easier to understand and master, and facilitating better interdisciplinary collaboration. between engineers of various specialties. The Reference Architecture Model for Industry 4.0 harmonizes major advances in enterprise concepts, life cycle concepts, and generally recognized engineering standards, serving as a navigator for designers. Application of the RAMI model allows organizing the necessary variability and flexibility of educational technologies in accordance with the requirements and expectations of the industry and personal requests of students [25]. This is ensured by the multilevel and multi-disciplinarity of the proposed model, combining the best experience and developments in the concepts of previous stages - the life cycle of the development of technical products and products, a hierarchical pyramid of automation, various specializations, and aspects of the design and manufacture of complex technical devices (from mechanics and kinematics of units to program logic and electronic components). A variant of using the proposed approach is shown in Fig. 1. This illustration shows the relationship between the reference architecture of the ICFS based on RAMI and curricula: RAMI provides navigation, acting as the “Atlas of competencies for an ICPS engineer” in the process of forming individual educational paths: from the type of specialization of the program to the choice of subjects and educational projects of students depending on the expectations of learning outcomes. The curriculum can be determined based on the tasks of the individual project activities of students. In the event of an understanding or a need to master a new subject area,

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Fig. 1. The choice of academic disciplines based on the reference architectural model RAMI

the student can change the order or structure of educational modules. For example, having received the task to implement the integration of software modules for monitoring equipment with an enterprise resource management system, the student will need skills in software engineering, network protocols, and system integration. These aspects can be studied by students by introducing additional modules by replacing them, rotating or expanding the curriculum. To consolidate the acquired knowledge in practice and test the hypotheses and solutions developed, the student will need a learning environment for prototyping and debugging. This task can be solved in several ways: building a prototype in an educational factory, pilot projects with industrial partners, etc.

5 Building an Educational Environment for Learning ICPS The role of technical universities today is to be a center for the creation of advanced technologies and innovative development projects for various industries, to act as the organizer of a new educational ecosystem, integrating the current and future needs of industry with advanced research, technologies and approaches in training specialists. This determines the key task of universities, namely, predicting the sources of future scientific breakthroughs and working “ahead of the curve”: preparing specialists for the requirements of future reality, considering the duration of the academic training cycle.

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For the successful implementation of this task, it is necessary to regularly monitor the world achievements in the field of education and production. It is the ability of universities to foresee, anticipate the future and build, organize academic and industrial cooperation is the key to its success today. It is important to note the development trends of global industrial technologies towards the creation of smart manufacturing in accordance with the concept of Industrie 4.0 (Fig. 2). The expansion of the use of information and communication technologies in industrial production, as well as the new role of information processing in management and decision-making processes, determine new requirements for the formation of the educational environment and the adaptation of new solutions. New approaches in building an educational environment for training specialists in the field of ICPS must meet the future needs of industry and provide the ability to study advanced technologies, state-of-the-art, as well as conduct innovative research and development.

Fig. 2. Development milestones for Industry 4.0 [23]

In particular, this approach was tested in the framework of academic cooperation between students and professors of ITMO University and the University of Applied Sciences, Emden-Leer, Germany, which made it possible to pay great attention to the use of new communication technologies, organization of system integration of components, human interaction with ICPS. using the educational approach T-CHAT presented in the learning process [6]. The implementation of the project for the development of an educational factory within the framework of a joint collaboration of students made it possible to obtain intermediate results presented by the authors in [19]. The created site is designed to familiarize students with the basics of modern industrial automation and Industrie 4.0 technologies, serving as a platform for new student research projects. This aspect is realized due to the modular organization of the site, as well as the ability to connect new components and systems. It should be noted that there is a description of this section in

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the RAMI 4.0 architecture notation, which ensures that the project relates to the proposed model for constructing an educational trajectory by students. In 2019, in collaboration with international industrial and academic partners, in collaboration with international industrial and academic partners, as part of the creation and provision of an educational environment, a new master’s program “Industrial CyberPhysical Systems” was created. This program is focused on training a new generation of specialists for the development of high-tech industry in the format of “smart” industries with a high degree of automation, flexibility, self-organization, the ability to quickly respond to individual consumer requests. The program covers such advanced information and production technologies as digital design and modeling, digital manufacturing, additive and hybrid technologies, robotics, industrial sensing, the industrial Internet of things (IIoT) and services, big data and artificial intelligence, and also information systems for production and enterprise management in general. This set of technologies meets and supports the development of new competencies among students in accordance with future technology development trends for Industrie 4.0 (Fig. 2). It should be noted that ITMO University has extensive experience in creating Engineering Centers that meet the challenges of building educational factories [24]. There is considerable experience based on the PLM paradigm - product life cycle management. So, in order to ensure a holistic educational approach and in order to develop a new direction for training masters at the university, together with industrial partners, a laboratory “Digital production technologies” was created based on the conceptual provisions of the RAMI reference architecture - the basis for future smart industries: • BiPitron JV LLC, as an expert in the field of PLM class information systems and digital Internet of things platforms. Partner of ITMO University in the ICPS corporate master’s program; • Schneider Electric, a global leader in providing digital solutions for power management and automation. Partner of ITMO University in the ICPS corporate master’s program; • PJSC Techpribor, a historical partner of the university, supports the concept of digital production in a joint laboratory. The company is engaged in the development, production, certification and maintenance of on-board avionics. The main task of partners is to support educational programs that consider technologies for the design, production and operation of competitive products through: • development of infrastructure and laboratories; • active international cooperation; • development and maintaining competencies in technology, automation and production organization; • the formation of a partner environment from high-tech companies; • development of a methodology for constructing ICPS in the Industry 4.0 paradigm.

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6 Results and Discussion The industry is always interested in competent specialists. New generation engineers are faced with the task of designing and creating new, modernized “smart” industries. Requirements for the complexity of these systems are growing - adaptability and selfcorrection, information integration and a high degree of process automation, reaction speed, mass personalization [3, 8, 11, 12, 20–23]. Industry requirements for young specialists graduating from technical universities are changing [8–10]. It was noted that the study of ICPS can be organized using multidisciplinary methods and approaches in education that have integrity and adaptability [4]. The connectivity of academic disciplines should be determined by the unity of the conceptual principles of educational trajectories and the flexibility of their choice, provided by their modularity and variability [5]. It is noted that one of the possible options for the conceptual principle in the formation of a methodology that ensures the coherence of academic disciplines and factories can be the enterprise’s reference architecture, which allows one to determine the stage-bystage study of technology by a student, build a logical connection between educational courses and research projects, and form a holistic view of modern technologies, as well as to provide a connection between the theoretical and experimental-applied aspects of the educational process, through the use of the educational approach T-CHAT in the training of ICPS specialists [6, 7, 26]. It is necessary to note a special place in the consolidation of students’ skills through work in educational factories - a special environment based on a didactic concept that focuses on experimental and problem-based learning and allows to provide many aspects of the proposed educational approach and the considered competency model [11–17, 19]. World experience shows that the adopted concept of developing the training of engineers through an educational factory and the connection it creates between academic education and the industrial world should become a common connecting foundation for a new educational ecosystem. It should be noted that the results will be used to develop a comprehensive model and engineering methodology for Smart Factory as part of the ongoing study, as well as updating curricula, work disciplines and materials within the framework of the ITMO University’s “Master Cyber Physical Systems” Master Program.

7 Conclusion The educational aspect of the engineering of ICPS is a complex task requiring the revision of classical approaches in the training of technical specialists in view of its multidisciplinary structure and the high expectations placed by the industry. The use of new combinations of methods, the rethinking of classical approaches along with the study of the achievements of leading international centers, can become the basis of a new intellectual philosophy and attitude towards the construction of the educational process in academic environments and universities.

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18. Erpenbeck, J.: Handbuch Kompetenzmessung: erkennen, verstehen und bewerten von Kompetenzen in der betrieblichen, pädagogischen und psychologischen Praxis, Schäffer-Poeschel (2007) 19. Tisch, C., Hertle, E., Abele, J.Metternich, Tenberg, R.: Learning factory design: a competencyoriented approach integrating three design levels. Int. J. Comput. Integr. Manuf. 29(12), 1355–1375 (2016). https://doi.org/10.1080/0951192X.2015.1033017 20. Glushchenko, F.A., Borzykh, V., Vermann, J., Colombo, A.V.: Development of a model of a training platform for the study of the production process in the concept of industry 4.0. Sci. Tech. Bull. Inf. Technol. Mech. Opt. 18(3), 554–560 (2018). https://doi.org/10.17586/22261494-2018-18-3-554-560 21. Metternich, J., Abele, E., Chryssolouris, G., Sihn, W., Maraghy, H.E., Tracht, K., Tolio, T., et al.: Definitions of “Learning Factories”. CIRP Collaborative Working Group on Learning Factories, CIRP General Assembly, Nantes, France, 28 August 2014 (2014) 22. Plattform Industrie 4.0. The background to Plattform Industrie 4.0. https://www.plattformi40.de/PI40/Navigation/EN/Home/home.html. Accessed 17 Jan 2020 23. Shu, G., Anderl, R., Gausemeier, Y., ten Hompel, M., Walster, V., et al.: Industry 4.0 maturity Index – managing digital transformation of companies (acatech research). Herbert Utz Verlag, Munich (2017) 24. Yablochnikov, E.I.: Creation of the engineering center “Integrated computer technologies of design and production of devices and systems”. In: Collection of Materials of the Fourth All-Russian Scientific and Practical Conference “Computer Integration of Production and IPI-Technologies” - Orenburg: IPK GOU OSU, pp. 104–110 (2009) 25. Radermacher, A., Walia, G., Knudson, D.: Investigating the skill gap between graduating students and industry expectations. In: Companion Proceedings of the 36th International Conference on Software Engineering, pp. 291–300. ACM (2014) 26. Mäkiö-Marusik, E., Colombo, A.W., Mäkiö, J., Pechmann, A.: Concept and case study for teaching and learning industrial digitalization. Procedia Manuf. 31, 97–102 (2019). https:// doi.org/10.1016/j.promfg.2019.03.016

System Analysis of Cold Axial Rotary Forging of Thin-Walled Tube Blanks L. B. Aksenov(B) , S. N. Kunkin, and N. M. Potapov Peter the Great St. Petersburg Politechnic University, St. Petersburg, Russia [email protected], [email protected], [email protected]

Abstract. The paper presents the results of a system analysis of the process of cold axial rotary forging of parts from thin-walled tube blanks with a conical roll. The technology is designed for manufacturing axisymmetric hollow parts of various shapes. A feature of the process is the possible loss of stability of the shape of the tube blanks, which does not allow getting the parts of the required geometry. On the basis of experiments and computer simulation of the process, six types of metal flow of the tube blank during rotary forging are determined, three of which are associated with the loss of stability of the tube blank. The main ten parameters that affect the technological process, including the kinematic characteristics of the machine and the geometric parameters of the tube blanks are systematized. Only three parameters can actually be used to control the metal flow. Recommendations are offered to reduce the time of adequate computer simulation of metal flow when forming tube blanks in the Deform 3D complex, which is used to determine the rational values of the parameters of a stable rotary forging process. Keywords: Axial rotary forging · System analysis · Tube blank · Stability · Conical roll · Angle of inclination · Computer simulation

1 Introduction Axial rotary forging is a representative of technology with local deformation of the formed metal. In this case, only a part of the blank is in contact with the deforming tool. That reduces the contact area and the amount of contact stresses, and, accordingly, the necessary forming force, which ensures the effectiveness of the process, especially in small-scale production. The capabilities of this relatively new technology are not yet fully established. At the present stage, considerable attention is paid to the development of new kinematic schemes of machines and the type of forming tool [1–3], as well as to the expansion of the application of rotary forging to obtain specific parts [4], including from new materials [5–7]. Parts made by axial rotary forging from thin-walled tube blanks can have a variety of shapes: with external and internal flanges, with conical bell mouth or spherical surfaces (Fig. 1). Manufacturing of parts from thin-walled tube blanks by axial rotary forging has a number of technological features. Non-wide flanges with a thickness of the flanged part © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 20–29, 2021. https://doi.org/10.1007/978-3-030-62062-2_3

System Analysis of Cold Axial Rotary Forging of Thin-Walled Tube Blanks

21

Fig. 1. Parts that can be produced by cold axial rotary forging from thin-walled tube blanks

about the thickness of the initial blank can be successfully obtained using the technology of rotary forging with flanging [8]. For thickened flanges with a small flange width (less than two wall thicknesses of the original blank), an effective technology is axial rotary forging with the upsetting of the exposed deformed part of the blank [9]. A particular problem is getting parts with a developed flange width. The size of the deformable part of the tube blank is determined by the stability of the blank and usually should not exceed two wall thicknesses of the tube. At this size of the deformed part its volume may not be enough to form the required part geometry. But the height of the deformed part of the tube blank cannot be increased due to the loss of stability of the tube, formation of folds and cracks on the inner part of the flange. The boundary of the transition from a stable deformation process to an unstable state is difficult to determine, which hinders the practical application of the process of axial rotary forging of thin-walled parts. Objective of the research: to determine the factors affecting the forming of thin-walled tube blanks by cold axial rotary forging, possibility of tube blanks forming when rotary forging out and to define the parameters of a computer model adequate to the process of deformation with a view to its use for design of sustainable processes.

2 Research Methods The ability to control the flow of metal in the rotary forging process is very limited. Various types of tools used (cross-roll, forging rolls with a rib, mandrels) can limit the flow of metal in certain directions and after forming a certain part of the flange, redirect the metal in the direction where the formation of the part has not yet been completed [9]. Such technologies require considerable forming load, since most of the metal, in the final stage of forming the part, is a rigid zone with a stress state close to 3D compression. It is more effective to influence on the direction of metal flow by using a change in the

22

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direction of the friction forces acting on the contact surface of the rotary forged blank with the forming roll [10]. To achieve a change in the direction of friction forces, you can shift the rolls relative to their traditional location. Thus, for the operation of forming internal flanges by flanging, the metal flow to inside the tube blank is required. To direct the friction forces in the desired direction, the forming roll should be positioned with a certain offset of the roll axis relative to the longitudinal axis of the tube blank. The displacement of the top of the conical roll significantly changes the direction of friction forces on the contact surface and facilitates the flow of metal in the radial direction. The recommended values of the offset of the forming conical roll for the implementation of stable flanging depend on the ratio of the thickness and diameter of the tube blank, as well as on the material of the blank. However, these technologies do not have sufficient stability and require strict compliance with the values of technological parameters, which makes it difficult to use them in industry. The inclination angle of the forming roll has a significant influence on the rotary forging process [11]. Rotary forging machines built on the basis of orbital mechanisms have a tool inclination angle of 1–2°. In machines with simpler kinematics, this angle is much larger and is 10–20°. In some cases, changing the inclination angle of the forming roll can significantly expand the technological capabilities of the process. Thus, when processing powder blanks by rotary forging it is possible to seal the initial powder at the first stage of the process in the absence of a tool tilt, and to rotary forge the part with an inclined tool at the second stage [12]. So, it is possible to combine the operations of compaction of the powder and its forming on the same machine. Two discrete tilt angles of the forming roll can be used for forming of wide flanges, when the roll without a tilt creates a bell-mouth on the tube, then the roll gets a tilt and the flange is rolled finally [13]. Modern rolling machines are machines with variable kinematics and CNC systems [14, 15] that allow you to implement software changes in inclination of the forming roll angle and open up new opportunities for the rotary forging processes [16]. The technology of axial rotary forging of tube blanks studied in this work is shown in Fig. 2. A rotary forging machine was used according to the scheme with rotation of the tube-blank and transmission of rotation from the blank to the forming roll due to friction forces on the contact surface between the tube blank and the forging roll. This type of machine is one of the machines with fairly simple kinematics, but allows you to change the angle of inclination of the forming roll over a wide range. In addition, in such machines, it is possible to change the position of the forming roll relatively to the axis of the tube blank, which is very difficult to implement at orbital rotary forging machines. Systems analysis methods were used to study the relationships in the axial rotary forging process. The primary task of the system analysis of the studied object (process, phenomenon, etc.) is to create a form of system description. This operation comes down to identifying the parameters (factors) influencing the technology results of the vector ¯ 1 , . . . , ak ), characterizing alternative options of process, and the of input parameters A(a vector of output parameters Y¯ (y1 , . . . , yn ), reflecting the objectives of the study and revealing the results of using the technology, i.e. representation of the technology in the form shown in Fig. 3a.

System Analysis of Cold Axial Rotary Forging of Thin-Walled Tube Blanks

a)

23

b)

Fig. 2. Scheme of the process of axial rotary forging of parts with a conical roll from thin-walled tube blanks: start (a) and end (b) of the process

Fig. 3. Systemic representations of the process under study (a) and of computer model of cold axial rotary forging (b)

The most common tool in system analysis for the study of new level objects is computer simulation, which is much easier, faster, cheaper to implement than full-scale experiments. Modeling involves the use of phenomenological mathematical models implemented in computational complexes. In relation to the technology of rotary forging of thin-walled tube blanks, a system representation of the process model is presented in Fig. 3, b. For the process under consideration, k = 10 and the vector consists of ten components ¯ 1 , . . . a10 ). For ease of process analysis, the vector of input parameters can be divided A(a

24

L. B. Aksenov et al.

¯ into two: A-vector of unchanged input parameters; X¯ -vector of input parameters that can be changed (control parameters), which can be used to influence the course of the process. When rotary forming out a specific part, the main number of the input parameters usually cannot be changed, and then only three control parameters remain to be changed, i.e. the input parameters will be: a1 = D – tube blank diameter; a2 = s – blank wall thickness; a3 = H – height of the deformable part of the blank; a4 = ω1 – rotation speed of the tube blank; a5 = ω2 – rotation speed of the forming roll; a6 = μ – coefficient of friction between the blank and the forming roll; a7 = material grade; x1 = δ – offset of the top of the conical forming roll relative to the longitudinal axis of the tube blank; x2 = α – forming roll inclination relative to the tube blank axis; x3 = V– the value of the axial feed of the forming roll per revolution of the tube blank; Y = y1 the output parameter, in this study, representing a verbal description of the shape change of the tube blank as a result of rotary forming. Not all input parameters are independent. So, the value of the axial feed of the forming roll per revolution of the tube blank is an important technological parameter, but depends on the speed of the translational movement of the forming roll (Vtrans ) and its rotary speed: V = Vtrans ω2 . In turn, the rotary speed of rotation of the forming roll depends on the technological parameters D, α, δ and ω1 : ω2 =

ω1 · D (D + 2δ) · cosα

Computer simulation of the rotary forging is a complex process due to the special kinematics of rotation of the blank and tool, which are connected by friction forces, the periodically changing local zone of plastic deformation of the tube blank, and other features. Therefore, such a simulation is time-consuming, as many researchers have noted [17–19]. To simulate the process, the DEFORM 3D calculation complex was chosen. The creators of this complex purposefully improve its capabilities for modeling processes of local deformation processes with cyclic load effects on the blank [20]. Numerous studies of the axial rotary forging process and its modeling have shown that a very diverse change in the shape of the initial blank can occur during this process (Fig. 4). Studies have identified six possible types of deformation of the tube blank during rotary forging. Two of them: flanging in and out (Fig. 4a and b), can be attributed to stable, predictable, monotonously developing processes. The third type (Fig. 4, c), in which the tube upsets in the deformation zone, is characterized by the flow of metal in two directions in and out of the tube blank. The process is stable, monotonous, but it is difficult to predict in which direction the most intense metal flow will occur. Three more types of blank deformation (Fig. 4 d, e, and f) relate to the unstable development of the metal flow caused by the loss of blank stability. The bending point

System Analysis of Cold Axial Rotary Forging of Thin-Walled Tube Blanks

a)

b)

c)

d)

e)

f)

25

Fig. 4. Types of tube blank forming during axial rotary forging: a - the formation of the outer flange; b - the formation of the inner flange; c - wall upsetting of the tube blank with the flow of metal in and out; d - loss of stability of the tube with eversion; e - loss of stability of the tube with inversion inward; f - periodic loss of stability

of the tube blank is located between the die and the roll and depending on the direction of the friction forces. Thus, the blank is bent instead of flanging (Fig. 4 d and e). In this case, the deformation zone is removed from the edge of the surface blank, contributing to the bending of the metal. The form change shown in Fig. 4f is the result of the overlapping of the processes shown in Fig. 4d and e, when the resulting frictional force on the contact surface periodically changes its direction and a waveform change of the tube take place. This problem can be partially solved by using an additional drive for rotation of a conical forming roll in the machine for rotary forming, which reduces the effect of friction forces on the contact surface. The possible values of the blank parameters and the forming process, at which different types of tube blank shaping are observed, are shown in Table 1. To obtain a complete picture of the possibilities of shaping the tube blank, the values of most of the input parameters were changed. It would be useful to build a statistical model of the process using computer modeling. However, with ten parameters influencing the process, building an adequate statistical model is extremely difficult. Equally difficult is the attempt to find a complex dimensionless criterion composed of the parameters ai and x j , the numerical value of which could be connected with the type of final shape of formed tube-blank. Therefore, the main tool for analyzing the process is computer modeling and purposeful change of control parameters until the desired result is obtained. The kinematic features of the rotary forming process: transfer of rotation from the tube-blank to the forming roll by means of friction forces, the presence of a local zone of plastic deformation, and a significant area of the blank being in an elastic state lead to long time of process modeling. The increased value of the coefficient of friction for the cold deformation process to 0.2 provides a stable transfer of rotation from the blank to the forming roll. The process simulation time can be significantly reduced (by 30–40%)

62.1 0.2

a5 = ω2

a6 = μ

a7

Rotation speed of rev/min the forming roll

Coefficient of – friction between the blank and the forming roll

Material grade

DIN-C45

60

a4 = ω1

Rotation speed of rev/min the tube blank

–

15

a3 = H

mm

Height of the deformable part of the blank

2

a2 = S

mm

Wall thickness

Figure 4 b

Figure 4 a

AISI-1045

0.2

33.6

60

10

2

100

Internal flange

Outer flange

DIN-C45

0.2

62.1

60

15

4

100

Figure 4 c

Wall upsetting

Type shape change of tube blank

100

mm

Diameter

Designation

a1 = D

Unit of measurement

Parameters of tube blank and rotary forming process

AISI-1045

0.2

77.6

60

20

1

100

Figure 4 d

Outward bending

AISI-1045

0.2

31.0

60

30

1

80

Figure 4 e

Inward bending

(continued)

AISI-1045

0.2

32.4

60

10

2

120

Figure 4 f

Cyclic loss of stability

Table 1. Values of parameters of the tube-blank and the rotary forming process, which lead to various types of tube blanks shaping

26 L. B. Aksenov et al.

Unit of measurement

mm

grad

mm/rev

Parameters of tube blank and rotary forming process

Offset of the top of the conical forming roll relative to the longitudinal axis of the tube blank

Forming roll inclination

Axial feed of the forming roll per revolution of the tube blank

15° 2.0

x2 = α

x3 = V 1.5

7°

40

Figure 4 b

Figure 4 a 0

Internal flange

Outer flange

2.0

15°

0

Figure 4 c

Wall upsetting

Type shape change of tube blank

x1 = δ

Designation

Table 1. (continued)

1.5

15°

-10

Figure 4 d

Outward bending

1.5

15°

40

Figure 4 e

Inward bending

1.0

15°

55

Figure 4 f

Cyclic loss of stability

System Analysis of Cold Axial Rotary Forging of Thin-Walled Tube Blanks 27

28

L. B. Aksenov et al.

without loss of modeling accuracy, if to limit yourself to three layers of finite elements in the thickness of the tube, and the calculated height of the blank to limit by two the height of the deformable part of the tube blank. But even in this case, with the number of finite elements 7.000–15.000 with a tetrahedral shape and taking into account the piecewise linear approximation of strain hardening, the simulation time for a single rotational forging process takes several hours.

3 Summary 1. It is shown that the formation of thin-walled tube blanks during the axial rotary forging is affected by ten factors. A significant part of them determine the design parameters of the formed part and the technical characteristics of the rotary forming machine, and therefore cannot be changed during the process. Thus, when manufacturing a specific part, only three parameters can be actively using to control the metal flow process: the offset of the top of the conical roll relatively to the longitudinal axis of the tube blank (δ), the angle of inclination of the conical forming roll (α) and the value of the forming roll axial feed per revolution of the tube blank (V). 2. At the axial rotary forging of thin-walled tube blanks, at least six types of blank shaping are possible. Two of them: flanging in and out can be attributed to stable, predictable, monotonously developing processes. The third type is stable, monotonous, but it is difficult to predict in which direction the most intense metal flow will occur. The last three are associated with loss of stability of the blank. 3. The time of computer simulation of the process can be significantly reduced (by 30–40%) without losing accuracy in modeling the shape change of the rotary forged tube blank, if to restrict ourselves to three layers of finite elements by the thickness of the tube, and to limit the calculated height of the tube to a height twice that is deformable part of blank.

References 1. Dong, L., Han, X., Hua, L., Lan, J., Zhuang, W.: Effects of the rotation speed ratio of double eccentricity bushings on rocking tool path in a cold rotary forging press. J. Mech. Sci. Technol. 29(4), 1619–1628 (2015). https://doi.org/10.1007/s12206-015-0333-5 2. Hamdy, M.M.: Rotary hammer forging, a new manufacturing process. Appl. Mech. Mater. 446–447, 1321–1329 (2014). https://doi.org/10.4028/www.scientific.net/AMM.446447.1321 3. Hosseini, S.V., Nategh, M.J., Agheti, M.M., Imani, H.: Optimum design of rotary forging machine with parallel mechanism. Modern Mechanical Engineering. In: Proceeding of the Advanced Machining and Machine Tools Conference, vol. 15, no.13, pp. 486–490 (2015). (in Persian) 4. Dang, Y., Yao, Y., Deng, X., Li, G., Jiang, C.: Compensation of springback error based on comprehensive displacement method in cold rotary forging for hypoid gear. J. Mech. Sci. Technol. 33(7), 3473–3486 (2019). https://doi.org/10.1007/s12206-019-0642-1 5. Mandal, P., Lalvani, H., Tuffs, M.: Cold rotary forging of inconel 718. J. Manuf. Process. 46, 77–99 (2019)

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6. Deng, X.B., Hua, L., Han, X.H.: Numerical and experimental investigation of cold rotary forging of a 20CrMnTi alloy spur bevel gear. Mater. Des. 32, 1376–1389 (2011) 7. Loyda, A., Reyes, L.A., Hernández-Muñoz, G.M., García-Castillo, F.A., Zambrano-Robledo, P.: Influence of the incremental deformation during rotary forging on the microstructure behavior of a nickel-based super alloy. Int. J. Adv. Manuf. Technol. 97, 2383–2396 (2018) 8. Aksenov, L.B., Kunkin, S.N., Elkin, N.M.: Tortsevaya raskatka flantsevykh detalej trubnykh soedinenij. Metalloobrabotka 3, 31–36 (2011). (in Russian) 9. Aksenov, L.B., Kunkin, S.N.: The combined cold axial rotary forging of thick hollow flanges. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 1–10. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53363-6_1 10. Aksenov, L.B., Kunkin, S.N.: metal flow control at processes of cold axial rotary forging. In: Evgrafov, A. (ed.) Advances in Mechanical Engineering. LNME, pp. 175–181. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29579-4_18 11. Standring, P.M., Appleton, E.: The kinematic relationship between angled die and workpiece in rotary forging. In: 1-st International Conference on Rotary Metalworking Processes, London, UK, 20–22 November, pp. 275-288 (1979) 12. Moon, J.R., Standring, P.M.: Rotary forging of metal powders. Powder Metall. 30(3), 153–158 (1987) 13. Standring, P.M.: The significance of nutation angle in rotary forging. Advanced Technology of Plasticity. In: Proceeding of the 6th ICTP, vol. III, pp. 1739–1744 (1999) 14. MJC Engineering&Technology. Rotary Forging Equipment. http://www.mjcengineering. com/. Accessed 13 Feb 2020 15. Global Metal Spinning Solutions Inc. CNC Wheel Forming and Rotary Forging Machines. http://www.globalmetalspinning.com/. Accessed 10 Feb 2020 16. Aksenov, L.B., Kunkin, S.N., Potapov, N.M.: Axial rotary forging of inner flanges at thin wall tube blanks. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 1–9. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39500-1_1 17. Krishnamurthy, B., Bylya, O., Muir, L., Conway, A., Blackwell, P.: On the specifics of modeling of rotary forging processes. Comput. Methods Mater. Sci. 17(1), 22–29 (2017) 18. Han, X., Hua, L.: Investigation on contact parameters in cold rotary forging using a 3D FE method. Int. J. Adv. Manuf. Technol. 62, 1087–1106 (2012) 19. Asfandiyarov, R., Raab, G.I., Aksenov, D.: Analysis of the stress-strained state of billets processed by rotary forging with special shape of the tool. J. Metastable Nanocrystalline Mater. 31, 16–21 (2019). https://doi.org/10.4028/www.scientific.net/JMNM.31.16 20. DEFORM Design Environment for FORMing. http://www.deform.com. Accessed 10 Jan 2020

Mechanical Challenges of Inspection Robot Moving Along the Electrical Line: Effect of Flexural Rigidity Mohammad Reza Bahrami(B) Innopolis University, Innopolis, Russia [email protected]

Abstract. The goal of this work is the mathematical modeling of the electrical line stretched and supported by two isolators while a robot inspector moves with a constant speed along with it. The model, involving the flexural rigidity of the conductor, solved by the expansion method and Galerkin method. As shown in results, the difference between model including flexural rigidity and not including flexural rigidity is negligible, and results obtained by the Galerkin method are in a good match with results obtained by expansion methods. Also, one can see from the results that the fluctuations occur although the speed of the robot is constant and stable. These vibrations may affect the performance of the robot inspector or even cause failure. Keywords: Vehicles mathematical modeling · Robot inspector · Electrical line · Galerkin method · Flexural rigidity

1 Introduction In many countries, ground inspection is a common method to check overhead electrical transmission lines [1, 2]. This type of inspection puts humans into a hazardous environment meanwhile it is time-consuming. By technology development, the trend is to replace human forces with semi or fully autonomous robots. In some countries, according to the regional standards, different types of electrical transmission line robot inspectors are designed and used [3–10]. One of these robots is presented in Fig. 1 [10]. As shown in [8–14] vibrations occur even by the stable motion of the robot along the line. These vibrations may affect the performance of the robot inspector or even cause failure. To prevent any unwanted situation, mathematical modeling is required [15–22]. This article, which is the extended work of [14], deals with the construction of mathematical models involving the flexural rigidity of the line. Here, as [14] the electrical line considered as a stretched string and supported by two isolators, one horizontal and the other one is vertical. It should be mentioned that in this article robot inspector considered as a concentrated force that moves with a constant speed on the taut string.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 30–37, 2021. https://doi.org/10.1007/978-3-030-62062-2_4

Mechanical Challenges of Inspection Robot

31

Fig. 1. Scheme of a proposed electrical transmission robot inspector [10]

2 Taut String Vibration While a Concentrated Load is Moving on it The target of this part is to investigate the dynamical behavior [22–35] of the robot inspector moving on the electrical line (Fig. 2). As shown in Fig. 2a the overhead power line is stretched between horizontal and vertical isolators which are modeled as pinned and slide supports. To construct the mathematical model of the system, the conductor is considered as a taut string and the robot as a concentrated load that moves along the conductor with the constant speed of v (Fig. 2b).

Fig. 2. (a) Schematic of an electrical line with a robot on it, and (b) concentrated load moves along the taut string

To conduct the statement for the deflection of the stretched string as the result of the movement of the point load on it, we use the expansion method. The kinetic energy (K) and the potential energy (U) of the string are known as:

(1)

32

M. R. Bahrami

where the tension in the string is labeled by T, ρ denoted to the mass per length, l is the string length, and EI is the flexural rigidity. In this article, prime and dot are used to show the differentiation with relative to the coordinate x and time t. Since our system is subject to generalized forces, here q(x, t), the extension of Hamilton’s principle through the virtual work expression δA = q(x, t)δu can be used. t Extension Hamilton’s principle is written as δ 0f (K − U + A)dt = 0. To obtain the equation of motion process as explained [23–25] is used: 

l

t

ρ u˙ δu|0f dx− · · · 0  tf    Tu + EIuIV δu − EIu δu |l0 dt+ · · · ··· − 

0

tf

··· + 0



l

(Tu − ρ u¨ − EIu  )δu dxdt + q(x, t) = 0.

(2)

0

The first of (2) is zero since at the time t = 0, and t = tf the variation δu is zero. Equating the summation of the third and fourth terms in (2) to zero yields to the dynamic equation for the string as: ρ u¨ + (EIu ) − Tu = q(x, t).

(3)

The boundary conditions are obtained by equating the second term in (2) to zero which are: x = 0 : u = 0, x = l : EIu = 0,

(4)

It should be noted that the initial conditions are u(x, 0) = u˙ (x, 0) = 0. As above mentioned, in this work, we consider the robot inspector as a point load P which is constant and moves stably with constant speed v. Therefore, to present the force located at point x from the origin we use Dirac delta function δ. Here, we have q(x, t) = P δ(x − vt). We rewrite (3) considering abovementioned force as: ρ u¨ + EIuIV − Tu = Pδ(x − vt).

(5)

To solve (5) we use two methods: method of eigenfunctions and Galerkin method. First, we start the solution by the method of eigenfunctions. The solution of (5) can be u(x, t) =

∞ 

un (t)ϕn (x),

n=1

ωn = λn c, ϕn = sin λn x, λn = nπ l

(6)

where un (t) are the unknown modal coordinates, ωn and ϕn are the eigenvalues and eigenfunctions respectively.

Mechanical Challenges of Inspection Robot

33

Calling the orthogonality properties of eigenfunctions,

l ϕn ϕk dx = δkn ≡ 0

1, k = n 0, k = n

l ⇒ un =

uϕn dx,

(7)

0

and multiplying both sides of (5) by ϕn ϕ n and integrating, ordinary differential equations (ODE) for un may obtain as: u¨ n + ω2n un = fn (t), ω2n = ( p=

T λ2n + EI λ4n ), ρ

P , ρ

fn (t) =

l 0

ϕn (x)q(x, t) dx 2 = p sin(λn vt). l 2 l 0 ϕn (x) dx

(8)

The solution of (8) can be written as: un (t) = C1 cos(ωn t) + C2 sin(ωn t) + · · · 2p sin(λn vt), ··· + l(ωn2 − λ2n v2 )

(9)

where C 1 and C 2 are arbitrary integration constants which can be found using initial values. Substituting (9) in (6) the solution of (5) can be found as:  u(x, t) = [C1 cos(ωn t) + C2 sin(ωn t)+ . . .  2p sin(λ vt) sin(λn x). (10) ... + n l(ωn2 − λ2n v2 ) Here, we can find the integration constants under zero initial conditions as C1 = 0 2pλn v and C2 = − lω (ω 2 −λ2 v 2 ) . Substituting in (10), we obtain the solution of problem (5): n

n

n

u(x, t) = =

 N 2p  sin(λn x) λn v sin(ω t) − sin(λ vt) . n n l λ2n v2 − ωn2 ωn

(11)

n=1

Solving (11) gives us the deflection of the stretched string (conductor) shown in Fig. 3. For calculation we use the following parameters: T = 10 kN, P = 1 kN, v = 2 m/s, ρ = 10 kg/m, l = 200 m, N = 1000, EI = 2.148 × 104 J. The results shown

34

M. R. Bahrami

(a)

(b)

Fig. 3. (a) General deflection, and (b) dynamic deflection of the electrical line with a moving load on it

in Fig. 3 are for two cases including EI and not including. As shown in results, the difference between model including EI and not including EI is negligible and results are in a good match with results obtained by expansion methods. As one can see from Fig. 3a graphs have a parabolic shape. This is because of the quasistatic deflection of the conductor which is also included. To omit this deflection, we need to set the time-functions in Eq. (3) to zero. This deflection from the results and show just the difference Δ in Fig. 3b. It is interesting to see although the speed of the robot is constant and stable, vibrations occur. The effect of flexural rigidity can be seen from Fig. 4. To validate the results obtained in the previous section, in this section, the approximate solution of (3) can be constructed by using Galerkin method [25] by rewriting of (6) in the matrix form: u(x, t) =

∞ 

un (t)ϕn (x) = UT (t)(x),

(12)

n=1

In (12) (x) which is known as comparison functions are orthogonal and orthonormal. Although (12) satisfies all boundary conditions, since there remains error, ¨ T Φ − T UT Φ  + EI UT   , e(x, t) = U

(13)

It will not satisfy (3) fully. On the other hand, one can make it have a zero projection on the chosen functions (x).  0

l

e(x, t)φ dx = 0.

(14)

Mechanical Challenges of Inspection Robot

35

Substituting (13) in (14), we have:

(15)

System (15) is an ordinary differential equation with constant coefficients which can be represented in the form of first-order equations:

˙ =Z U . (16) −1 Z˙ = M (F − (C + D)U) The result of (16) using parameters as before is shown in Fig. 4 (comparison of two methods). As one can see, results obtained by the Galerkin method are in a good match with results obtained by expansion methods. One of the most advantages of using the Galerkin method is one can use a different function to describe the motion of the robot inspector, i.e. not just using motion with the constant speed.

(a)

(b)

Fig. 4. (a) General deflection and (b) dynamic deflection of the electrical line with a moving load on it presented by two methods

3 Conclusion The mathematical modeling of the electrical transmission lines robot inspector while moving through the overhead power line with a constant speed including the flexural

36

M. R. Bahrami

rigidity of the line has been done. The expansion method and Galerkin method have been used to solve the model. Results show that the difference between model including EI and model not including EI is negligible. Also, results obtained by the Galerkin method are in a good match with the results obtained by expansion methods. The other important result of mathematical modeling is the appearance of vibrations of the line while the robot travels with a constant speed along with it. The mathematical modeling results are valuable for designers to select proper parameters for designing and constructing the robot.

References 1. Seok, K.H., Kim, Y.S.: A state of the art of power transmission line maintenance robots. J. Electr. Eng. Technol. 11(5), 1412–1422 (2016) 2. Aggarwal, R., Johns, A., Jayasinghe, J., Su, W.: An overview of the condition monitoring of overhead lines. Electr. Power Syst. Res. 53(1), 15–22 (2000) 3. Dong, G., Chen, X., Wang, B., Zhang, J., Liu, L., Wang, Q., Wei, C.: Inspecting transmission lines with an unmanned fixed-wings aircraft. In: 2012 2nd International Conference on Applied Robotics for the Power Industry (CARPI), pp. 173–174. IEEE (2012) 4. Toussaint, K., Pouliot, N., Montambault, S.: Transmission line maintenance robots capable of crossing obstacles: State-of-the-art review and challenges ahead. J. Field Robot. 26(5), 477–499 (2009) 5. Sawada, J., Kusumoto, K., Maikawa, Y., Munakata, T., Ishikawa, Y.: A mobile robot for inspection of power transmission lines. IEEE Trans. Power Delivery 6(1), 309–315 (1991) 6. Higuchi, M., Maeda, Y., Tsutani, S., Hagihara, S.: Development of a mobile inspection robot for power transmission lines. J. Robot. Soc. Japan 9(4), 457–463 (1991) 7. Tsujimura, T., Morimitsu, T.: Dynamics of mobile legs suspended from wire. Robot. Auton. Syst. 20(1), 85–98 (1997) 8. Alhassan, A.B., Zhang, X., Shen, H., Jian, G., Xu, H., Hamza, K.: Investigation of aerodynamic stability of a lightweight dual-arm power transmission line inspection robot under the influence of wind. Math. Probl. Eng. 2019, 16 (2019) 9. Miller, R., Abbasi, F., Mohammadpour, J.: Power line robotic device for overhead line inspection and maintenance. Ind. Robot Int. J. 44, 75–84 (2017) 10. Bahrami, M.R.: A novel design of an electrical transmission line inspection machine. In: Advances in Mechanical Engineering, pp. 67–73. Springer, Cham (2016) 11. Bahrami, M.: Mechanics of diagnostic machine on electrical transmission lines conductors. In: Matec web of conferences., vol. 224, pp. 02021 (2018) 12. Bahrami, M.R., Abed, S.A.: Mechanics of robot inspector on electrical transmission lines conductors: performance analysis of dynamic vibration absorber. Vibroeng. Proc. 25, 60–64 (2019) 13. Bahrami, M., Abed, S.: Mechanical challenges of electrical transmission lines inspection robot. In: IOP Conference Series: Materials Science and Engineering, vol 709, p 022099. IOP Publishing (2020) 14. Bahrami, M.R.: Mechanics of robot inspector on electrical transmission lines conductors with unequal heights supports. Vibroeng. Proc. 30, 20–25 (2020) 15. Bahrami, M., Abed, S.: Mechanical challenges of electrical transmission lines inspection robot. In: IOP Conference Series: Materials Science and Engineering, vol. 709, p. 022099. IOP Publishing (2020) 16. Asheichik, A.A., Zorin, D.K.: Research of the antifriction properties of carbon. In: Advances in Mechanical Engineering, pp. 17–25. Springer, Cham (2020)

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17. Evgrafov, A.N., Petrov, G.N.: Self-braking of planar linkage mechanisms. In: Advances in Mechanical Engineering, pp 83–92. Springer, Cham (2018) 18. Asheichik, A., Samsonenko, A.: Research of the driven capacity of pushers with rubber pads for oil equipment. In: IOP Conference Series: Materials Science and Engineering, vol. 666, p. 012002. IOP Publishing (2019) 19. Semenov, Y., Semenova, N.S.: Analysis of machine tool installation on the base. In: Advances in Mechanical Engineering, pp. 105–114. Springer, Cham (2017) 20. Evgrafov, A.N., Karazin, V.I., Khisamov, A.V.: Research of high-level control system for centrifuge engine. Int. Rev. Mech. Eng. 12(5), 400–404 (2018) 21. Evgrafov, A., Petrov, G.: Calculation of the geometric and kinematic parameters of a spatial leverage mechanism with excessive coupling. J. Mach. Manuf. Reliabil. 42(3), 179–183 (2013) 22. Evgrafov, A.N., Petrov, G.N.: Drive selection of multidirectional mechanism with excess inputs. In: Advances in Mechanical Engineering, pp. 31–37. Springer, Cham (2016) 23. Evgrafov, A.N., Petrov, G.N.: Computer simulation of mechanisms. In: Advances in Mechanical Engineering, pp. 45–56. Springer, Cham (2017) 24. Kamper, M., Bekker, A.: Non-contact experimental methods to characterize the response of a hyper-elastic membrane. Int. J. Mech. Mater. Eng. 12(1), 15 (2017) 25. Oden, J.T., Reddy, J.N.: Variational Methods in Theoretical Mechanics. Springer, Heidelberg (2012) 26. Asheichik, A., Bahrami, M.: Investigation of graflon antifriction feature under friction in seawater. Proc. Eng. 206, 642–646 (2017) 27. Bahrami, M.R., Ramezani, A., Osquie, K.G.: Modeling and simulation of non-contact atomic force microscope. Eng. Syst. Des. Anal. 49194, 565–569 (2010) 28. Bahrami, M.R., Abeygunawardana, A.B.: Modeling and simulation of tapping mode atomic force microscope through a bond-graph. In: Advances in Mechanical Engineering, pp. 9–15. Springer, Cham (2018) 29. Bahrami, M.R, Abeygunawardana, A.B.: Modeling and simulation of dynamic contact atomic force microscope. In: Advances in Mechanical Engineering, pp. 109–118. Springer, Cham (2019) 30. Bahrami, M.R.: Mechanics of electrical transmission line robot inspector: pendulum as a dynamic vibration absorber. In: E3S Web of Conferences, EDP Sciences, vol. 162, p. 03004 (2020) 31. Abed, S.A., Bahrami, M.R., Thijel, J.F.: Effect of two cracks in a rotor on stiffness using the theory of fracture mechanics. In: International Conference on Industrial Engineering, pp. 921–929. Springer, Cham (2019) 32. Bahrami, M.R., Suvorov, V.: Virtual non-contact atomic force microscope. In: IOP Conference Series: Materials Science and Engineering, IOP Publishing, vol. 666, p. 012008. Springer, Cham (2019) 33. Evgrafov, A.N., Karazin, V.I., Petrov, G.N.: Analysis of the self-braking effect of linkage mechanisms. In: Advances in Mechanical Engineering, pp. 119–127. Springer, Cham (2019) 34. Semenov, Y.A., Semenova, N.S.: Study of mechanisms with allowance for friction forces in kinematic pairs. In: Advances in Mechanical Engineering, pp. 169–179. Springer, Cham (2019) 35. Zlatanov, V.D., Nikolov, S.G.: Vibrations of a chain in the braking regime of the motion mechanism in load-lifting machines. In: Advances in Mechanical Engineering, pp. 221–232. Springer, Cham (2019)

Development and Research of Mechatronic Spring Drives with Energy Recovery for Rod Depth Pumps Valentina P. Belogur1 , Victor L. Zhavner2 , Milana V. Zhavner2(B) , and Wen Zhao2 1 LLC FIRM “SPRING-CENTER”, Moscow, Russia

[email protected] 2 Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

[email protected]

Abstract. When developing the theoretical basis for the design of technological equipment and its individual units that perform reciprocating movements, the development of a fundamentally new type of drive for rod depth pumps is particularly promising in terms of reducing energy consumption, which will also reduce the weight, dimensions and cost of foundations for their installation. The development of theoretical foundations for designing equipment with minimal energy costs will allow us to develop both design algorithms and evaluation of technological equipment in terms of energy costs. The relevance of the task of reducing energy consumption by technological equipment is that there is currently no systematic approach to identifying and evaluating the components of energy costs. Reducing energy costs, for example, by three times, will reduce the required drive power by nine times. This applies primarily to oil production and loading and unloading operations. Keywords: Energy saving · Mechatronics · Spring drive · Energy recovery · Cyclic operation · Rod depth pumps · Inertia forces · Balancing devices · Performance · Speed

1 Introduction Reducing energy consumption while ensuring the growth of the world economy requires an integrated approach to solving environmental problems, ensuring food security while reducing energy costs. It must be recognized that the main consumer of energy is technological equipment in any industry. Considering the life cycle of technological equipment and the infrastructure that ensures its operation, it can be noted that the main characteristics that determine the energy consumption at the stage of its manufacture is the mass of equipment. During operation, the mass of equipment determines the energy consumption for the construction of buildings and maintaining their working condition. The mass of its moving parts, © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 38–51, 2021. https://doi.org/10.1007/978-3-030-62062-2_5

Development and Research of Mechatronic Spring Drives

39

overall dimensions, drive system and installed power determine the current energy consumption. In a comparative analysis of various types of technological equipment of the same functional purpose, all of the above characteristics, referred to a unit of production, can serve as criteria for assessing their energy efficiency [1–12]. Energy consumption by technological equipment is determined by the work spent on performing technological operations, work associated with dissipative losses, work caused by inertial forces, work spent on overcoming gravity forces and energy consumption for control systems [1]. In Fig. 1 shows a diagram of energy consumption in modern technological equipment.

Fig. 1. Scheme of energy consumption in process equipment

For a given technological process, a significant part is made up of energy costs associated with overcoming gravity and inertial forces, and in the presence of working operations associated with reciprocating and reciprocating movements of the working masses, these operations mainly determine the energy consumption. In work [1] it is shown that with an increase in productivity, for example, 2 times, energy consumption increases by 4 times, and the required power by 8 times. In [11, 13], a relationship was established between the coefficient of friction, time and the amount of displacement, which determines the equality of the work spent on acceleration and deceleration of inertial masses and on overcoming dissipative losses. The first mechatronic spring drive with energy recovery can be considered a device patented in the USSR in 1977 by the Swedish company Electrolux [2]. The spring accumulator in this device is similar to the spring accumulator described in the description of the German patent obtained by L. Szilard and A. Einstein “Electromechanical device for obtaining oscillatory movements” [3]. The device was intended for refrigeration compressors and used a linear electric motor and a linear spring accumulator. This patent was later purchased by Electrolux. When considering the Swedish patent device, it can be considered as a mechanical device, which includes the concept of “mass” according to the work [4]. The contents and equations of “energy” and “information” should be designed to take into account the relationship between existing information, mass and energy. After the demonstration of this device in Moscow at the robotics symposium, many technical solutions appeared for the development of drives for industrial robots [7–12]. Works [11, 12] summarize theoretical work on the design of mechatronic spring drives with energy recovery for industrial robots. It should be noted that industrial robots consume high energy. First of all, this is determined by the transfer of products along complex trajectories with given laws of motion and with large masses of manipulation

40

V. P. Belogur et al.

systems. In this regard, spring balancing devices that reduce workloads in industrial robot drive systems are widely used [14]. It is necessary to note a variety of terms in the development of the above drives. In this paper, the authors use the term “mechatronic spring drive with energy recovery”, which clearly defines its essence: 1. The main engine is a spring accumulator, which accelerates the working masses due to stored energy, and then slows them down, accumulating potential energy; 2. The presence of a control system that ensures the switching on and off of motors to compensate for dissipative losses and control of the clamps to ensure technological stability; 3. There is a motor for compensation of dissipative losses, the power of which is several times less than the power of the spring accumulator and which is determined by the magnitude of the dissipative losses. In addition to industrial robots, mechatronic spring drives have become widespread for auxiliary work operations [14–21]. An interesting direction is the use of standard pneumatic cylinders with return springs for mechatronic spring actuators with energy recovery. With this construction of the mechatronic drive, the springs determine the operating modes, and the pneumatic cylinders provide compensation for dissipative losses and fixing the output link in the extreme positions [22–29]. In [30–32], a comparative analysis of the characteristics of electromechanical, pneumatic, hydraulic and spring drives with energy recovery, intended for working operations associated with the reciprocating movement of the working masses, is carried out. The maximum displacement value was 2000 mm, the maximum weight was 8000 kg. Mechatronic spring actuators with energy recovery have the lowest energy consumption compared to all other types of actuators, have the best power density (the power developed by the actuator in relation to its weight - W/N). In addition, they have the following advantages: 1. 2. 3. 4. 5. 6.

High reliability; Long service life; Low noise; Good fire resistance and explosion safety; Insensitivity to temperature changes; Favorable dynamic modes.

In terms of maximum overall dimensions, along the direction of travel, they can compete with electromechanical modules for linear movements using a screw drive. Energy recovery linear spring actuators have a smaller linear dimension in the direction of travel compared to actuators, pneumatic cylinders and hydraulic cylinders, and have the same performance as linear modules with electrically driven movable tables. The paper sets the task of developing a technical ideology for designing a mechatronic spring drive when moving working bodies in a vertical plane with a displacement of up

Development and Research of Mechatronic Spring Drives

41

to 2 meters and a moving mass of up to 8000 kg. These characteristics have rod deep (borehole) pumps, which are operated in the world at least 10000 pieces. Considering the operating modes of the sucker rod pump, it should be noted that the working movements occur in the vertical plane, as a rule, within the range of up to 2 m and with a load mass of up to 8000 kg. The number of cycles per minute is approximately 15. Unlike the mechatronic drives discussed above, where movements are carried out in the horizontal plane, in the rod deep (borehole) pump, the movement occurs in the vertical plane and the main loads are gravity and inertial loads. The classic version of the sucker rod pump (borehole) pump is a drive with an output link that performs a reciprocating movement, and on one side of this link a block segment with a flexible element is fixed, which in turn is connected to the pump rod. On the other hand, the output link is connected to the counterweight. The vast majority of oil pumping units are made according to this scheme. In recent years, other types of sucker rod (borehole) pump drives have appeared, of which we will single out the following types: 1. 2. 3. 4.

Actuators with pneumatic balancing systems; Drives using hydraulic cylinders without balancing system; Actuators with electric and cargo balancing; Drives with linear electric motors with load balance [33].

2 Methods and Materials The paper considers a mechatronic spring drive with energy recovery, the diagram of which is shown in Fig. 2. The drive consists of a spring accumulator, a motor to compensate for dissipative losses, a balancing device and a control system with information and measurement components.

Fig. 2. Block diagram of the mechatronic spring drive

As a motor to compensate for dissipative losses, it is possible to use standard motors with a reciprocating output link. In this work, it is proposed to use linear motors with a reciprocating rotor [33].

42

V. P. Belogur et al.

Energy costs in mechatronic spring drives are determined mainly by the losses associated with internal friction in the springs [11] and are approximately five times less than the energy costs of standard classical drives. In this regard, for initial calculations, the work spent on overcoming dissipative losses can be taken as equal to 20% of the maximum potential energy of the spring battery. When creating mechatronic spring drives, standard compression springs are used, the production of which has long been mastered in Russia. A study of the characteristics of spring drives with a central location relative to the wellhead was carried out. Figure 3 shows the scheme of the spring drive with the Central location of the rod and load balancing.

Fig. 3. Diagram of a spring actuator with a center bar and balancing: 1 - output link, 2 - springs, 3 - base, 4 - plate, 5 - couplers, 6 - rod, 7 - traverse

The dynamic drive model is shown in Fig. 4. The relationship between the main parameters of the spring drive is determined by the equation that determines the travel time of the output from one extreme position to another. The time of moving the working body from one extreme position to another,

Development and Research of Mechatronic Spring Drives

43

Fig. 4. Dynamic model of a spring accumulator with two compression springs

for a linear spring accumulator with two springs, without taking into account dissipative losses, is determined from the expression:  m (1) t = π 2c Time t and mass m are the main characteristics of the working operation performed by the drive. The boom travel is also a characteristic of the drive. The ideology of the spring drive allows it to be used while maintaining the speed of the drive at shorter strokes. When designing the actuator, the required spring rate c is determined from Eq. (1). c =

π2 · m/2 t2

(2)

Knowing the required displacement h, the maximum working force of the spring is determined: F2 = ch

(3)

h , s3

(4)

And the operating number of turns: n=

where s3 is the accepted maximum deflection of one turn. Table 1 shows the characteristics of spring accumulators, which are proposed to be used in the development of drives for deep rod pumps. The energy characteristic of a spring is the maximum potential energy accumulated by the spring in the maximum compressed position: V = 0, 5ch2

(5)

Then the maximum work that should be spent on compensating for dissipative losses will be equal to: AD = 0, 2 V

44

V. P. Belogur et al. Table 1. Characteristics of spring accumulators m, kg

2000 3000 4000 6000

8000

h, mm

1000 1000 1500 2000

2000

F2 , N

3500 3550 5760 13670 14780

c, N/m

2580 3550 3840 6800

7390

t, s

1,96 2,10 2,27 2,08

2,31

n

25

20

38

25

25

D, mm

125

140

180

260

320

d, mm

10

11

16

22

L, mm

2200 2500 3100 3400

4mspring , kg 30

76

80

140

25 3400 180

Determine the required drive characteristics to compensate for dissipative losses. In most cases, the compensation of dissipative losses for self-oscillating systems is carried out in accordance with the Erie theorem [33–35]. An indispensable requirement in the construction of systems for compensating dissipative forces for a mechatronic regenerative drive is the following condition: the motor operation to compensate for the dissipative forces of a linear spring accumulator ALP must be equal to or slightly exceed the work it spends on overcoming the dissipative forces of AD . ALP ≥ AD In accordance with the Erie theorem, when a compensatory force is applied at the beginning of a period, the travel time for a given distance decreases, at the end of the period increases, and when an impact occurs in a time interval symmetrical to the point of stable equilibrium, the time is determined by the half-period of the oscillating system. In the same way, the half-period of an oscillating system will be preserved if two identical pulses are given at the beginning and at the end of the half-period. Graphically, the Erie theorem is shown in Fig. 5.

Fig. 5. Graphic representation of the Erie theorem

From the point of view of reducing the power of the compensation engine, it is preferable to include the force action either during the entire half-cycle or at the beginning

Development and Research of Mechatronic Spring Drives

45

and end of the stroke. The limit in this case is the maximum speed developed by the compensation engine. If the maximum speed of the compensation motor x˙ ld is less than the maximum speed of the spring accumulator, it is advisable to use the option with constant activation during movement. The maximum speed of a spring accumulator without taking into account dissipative losses is determined from the expression [32]. x˙ max =

π xmax t

(6)

If the maximum speed of the compensation motor x˙ ld is less than the maximum speed of the spring battery, then switching it off and on must be at the points where their speed characteristics intersect (Fig. 6).

Fig. 6. Speed characteristics of linear and spring drives

Traditionally, in drives of sucker rod (borehole) pumps, cargo balancing is used [36, 37]. To balance systems with a constant load characteristic, a device can also be used in which the load is applied to a cam of a variable radius, rigidly connected to a block of constant radius, on which a cable is fixed, connected on the other side with a cylindrical helical spring [12, 38]. In addition, the balancing of a constant static load can be provided with the help of a crank-slider mechanism, in which the length of the connecting rod is equal to the length (radius) of the crank, and the crank is equipped with a spring accumulator with a torque characteristic that changes according to a sinusoidal law [13]. The implementation of such a device is ensured by the serial connection of the crankslider mechanism with the crank-rocker mechanism, and the crank is their common element, and the length of the crank is equal to the length of the connecting rod and the distance between the axes of the hinge joints of the rocker mechanism with the base. This size is designated l. In addition, the rocker is made in the form of a flexible element enveloping two blocks, one of which is installed on the crank axis, and the second on the base, and the distance between the block axis and the axis of the crank connection with the base is equal to the crank length (Fig. 7). In the configuration of the balancing device, when the axes O1 and O2 coincide, the force of the springs should be zero, and in the configuration when the distance between

46

V. P. Belogur et al.

Fig. 7. Diagram of a balancing device for driving an oil pump: 1 - swing arm, 2 - block, 3 - flexible element, 4a, 4b - blocks fixed on the base, 5,6 -springs

the axes O2 and O3 is maximum and equal to 3l, the maximum force of the two springs is equal to: F2 = 4mg

(7)

and the total stiffness of the two springs should be equal to: c =

2mg l

(8)

With such characteristics of the springs, we obtain a sinusoidal moment characteristic that ensures the balance of the mass of the oil pumping rod (Fig. 8).

Fig. 8. Sinusoidal moment characteristic of the balancing device

We determine the reduced mass to the axis of the pivot joint of the rod with the balancing device from the mass of its links. The masses of the links of the balancing mechanism can be replaced by three concentrated masses [39]:

Development and Research of Mechatronic Spring Drives

47

1. Mass m1 – in the crank pin (O1 axis). 2. Mass m2 – progressive moving elements of the rocker mechanism: flexible elements, springs, and a mass of other elements moving at the speed of flexible elements. 3. The mass of the balancing devices swivel joint with the rod. The masses of the articulated joint with the rod must be taken into account in the verification calculations of the drive The mass m1 depends on the mass of the structural elements of the swivel joint: bearings, axles, covers, etc., the mass of the blocks and the mass of the flexible elements enveloping the block. The kinetic energy equation for determining the reduced mass to the output link of the spring drive is: ˙2

m2 · S3 J q˙ 2 mreduced Z˙ 2 = + 2 2 2

(9)

Where Z˙ – speed of the connection point of the spring accumulator with the rod, J – is the moment of inertia of the crank, q˙ – the angular speed of the crank, S˙ – is the rate of spring deformation. The moment of inertia of the crank is equal to J = m1 l 2

(10)

The dependence of the angular speed of the crank and the speed of the rod has the form q˙ =

Z˙ 2l sin q

(11)

The relationship between crank angular velocity and spring deformation velocity is determined. The equation used to determine the current distance between O1 and O2 is S = 2l cos

q 2

(12)

The variables in this equation are S and q. The relationship between spring deformation velocity and crank angular velocity is obtained by derivative q S˙ = q˙ l sin 2

(13)

Taking into account Eqs. (9–11), the reduced mass of the balancing device to the axis of the articulated joint with the rod is determined from the following expression: mreduced =

m1 4 · (sin q)

2

+

m2 2  16 · cos 2q

(14)

48

V. P. Belogur et al.

The main part of the estimated mass is the converted component of m2 mass of the spring mass. The mass of the spring has an amount much greater than the mass of all elements of the balancing device. The weight of the rod m determines the maximum force of the spring, in accordance with Eq. (7), and the maximum working stroke allows to determine the mass of the spring, depending on the accepted dimensions of the balancing device. For example, for a bar with a mass of m = 8000 kg, two balancing devices must be used (Fig. 7) and the total force of the four springs must be equal to 320000 N, which ensures the exact sinusoidal torque characteristic shown in Fig. 8. In this case, with a working elongation of the spring equal to 2 l, the total mass of the springs will be 4640 kg. In Fig. 9 shows a graph of the change in the coefficient K, which determines the component of the reduced mass from m2 . Its variation ranges from 0.0625 to 8.228.

Fig. 9. The law of change of the coefficient K depending on the angle q

Consider the mass range - 2000, 3000, 4000, 6000 and 8000 kg. The maximum spring forces for this series of masses are, respectively, 80000, 120000, 160000, 240000 and 320000 N. When calculating the reduced mass, one third of the spring mass is taken into account. For a bar weight of 8000 kg, the total weight of the springs is 2320 kg. Table 2 shows the main characteristics of the springs of balancing devices using springs in accordance with GOST 13773-86 (ISO/DIS 22705). Table 2. The main characteristics of the springs of balancing devices Maximum spring force, N

Weight of four springs, kg

l, mm

h, mm

m, kg

1

320 000

2320

1250

2000

8000

2

240000

1720

1250

2000

6000

3

160 000

1290

1000

1500

4000

4

120 000

750

1000

1500

3000

5

80 000

530

1000

1500

2000

Development and Research of Mechatronic Spring Drives

49

3 Results and Conclusion The result of the work is the determination of the characteristics of spring accumulators and spring balancing devices, on the basis of which, in various combinations, mechatronic spring actuators with energy recovery can be realized with a significant reduction in both energy consumption and installed power. The authors would like to draw the attention of both oil producers and designers of oil pumping units to the advantages of mechatronic spring drives with energy recovery. As part of further research, a design ideology will be developed that allows minimizing the dimensions of the links and ensuring a decrease in dissipative losses. The only major challenge is the choice of motors to compensate for dissipative losses. The authors hope that the engineering community will pay attention to the research data. The use of mechatronic spring drives in rotators and lifting platforms is also a fairly promising direction.

References 1. Levin, A.I.: Mathematical modeling in research and design of machine tools. Engineering, 184 (1978) 2. Patent No. 568346 (USSR) Mechanical arm / Geran Arvid Henning Readerstram.Application. 02/18/74, B.I. №29 (1977) 3. Patentschrift №562040 Dr. Leo Szilard und Dr. Albert Einstein. Elertromagnetische Vorrichtung zur Erztugung einer oszillierenden Bewegungen. 1933 Prioritatsdatum 1.Juni (1928) 4. Mechatronics: Per. from Japanese/ T. Ishii, I. Simoyama, H. Inoue et al.-M.: Mir, p. 318 (1988) 5. Patent RF 1006208. Mechanical arm/ L.M. Bolotin, A.I. Korendyasev, B.L. Salamander, L.I. Tyves. B.I. No. 11 (1983) 6. Patent R.F. No. 1110623. The mechanical arm/ L.M. Bolotin, A.I. Korendyasev, B.L. Salamander, L.I. Tyves. BI. № 32 (1984) 7. Patent R.F. No. 1323378. The mechanical arm/ L.M. Bolotin, A.I. Korendyasev, B.L. Salamander, L.I. Tyves. BI. –1987. -№ 11. B.I. No. 11. (1987) 8. Patent R.F. 1544550. Resonant drive/V.I. Babitsky, A.A. Kotlyachkov, B.L. Salamander, V.A. Checherov, A.V. Shipilov, V.N. Panin. B.I № 7 (1990) 9. Akinfiev, T.S.: Resonant handling systems with electric drive. Eng. Sci. 6, 18–23 (1983) 10. Akinfiev, T.S., Pozharinsky, A.A.: Dynamics and stability of a resonant manipulation system, with double-sided pneumatic drive. Engineering, pp. 279–286 (1984) 11. Korendyasev, A.I., Salamander, B.L., Tyves, L.I.: Theoretical foundations of robotics. In: 2 book, Science (2006) 12. Korendyasev, A.I., Salamander, B.L., Tyves, L.I., et al.: Manipulation systems of robots. Mechanical Engineering, pp. 279–286 (1989) 13. Pelupessi, D.S., Zhavner, M.V.: Spring accumulators with an output rotary link for step movements. News of higher educational institutions. Engineering, 10(679), 9–17 (2016) 14. Zhavner, M.V.: Methods for calculating and designing actuators of robotic systems based on spring mechanisms: abstract of diss. Cand. tech. sciences. SPB.: Publishing house of SPbSPU (2003). 18 p 15. Salamandra, B.L.: Analysis of ways to stabilize the position of the label on automatic packaging machines. Probl. Mech. Eng. Mach. Reliab. 2, 106–112 (2017)

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16. Nadezhdin, I.V., Mochanov, A.A.: Dynamics of mechatronic regenerative drives of loading devices of automated assembly systems. Bulletin of the Rybinsk State Aviation Technological Academy named after P. A. Soloviev, № 1(32), 19–24 (2015) 17. Nadezhdin, I.V.: Executive mechanisms of cyclic automatic machines and mechatronic systems. In: Nadezhdin, I.V (ed.) LAP LAMBERT Academic Publishing, Deutschland (2015). 280 p 18. Nadezhdin, I.V.: Highly dynamic mechanisms of auxiliary operations of automated assembly plants. Mechanical Engineering, Moscow (2008). 270 p 19. Nadezhdin, I.V., Mochanov, A.A.: Experimental studies of the dynamics of mechatronic loading devices with energy recovery. Fundament. Mech. 2, 54–58 (2017) 20. Sysoev, S.N., Glushkov, A.A.: Cyclical drives of oscillatory type: monograph. Feder. Education Agency, Vladimir. State un-t - Vladimir: Publishing house Vladimir. State University, p. 184 (2010) 21. Semenozhenkov, V.E.: Development of the theory, design methods and the creation of recuperative means of mechanization of forging and stamping production; dis. Doctor of Technical Sciences. 03/05/05./ Semenozhenkov V.S. Voronezh, (1999). 366 p 22. Zhavner, V.L., Matsko, O.N.: Spring drives for reciprocal motion. J. Mach. Manuf. Reliab. 45(1), 1–5 (2016) 23. Zhavner, V.L., Zhao, V., Yan, Chuanchao, Wu, Lun: Mechatronic regenerative drives for reciprocating movements based on pneumatic cylinders with return springs. Modern Eng. Sci. Educ. 4, 476–486 (2019) 24. Zhavner, M.V., Yan, Chuanchao, Zhao, Wen: Mechatronic regenerative drives for step movements based on pneumatic cylinders with return springs. Bull. Int. Acad. Refrig. 2, 22–28 (2019) 25. Zhao, V., Zhavner, V.L., Smirnov, A.B., Yan, C.: Application of pneumatic cylinders with return springs in mechatronic regenerative drives, scientific and technical statements of SPbPU. Nat. Eng. Sci. 25(1), 111–123 (2019) 26. Zhao, W., Zhavner, V.L.: The use of pneumatic cylinders with a return spring to compensate for balance losses in mechanical regenerative drives for reciprocating movements. In: 6th International BAPT Conference “POWER TRANSMISSIONS 2019” VARNA, vol. 1, p 107–112 (2019) 27. Zhavner, V.L., Zhao, W., Long, W.: Investigation of a mechatronic device for unwinding and pulling rolled materials in discrete mode. Izv. Universities. Instrument making. T63, No. 4. pp. 322–329 (2020) 28. Chuanchao, Y., Wen, Z.: The use of pneumatic cylinders with return springs when creating mechanical drives with recuperative energy, 6-th international BAPT conference “POWER TRANSMISSIONS 2019”. VARNA 1, 163–167 (2019) 29. Zhavner, V.L., Zhao, W., Yan, C., Long, W.: Research and development of a spring drive with recovery energy in the presence of a variable inertial load. In: Advances in Mechanical Engineering. Selected Contributions from the Conference, « Modern Engineering: Science and Education » , Saint Petersburg, Russia, June 2019, vol. 1, pp 209–220 (2019) 30. Sarvarov, A.S., Vasiliev, A.E., Danilenko, K.V., Menshikova, E.V.: Comparative analysis of mechatronic systems drives. Theory Pract. Automat. Prod. 4(25), 21–24 (2014) 31. Brylina, O.G., Languages, Yu.S.: To a comparative analysis of pneumatic, hydraulic and electric drives. Energy and resource saving. Tr. 4. No. 1, South Ural State University (2016) 32. Zhavner, V.L., Matsko, O.N., Zhavner, M.V.: Comparative analysis of mechatronic drives for reciprocal motion. Int. Rev. Mech. Eng. (I.RE.ME.) 12(9), 784–789 (2018) 33. La formule, K.G.: dˆAiry.-Mem. Acad. Bel., vol. 5 №. 11 (1897) 34. Bautin, N.N.: Dynamic clock theory. Nauka (1986). 192 p

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35. Zhavner, M.V., Li, S., Yan, C.: Research of Spring Accumulators with Output Rotary Link Used in Technological Equipment to Reduce Energy Consumption// Advances in Mechanical Engineering Selected Contributions from the Conference “Modern Engineering: Science and Education”, pp. 221–232. Russia, June, Saint Petersburg (2019) 36. Shen, F., Sun, P., Wang, T.: CN2284867U Y, 1998. 06. 24/ Linear motor driven beam-pumping unit (1998) 37. Zhang, R., Zhao, K., Zhao, X.: CN-0561759, 2015-09-07, Submersible oil well pump set having underground drive conversion device (2015) 38. RF patent No. 529074 from 25. 09. 1976. Device for vertical movement/ Ivkin A. M., Petrov L. N., Polishchuk N. F. (RU) 39. Mechanics of machines. T. 2. Kinetostatics and dynamics of machines. Friction in cars. Kolchin N.I. L., Mechanical engineering (1972), 456 p

Solving the Problem of Decomposition of an Orthogonal Polyhedron of Arbitrary Dimension Vladislav A. Chekanin(B) and Alexander V. Chekanin Moscow State University of Technology «STANKIN», Moscow, Russia [email protected]

Abstract. The paper is devoted to the problem of decomposition of an orthogonal polyhedron of arbitrary dimension into a set of large orthogonal objects. To solve this problem, an algorithm has been developed based on the application of the model of potential containers which is used in solving of orthogonal packing problems of arbitrary dimension. Examples of decomposition of two-dimensional and three-dimensional orthogonal polyhedrons are given. Keywords: Orthogonal polyhedron · Decomposition · Model of potential containers · Orthogonal object · Packing problem

1 Introduction When solving the problem of packing D-dimensional objects of arbitrary geometric shape, the placed objects can be represented as orthogonal polyhedrons, which are sets of orthogonal objects (D-dimensional parallelepipeds) with a fixed position relative to each other [1]. This problem has a large number of practical applications in various fields, including the problems of cutting industrial materials and the layout problems of geometrically complex objects [2–5]. It is obvious that as the number of orthogonal objects composing an orthogonal polyhedron increases, the time spent on checking the possibility of placing this orthogonal polyhedron in a given area of a container also increases [1, 6]. This problem is most acute when using the voxel model [7, 8], which provides a description of an orthogonal polyhedron with a large number of equal-dimensional cubes. Therefore, to increase the speed of formation a packing, it is necessary to decompose an orthogonal polyhedron, which results in a set of large orthogonal objects that fill the same area as the original set of orthogonal objects of the orthogonal polyhedron. The problem of decomposition of an orthogonal polyhedron into a minimal set of large orthogonal objects is a special case of the NP-hard problem of covering an orthogonal polyhedron with orthogonal objects [9], and it is NP-hard even for an orthogonal polyhedron without holes [10]. The following algorithms are used to decompose an orthogonal polyhedron. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 52–59, 2021. https://doi.org/10.1007/978-3-030-62062-2_6

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1. Level selection of areas The algorithm described in articles [11, 12] performs several iterations of passing through the cells of the matrix describing the area in the form of a set of levels occupied by an orthogonal polyhedron [13] (for the horizontal direction of union, the matrix cells are analyzed by rows, for the vertical direction – by columns, for the diagonal direction – both by rows and columns), with their subsequent union into large rectangular objects. The disadvantage of this algorithm is that it gives different results when using different directions of union the cells of the matrix, while there is no information about the effectiveness of the use of individual directions of the union. This algorithm is presented in [11] only in the two-dimensional case. 2. Projections creation The algorithm presented in the article [14] is based on obtaining orthogonal objects as a result of constructing projections of the faces of an orthogonal polyhedron on a certain base plane. The problem of histogram decomposition in the two-dimensional case is also discussed in the article [15], in the three-dimensional setting – in the article [16]. The algorithm is implemented only for orthogonal polyhedrons, which are represented as sets of orthogonal objects that have a common base (so-called histograms), which significantly limits the scope of its practical application. 3. Cutting off orthogonal objects with subsequent merging The article [17] proposes an algorithm based on cutting off orthogonal objects from an orthogonal polyhedron by secant planes with subsequent merging of these objects into large ones during a series of passes along each coordinate axis. In the process of cutting off, several secant planes perpendicular to different coordinate axes are used. The main disadvantage of this algorithm is the appearance of a large number of small intermediate orthogonal objects, which slows down the algorithm and requires additional computing resources to process them. Another disadvantage of this algorithm is that it is a multi-pass algorithm and it requires passing along each coordinate axis to merge orthogonal objects. 4. Using graphs The algorithms of this group are based on dividing an orthogonal polyhedron into a series of rectangular areas, for which a graph is subsequently compiled, that determines neighborhood of these areas to each other. The solution to the problem of minimizing the number of orthogonal objects in an orthogonal polyhedron is reduced to solving the problem of finding a graph with the best characteristics. Decomposition algorithms using graphs and trees are found in a large number of publications [15, 18, 19], but all of them are used only for decomposition of two-dimensional orthogonal polyhedrons. As a result of the analysis of the basic algorithms used for the decomposition of an orthogonal polyhedron, the following conclusions were drawn:

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– most algorithms are focused on applying to two-dimensional orthogonal polyhedrons (orthogonal polygons); – many algorithms perform the decomposition of an orthogonal polyhedron, for which a number of restrictive conditions are mandatory (for example, the presence of a common base for all objects, the absence of any holes, etc.); – there is no algorithm providing decomposition of an orthogonal polyhedron of arbitrary dimension. In this paper, we propose an algorithm for decomposing an orthogonal polyhedron of arbitrary dimension into a set of large orthogonal objects, based on solving the orthogonal packing problem using the developed model of potential containers [20, 21]. This model describes the entire free space of a container with packed objects as a set of all possible orthogonal areas with the maximum overall dimensions (so-called potential containers) that can be used for subsequent placement of objects inside them.

2 The Developed Decomposition Algorithm of an Orthogonal Polyhedron of Arbitrary Dimension We will consider an orthogonal polyhedron V consisting of m orthogonal objects in the form parallelepipeds ok , k ∈ {1, . . . , m} with the overall dimensions   1 of2 D-dimensional wk ; wk ; . . . ; wkD (the superscript in the formulas is used to denote the dimension), the position of which  the local coordinate system of the orthogonal polyhedron  is specified in using vectors zk1 ; zk2 ; . . . ; zkD . The decomposition algorithm of a D-dimensional orthogonal polyhedron V into a set of large orthogonal objects includes the following steps. Step 1. Create orthogonal container No. 1 with the over an empty D-dimensional  all dimensions W11 , W12 , . . . , W1D matching the dimensions of a D-dimensional parallelepiped bounding the original orthogonal polyhedron V (W1d = S d , ∀ d ∈ {1, . . . , D}, where S d is the length of the packing measured along the axis d (S d =  d d max zk + wk , ∀d ∈ {1, . . . , D}, k ∈ {1, . . . , m}). Step 2. Place the orthogonal polyhedron V into container No. 1. dimensions   As a result, a set of potential containers 1 with the overall 1 2 D 1 2 D wk1 , wk1 , . . . , wk1 , k1 ∈ 1 located at points pk1 , pk1 , . . . , pk1 will be formed in container 1. The set of potential containers 1 describes the space of container No. 1, which does not belong to the placed orthogonal polyhedron V . Step 3. Create an empty D-dimensional orthogonal container No. 2 with the overall dimensions that match the overall dimensions of container No. 1 (W2d = W1d , ∀d ∈ {1, . . . , D}). Step 4. Place in container No. 2 a set of D-dimensional orthogonal objects with parameters matching the parameters of potential containers from container No. 1 (when placing objects, their mutual overlap is allowing): xid = pkd1 and wid = wkd1 ∀d ∈ {1, . . . , D}, i = 1, . . . , K1 , where K1 – cardinality of the set 1 (the number of potential containers in the container No. 1). As a result  2, a set of potential  in the container No. 1 2 D containers 2 with the overall dimensions wk2 , wk2 , . . . , wk2 , k2 ∈ 2 , located at

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  points pk12 , pk22 , . . . , pkD2 , will be formed. The set 2 describes the space of container No. 2, which belongs to the orthogonal polyhedron V . Step 5. Create a D-dimensional orthogonal polyhedron V  , consisting of orthogonal objects with parameters matching the parameters of potential containers from the container No. 2: zid = pkd2 and wid = wkd2 ∀d ∈ {1, . . . , D}, i = 1, . . . , K2 , where K2 – cardinality of the set 2 . The orthogonal polyhedron V  will contain the set of all orthogonal objects into which it can be decomposed. Step 6. Apply the set-theoretic addition operation [22] to all orthogonal objects that are part of the orthogonal polyhedron V  . As a result, an orthogonal polyhedron O will be obtained, consisting of orthogonal objects that do not overlap each other. Since the addition operation is applied to the set of all possible orthogonal objects into which the original orthogonal polyhedron V can be decomposed, ordered in descending order of the volumes occupied by them, the resulting orthogonal polyhedron O will contain the largest possible orthogonal objects.

3 Results of the Application of the Developed Algorithm Consider examples of decomposition of orthogonal polyhedrons of various dimensions created using the voxel model. Example 1. The orthogonal polyhedron in the form of an ellipse with a hole, shown in Fig. 1a, consists of 1398 identical orthogonal objects with the overall dimensions {1; 1} and is described by a set of inequalities:  2(x − 50)2 + (y − 25)2 − 150 ≥ 0; (x − 50)2 + 2(y − 25)2 − 800 < 0, where x ∈ [0; 100], y ∈ [0; 50]. In Fig. 1b and Fig. 1c the contents of containers No. 1 and No. 2 are shown (potential containers are shown in gray without filling). As a result of the developed decomposition algorithm, an orthogonal polyhedron was obtained, consisting of 72 orthogonal objects, shown in Fig. 1d. Example 2. The orthogonal polyhedron in the form of a pyramid, shown in Fig. 2a, consists of 2653 identical orthogonal objects with overall dimensions {1; 1; 1} and is described by the inequalitiy x2 + y2 − 2z 2 ≤ 0, where x ∈ [−10; 10], y ∈ [−10; 10], z ∈ [−10; 0]. As a result of decomposition, an orthogonal polyhedron was obtained, consisting of 89 orthogonal objects, shown in Fig. 2b. Example 3. The orthogonal polyhedron in the form of a ball, shown in Fig. 3a, consists of 5233 identical orthogonal objects with the overall dimensions {1; 1; 1} and is described by the inequalitiy x2 + y2 + z 2 − 100 ≤ 0, where x ∈ [−10; 10], y ∈ [−10; 10], z ∈ [−10; 10].

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Fig. 1. Decomposition of a two-dimensional orthogonal polyhedron: a original orthogonal polyhedron; b container No. 1; c container No. 2; d resulting orthogonal polyhedron

Fig. 2. Decomposition of a three-dimensional orthogonal polyhedron in the form of a pyramid: a original orthogonal polyhedron; b resulting orthogonal polyhedron

As a result of decomposition, an orthogonal polyhedron was obtained, consisting of 181 orthogonal objects, shown in Fig. 3b. Example 4. In Fig. 4a an example of an orthogonal polyhedron of a geometrically complex shape, consisting of 14286 identical orthogonal objects with the overall dimensions {1, 1, 1}, is presented. As a result of decomposition, an orthogonal polyhedron was obtained, consisting of 114 orthogonal objects, shown in Fig. 4b.

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Fig. 3. Decomposition of a three-dimensional orthogonal polyhedron in the form of a ball: a original orthogonal polyhedron; b resulting orthogonal polyhedron

Fig. 4. Decomposition of a three-dimensional orthogonal polyhedron of geometrically complex shape: a original orthogonal polyhedron; b resulting orthogonal polyhedron

4 Discussion The use of the orthogonal polyhedron decomposition algorithm allows significantly increase the speed of object placement, which is confirmed by the results of a computational experiment (Table 1). The following notation is used in the table: T0 – time spent on compact placement of the original orthogonal polyhedron, T1 – time spent on compact placement of the decomposed orthogonal polyhedron. The computational experiment was carried out on a personal computer (CPU – Intel Core i5-8400, 2.8 GHz; 8 GB RAM) using the developed software Packer [23].

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V. A. Chekanin and A. V. Chekanin Table 1. Time spent on compact placement of an orthogonal polyhedron into a conatiner Orthogonal polyhedron T 0 , s T 1 , s Example 1 (Fig. 1)

0.30 0.04

Example 2 (Fig. 2)

1.89 0.12

Example 3 (Fig. 3)

14.75 0.76

Example 4 (Fig. 4)

56.51 0.52

5 Conclusion An algorithm has been developed that decomposes an orthogonal polyhedron into a set of orthogonal objects with the largest areas (volumes) based on solving the packing problem using a model of potential containers. Unlike other known decomposition algorithms, the proposed algorithm is implemented programmatically invariant with respect to the dimension of the compound object. The developed algorithm is applicable for the decomposition of orthogonal polyhedrons of any geometric shape, those with holes or internal cavities.

References 1. Chekanin, V.A., Chekanin, A.V.: Algorithm for the placement of orthogonal polyhedrons for the cutting and packing problems. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 41–48. Springer, Cham (2020). https://doi.org/10.1007/9783-030-39500-1_5 2. Chekanin, V.A., Chekanin, A.V.: Development of algorithms for the correct visualization of two-dimensional and three-dimensional orthogonal polyhedrons. In: Advances in Automation. Proceedings of the International Russian Automation Conference, RusAutoCon 2019, 8-14 September 2019, Sochi, Russia RusAutoCon 2019. Lecture Notes in Electrical Engineering, vol. 641, pp. 891–900. Springer, Cham (2020). https://doi.org/10.1007/978-3-03039225-3_96 3. Shlepetinskiy, A.Y., Manzhula, K.P.: Size of a zone dangerous by damage at the root of cruciform weld joint. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 181–189. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-119812_16 4. Godlevskiy, V.A., Markov, V.V., Usoltseva, N.V.: Principle of compatibility of heterogeneous additives in triboactive metalworking fluids for edge cutting of metals. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 65–71 (2017). https://doi.org/10.1007/978-3-319-53363-6_8 5. Chekanin, V.A., Chekanin, A.V.: Packing compaction algorithm for problems of resource placement optimization. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 1–9. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-119 81-2_1 6. Chekanin, V.A., Chekanin, A.V.: Deleting objects algorithm for the optimization of orthogonal packing problems. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 27–35. Springer (2017). https://doi.org/10.1007/978-3-319-53363-6_4

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7. Tolok, A.V., Tolok, N.B.: Mathematical programming problems solving by functional voxel method. Autom. Remote Control 79(9), 1703–1712 (2018). https://doi.org/10.1134/S00051 17918090138 8. Byholm, T., Toivakka, M., Westerholm, J.: Effective packing of 3-dimensional voxel-based arbitrarily shaped particles. Powder Technol. 196(2), 139–146 (2009). https://doi.org/10. 1016/j.powtec.2009.07.013 9. Masek, W.J.: Some NP-Complete Set Covering Problems: Manuscript. MIT, Cambridge (1979) 10. Culberson, J.C., Reckhow, R.A.: Covering polygons is hard. J. Algorithms 17(1), 2–44 (1994) 11. Khasanova, E.I., Valeev, R.S.: The matrix decomposition method of multicoherent polygon into the set of rectangular areas of minimum capacity. Vest. UGATU 14(2), 183–187 (2010). (in Russian) 12. Valiakhmetova, Yu.I., Telitskiy, S.V., Hasanova, E.I.: Gibridnyi algoritm na osnove posledovatel’nogo utochneniya otsenok dlya zadach maksimal’nogo ortogonal’nogo pokrytiya. Vest. Bashkirskogo Univ. 17(1), 421–424 (2012). (in Russian) 13. Lodi, A., Martello, S., Vigo, D.: Recent advances on two-dimensional bin packing problems. Discrete Appl. Math 123(1–3), 379–396 (2002). https://doi.org/10.1016/S0166-218X(01)003 47-X 14. Floderus, P., Jansson, J., Levcopoulos, C., Lingas, A., Sledneu, D.: 3D rectangulations and geometric matrix multiplication. Algorithmica 80(1), 136–154 (2016). https://doi.org/10. 1007/s00453-016-0247-3 15. Durocher, S., Mehrabi, S.: Computing conforming partitions of orthogonal polygons with minimum stabbing number. Theor. Comput. Sci. 689, 157–168 (2017). https://doi.org/10. 1016/j.tcs.2017.05.035 16. Biedl, T., Derka, M., Irvine, V., Lubiw, A., Mondal, D., Turcotte, A.: Partitioning orthogonal histograms into rectangular boxes. In: Bender, M.A., Farach-Colton, M., Mosteiro, M.A. (eds.) LATIN 2018: Theoretical Informatics. Lecture Notes in Computer Science, pp. 146–160. Springer, Cham (2018) 17. Cruz-Matías, I., Ayala, D.: Compact union of disjoint boxes: an efficient decomposition model for binary volumes. Comput. Sist. 21(2), 275–292 (2017) 18. Wu, S.Y., Sahni, S.: Fast algorithms to partition simple rectilinear polygons. VLSI Design 1(3), 193–215 (1994). https://doi.org/10.1155/1994/16075 19. Györi, E., Hoffmann, F., Kriegel, K., Shermer, T.: Generalized guarding and partitioning for rectilinear polygons. Comput. Geomet. 6(1), 21–44 (1996). https://doi.org/10.1016/0925-772 1(96)00014-4 20. Chekanin, V.A., Chekanin, A.V.: An efficient model for the orthogonal packing problem. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, vol. 22, pp. 33–38. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15684-2_5 21. Chekanin, V.A., Chekanin, A.V.: New effective data structure for multidimensional optimization orthogonal packing problems. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 87-92. Springer, Cham (2016). https://doi.org/10.1007/9783-319-29579-4_9 22. Chekanin, V.A., Chekanin, A.V.: Algorithms for the formation of orthogonal polyhedrons of arbitrary dimension in the cutting and packing problems. Vest. MSTU «STANKIN» 3, 126–130 (2018). (in Russian) 23. Chekanin, V.A., Chekanin, A.V.: Design of library of metaheuristic algorithms for solving the problems of discrete optimization. In: Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 25–32. Springer, Cham (2018). https://doi.org/10.1007/9783-319-72929-9_4

Shock Spectra in Non-linear Interactions Alexander N. Evgrafov(B) , Vladimir I. Karazin, Valery A. Tereshin, and Igor O. Khlebosolov Peter the Great Saint-Petersburg Polytechnic University, St.Petersburg, Russia [email protected]

Abstract. The paper is devoted to the synthesis of the load characteristics of pulse shapers with shock test stands. For reproduction of shock acceleration spectra, shapers (gaskets) of different sizes and shapes made of a variety of materials are used. With new requirements to shock spectra, there is a need for quick calculation of shaping gaskets. It would be very convenient to have analytical expressions that link the spectrum parameters to the gasket parameters. There are already some good analytical dependencies in the literature which link the static load characteristics of a shaper with its geometrical and physical parameters. There are also very simple expressions to calculate the shock spectrum on the basis of the time function of acceleration. The problem of obtaining acceleration depending on the nonlinear characteristic of the shaper remains unsolved. This paper is devoted to the solution of this particular problem. Keywords: Shock spectrum · Test stand · Acceleration pulse

1 Introduction Reproduction of shock actions with test stands is a relevant task [21, 22]. Figure 1 shows a shock test stand with shapers, designed to reproduce shock pulses with a maximum acceleration of several thousand “g”. With the record of acceleration of the table with the test item, the shock spectrum as the maximum acceleration of a single-degree-offreedom (SDOF) oscillator depending on its natural frequency is calculated [15, 16]. Figure 2 shows an example of such a record [6], in which the maximum acceleration of 3000 g is achieved in 10 ms, after which it is sharply reduced. The graph clearly shows the presence of an intense high-frequency component with a period of less than 1 ms, corresponding to the travel time of the wave in the table or the first natural frequency of longitudinal oscillations. Free oscillations of the stand with a frequency of about 100 Hz can also be detected. Of particular interest is the formation of a slow component of shock acceleration [2, 3], corresponding to the average movement of the table. This acceleration is well described by the interaction of two absolutely solid bodies connected by a nonlinear conservative spring. Using the average movement, it is possible to determine the oscillations of the test item location based on the dynamic properties

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 60–70, 2021. https://doi.org/10.1007/978-3-030-62062-2_7

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Fig. 1. Stand for reproduction of high-intensity shock pulses

Duration, s

Fig. 2. Shock pulse acceleration (in “g”)

of the stand [4]. Figure 3 shows the spectrum of shock acceleration, shown in Fig. 2, calculated at the time interval of 40 ms. As can be seen, the spectrum was unsuccessful because it did not fit in the required range between the extreme piecewise linear functions. The main reason for such errors is a wrong choice of shapers. This paper is largely devoted to the method of selecting the load characteristics of a shaper.

2 Mathematical Model For obtaining the analytical dependence of acceleration on the load characteristics of the shaper, let us examine the dynamic model of the shock test stand in the form of a progressively moving system of two masses connected by a nonlinear spring, shown in Fig. 4.

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Frequency, Hz

Fig. 3. Shock spectrum of acceleration

mS F(xS – xt) m t xS xt Fig. 4. Two-mass model of the shock test stand: mS , x S , mt , x t are the mass and the absolute coordinate of the striker and the table respectively; F(x S – x t ) is the force of nonlinear elastic (conservative) interaction

Let us write down the equations of motion of this system.  mS x¨ S = −F(xS − xt ) mt x¨ t = F(xS − xt )

(1)

The dots above the variables denote time derivatives. After finding the difference of the second derivatives, 1 x¨ S − x¨ t = − F(xS − xt ), μ where μ =

mS mt mS + mt ,

(2)

and deformation designations of the elastic element x = xS − xt

(3)

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Equation (2), describing the relative movement of masses, takes the form of a nonlinear, second order differential equation x¨ = f (x),

(4)

1 f (x) = − F(xS − xt ). μ

(5)

where

3 Solving the Nonlinear Equation We obtain an approximate solution of Eq. (4) in the general form [8] in small time intervals at the given sufficiently smooth function f (x) under initial conditions f0 = 0, x0 = 0, x˙ 0 = 0.

(6)

We will look for the solution in the form of the final sum of a Maclaurin series x(t) =

N (n)  x 0

n=0

n!

tn,

(7)

(n)

where t is time; x0 is the time derivative of the n-th order at the initial moment of time; N is the number of summands taken into account. Let us differentiate Eq. (4) several times by time. We will limit ourselves to N = 7. Roman numerals in indices denote derivatives by x. Let us write down the system of equations obtained through differentiation. ⎧ ⎪ x(2) = f ⎪ ⎪ ⎪ x(3) = f  x˙ ⎪ ⎪ ⎪ ⎨ (4) x = f  x˙ 2 + f  x(2) (8) ⎪ x(5) = f  x˙ 3 + 3f  x˙ x(2) + f  x(3) ⎪ ⎪ ⎪ ⎪ x(6) = f v x˙ 4 + 6f  x˙ 2 x(2) + 3f  x(2)2 + 4f  x˙ x(3) + f  x(4) ⎪ ⎪ ⎩ (7) x = f v x˙ 5 + 10f v x˙ 3 x(2) + 15f  x˙ x(2)2 + 10f  x˙ 2 x(3) + 5f  x˙ x(4) + fx(5) Let us express all the time derivatives in the right parts through the first derivative. ⎧ ⎪ x(2) = f ⎪ ⎪ ⎪ ⎪ x(3) = f  x˙ ⎪ ⎪ ⎨ (4) x = f  x˙ 2 + f  f (9) ⎪ x(5) = f  x˙ 3 + 3f  x˙ f + f 2 x˙ ⎪ ⎪ ⎪ ⎪ x(6) = f v x˙ 4 + 6f  x˙ 2 f + 3f  f 2 + 5f  x˙ 2 f  + f 2 f ⎪ ⎪ ⎩ (7) x = f v x˙ 5 + 10f v x˙ 3 f + 15f  x˙ f 2 + 11f  x˙ 3 f  + 5f 2 x˙ 3 + 18f  f  f x˙ + f 3 x˙

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For the initial moment of time with allowance for (6), the system will take the following form ⎧ (2) x0 ⎪ ⎪ ⎪ (3) ⎪ ⎪ x0 ⎪ ⎪ ⎨ (4) x0 (5) ⎪ x0 ⎪ ⎪ ⎪ (6) ⎪ ⎪ ⎪ x0 ⎩ (7) x0

= = = = = =

0 f0 x˙ 0 f0 x˙ 02 f0 x˙ 03 + f02 x˙ 0 f0v x˙ 04 + 5f0 f0 x˙ 02 f0v x˙ 05 + 11f0 f0 x˙ 03 + 5f02 x˙ 03 + f03 x˙ 0

(10)

Expressions (10) are the desired multipliers for the solution of (7). If the solution of x(t) is analytically given, then function f (x) can be approximated as a final sum of the series of degrees of x with the multipliers found from (10). In applications, function f (x) is often calculated as the result of a serial connection of nonlinear elements and the inverse functions are formed. In this case, dependence x (f ), which contains parameters in the form of alphabetical designations that will be included in the analytic representation of the solution of Eq. (4), should be given analytically. Let us differentiate function f (x) and express its derivatives through the known derivatives of function x (f ). ⎧ ⎪ f  = x−1 ⎪ ⎪ ⎪ ⎪ ⎨ f  = −x−2 x f  = −x−3 x (11) f  = 3x−4 x f  x − x−3 x f  = 3x−5 x2 − x−4 x ⎪ ⎪ ν = −15x −7 x 3 + 10x −6 x  x  − x −5 x ν ⎪ f ⎪ ⎪ ⎩ f ν = 105x−9 x4 − 105x−8 x2 x + 10x−7 x2 + 15x−7 x xν − x−6 xν By determining a sufficient number of derivatives at the initial moment of time from (11), in a similar way using (10), a solution in the form of series (7) can be obtained. After deriving the law of relative motion x(t) and its acceleration x¨ (t), we can determine the absolute table acceleration from the second equation of (1) allowing for (5) and (4). x¨ t (t) = −

mS x¨ (t) mS + mt

(12)

4 Shock Spectrum of Acceleration Let us obtain an approximate expression for the shock spectrum of acceleration of (12). This is of practical importance only if the characteristic time of free oscillations of the stand is much shorter than the duration of the strike. We will look for acceleration of an SDOF oscillator in the form of Duhamel’s integral t x¨ P (t, ω) = ω

x¨ t (τ) sin ω(t − τ) d τ 0

(13)

Shock Spectra in Non-linear Interactions

65

Relative acceleration x¨ (t) after differentiating (7) is representable as well as x(t) in the form of the final sum of Maclaurin series x¨ (t) =

N −2 (n+2)  x0 n n=1

n!

t ,

(14)

All coefficients of which are determined in (10). Let us substitute (14) into (12) and (13) (Table 1). Table 1. Duhamel’s integrals of degree functions n

t n τ sin ω(t − τ) d τ

Final sum to a precision of the values of the order of vanishing (ωt)5

0

1 2 3

−(sin(ωt) − ωt)/ω2

(ωt)3 /(6ω2 ) – (ωt)5 /(120ω2 )



 2 cos(ωt) + (ωt)2 − 2 /ω3

(ωt)4 /(12ω3 )



 6 sin(ωt) + (ωt)3 − 6ωt /ω4

(ωt)5 /(120ω4 )

4

 − 24 cos(ωt) − (ωt)4 + 12(ωt)2 − 24 /ω5

0

5

 − 120 sin(ωt) − (ωt)5 + 20(ωt)3 − 120ωt /ω6

0

6 7



 720 cos(ωt) + (ωt)6 − 30(ωt)4 + 360(ωt)2 − 720 /ω7



 5040 sin(ωt) + (ωt)7 − 42(ωt)5 + 840(ωt)3 − 5040ωt /ω8

0 0

⎛ ⎞ N −2 (n+2)  t  x0 mS ⎝ x¨ P (t, ω) = − ω τn sin ω(t − τ) d τ⎠ mS + mt n! n=1

(15)

0

The table shows the exact solutions to the required integrals and the final sums of their decomposition into degree series. Since we did not find a closed formula for the integral in (15) at arbitrary n, we will limit ourselves to N = 7, as before. Prior to finding the maximum shock acceleration of oscillator w(ω) = max x¨ p (t, ω), it is necessary t to determine maximum time tmax , until which this acceleration will appear. First, let us estimate maximum deformation xmax of the nonlinear element and then, using the harmonic linearization method, find tmax as a quarter of the free oscillation period corresponding to the amplitude of xmax . Let us lower the order of Eq. (4) and write it

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down for the moment when the deformation stops. −˙x02 = 2

x max

f (x)dx

(16)

0

Taking into consideration (5) and (3), it takes its final form x max

F(x)dx = μ

x˙ 02 . 2

(17)

0

Having determined the value of xmax from (17), it is possible to estimate tmax from the following equality 

2π 4tmax

2 =−

f (xmax ) , xmax

(18)

obtained from (4). Thus, tmax =

 xmax π − . 2 f (xmax )

(19)

For low frequencies in function (15), it is possible to use substitution of Duhamel’s integrals with final sums of decomposition into degree series and to limit oneself to the values of the fifth order of vanishing (ωt)5 . With ωt ω1 ⎪ 2 gmt ⎪ t ⎪ ⎪ ⎪ ⎪ ⎪ 4F0 max ⎪ ⎪ ⎪ ⎩ 120 ⎭ ⎩

(25)

Let us consider an example of solving the system of Eqs. (1) at cubic nonlinearity F(x) = cx + ex3 ,

(26)

at c = 5·104 N/m, e = 5·109 N/m3 , ms = mt = 10 kg, μ = 5 kg, x˙ 0 = 1 m/s. Let us determine all the required derivatives in (10) with allowance for (5). F0 = c, F0 = 0, F0 = 6e, F0v = 0, F0v = 0.

(27)

Then (2)

x0 = 0,

(4)

3 x0(5) = − 6e μ x˙ 0 + (7)

x0 =

(6)

x0 = 0,

66ce 3 x˙ μ2 0

−

c2 x˙ μ2 0 c3 μ3

x0 = 0,

(3)

9

= − 6·5·10 ·1+ 5

x˙ 0 =

4

3 4 x0 = − μc x˙ 0 = − 5·10 5 · 1 = 10 m/s ,

66·5·104 ·5·109 52

5·104 52

·1−

!2

· 1 = −59 · 108 m/s5 ,

5·104 53

!3

(28)

· 1 = 659 · 1012 m/s7

Let us substitute (28) into (7)     6e 1 c 3 1 c2 1 66ce 3 c3 7 x˙ 0 t + − x˙ 03 + 2 x˙ 0 t 5 + x(t) = x˙ 0 t − x ˙ − x ˙ 0 t 3! μ 5! μ μ 7! μ2 0 μ3 = t − 1, 67 · 103 t 3 − 4, 92 · 107 t 5 + 1, 31 · 1011 t 7 ,

(29)

Let us determine x max from Eq. (17) for nonlinear function (26). After taking the integral, we obtain biquadratic equation μ˙x2 c 2 4 + 2 xmax − 2 0 = 0, xmax e e

(30)

The solution of which  ⎛" ⎞  ⎛ ⎞      μe˙x02 5 · 5109 1 c⎝  5 · 104 ⎝ ⎠  xmax =  1+2 2 −1 = · 1+2· !2 − 1⎠ = 0.00598 m e c 5 · 109 5 · 104

(31)

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x(t)

Fig. 5. Solutions to nonlinear Eq. (4): exact - solid line, approximate - dashed line

corresponds well with the graphs in Fig. 5. Let us determine stopping time t max corresponding to x max from the equation obtained from (19) " √  π μxmax μxmax μ/c π π tmax = = 0.00734 s (32) = # = 3 2 F(xmax ) 2 cxmax + exmax 2 4 μe˙x2 1 + 2 c2 0 This result is also confirmed by Fig. 5. Consequently, the method proposed in this paper can be applied to an approximate analytical solution of nonlinear differential equations. Let us write down the analytical expression for the shock spectrum in the example under consideration and obtain numerical values. For this purpose, let us substitute (27) into (25) allowing for (23).   ⎧  3

 2 x˙ 0 tmax ⎪ 1 c2 tmax ⎪ c 16 + 6e˙x02 − μ + 2 lg ω = −4.37 + 2 lg ω, ω < ω1 ⎪ lg gm 720 ⎪ t ⎪ ⎪ ⎧  2 ⎫ ⎪ ⎨

 t2 ⎪ ⎪ ⎪ ⎪ 1 1 2 2 max ⎪ ⎪ c 6 + 6e˙x0 − μ c S(ω) = ⎬ ⎨ 3 720 ⎪ x ˙ t 0 max · ⎪ ⎪ lg = 0.999, ω > ω1 ⎪ 2 gm t ⎪ t ⎪ ⎪ ⎪ ⎪ ⎪ 2c max ⎪ ⎪ ⎪ ⎩ 120 ⎭ ⎩

 2

  c 1 + 6e˙x2 − 1 c2 tmax  6 0 μ 720 ω1 =  = 484 c−1 = 77.1 Hz, lg(ω1 ) = 2.69 2 tmax 2c 120

(33)

(34)

Figure 6 presents two graphs. One is plotted on the basis of approximate analytical dependencies (33), (34), and the second is the exact solution of Eqs. (4) and (12) with finding max x¨ p (t, ω) (13) without determining the maximum deformation and stopping t time. Despite the considerable discrepancy between the graphs, the proposed method is very convenient for preliminary selection of the shock pulse shaper [5]. It should be noted that the slow function of acceleration during a strike is overlaid with high-frequency components [11], which are usually of random nature due to non-ideal repeatability and different nonlinear phenomena. Random high-frequency processes significantly distort the shock spectrum, therefore, higher manufacturing and operating requirements are imposed on shock test stands.

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Fig. 6. Shock spectra of acceleration: exact solution - solid line, approximate - dashed line

5 Conclusion Working with shock test stands requires two problems to solve [1]: one is to reproduce the required shock pulse, the other is to form a shock spectrum. These seemingly related tasks are in fact almost independent [9]. The matter is that duration of a shock pulse usually considerably exceeds the maximum free oscillation period of a stand [10]. Therefore, its reproduction is determined in many respects by the shaping gasket. The shock spectrum characterizes the maximum acceleration of an SDOF oscillator located on the table during the strike. The maximum that needs to be found should be global, and if we take into consideration that the strike causes intensive high-frequency oscillations of the whole structure [7, 13], then the search for the maximum is possible only when analyzing experimental data [14]. Probably, adequate mathematical models, which allow calculating the shock spectrum well in advance, can be constructed only for each particular stand on the basis of numerous experiments [12]. In this paper, we propose an algorithm for obtaining an approximate analytical expression to accelerate the shock test stand table at an arbitrary smooth nonlinear characteristic of the shaper. This paper provides the method for calculating the shock spectrum of acceleration. Consequently the analytical dependence between the spectrum parameters and the main shock test stand parameters is obtained.

References 1. Andrienko, P.A., Karazin, V.I., Khlebosolov, I.O.: On tests for combined effects. Mod. Mech. Eng. Sci. Educ. 2, 142–149 (2012) 2. Evgrafov, A.N., Karazin, V.I., Smirnov, G.A.: Rotary stands for reproduction of motion parameters. Sci. Techn. J. St. Petersburg State Polytechnic Univ. 3, 89–94 (1999) 3. Evgrafov, A.N., Karazin, V.I., Khlebosolov, I.O.: Reproduction of motion parameters on rotary stands. Theor. Mech. Mach. 1(1), 92–96 (2003)

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4. Karazin, V.I., Kolesnikov, S.V., Litvinov, S.D., Sukhanov, A.A., Khlebosolov, I.O.: Features of modeling and reproduction of vibratory impact. Theor. Mech. Mach. 11(22), 55–64 (2013) 5. Karazin, V.I., Kolesnikov, S.V., Litvinov, S.D., Sukhanov, A.A., Khlebosolov, I.O.: Optimization of parameters of a broadband vibratory shock mechanical stand. In: Modern Mechanical Engineering. Science and Education, p. 752 (2013) 6. Komarov, I.S.: Ground-based experimental testing of rocket and space technology products for impact force from pyrotechnic separation means. In: Central Research Institute of Machine Building (TsNII Mash). Electronic Journal “Trudy MAI”, Issue 71, p. 22 www.mai.ru/science/ trudy/ 7. Tereshin, V.A.: Shock spectra in linear interaction. In: Evgrafov, A.N., Popovich, A.A. (eds.) Modern Mechanical Engineering: Science and Education: Proceedings of the 6th International Scientific-Practical Conference, p. 818, pp. 202–213. Polytechnic University Publishing House, St. Petersburg (2017) 8. Tereshin, V.A., Shemiakina, T.A.: Integrating nonlinear autonomous differential equations. In: Proceedings of the XX International Conference on Computational Mechanics and Modern Applied Software Systems (VMSPPS 2019), 24–31 May 2019, p. 816, pp. 118–120. MAI Publishing House, Alushta, Moscow (2019). il. ISBN 978-5-4316-0589-5 9. Yarovitsyn, V.S., Litvinov, S.D., Karazin, V.I., Sukhanov, A.A., Khlebosolov, I.O.: Device for testing products for vibratory shock loads. Invention patent RUS 2348021 14.05.2007 10. Chang, K.Y.: Pyrotechnic Devices, shock levels and their applications. In: 9th International Congress on Sound and Vibration Orlando, USA, July 2002, p. 19 (2002) 11. Karazin, V.I., Kozlikin, D.P., Sukhanov, A.A., Tereshin, V.A., Khlebosolov, I.O.: Some ways of stable counterbalancing in respect to moving masses on centrifuges. In: Lecture Notes in Mechanical Engineering, pp. 73–85. Springer International Publishing, Cham (2017). https:// doi.org/10.1007/978-3-319-53363-6_3 12. Lee, J.-R., Chia, C.C., Kong, C.-W.: Review of pyroshock wave measurement and simulation for space systems. J. Measur. 45, 631–642 (2012) 13. Tereshin, V.A.: Shock response spectra as a result of linear interactions. In: Lecture Notes in Mechanical Engineering, Part F5, pp. 151–161. Springer, Cham (2018) 14. Zulkarnay, U., Kaverzneva, T.T., Tarkhov, D.A., Tereshin, V.A., Vinokhodov, T.V., Kapitsin, D.R.: A two-layer semi empirical model of nonlinear bending of the cantilevered beam. In: Joint IMEKO TC1-TC7-TC13 Symposium, Rio de Janeiro, Brazil, 3–31 July 2017 (2017) 15. GOST R (Russian National Standard) 51371-99. Methods of testing for resistance to mechanical external influencing factors of machines, devices, and other technical products. Shock tests. – Moscow, p. 24 (2000) 16. Evgrafov, A.N., Karazin, V.I., Kozlikin, D.P., Khlebosolov, I.O.: Some characteristics of linear acceleration reproduction with flexible harmonical component. In: Lecture Notes in Mechanical Engineering, PartF5, pp. 71–81 (2018) 17. Andrienko, P.A., Karazin, V.I., Khlebosolov, I.O.: Bench tests of vibroacoustic effects. In: Lecture Notes in Mechanical Engineering, pp. 11–17 (2017) 18. Evgrafov, A.N., Karazin, V.I., Khisamov, A.V.: Research of high-level control system for centrifuge engine. Int. Rev. Mech. Eng. 12(5), 400–404 (2018) 19. Evgrafov, A.N., Karazin, V.I., Petrov, G.N.: Analysis of the self-braking effect of linkage mechanisms. In: Lecture Notes in Mechanical Engineering, pp. 119–127 (2019) 20. Evgrafov, A., Khisamov, A., Egorova, O.: Experience of modernization of the curriculum TMM in St. Petersburg state polytechnical university. In: Mechanisms and Machine Science, vol. 19, pp. 239–247 (2014) 21. Popov, A.N., Polishchuck, M.N., Vasiliev, A.K.: Method for reproducing shock acceleration in mechanical testing. Int. Rev. Mech. Eng. 14(2), 105–110 (2020) 22. Popov, A.N., Polishchuck, M.N., Pulenets, N.Y.: Test centrifuge arrangement analysis. In: Lecture Notes in Mechanical Engineering, pp. 139–151 (2019)

Algorithm of Determining Reactions in Hinge Joints of Linkage Mechanisms with Non-ideal Kinematic Pairs Alexander N. Evgrafov1(B) , Gennady N. Petrov1 , and Sergey A. Evgrafov2 1 Peter the Great Saint-Petersburg Polytechnic University, St. Petersburg, Russia

[email protected] 2 LTd “BaltStream”, St. Petersburg, Russia

Abstract. The authors consider the kinetostatic calculation of flat-faced linkage mechanisms with allowance for friction in kinematic pairs exemplified by an Assur group with three rotational pairs (RRR). The paper provides examples. In order to determine reactions in hinge joints, the authors propose the algorithm of searching for a structural group with replacement ideal kinematic pairs. Keywords: Structural group · Assur group · Mechanism · Replacement ideal kinematic pair · Friction

1 Introduction It is essential to take into account friction forces in hinge joints of linkage mechanisms when some structural groups reach specific positions [5, 7, 9, 11, 13–16, 18–23]. This can cause the situation when the mechanism jams [10] (one of the structural groups brakes itself [3, 4, 6, 8, 12, 17]). Publications [1, 2] show that it can occur, if among numerous structural groups with replacement ideal kinematic pairs there is at least one pair in the specific position. Similar to the replacement mechanism, where linkage chains move in the same way as in the original mechanism, the authors introduced the term “replacement ideal kinematic pair” (RIKP). Such a kinematic pair will have the same reactions as in the kinematic pair with allowance for friction forces. Figure 1 shows RIKP axes A , B , C  for the Assur group RRR (with three rotational kinematic pairs). A lot of these RIKPs are inside the circles of friction ρA , ρB , ρC with no relative movements of the links connected by the hinge, or, otherwise, on the circles of these radii. If to determine positions of the points A , B , C  for the given external forces and friction, the task of the kinetostatic calculation is considerably simplified, reducing to determination of reactions in hinge joints with ideal kinematic pairs. On the example of the RRR group the authors developed the RIKP search algorithm.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 71–79, 2021. https://doi.org/10.1007/978-3-030-62062-2_8

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Fig. 1. Determination of replacement IKP for the RRR dyad

2 RRR Group with Ideal Kinematic Pairs The kinetostatic calculation of flat-faced linkage mechanisms is regarded as determination of driving forces, momentum and reactions in the hinges, lying in the plane of link movement. The task is solved by the structural groups, starting with the groups of the last layer [10]. If friction is not taken into account (kinematic pairs are considered ideal), the task has a rather simple solution. Let us analyze the example of the Assur group RRR with three rotational kinematic pairs (Fig. 2).

Fig. 2. RRR group with ideal kinematic pairs

We will introduce the notation: P¯ 1 , M1 are the main vector and the main momentum of external forces and inertia forces relative to point A, acting on the first link; P¯ 2 , M2 are the main vector and the main momentum of external forces and inertia forces relative to point C, acting on the second link; h is the distance from point B to AC line (vectors AB and CB projected onto the axis y);

Algorithm of Determining Reactions in Hinge Joints

73

a, b are vectors AB and CB projected onto the axis x); In the figure h > 0, a > 0, b < 0. Let us set the coefficient of the ratio of the main momentum, acting on the links kM = −

M1 . M2

We will introduce the coordinate system x1 y1 . The x-axis is directed parallelly towards vector R¯ B (reaction in the central hinge). We can find vector projections AB and CB onto the axis y1 h1 = h · cos θ − a · sin θ, h2 = h · cos θ − b · sin θ, where θ is the angle between the axes x and x1 . It is obvious, that kM = −

M1 h1 h · cos θ − a · sin θ . = = M2 h2 h · cos θ − b · sin θ

(1)

Figure 2 presents the case, when M1 > 0, M2 < 0, h1 > 0, h2 > 0. Resulting from (1), we can obtain angle θ tan θ =

h − kM h . a − kM b

Assume, that angle θ is determined in I and IV quarters, therefore,   h − kM h θ = arctan a − kM b Reactions in the hinge B, acting on the first and second links, are the following:   M1 1 RB1 = − M h1 = RB sign h1 , (2) M2 2 RB2 = − M h2 = RB sign h2 = −RB1 .     M2 1 In figure we have sign M = 1, sign h1 h2 = −1 Let us determine reactions in the hinges A and C R¯ A1 = −R¯ B1 − P¯ 1 , R¯ C2 = −R¯ B2 − P¯ 2 .

(3)

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3 Allowance for Friction Forces in the Central Hinge of the RRR Group We will allow for friction in the central hinge B (Fig. 3). Consider the case, when the friction circle with radius ρB does not cross line AC (Fig. 3, a). Then determine the position of hinge C  of RIKP. The action line of vector R¯ C  ) is characterized has to touch the radius of the friction circle. Tangency point B (B∗ or B∗∗ by the fact that the frictional moment in the hinge is opposite to the relative angular velocity of the connected links. Radius sign ρB for point B∗ is considered positive, for  it is negative, which corresponds to the sign of vector BB projected onto the point B∗∗ axis y1 . Insert the notion, where ρ

εB = sign(ρB ) is radius ρB , εVB1 = sign(ω12 ) is a sign of the angular velocity of the first link relative to the second link, εVB2 = sign(ω21 ) is a sign of the angular velocity of the second link relative to the first link, εRB1 = sign(RB1 ), εRB2 = sign(RB2 ) are signs of vectors R¯ B1 and R¯ B2 projected onto the axis x1 . Define radius ρB as sign(ρB ) = sign(RB1 ) · sign(ω12 ), or ρ

εB = εRB1 εVB1 = εRB2 εVB2 . ρ

Exemplify εRB1 = −εRB2 = 1, εVB1 = −εVB2 = −1, εB = −1 (in the scheme ω12 is directed clockwise).

Fig. 3. Allowance for friction in the hinge B of the BBB group a) current position, b) self -braking mode

Algorithm of Determining Reactions in Hinge Joints

75

Resulting new distance values h1 and h2 from (1), insert them into coefficient kM kM = −

M1 h1 h · cos θ − a · sin θ + ρB = = . M2 h2 h · cos θ − b · sin θ + ρB

Find angle θ sin θ · (a − kM b) − cos θ · (h − kM h) = ρB (1 − kM ), 

h − kM h θ = arctan a − kM b



 + arcsin 

ρB (1 − kM ) (h − kM h)2 +(a − kM b)2

 .

Reactions RB1 , RB2 , RA1 , RC2 are determined from ratios (2) and (3). If the friction circle touches or crosses line AC (Fig. 3, b), there are Assur groups with RIKP in a unique position (when hinges AB C are located on the same line). Therefore, with a definite set of external forces, inertia and angular velocities of links, as it is shown in [1, 2], a self-braking mode of the structural group can occur. With allowance for friction forces in all the hinges, determination of points A , B ,  C of the group with RIKP is a more complicated task. The accurate solution is possible, if initially we can find angle θ of the action line of vector R¯ B .

4 Example of a Task Solution with Allowance for Friction Forces in Hinges Let us consider the example. For the mechanism presented in Fig. 4 we will find out what the value of angle γ under the gravity forces for the first and second links G1 = G2 = G when motion starts (frictional forces are overcome).

Fig. 4. Example: a) friction in progressive pairs, b) friction in all hinges

Masses of the third and fourth links are neglected. The lengths of the links are AC = CB = l, distances to the centers of masses are AS1 = BS2 = l/2. First of all, it is crucial to take into account friction only in progressive pairs (Fig. 4, a). The mechanism is symmetric, that is why the direction of the reaction in the central hinge B must be horizontal.

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Designate the following: RA1 , RB1 are reactions in the hinges A and B, acting on the first link, RC2 , RB2 are reactions in the hinges C and B, acting on the second link. The intersection points of the line that show the reaction action in the hinge B, and gravity forces of the first and the second links will be designated as K1 and K2 respectively. Reactions RA1 and RC2 go through these points. If the contact in sliders 3 and 4 occurs on the same surface, the mechanism starts moving under gravity action when angle β is larger than the friction angle β > arctan f ,

(4)

where f is a coefficient of sliding friction. Express angle β as γ tan β =

1 l/2 · cos γ = . l · sin γ 2 · tan γ

(5)

Insert (5) in (4), and we will obtain the condition of starting the mechanism movement γ < arctan

1 . 2f

(6)

Friction is taken into account not only in progressive, but in all rotational pairs (radius of friction circle ρ is considered equal for all hinges). The task can be solved as we know a horizontal direction of the reaction in the internal hinge B (Fig. 4, b). Determine directions of relative angular velocities of the links in the initial movement. εVA1 = sign(ω13 ) = −1 is a sign of the relative angular velocity of the first and the third links (clockwise), εVB1 = sign(ω12 ) = −1 is a sign of the relative angular velocity of the first and the second links (clockwise), εVB2 = sign(ω21 ) = 1 is a sign of the relative angular velocity of the second and the first links (counterclockwise), εVC2 = sign(ω24 ) = 1 is a sign of the relative angular velocity of the second and the fourth links (counterclockwise). Let us highlight that the frictional moment, caused by reactions RA1 , RB1 , RB2 , RC2 relative to points A, B, C, should be in the direction opposite to relative angular velocities ω13 , ω12 , ω21 , ω24 respectively (like in Fig. 4) Find angle β tan β =

l/2 · cos γ − ρ · cos β . l · sin γ + ρ · sin β

The start of movement is expressed as it follows f 1 tan β = f , sin β =  , cos β =  . 2 1+f 1+f2

(7)

Algorithm of Determining Reactions in Hinge Joints

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Insert these expressions in (7). After simple transformations it is possible to obtain the condition of starting the mechanism movement  1 2ρ 1 + f 2 γ < arctan − arcsin  2f l 1 + 4f 2 The second summand clarifies the condition obtained before (6).

5 Allowance for Friction in All Hinges of the RRR Group If friction is allowed for in all the hinges of the RRR group (Fig. 5), generally it is necessary to use an iterative procedure. Let us consider one of the possible algorithms.

Fig. 5. Determination of RIKP for the RRR dyad

Construct RIKP A(0) , B(0) , C (0) with zero approximation, where ψ1 is the angle between the axis x and main vector P¯ 1 of the forces, acting upon the first link. At this angle we can draw tangents to the circle of radius ρA (friction circle in the hinge A). Out of points A∗ and A∗∗ , we select the point, where vector −P¯ 1 creates the momentum relative to point A, opposite to the direction of the relative angular velocity ωA1 (angular velocity of the first link relative to the connecting link). In the example (Fig. 5) εVA1 = sign(ωA1 ) = 1, select A(0) ≡ A∗∗ . Similarly, on circle radius ρc . we select point C (0) ≡ C ∗∗ (if εVC2 = sign(ωC2 ) = −1). Bring the momentum of active forces and inertia, acting on the first and second links, to points (0) (0) A(0) and C (0) (M1 , M2 ) respectively. Hinges A0 and C0 are regarded as ideal. Taking advantage of the algorithm, described above, we determine position B(0) and reactions (0) (0) (0) (0) in the kinematic pairs of R¯ A1 , R¯ B1 , R¯ B2 , R¯ C2 group, allowing for friction only in the central hinge. At the intersection of the perpendicular drawn from point A to the line of vector (1) to a first approximation. Similarly, we obtain C (1) . R¯ (0) A1 and circle ρA , find RIKP A Determine reactions to a first and sequent approximations until points A(n) , B(n) , C (n) and A(n−1) , B(n−1) , C (n−1) coincide at the n-th iteration with a given error.

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Fig. 6. Mode of self-braking for the RRR dyad

6 Conclusion The authors propose the algorithm that enable simplifying the kinetostatic calculation of the Assur flat-faced structural group with non-ideal connections. The present search algorithm might not converge, if among numerous groups with replacement ideal kinematic pair (RIKP) there are groups in the specific positions (when hinges A , B ,C  lie on the same line – Fig. 6). Therefore, in certain conditions it can lead to the self-braking mode (no solutions) or braking (several solutions available).

References 1. Evgrafov, A.N., Karazin, V.I., Petrov, G.N.: Analysis of the self-braking effect of linkage mechanisms. In: Lecture Notes in Mechanical Engineering, pp. 119–127 (2019) 2. Evgrafov, A.N., Petrov, G.N., Evgrafov, S.A.: Consideration of friction in linkage mechanisms. In: Lecture Notes in Mechanical Engineering, pp. 75–82 (2020) 3. Kolovsky, M.Z.: Criterion of the position quality in multidirectional linkage mechanisms. Mech. Eng. Issues Reliab. (2) (1997) 4. Kolovsky, M.Z., Petrov, G.N., Sloushch, A.V.: On movement control of closed linkage mechanisms with several degrees of freedom. Mech. Eng. Issues Reliab. (4) (2000) 5. Evgrafov, A.N., Petrov, G.N.: Geometric and kinetostatic analysis of flat linkage mechanisms of 2nd class. Theory Mech. Mach. (2), 50–63 (2003) 6. Petrov, G.N.: Computer modeling of mechanical systems in the «ModelVision» environment. Theory Mech. Mach. 2(1), 75–79 (2004) 7. Khlebosolov, I.O.: Computer-aided graphical methods of calculating mechanisms. Theory Mech. Mach 2(2), 40–44 (2004) 8. Evgrafov, A.N., Petrov, G.N.: Computer animation of kinematic schemes in Excel and Mathcad. Theory Mech. Mach. 6(1), 71–80 (2008) 9. Verkhovod, V.P.: Using Mathcad for the synthesis of linkage transmission mechanisms. Theory Mech. Mach. 9(1), 69–76 (2011) 10. Evgrafov, A.N.: Theory of mechanisms and machines: training manual. In: Evgrafov, A.N., Kolovsky, M.Z., Petrov, G.N.(eds.) 3-D edition. SPbPU Publishing, St. Petersburg (2015). 248 p.

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11. Peisakh, E.E.: System of designing flat linkage mechanisms. In: Peisakh, E.E., Nesterov, V.A. (ed.) Mashinostroenie, Moscow (1988). 232 p. 12. Evgrafov, A.N.: On pressure angle as a quality criterion for transfer of forces. In: Mechanical Engineering. Collection of scientific works, pp. 84–90. SPbSTU Publishing, St. Petersburg (1997) 13. Dimentberg, F.M.: Theory of Spatial Jointed Mechanisms. Nauka, Moscow (1982). 36 p. 14. Zinoviev, V.A.: Spatial Mechanisms with Lower Pairs. Gostehizdat, Moscow (1952). 432 p. 15. Mudrov P.G. Spatial mechanisms with rotational pairs. - Kazan: Kazan University publishing, 1976. - 264 p 16. Petrov, G.N.: Algorithm of computer-aided kinetostatic calculation for closed linkage mechanisms. Mech. Eng. Issues Reliab (3) (1993) 17. Khrostitsky A.A., Evgrafov A.N., Tereshin V.A. Geometry and kinematics of a spatial six-link mechanism with excessive coupling. St. Petersburg State Polytech. Univ. J. (123), 170–176. SPbSTU Publishing, St. Petersburg (2011) 18. Ovakimov, A.G.: Analysis of the passive connection of the spatial six-link mechanism with rotational pairs. BMSTU J. Mech. Eng. (2), 58–61 (1970). N.E. Bauman Moscow State Technical University Publishing 19. Semenov, Y.A., Semenova, N.S.: Determination of Dynamic Errors in Machines with Elastic Links. In: Lecture Notes in Mechanical Engineering, pp. 163–174 (2020) 20. Evgrafov, A., Khisamov, A., Egorova, O.: Experience of modernization of the curriculum TMM in St. Petersburg state polytechnical university. Mechanisms and Machine Science, vol. 19, pp. 239–247 (2014) 21. Semenov, Y.A., Semenova, N.S.: Calculation of forces and moments for mechanisms with allowance for friction. Theory Mech. Mach. 14(3(31)), 135–144 (2016) 22. Evgrafov, A.N., Petrov, G.N.: Calculation of the geometric and kinematic parameters of a spatial leverage mechanism with excessive coupling. J. Mach. Manuf. Reliab. 42(3), 179–183 (2013) 23. Semenov, Y.A., Semenova, N.S.: Study of mechanisms with allowance for friction forces in kinematic pairs. In: Lecture Notes in Mechanical Engineering, pp. 169–179 (2019)

Axisymmetric Vibrations of the Cylindrical Shell Loaded with Pointed Masses George V. Filippenko1,2(B) and Tatiana V. Zinovieva1 1 Institute for Problem in Mechanical Engineering, St. Petersburg, Russia

[email protected] 2 Saint Petersburg State University, St. Petersburg, Russia

[email protected]

Abstract. In the paper the oscillations of a circular cylindrical shell of the Kirchhoff – Love type with additional inertia in the form of a “mass belt” of zero width is considered. The dispersion equation is obtained and the nature of the dispersion curves in the vicinity of singular points is investigated. Additionally, a modal analysis of the system was carried out in ANSYS program by the finite element method. Keywords: Cylindrical shell · Shell oscillations

1 Introduction Cylindrical shells in liquids have long been researched in connection with rich applications in engineering and construction. It is one of the most important elements in the simulation of acoustic waveguides, various pipelines, supports of offshore drilling rigs and other hydraulic structures [1–5]. In this paper the variant of the classical theory of Kirchhoff-Love shells, constructed on the basis of analytical mechanics of Lagrange [6–9], is used. This theory, in particular, was used in [10–12] to calculate the statics of shells of rotation, and in [13, 14] to study the wave processes and oscillations of shells of rotation with an arbitrary meridian. The above theory of the shells was also applied in works on the loaded shell: the pressure of a steam jet [15], an acoustic fluid [16–20], a load in the form of Winkler foundation [21]. Load by inertial forces generated by massive “belts” [22] can act as a load also. Axially symmetric wave processes are of special interest [23] among them, because of both their frequent occurrence and convenience of analytical analysis. These oscillations are excited by axisymmetric source. Concentrated mass significantly effects on the shell oscillations. The purpose of this work is to investigate the nature of free oscillations dependence of the cylindrical shell with concentrated mass on the value of this mass. The free oscillations of a finite circular cylindrical shell of Kirchhoff-Love type [6] are explored. Let’s consider a cylindrical system of coordinates (r, ϕ, z), where the z axis coincides with the axis of the cylinder, and a local system of coordinates (t, n, k), where © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 80–91, 2021. https://doi.org/10.1007/978-3-030-62062-2_9

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the vectors t, n are tangent and normal unit vectors to the shell, respectively, and the vector k is a unit vector along the axis z. The shell occupies the space −l/2 ≤ z ≤ l/2. On line z = 0 the concentrated loading in the form of “mass belt” of zero width is located. As the variables describing the vibration field in the system, let’s choose the vector of displacements of the shell u = (ut , uz , un )T (T is sign of transposition operation). Since only axially symmetric vibrations will be considered further, there is no dependence of all processes on the angle ϕ, so it will be omitted the function arguments.  in    Let’s enter following designations: cs = Eh/ 1 − ν2 ρ is the deformation waves velocity of the median surface of cylindrical shell, E, ν and ρs are Young’s modulus, Poisson’s factor and volume density of a material of a shell accordingly, h is thickness of a shell, ρ = ρs h is its surface density, R is radius of a cylindrical shell. The mass “belt” is characterized by surface densities μt , μz , μn acting as inertial masses in the corresponding directions (each of them is equal to the corresponding inertial mass of the whole “belt”, divided by the length of the circle). The dependence of all processes on time is assumed to be harmonic with frequency ω, while the time factor exp{−iω t} is assumed to be omitted everywhere. In addition, we set the dimensionless parameters: dimensionless frequency w = ωR/cs and the parameter characterizing the relative thickness of the shell α2 =      2 1 12 h R . The shell length l, coordinate z and displacement vector u are normalized to the shell radius R, keeping the previous designations behind them. Also the δ-function becomes dimensionless. In dimensionless form the balance of forces acting on the cylinder in axially symmetric case [22] has the view:   L + w2 (I + l M)δ(z) u = 0, −l/2 < z < l/2 (1) ⎞ 0 0 w2 + d 2 ∂z2 ⎠, I = diag{1, 1, 1}, L=⎝ ν ∂z 0 w2 + ∂z2 2 2 2 4 0 −ν ∂z w − α (1 − 2ν∂z + ∂z ) − 1   μt,z,n ∂ , d 2 = 1 + 4α 2 ν− , ∂z = R M = diag{Mt , Mz , Mn }, Mt,z,n = ρlR ∂z ⎛

μ

2π Rμ

t,z,n t,z,n Here Mt,z,n = ρlR = 2π R2 lhρ . is the ratio of the corresponding inertial mass of s the whole “belt” to the mass of the shell without it. Equation (1) is supplemented by boundary conditions (BC) on lines z = ±l/2. Four types of BC are possible: rigid BC, hanged BC, free BC and sliding BC.

2 Getting the View for the Vibration Field The method of solution is similar to the one considered in [22]. The solution of Eq. (1) is searched separately on the left and on the right of the line z = 0 (ut , uz , un )T = Aeiλz (ς, ξ, γ)T .

(2)

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Here A, ς, ξ, γ are arbitrary dimensionless constants; λ is required wave number, which is considered to be dimensionless to the radius of the shell R. As a result we get two homogeneous algebraic systems of the species: Lλ x = 0,

(3)

the where x ≡ (ς, ξ, γ)T (this vector will be normalized to a single length). Here

matrix Lλ is the image of the Fourier differential operator L + w2 (I + l M) in (1). The condition of existence of a nontrivial solution of the system (3) leads to the equation det Lλ (w, λ) = 0. The solution to this equation can be written in the form of λj = λj (w), j = 1, 6. After substitution of these roots in the system (3) we find our own vectors x j and, with the accuracy to six undefined constants, the solutions on the left and right of the line z = 0. Then we satisfy the BC and conditions on the line z = 0. The condition of existence of the nontrivial solution of this system leads to the equation det A(w, λ(w), Mt , Mz , Mn ) = 0,

(4)

whose solution is the function w = w(Mt , Mz , Mn ). The corresponding curves, in the future we will call “branches of the dispersion curve” or simply “branches”. After substitution of the corresponding roots wm , m = 1, 2, . . . in the system, we find the coefficients Aj , j = 1, . . . , 6 in (2) and, consequently, the type of vibration field in the system. Let us restrict ourselves to the consideration of the shell motions that do not contain a rotational component ut . In this case, the problem (1) is reduced to finding the vector (uz , un ). In addition to the analysis of the components of this vector, we will analyze the components fz , fn of the generalized force vector [22].

3 Numerical Results and Analysis To calculate the presented graphical dependencies in Fortran [24], the dimensionless parameters of the shell are taken as follows: ν = 0.3, h/R = 0.01, l = 4. Additionally, for calculations in ANSYS package, the following dimensional parameters of the shell are taken: R = 1 m, E = 2.1 · 1011 N/m2 , ρs = 7850 kg/m3 . In the future, we will consider the mass matrix M = diag{Mt , Mz }, where either Mz = 0, Mn ≡ M , or Mz ≡ M , Mn = 0. The corresponding oscillations of the shell will be denoted by the letters “N” and “Z” and will be called “N-type” and “Z-type” correspondingly. We will also consider the special case when Mn ≡ M , Mz = 0.7M . This type of shell oscillations will be denoted by the abbreviation “ZN”. For illustration we will consider the vibrations with free BC on both sides of the shell (Fig. 2) and with fixed BC (all other figures). The dependencies of two types is explored: the dependence of the dimensional frequency f in Hertz (Hz) on the dimensionless concentrated mass M (Fig. 1, 2, 4, 5a, 6), and the dependence of component of dimensionless displacement vectors and forces on the longitudinal dimensionless coordinate z (Fig. 5b, 7, 8, 9, 10 and 11). In this case, the displacement vector (uz , un ) is normalized to the largest value module of the vector component in the range −l/2 < z < l/2. The force vector (fz , fn ) is normalized in the same way. For first type dependencies, the branches

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corresponding sequentially increasing frequencies f1 , f2 , . . . are marked with numbers 1, 2,… respectively. For example Fig. 1a shows branches 1, 2 and Fig. 1b shows branches 2–5 of the dispersion curve for the case Mn = 0. This corresponds to the interaction of the mass belt with the shell only along the z axis. The characteristic feature of this case is the absence of branch intersection. The dispersion curve calculations were controlled by finite element method in ANSYS package in reference points M = 0.0, 0.1, 1.0, 10.0. For modeling of the shell the axially symmetric element SHELL209 was used, the concentrated mass was determined by the specification of inertial element characteristics MASS21. The dispersion curves of solid lines type show the results of counting in FORTRAN program and circles show the results of control calculations in ANSYS. The comparison of results shows that in the considered range of parameters Kirchhoff-Love model appears to be a good approximation (Table 1 shows the values of the first five frequencies calculated for the largest mass M = 10.0 by the both methods). The relative error of the data presented in the table does not exceed 0.4%, therefore, on the scale of the presented graphs the points of dispersion curves founded by these two methods are indistinguishable. Table 1. Frequencies. Branch Z Fortran

N ANSYS Fortran 51.368

ZN ANSYS Fortran 51.175

51.368

ANSYS

1

128.231 128.228

2

804.582 804.541 617.509 617.480 128.231 128.228

51.175

3

807.003 806.929 808.351 808.277 807.003 806.929

4

820.964 820.876 817.322 817.265 808.351 808.277

5

822.331 822.208 823.190 823.061 822.331 822.208

Figures 2, 4, 5 show the dispersion curve branches for the case Mz = 0. This case corresponds to the interaction of the mass belt with the shell only in normal direction. In contrast to the previous case, the intersection of branches is observed. It should be noted that if at least one of the BC is of sliding fixing type or of free fixation type then observed quasi-intersection (but not their intersection) of the first two dispersion branches (curves 1 and 2 on Fig. 2a). Thus, two new “virtual” curves AD and CB (Fig. 2a) appear along which the oscillation type is preserved (Fig. 3a, b, c, d). Note that even in the case of visual intersection of curves, we can not say for sure whether they do intersect or it is a quasi-intersection. Although the calculations show that even in the “small rectangle” (Fig. 4b) M ∈ [0.0372214, 0.0372216], f ∈ [617.5080, 617.5083], (i.e. M = 2.0E−7, f = 3.0E−4) from the viewpoint of the calculating person, the curves can be considered really intersecting and then the point of their intersection is the point where both types of oscillations (Z and N-type) are possible.

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A characteristic feature of the above figures is alternation of branches that “feel” the increase in mass, with branches that practically do not “notice” it, which is due to the presence of node or maximum of the standing wave at the point of mass attachment. Thus, oscillation of the component uz at the point z = 0 increases on branches with odd numbers, and the nodes are observed on branches with even numbers (Fig. 4b). Therefore, in case of Z-vibrations on odd-numbered branches, the frequency naturally drops with increasing mass [25], infinitely approaching to the next eigen frequency, where the system can realized the node at the point of mass location (z = 0). Thus, the first branch approaches the frequency f0 = 0, the third branch approaches f2 , etc. (Fig. 1a, b). A well-known physical principle works – the system minimizes its influence on it. Note that in the case of Z-vibrations, the concentrated mass interacts with the shell like a mass interacts with the rod [21, 25]. For the component un the opposite situation is true: maximum of vibrations at the point z = 0 are observed on branches with even numbers, while the nodes are observed on branches with odd numbers. Therefore, in the case of N-vibrations on even-numbered branches the frequency decreases when mass is increasing, but in this case the intersection of branches is observed. Thus, the second branch crosses the first one at the point M ≈ 0.04 (Fig. 4a, b) where a node of this type of oscillations is realized and then tends to zero frequency. The fourth branch crosses the third one at the point M ≈ 0.02 (Fig. 5). The calculations show that curves 5 and 6 in Fig. 5 also intersect (a point M ≈ 0.0274 is outside the range in this figure). In this case, the sixth branch remains above the frequency of the fourth branch of the unloaded shell, and the fourth branch is above the frequency of the second branch of the unloaded shell, as it would be in the case of oscillations of the beam [25]. Figures 6a, b show branches 1–3 for the case when Mn ≡ M , Mz = 0.7 M (ZN-type of oscillations), which corresponds to the interaction of the mass belt with the shell, both on the normal and along its axis (the belt is rigidly attached to the shell). This case inherits the properties of both Z- and N-type oscillations. As in the case of N-type, branches 1 and 2 intersect of at the point M ≈ 0.04, but branch 1 behaves as in the case of Z-type oscillations (it decreases with mass growth). Let us stop in detail on the analysis of the vibration field in the vicinity of the intersection point of branches 1 and 2 for the case of N-vibrations (Fig. 4b). Profiles of displacement vector components and forces (Fig. 7a, 7b) retain their view throughout the first branch, in accordance with the analysis (the component un implements a node at the point z = 0), and the relative contribution of the normal force component is almost equal to zero, and its longitudinal component is in antiphase with the longitudinal displacement component. Let us consider the nature of oscillations from the second branch at M = 0.005, 0.035, 10.0. As the mass increases from M = 0.005 to M = 0.035, the relative contribution of the longitudinal displacement component decreases (Fig. 8a) and the contribution of normal displacement component increases (Fig. 8b). In this case there is a rapid localization of oscillations in the vicinity of z = 0. The break of the normal force component increases (Fig. 9b), but the relative contribution of the longitudinal force component decreases sharply to some constant value (Fig. 9a). And, as the calculations

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show, further, with the growth of mass from M = 0.035 to M = 10.0, the graphs for displacements and forces practically do not change (Fig. 9b). Let us consider a similar crossing of branches 1 and 2, but for ZN-vibrations (Fig. 6). The qualitative difference is that now the oscillations from branch 1 essentially interact with the concentrated mass through longitudinal displacements similar to the case of Z-vibrations. The growth of mass from M = 0.02 to M = 10.0 is expected to increase the relative contribution of the longitudinal displacement component (Figs. 10a, 11a) and force (Figs. 10b,11b), while the relative contribution of the normal force component remains almost equal to zero. There is also an increase in the localization of oscillations, but not to such a fast degree as for the localization of normal oscillations from the second branch. In general, despite the noted differences from the N-case, we can note a clear dominance of longitudinal processes on the first branch.

(а)

(b)

Fig. 1. Z-type dispersion curves for different frequency ranges.

(а)

(b)

Fig. 2. Free BC. N-type dispersion curves for the different frequency and mass ranges.

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(a)

(b)

(c)

(d)

Fig. 3. Profile of displacements Uz (1), Uz (2) for the points A (a), B (b), C(c) and D(d)

(а)

(b)

Fig. 4. N-type dispersion curves for different frequency and mass ranges.

Axisymmetric Vibrations of the Cylindrical Shell

Fig. 5. N-type dispersion curves for different frequency and mass ranges.

(а)

(b)

Fig. 6. ZN-type dispersion curves for different frequency and mass ranges.

(а) Profile of displacements U z (1), U n (2)

(b) Profile of forces Fz (1), Fn (2)

Fig. 7. N-type oscillations, first branch, M = 10.0.

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(а) Profile of displacements u z

(b) Profile of displacements un

Fig. 8. N-type oscillations, second branch, M: 0.005 (1), 0.035 (2), 10.0 (3).

(а) Profile of force Fz

(b) Profile of force Fn

Fig. 9. N-type oscillations, second branch, M: 0.005 (1), 0.035 (2), 10.0 (3).

(а) Profile of displacements U z (1), U n (2)

(b) Profile of forces Fz (1), Fn (2)

Fig. 10. ZN-type oscillations, first branch, M = 0.02.

Axisymmetric Vibrations of the Cylindrical Shell

(а) Profile of displacements U z (1), U n (2)

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(b) Profile of forces Fz (1), Fn (2)

Fig. 11. ZN-type oscillations, first branch, M = 10.0.

4 Conclusion In the work the character of oscillations of the cylindrical shell of Kirchhoff-Love with additional inertia in the form of “mass belt” is investigated. It is shown that the case of fixation of the concentrated mass in the direction of the axis of the cylindrical shell qualitatively differs from the other types of fixation, in which there are points of intersection of dispersion branches. In the vicinity of these points there is a sharp change of the oscillations character. These oscillations, with the growth of the concentrated mass, quite quickly transform from oscillations of mainly longitudinal character to clearly expressed normal oscillations, strongly localized in the vicinity of the mass belt. In the case of different constructions it can be dangerous for their integrity on certain modes. On the other hand, by varying the nature of the load securing, it is possible to achieve a more stable operating mode of the structure in a given range of parameters. Calculations under the stated variant of Kirchhoff-Love theory are well coordinated with the calculations carried out by the finite element method. It is necessary to notice that Kirchhoff-Love theory allows to consider multilayered shells also, under condition of reduction of characteristics of several layers to characteristics of a certain one average layer [26]. It provides an opportunity to analyze the investigated effects for such important applied problems as oscillations of cylindrical supports and pipelines weakened by hydrogen corrosion.

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25. Filippenko, G.V.: The banding waves in the beam with periodically located point masses. Comput. Continuum Mech. 8(2), 153–163 (2015). https://doi.org/10.7242/1999-6691/2015. 8.2.13 26. Kossovich, L.Y., Wilde, M.V., Shevzova, Y.V.: Asymptotic integration of dynamic elasticity theory equations in the case of multilayered thin shell. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform. 12(2), 6–64 (2012). https://doi.org/10.18500/1816-9791-2012-12-2-56-64

Gear Shift Analysis in Dual-Clutch Transmissions Using Impact Theory Georgy K. Korendyasev and Konstantin B. Salamandra(B) Mechanical Engineering Research Institute of the Russian Academy of Sciences (IMASH RAN), Moscow, Russia [email protected]

Abstract. The article discusses a dynamic model of a transmission containing a dual-clutch gearbox. The absence of a torque converter in such gearboxes with taking into account the short duration of gear shifting (0.2–0.5 s) allows to present the gear shifting process as an impact. Gear shifting excites the output shaft of the gearbox. Thus, the transmission of the vehicle can be considered as a vibro-impact self-oscillating system. Keywords: Gearbox · Gear shift · Transmission · Shift control · Dynamic analysis

1 Introduction Requirements to improve efficiency and reduce harmful substances emission from vehicle engines have led, in particular, to the fact that in modern automatic transmissions gear shifting is carried out in 0.2–0.5 s [1, 2]. The gear shifting process occurs without interrupting the power flow, by simultaneously disengaging one control element (friction clutch) and switching on another control element. The fast shifting process, even with a small difference between the velocities of the engine and the output shaft of the gearbox, leads to a sharp change in inertial torques, which in turn can worsen the comfort of vehicle passengers, reduce the service life and reliability of the transmission. A torque converter is installed between the engine output shaft and the gearbox in automatic transmissions consisting of planetary gear trains [1–5]. The rotation of the turbine wheel connected to the engine output shaft in a torque converter is transmitted to the impeller wheel connected to the gearbox input shaft using a fluid (oil). The lack of a rigid connection between the engine and the gearbox allows the torque converter to dampen sudden changes in inertial torque caused by short gear shifts. Low efficiency is the main disadvantage of the torque converter. Dual-clutch transmissions [6–9] have become widespread and well developed at present. There is no torque converter, and the connection of the engine output shaft and the gearbox input shaft is carried out by multi-plate friction clutches controlled by servo drives. The use of clutches instead of a torque converter allows to increase the efficiency, © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 92–101, 2021. https://doi.org/10.1007/978-3-030-62062-2_10

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but it leads to impact interaction of the transmission parts due to the short duration of the gear shifting process. The kinematic diagram of a 6-speed dual-clutch gearbox [10] is shown in Fig. 1. In the diagram (Fig. 1): I - input shaft; O1, O2 - gears meshing with the output shaft of the gearbox (not shown in the diagram); c1, c2 - multi-plate friction clutches; s31, s24, s5 and sR6 are synchronizers that engage the appropriate gear.

Fig. 1. Kinematic diagram of a 6-speed dual-clutch gearbox.

The dual-clutch gearboxes have branched arrangement. It allows shifting in an unloaded branch, that is, the next speed can be prepared in advance before it is directly engaged. Conventionally, such a gearbox can be represented in the form of two gearboxes, for the alternate connection of which to the input shaft is responsible the corresponding multi-plate friction clutch located at the input of the gearbox. Both clutches are made as single unit structurally. For example, if the vehicle moves in the fourth speed and continues to accelerate, then the synchronizer responsible for the fifth speed can be switched on in advance and, as soon as the required velocity to shift the next speed is reached, the control system disengages clutch c2 and switches on clutch c1 to engage fifth speed.

2 Formulation of the Problem The dynamic analysis of gear shifting process is currently carried out sequentially by dividing it into stages [11–14]. Differential equations are drawn up for each stage, the equations solutions determine the overall pattern of velocities and torques changes. This approach is fully justified for dynamic analysis of processes in manual transmissions, the duration of gear shifts in which is seconds, and all phases of the process can be correctly described.

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The dynamic analysis of fast shifting processes in modern automatic transmissions leads to useful and interesting results but is associated with a number of assumptions about the interactions of gearbox elements (friction clutches, brakes, or synchronizers) switched on and off. It is often assumed to simplify the problem that the reduced moment of inertia of the rotating links of the vehicle after the gearbox is incommensurably large in comparison with the moment of inertia of the links of the engine. This assumption halves the order of the equations being solved but introduces errors in the analysis results. It is rather difficult to divide the short time period of gear change even into two stages for sequential analysis in addition. It is also obvious that gear shifting actually acquires the properties of impact interaction at small values of the duration of the process. The inertial moments caused by a sharp change in the velocities of the transmission elements in this case are significantly exceed the engine moments and the reduced moments of resistance to motion. It should be noted that the angular momentum conservation theorem was used [15, 16] to determine the duration of the gear shifting process and the effect of inertial parameters and velocities of links on impact loads. The accuracy of quantitative assessment of the impact torque for a manual gearbox was not great at the same time, since manual gear shifting is calculated in seconds, and the unaccounted driving torques and torques of resistance to motion are practically of the same order as the moments of inertia forces of rotating links. It is proposed to return to the use of the angular momentum conservation theorem of a mechanical system upon impact for a dynamic analysis of this process in this work in a view of the short duration of the gear shifting process in modern dual-clutch transmissions.

3 Transmission Dynamic Model A simplified transmission diagram to be considered are shown in Fig. 2. It consists of an engine, a two-speed dual-clutch gearbox (clutches c1, c2), an output shaft of the gearbox connected to the differential by an elastic link with a stiffness coefficient c and a damping coefficient b. The differential distributes the rotation to the drive wheels connected to the vehicle body. The rotation is transmitted to the intermediate shaft through the gear wheel of the first speed with the gear ratio i1 when the clutch c1 is switched on; and to the intermediate shaft through the gear wheel of the second speed with the gear ratio i2 when the clutch c2 is switched on. Rotation is transmitted from the intermediate shaft to the output shaft with the gear ratio i. The moment of inertia of the moving parts of the engine reduced to the body of the clutch being switched on will be denoted by JI ; the moment of inertia of the moving parts of the gearbox reduced to the output shaft before the link with the maximum flexibility will be denoted by JO ; element JV - the moment of inertia of the transmission links after the link with the maximum flexibility, reduced to the output shaft, including the differential, half-shafts, drive wheels and the vehicle body. Thus, the resulting dynamic model contains three inertial elements: JI and JO are rigidly connected using clutches c1 or c2, and the JO element is connected to the JV element by an elastic connection. The resulting model is a self-oscillating vibro-impact system as shown below. Suppose that the movement of the model (Fig. 2) occurs in first speed and we need to consider shifting into the second speed. The driving torque of the engine and the

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Fig. 2. Transmission diagram with two-speed dual-clutch gearbox.

resistance forces to the vehicle movement do not have a significant effect on the changes in the velocities of the model inertial links and excluded from consideration because a short time interval of the switching on the clutch c2. We also assume that the pressure relief in the hydraulic drive of the clutch c1 to be switched off occurs instantly, and the residual torque transmitted by it is negligible and therefore is also not taken into account. Thus, only clutch c2 is involved in the considered gear shifting process. There is a difference in the rotation velocities of the inertial elements JI and JO at the initial moment of time of the clutch c2 switching on process. The frame and discs velocities of the clutch c2 are equalized in a short time when the clutch is switched on. The torque of friction forces between the clutch discs and the frame (depends on the compression force and the coefficient of friction) excludes slippage between the discs and is internal in this case, applied to both parts of the model with different signs. The model under consideration on a little time interval in view of the above is a closed system for which the angular momentum conservation theorem upon impact is applicable. The vehicle speed sensor is located on the transmission output shaft. The magnitudes of the output shaft velocities ωO are set in the gear shift program. The gear shift from (12) the first speed to the second occurs at output shaft velocity ωO and from second speed (21) to the first at ωO . It is necessary that the acceleration for gear shifting from the first to the second speed in this case has to be ω˙ O > 0 and for negative gear shifting has to be ω˙ O < 0. The angular velocity ωI of the gearbox input shaft is related to the angular velocity (12) (12) = i1 iωO at gear shifting from the first ωO of the output shaft by the gear ratios: ωI (21) (21) speed to the second and ωI = i2 iωO at shifting from second to first speed. The velocity of the JO element before gear shifting, for example, from the first speed to the (12) (12) second is ωO , and the velocity of the JI element at this moment is i1 iωO . The element JI moment of inertia reduced to the output shaft before gear shifting on the second speed is equal to JI i22 i2 .

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The angular momentum conservation theorem for the inertial elements JI and JO before and after engaging the second gear:   ωI (12) (2) + JO ωO = JI i22 i2 + JO ωO i2 (2)

JI i22 i2 (2)

ωO in the equation is the output shaft angular velocity after gear shifting to the (2) (12) second speed. Since ωI = i1 ωO , then we get: JI i1 i2 i2 + JO (12) ω JI i22 i2 + JO O

(2)

ωO = (2)

Change in angular velocity ωO at the output of the gearbox after shifting from the first gear to the second: (2)

(12)

ωO = ωO − ωO

=

i 1 − i2 (12) JI i2 i2 ωO JI i22 i2 + JO

(1)

As can be seen from Eq. (1), the change in the velocity of the output shaft when the speeds are shifted from the lowest to the highest is positive, since i1 > i2 . In other words, the gearbox output speed will increase during the shift in question. Similarly, the angular momentum conservation theorem for the inertial elements JI and JO before and after engaging the first gear:   ωI (21) (1) + JO ωO = JI i12 i2 + JO ωO i1 (1)

JI i12 i2 (1)

ωO in the equation is the output shaft angular velocity after gear shifting to the first (1) (21) speed. Since ωI = i2 ωO , then we get: (1)

ωO =

JI i1 i2 i2 + JO (21) ω JI i12 i2 + JO O

(1)

Change in angular velocity ωO at the output of the gearbox after negative shifting from the second gear to the first: (1)

(21)

ωO = ωO − ωO

=

i 2 − i1 (21) JI i1 i2 ωO JI i12 i2 + JO

The velocity of the output shaft when shifting from second to first gear will decrease accordingly. Thus, the ratios of the velocity change in the impact part of the model (Fig. 2) have been derived.

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4 Self-oscillation of the Transmission Output Shaft The self-oscillations of the transmission output shaft caused by a fast change in its velocity at gears shifting with taking into account the presence of elastic and dissipative connections will be considered further. At the intervals between shifting we can  consider the total moment of inertia of the  2 2 engine and gearbox elements JI ij i + JO (where j is the number of the engaged gear) after reducing to the output shaft the moment of inertia JI . Thus, one of the three degrees of freedom in the considered model (Fig. 2) is eliminated. The movement of the model in j-th speed between gear shifts is described by a system of two linear differential equations:       (j) (j) JI ij2 i2 + JO ω˙ (j) + b ω(j) − ωV + c ϕ (j) − ϕV = iMI     (2) (j) (j) (j) JV ω˙ V + b ωV − ω(j) + c ϕV − ϕ (j) = −MR   In system (2): ϕ is the angular coordinate of the inertial element JI ij2 i2 + JO ; ϕV (j) is angular coordinate of the output shaft after  the maximum flexibility; ω  the link with - angular velocity of the inertial element JI ij2 i2 + JO . Frequencies the model in the absence of damping: k1 =   of free vibrations of   (j) 0; k2 = c JI ij2 i2 + JO + JVT / JI ij2 i2 + JO JV . Damped oscillation frequency:      2 ∗(j) (j)2 k2 = k2 − h(j) , where h(j) = b JI ij2 i2 + JO + JV /2 JI ij2 i2 + JO JV .

5 Movement of Model Elements Between Gear Shifts The movement ofthe model elements between gear shifts is a torsional vibration of  2 2 inertial elements JI ij i + JO and JV relative to the center of mass rotating under the action of the engine torque and the torques of resistance forces MI = const = 0, MR = const = 0. The movement of the center of mass of the model is anabsolute movement of the model. The relative motion of the elements JI ij2 i2 + JO and JV are their synchronous torsional oscillations in antiphase, and the magnitudes of the oscillation amplitudes are inversely proportional to the moments of inertia [17]. The general solution of the obtained Eqs. (2) has the form:   (j) ∗(j) (j) (j) ϕ (j) = A(j) e−h t sin k2 t + β (j) + C1 + C2 t + a(j) t 2 /2   (3) (j) ∗(j) (j) (j) (j) ϕV = μ(j) A(j) e−h t sin k2 t + β (j) + C1 + C2 t + a(j) t 2 /2 (j)

In (3): μ(j) = AV /A(j) = −[(JI ij2 i2 + JO )/JV ], a(j) = (iMI − MR )/(JI ij2 i2 + JO + JV ). The constants A, β, C1 and C2 are determined through the initial conditions: the initial angular coordinates and velocities of the inertial elements of the model. The parameters

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C1 , C2 and a are the characteristics of the absolute motion in (3), that is, the motion of the center of mass of the model. The parameters A and β are the characteristics of the relative motion. The inertial elements velocities in the j-th gear between the gear shifts in general form could be obtained by differentiating the Eqs. (3):      (j) ∗(j) ∗(j) ∗(j) (j) ω(j) = A(j) e−h t k2 cos k2 t + β (j) − h(j) sin k2 t + β (j) + C2 + a(j) t      (j) ∗(j) ∗(j) ∗(j) (j) (j) ωV = μ(j) A(j) e−h t k2 cos k2 t + β (j) − h(j) sin k2 t + β (j) + C2 + a(j) t (j)

Absolute velocity C2 is initial velocity of the center of mass of the model in j(j) th gear.  Expressions for the velocities ofinertial elements after substitution sin α = ∗(j)2

h(j) / h(j) + k2 2

(j)

2

∗(j)2

are transformed to the form:



  2 ∗(j)2 ∗(j) (j) h(j) + k2 cos k2 t + β (j) + α (j) + C2 + a(j) t    (j) 2 ∗(j)2 ∗(j) (j) = μ(j) A(j) e−h t h(j) + k2 cos k2 t + β (j) + α (j) + C2 + a(j) t

ω(j) ωV

∗(j)

and cos α (j) = k2 / h(j) + k2

=

(j) A(j) e−h t

6 Numerical Simulation A numerical simulation of the model movement during acceleration of a vehicle with a dual-clutch 6-speed gearbox using the obtained dependences was carried out. The speed ratios of the gearbox considered as an example and the corresponding velocities of the output shaft of the gearbox ω(j) at the assumed maximum rotation velocity of the engine shaft ωI = 6000 rev./min. = 628.3 rad./s (shifting velocity) are given in Table 1. Table 1. Gear ratios of a dual-clutch 6-speed gearbox. Speed i

Step ω(j) at ωI = 628.3 rad./s

1.

4.12

152.5

2.

2.48 1.66 253.2

3.

1.64 1.51 382.2

4.

1.17 1.40 535.2

5.

0.91 1.29 690.5

6.

0.75 1.22 842.3

The rest of the model’s parameters correspond to a modern middle-class passenger car. The graph of the dependence of the angular velocity of the output shaft over time is shown in Fig. 3.

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Fig. 3. Graph of changes in the output shaft angular velocity of the gearbox.

The obtained numerical simulation results show that very significant impact loads arise in the transmission at an instantaneous gear shift. For example, the angular speed with the selected parameters of the model increases for a short time by 133.3 rad./s. when shifting from second to third speed, and by 121.6 rad./s. when shifting from fifth speed to sixth. The amplitude of the velocity change depends on the gear ratio step and decreases with decreasing step accordingly. It should be noted that an extreme shift process is considered here, which makes it possible to estimate the limiting velocity changes in the case of an instant gear shift. Unacceptably high torques with such sharp jumps in velocity act for a short time on the transmission links, which can lead to premature wear of gears, their breakdown and failure of transmission elements. Therefore, the problem of optimizing shift control in dual-clutch gearboxes remains relevant, which is confirmed by a large number of scientific works devoted to these processes.

7 Conclusion The short duration of gear shifts in modern automatic transmissions allows to consider this process as an impact. And the vehicle transmission on a short time interval is like a closed mechanical system for which the angular momentum conservation theorem upon impact is applicable. The dependences of the transmission elements velocities before and after gear shift were obtained with its use.

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It was shown that transmission dynamic model structure as a result of its description corresponds to a vibro-impact self-oscillating system in which the impact caused by gear shifting leads to fluctuations in the angle of rotation of the gearbox output shaft. The numerical simulation of the acceleration process in a vehicle equipped with a 6-speed dual-clutch gearbox carried out. It shows that the change in the velocity of the gearbox output shaft in the case of an instant gear shift is very significant and can exceed the value of the shift velocity to the next gear. It should be noted that the accepted assumption about the absence of slippage between the disks of the clutch being switched on greatly simplifies the simulation, but at the same time allows one to estimate the ultimate loads that occur in the vehicle transmission during gear shifts. Acknowledgments. The research was supported by Russian Science Foundation (Project No. 19-19-00065).

References 1. Naunheimer, H., Bertsche, B., Ryborz, J., Novak, W.: Automotive Transmissions. Springer, Berlin (2011) 2. Fischer, R., Küçükay, F., Jürgens, G., Najork, R., Pollak, B.: The Automotive Transmission Book. Powertrain. Springer, Cham (2015) 3. Förster, H.J.: Automatische Fahrzeuggetriebe. Springer, Berlin (1991) 4. Genta, G., Morello, L.: The Automotive Chassis. Mechanical Engineering Series. Springer, Dordrecht (2009) 5. Salamandra, K.: Perspective planetary-layshaft transmissions with three power flows. In: Evgrafov, A. (ed.) Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11981-2_14 6. Matthes, B.: Dual Clutch Transmissions - Lessons Learned and Future Potential. SAE Technical Paper # 2005-01-1021 (2005) 7. Wheals, J.C., Turner, A., Ramsay, K., O’Neil, A., Bennet, J., Fang, H.: Double Clutch Transmission (DCT) using Multiplexed Linear Actuation Technology and Dry Clutches for High Efficiency and Low Cost. SAE Technical Paper # 2007-01-1096 (2007) 8. Liu, Y., Qin, D., Jiang, H., Zhang, Y.: Shift control strategy and experimental validation for dry dual clutch transmissions. Mech. Mach. Theory 75, 41–53 (2014) 9. Kim, S., Oh, J., Choi, S.: Gear shift control of a dual-clutch transmission using optimal control allocation. Mech. Mach. Theory 113, 109–125 (2017) 10. Schreiber, W., Becker, V.: Doppelkupplungsgetriebe: Patent DE 19821164. 18.11.1999 (1999) 11. Basalaev, V.N., Kovalenko, A.V.: Investigation of the gearshift process under load and optimization controlling the friction clutches of mechanical transmission. Mechanica mashin, mechanismov i materialov # 2(15), 24–32 (2011) 12. Sharipov, V.M., Dmitriev, M.I., Shevelev, A.S.: Gear shifting with varying degrees of overlap in the gearboxes of cars and tractors. Eurasian Sci. Assoc. 1(6(6)), 67–70 (2015) 13. Bai, S., Maguire, J., Peng, H.: Dynamic Analysis and Control System Design of Automatic Transmission. SAE International, Warrendale (2013) 14. Pfeiffer, F.: Mechanical System Dynamics. Corrected Second Printing. Springer, Berlin (2008) 15. Chudakov, E.A.: Design and Calculation of the Car, 3rd edn. Mashgiz, Moscow (1953)

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16. Salamandra, K., Tyves, L.: Application of the Angular Momentum Conservation Law at the Dynamic Analysis of Gearshifts (Paper # F2016-THBG-002). In: FISITA 2016 World Automotive Congress – Proceedings (2016) 17. Panovko, J.G.: Introduction to the Theory of Mechanical Vibrations: Textbook. Manual for Universities, 3rd edn. Nauka, Moscow (1991)

Technological Schemes for Elongated Foramen Internal Surface Finishing by Forced Electrolytic-Plasma Polishing Mihail Mihailovich Radkevich and Ivan Sergeevich Kuzmichev(B) Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia [email protected]

Abstract. This work dedicated to the electrolytic-plasma polishing technology development for elongated foramen internal surface finishing and apparatus design for implementation the proposed technology. Determined main technological interdependence factors, such as flow speed, electric field tension, electrolyte concentration for internal elongated surface quality received by electrolytic-plasma polishing process. Developed local forced electrolytic-plasma polishing technology and obtained rational processing parameters allows to reaching desired internal surfaces quality for copper tubular workpieces. Keywords: Forced · Plasma · Polishing · Technology · Copper · Internal surface · Elongated foramen · Electrode-cathode · Flow speed · Quality · Roughness · Evenness · Precision

1 Introduction In modern manufacture most of efficient way for internal surface finishing is galvanic technologies, due to low labor intensity and high technological efficiency. However, the industrial usage of galvanic polishing constrained by toxic electrolytes that are difficult to dispose without any environmental harm. Despite this, galvanic technologies are used at many stages of precision workpieces manufacturing, starts from blanks production till their finishing. Due to intensive development of machine-building science and technologies, all crucial parts must be processed with higher accuracy and surface quality. Based on increasing productivity and environmental friendliness requirements the development of alternative finishing technologies is an urgent task [1]. One of most relevant alternative finishing technologies is the electrolytic-plasma polishing (EPP). EPP devoid of galvanic technologies realization difficulties, due to the usage of nonaggressive electrolytes. Using the EPP allowed significantly improve the surface quality parameter, defined by roughness Ra (µm). In this way, EPP provides superior effectiveness against now-common finishing technologies. Among all positive sides of EPP it has one major downside – inability to process elongated (L > d) internal surfaces of tubular workpieces. Therefore, the main purpose © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 102–111, 2021. https://doi.org/10.1007/978-3-030-62062-2_11

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is to improve quality of an internal surface, defined by roughness Ra, in extended tubular workpieces with a linear axis and ensure the uniformity of polishing quality parameter over the length of processed tubular workpiece. Developed at TKMM department, of IMMT institute, Peter the Great St. Petersburg Polytechnic University, the kind of EPP technology has received the name - the forced electrolytic-plasma polishing (FEPP). This method allowed to improve quality of an internal surface layer to Ra = 0.06 µm, with processing time taking no longer than 5 min. Possibilities of the forced electrolytic-plasma polishing (FEPP) method and different ways implementation of FEPP in practice show advantages of technology features [10].

2 Methods and Materials Literature sources analysis and laboratory experiments made possible to define the technological features of the electrolytic-plasma polishing and to determine main technological factors that ensure high processing quality. Obtained scientific and technical data allowed us to design fundamentally new scheme for the electrolytic-plasma polishing, which takes into account the vapor-plasma cover formation principles needed to process metal surface. For this purpose, special FEPP device was designed and manufactured, which allows to create electric field density along processing foramen of tubular workpieces. Forced electric field and certain electrolyte flow speed provides the vapor-plasma cover initiation mechanisms exclusively on the processed internal surface [2–6]. The scheme of forced electrolytic-plasma polishing for the elongated workpieces processing shown in Fig. 1.

Fig. 1. The scheme of forced electrolytic-plasma polishing for elongated workpiece internal surface: 1 – wire; 2 – electrolyte; 3 – electrode-cathode; 4 – workpiece-anode; 5 – framework

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Therefore, the main target is to improve quality of the workpiece internal surface and obtain uniformity of polishing along the processed surfaces. The FEPP scheme differs from EPP technology by electro-physical realization principles and includes the electrode-cathode, which equidistant to workpiece-anode cross-section, to achieve required quality of the internal elongated surface. Electrolyte is pumping through the interelectrode gap, formed by the anode and cathode surfaces, at flow speed rate that ensures film-boiling process on the polishing internal surface. During experiments a drawback of this scheme was revealed. The drawback of the FEPP prototype resulted in decreased quality of a middle part of tubular workpiece due to voids of the vapor-plasma cover [7, 8]. A series of experiments were accomplished in order to obtain correlation of the FEPP technological parameters for quality of copper workpiece internal surface. Samples for the experiments has following parameters: length = 100 mm; foramen diameter = 24 mm; input internal surface roughness Ra ≈ 0.32 µm. For this case following technological conditions were used: electrolyte concentration 4%; flow speed range from 0.15 to 0.2 l/s; electrolyte temperature form 70 to 80 °C; voltage 300 V; the FEPP process duration 120 s; electrode-cathode diameter chosen 2/8 from workpiece foramen diameter. According to the results of experiments, output roughness Ra ranged from 0.27 to 0.15 µm along the internal surface. Uniformity coefficient of polishing quality along the internal surface of the workpiece is K = 0.55 (higher is better). Sample’s wall thickness value is  = 1 mm. Most intensive wall thickness decreasing effect ( = 1 mm) detected near the input and output of the electrolyte flow. The cause of workpiece’s wall thickness reduction is uneven anodic dissolution effect. Therefore, to maintain accuracy of the internal surface original geometry, it is necessary find a way to minimize the anodic dissolution process on both sides of the workpiece. Figure 2 Internal surface of the workpiece before and after polishing process. a)

b)

c)

Fig. 2. Experiment stages: a – internal surface of the workpiece before processing; b – The FEPP technological process; c – internal surface of the workpiece after processing.

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Implementing FEPP scheme with electrolyte being fed into the device at a 90-degree angle results in geometry deformity at the workpiece’s ends, which is due to uneven process of anodic dissolution on the sample’s wall [9]. To eliminate uneven anodic dissolution effect, a new scheme of the FEPP with counter-flow was designed. Uneven dissolution process on both sides of workpiece caused by wrong method of electrolyte pumping into the interelectrode gap. The electrolyte entry direction implemented at a 90-degree angle, it caused localization of the vapor-plasma in particular sections on the internal surface of the tubular workpiece and uneven anodic dissolution effect, due to this, uniformity of polishing quality is at a low level [11]. The modified FEPP scheme with counter-flow eliminates uneven anodic dissolution effect. Low uniformity of polishing quality along the workpiece’s internal surface and wall dissolution effect was managed by splitting the flow of electrolyte into counter-flow by the new special device. The electrolyte flows feeding towards each other through opposite nozzle elements. In addition, proposed to pump electrolyte parallel to the axis of the processed sample (at a 0-degree angle) to minimize uneven anodic dissolution effect along the internal surface of a workpiece, thereby increasing processing accuracy. The modified FEPP scheme shown in Fig. 3.

Fig. 3. Functional elements of the special device for FEPP with counter-flow: 1, 6 – input nozzles; 2 – left dielectric body; 3, 9 – fixing flanges; 4 – electrode-cathode; 5 – right dielectric body; 7, 12 – output nozzles; 8, 11 – adapter sleeves; 10 – workpiece-anode.

A new series of experiments with copper tubular workpieces, with the same input characteristics and technological conditions, using the modified FEPP with counter-flow scheme was accomplished [12]. The internal surface quality parameters obtained during experiments: roughness Ra from 0.21 to 0.13 µm along the internal surface; uniformity coefficient of polishing quality along the internal surface of the workpiece is K = 0.62; sample’s wall thickness value is  = 0.3 mm. Figure 4. Internal surface of the workpiece before and after polishing process. During experiments and conditions testing of the forced electrolytic-plasma polishing with counter-flow technology the next limitation of the modified scheme was

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a)

b)

c)

Fig. 4. Experiment stages: a – internal surface of the workpiece before processing; b – The FEPP technological process; c – internal surface of the workpiece after processing.

revealed. When the length of the workpiece exceeds the foramen diameter by more than five times (L > 5d), uniformity of polishing quality noticeable got worse. Previously developed FEPP schemes, which main idea is coaxial positioning of the electrode-cathode all along the tubular workpiece-anode (seen Fig. 1, 3 and 5), not enough to provide the desired internal surface quality parameters. To provide better quality of elongated internal surface was proposed the alternative FEPP scheme, it is shown in Fig. 6 and 7. To provide better uniformity coefficient along the internal surface the localized electric field is required. Thus, was designed a new scheme - the local FEPP with counter-flow, which focuses electric field into one section of the workpiece internal surface [13].

Fig. 5. Illustration of the uneven electric field strength distribution along the internal surface.

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Fig. 6. Scheme of the local FEPP with counter-flow.

The local FEPP scheme implementing in several technological stages. The quantity of technological stages is assigned depending of the workpiece length. The electrodecathode local working zone moves from the beginning of the internal surface towards its end, so this way the polishing process is realized. A new alternative scheme - the local FEPP with counter-flow allowed to implement internal surface fine finishing with a high level of internal surface uniformity along the entire length of the tubular workpiece. The local FEPP provides a stable formation of the vapor-plasma cover in the interelectrode gap because the working surface of electrode-cathode is insulating by the ceramic bushings, which reduce the working zone to 10–50 mm. It is made to achieve high uniformity of internal surface quality by step-by-step processing of the internal surfaces along the workpiece. The local FEPP allows to fine finishing the internal surface of any length due to the moving ceramic bushings, with each step, a new local working area of the electrodecathode revealing.

3 Results Designed special device for realization the local FEPP technology allowed to minimize uneven of electrolyte flows by fine distributing them across the volume of the processing area. Electric field strength constancy and intensity level on the local section of processed internal surface ensured by using the special device for the local FEPP. Realization of these requirements provides stability and maintenance initialization of the vapor-plasma cover in the local processing area. The local FEPP with counter-flow technology shown in Fig. 7.

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Fig. 7. The special device for internal surface fine finishing by a local FEPP with counter-flow.

The special device consists: body elements and fixing flanges that provides electrodecathode and workpiece-anode fixation. Adapter sleeves for basing workpieces with different dimensions and ceramic bushings to adjust the length of electrode-cathode local working zone. The internal surface quality parameters obtained during experiments: roughness Ra from 0.06 to 0.08 µm along the internal surface; uniformity coefficient of polishing quality along the internal surface of the workpiece is K = 0.75; sample’s wall thickness value is  = 0.06 mm. Figure 8. Internal surface of the workpiece before and after polishing process. a)

b)

Fig. 8. Workpiece internal surface quality: a – before processing; b – after the local FEPP.

Figure 9. Three technological stages of the tubular workpiece internal surface processing by the local forced electrolytic-plasma polishing with counter-flow.

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a)

b)

109

c)

Fig. 9. The local FEPP technological steps: a – first pass; b – second pass; c – third pass.

4 Conclusion The local FEPP with counter-flow technological process allowed to remove the restriction on the length of the workpiece, while providing uniformity coefficient of internal surface quality K ≈ 0.75, regardless of the workpiece length. In this way, the local FEPP with counter-flow considerably surpasses effectiveness in comparison to the previous FEPP schemes. Currently, the automatic electrode-cathode feeding system and the new special device for local FEPP are developing for better polishing quality and higher uniformity criterion of the processed elongated internal surface. Implications. 1. A fundamentally new technology of the electrolytic-plasma polishing has been developed, as well as the special device that ensures the processing of internal surface by the forced electrolytic-plasma polishing. After FEPP of copper tubular workpiece, with an initial roughness Ra ≈ 0.32 µm, obtained roughness Ra in the range from 0.27 to 0.15 µm. Uniformity coefficient of polishing quality along the internal surface of the workpiece is K = 0.55. 2. A modified with counter-flow the FEPP technology has improved quality results. The values of Ra in range from 0.21 to 0.13 µm have been reached. Uniformity coefficient of polishing quality along the internal surface of the workpiece is K = 0.62. 3. The current FEPP scheme - the local FEPP with counter-flow technology eliminates workpieces length limitation, while providing internal surface quality parameters that are significantly superior to the results obtained using the previous FEPP schemes. New designed special device for the local FEPP ensures polishing process for the tubular workpieces with a wide range of dimensions. The internal surface quality

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parameters obtained during experiments: roughness Ra along the internal surface in range from 0.06 to 0.08 µm; uniformity coefficient of polishing quality of the workpiece is K = 0.75. Sample’s wall thickness value is  = 0.06 mm.

References 1. Kulikov, I.S., Vashchenko, S.V., Kamenev, A.Y.: Electrolitho-plazmennaya obrabotka materialov, p. 232. Belaruskaya Navuka, Minsk (2010). (rus.). ISBN 978-985-08-1215-5 2. Slovetsky, D.I, Terentyev, S.D.: Parametry elektricheskogo razryada v electrolitakh I fizikokhimicheskiye protsessy v electrolitnoy plasma. Khimiya vysokikh energy 37(5), 355–362 (2003). (rus.) 3. Arora, P.K., Haleem, A., Singh, M.K., Kumar, H., Singh, D.P.: Optimal design of a production system. In: Mandal, D.K., Syan, C.S. (eds.) CAD/CAM, Robotics and Factories of the Future. Lecture Notes in Mechanical Engineering, pp. 697–703. © Springer, India (2016). https://doi. org/10.1007/978-81-322-2740-3_67. ISBN 978-81-322-2738-0, ISBN 978-81-322-2740-3 (eBook) 4. Manna, S.K., Pradhan, S.K., Mandal, D.K.: Design and development of a low cost automation injection molding machine of 250 g capacity using microcontroller. In: Mandal, D.K., Syan, C.S. (eds.) CAD/CAM, Robotics and Factories of the Future. Lecture Notes in Mechanical Engineering, pp. 1–12. © Springer, India (2016). https://doi.org/10.1007/978-81-322-274 0-3_1. ISBN 978-81-322-2738-0, ISBN 978-81-322-2740-3 (eBook) 5. Suminov, I.V., Belkin, P.N., Epelfeld, A.V., Lyudin, V.B., Crete, B.L., Borisov, A.M.: Plasmenno-electroliticheskoye modifitsirovaniye poverkhnosti metallov i splavov, p. 464. Technosfera, Moskva (2011). (rus.) 6. Kuzmichev, I.S., Ushomirskaya, L.A., Shmelkov, A.V., Sysoev, I.A.: Linear, inner, lengthy surfaces finishing using the forced-flow electrolyte-plasma process for metal products. Nauchno-proizvodstvennii jurnal “Metalloobrabotka”. AO «Izdatel’stvo “Politehnika”». №3 (99)/2017. – 70 str. (rus.). ISSN 1684-6702 7. Pooranachandran, K., Deepak, J., Hariharan, P., Mouliprasanth, B.: Effect of flushing on electrochemical micromachining of copper and Inconel 718 alloy. In: Vijay Sekar, K., Gupta, M., Arockiarajan, A. (eds.) Advances in Manufacturing Processes. Lecture Notes in Mechanical Engineering, pp. 61–69. © Springer Nature Singapore Pte Ltd., Singapore (2019). https:// doi.org/10.1007/978-981-13-1724-8_6. ISBN 978-981-13-1723-1, ISBN 978-981-13-1724-8 (eBook) 8. Dutta, P., Barman, A., Kumar, A., Das, M.: Design and fabrication of electrochemical micromachining (ECMM) experimental setup for micro-hole drilling. In: Biswal, B., Sarkar, B., Mahanta, P. (eds.) Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering, pp. 561–573. © Springer Nature Singapore Pte Ltd., Singapore (2020). https://doi. org/10.1007/978-981-15-0124-1_51. ISBN 978-981-15-0123-4, ISBN 978-981-15-0124-1 (eBook) 9. Gysin, Al.F., Gysin, A.F., Gysin, F.M.: Mnogokanalnyy razryad mezhdu therdym i elektroliticheskim elektrodami v protsessakh modifikatsii materialov pri atmosfernom davlenii. «Nizkotemperaturnaya plazma v protsessakn naneseniya funktsionalnykh pokrytiy». Naucchno-tekhnicheskaya konferentsiya s elementami shkoly: sbornik statey, pp. 9– 20. Kazanskiy gosudarstvennyy tekhnologicheskiy universitet. Izd-vo KSTU, Kazan (2010). (rus.) 10. Radkevich, M.M., Kuzmichev, I.S.: Technological principles of internal surfaces finishing by forced electrolytic-plasma polishing. In: Key Engineering Materials, vol. 822, pp. 634–639. https://doi.org/10.4028/www.scientific.net/KEM.822.634. ISSN 1662-9795

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11. Popov, A.I., Tyukhtyaev, M.I., Radkevich, M.M., Novikov, V.I.: The analysis of thermal phenomena occuring under jet focused electrolytic plasma processing. Nauchno-tehnicheskie vedomosti Sankt-Peterburgskogo gosudarstvennogo politehnicheskogo universiteta 4(254) 141 (2016). https://doi.org/10.5862/jest.254.15. UDK.621.78 (rus.) 12. Ushomirskaya, L.U., Baron, Yu.M., Kuzmichev, I.S.: Design of special device for the forced electrolytic-plasma polishing of internal surfaces by counter flows. In: Key Engineering Materials, vol. 822, pp. 610–616. https://doi.org/10.4028/www.scientific.net/KEM.822.610. ISSN 1662-9795 13. Gaysin, Al.F., Son, E.Ye.: Parovozdushnyye razryady mezhdu struynym elektroliticheskim katodom I metallicheskim anodom pri ponizhennykh davleniyakh. Teplofizika vysokikh temperatur 48(3), 470–472 (2010). (rus.)

Local Buckling of Curvilinear Plates in Axial Compression Konstantin P. Manzhula and Anastasia A. Valiulina(B) Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia [email protected], [email protected]

Abstract. In this work the local buckling issues of curved on radius plates, compressed by axial force are considered. Were investigated instability modes, influence of geometrical parameters and boundary conditions of unloaded sides on local buckling and buckling stresses. Buckling stress dependencies are obtained depending on the boundary conditions and the geometrical parameters of the plate. Keywords: Curved on radius plate · Local buckling · Buckling stresses · Boundary conditions

1 Introduction The most common elements of steel structures are compound beams, they have walls and flanges in their composition. Generally, the beam walls are made thinner in order to reduce weight. Ensuring of local buckling resistance is the determining criterion of structural ability of beams under compression or compression with bending [1, 2]. To increase the local buckling resistance of walls and flanges they are supported by longitudinal and transverse ribs, which increases weight and labor intensity of manufacturing of all structure [3–6]. One of the ways to reduce steel intensity of the beams is corrugation of their walls. Such walls are widely used in building structures [7, 8]. The thickness of corrugate walls is assumed within 2…8 mm, which ensures them all advantages determined by thinness. The disadvantage of them is that they increase stiffness only in the transverse direction, but do not give an effect with a preferential longitudinal compression. In the last case of loading, the walls with longitudinal corrugations are proposed [9], which have found wide application, for example, in the wagons. Walls with corrugation aren’t widely used in steel structures of cranes, because of complex loading conditions, wide list of geometrical parameters and forms, low series production (walls with longitudinal corrugations are made by rolling) [10]. One of the effective ways to increase local buckling resistance of the walls is put them into curvilinear shape relative to the longitudinal axis. The effectiveness of using walls curved on the radius in box-section beams is proved in [11–14]. Buckling stresses in such beams are increased by 2–3 times [11]. In the scientific literature, there are practically no studies of local buckling of plates curved on the radius. The exception is the standard [15]. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 112–121, 2021. https://doi.org/10.1007/978-3-030-62062-2_12

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In the analytical solution of local buckling, a plate is considered to be simply supported or clamped along the contour, or rest upon a combination of these boundary conditions on the sides. In this work the influence of plate boundary conditions on the critical buckling stresses, as well as the influence plate geometrical parameters on them is investigated.

2 Local Buckling of Simply Supported Plates At the first stage, flat and curved on radius R along longitudinal axis X plates were investigated. All considered plates were simply supported with compressive loads at the but ends (Fig. 1).

Fig. 1. The diagram of plates: a – flat plate; b – curved on radius plate

Local buckling of plates was investigated in ANSYS software. Geometrical parameters of the considered plates are presented in Table 1. Table 1. Variable geometrical parameters of the plates The geometrical parameter The values of the variable parameters d, m

0.5, 1, 1.5, 2

a, m

0.5, 1, 2, 3, 4

t, mm

4, 5, 6, 7, 8, 10, 12, 16

R, m

0, 1, 2, 5

640 calculation models were made and investigate by varying the parameters presented above.

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3 Influence of Geometry on Instability Modes During the analysis the first buckling mode and value of buckling stress recorded. The analysis results allowed to form the following regularities. In flat plate, the classical buckling form is persist regardless of variation of plate height d, length a and the wall thickness t (Fig. 2, a). With a small curvature of plate (R = 5 m, d = 0.5 m; R/d = 2.5) the buckling form, shown in Fig. 2, b, is persisted as the length of the plate increases, but there is attenuation of the buckling to the fixed end.

Fig. 2. Forms of plates buckling at d = 0.5 m i t = 0.008 m: a – flat plate; b – curved-shaped on radius R = 5 m plate

With the height of the plate increases (R = 5 m, a = 1 m) to the height d = 1 m, the influence of the boundaries decreases and the plate takes the buckling form shown in Fig. 3, a. A similar buckling form is observed in the plate with R = 1 m, d = 0.5 m, a = 1 m i t = 0.004 m (R/d = 2) in Fig. 3, b. With the height of the plate increases the area of local buckling is localized by the side of the load application (Fig. 3, c), which can be seen in the plate with the parameters R = 2 m, d = 2 m, a = 3 m i t = 0.008 m (R/d = 1). The same forms of local buckling are observed in plates with R = 2 m, d = 1.5 m (R/d = 1.3) and R = 1 m, d = 1.5 m (R/d = 0.7).

Fig. 3. Buckling forms of curved plate: a – R = 5 m, d = 1 m, a = 1 m, t = 0.008 m; b – R = 1 m, d = 0,5 m, a = 1 m, t = 0.004 m; c – R = 2 m, d = 2 m, a = 4 m, t = 0.008 m; d – R = 1 m, d = 2 m, a = 1 m, t = 0.008 m

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Finally, at the combination d = 2 m and R = 1 m (R/d = 0.5) the plate loses overall buckling at any considered thickness and length, as shown in Fig. 3, d. Analysis of buckling forms of simply supported plate shows that buckling forms of curvilinear plates are mainly influenced by the ratio of curvature radius of a plate to its height R/d. At R/d ≥ 4 the plate with a/d = 1 has one bulge, the number of which grows in proportion to the growth of ratio a/d. At 0.5 < R/d ≤ 3.3 the shape of bulge shifts from the central position along the length to the end. At R/d = 0.5 (the high plate) the overall buckling of the plate is lost.

4 Influence of Geometrical Parameters on Buckling Stress Changing the geometry of the plate in different ways affects the critical buckling stress σ cr of the simply supported plates. The length of the plate in form of a/d ratio, as expected, practically doesn’t affect the buckling stress σ cr . Figure 4 and 5 show the influence of relative plate length a/d for different thicknesses t from 4 to 16 mm.

Fig. 4. Dependence of local buckling stresses on relative length of the plate a/d at d = 1 m

The dependences have a similar form and close stress values for plates with heights d = 0.5, 1 and 1.5 m. For the plate height d = 2 m a significant drop stress occurs (Fig. 5), which is a consequence of weak influence of boundary conditions. A slight decrease of buckling stresses within 3–7% is observed with the growth of the plate length at curvature radius R = 1 m and R = 2 m. For comparison, Fig. 4 and 5 show buckling stresses in flat plate. As can be seen from the diagrams, curvature of the plate along the radius allows increasing buckling stresses from 2 to 5 times in comparison with a flat plate. The thickness t of the plate has a significant impact on buckling stresses. In this case, the influence of thickness noteworthily increases with the growth of the curvature radius of the plate. Figure 4, 5 and 6 show the effect of thickness on buckling stresses at plate height d = 1, 1.5 i 2 m. As the curvature increases, the dependence becomes more nonlinear.

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Fig. 5. Dependence of local buckling stresses on relative length of the plate a/d at d = 2 m

Fig. 6. Dependence of local buckling stresses on relative thickness t/d, relative length a/d of the plate and curvature radius R = 0, 1, 1.5 and 2 m at d = 1.5 m

Based on the results of the numerical experiment by the method described in [13], a calculated dependence for determining the values of buckling stresses was obtained  t 1.007 σcr = 0.084 · Ks0 ·  0.992d R d

where Ks0 =

π 2 ·E 3(1−μ2 )

≈ 760000 MPa.

·

 a 0.004 , d

(1)

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The range of variable parameters included processing corresponds to Table 1, from which the parameters R = 0, a = 0.5 m i d = 0.5 m are excluded. Comparison of the numerical calculation results and calculation by dependence (1) is shown on the example of the plate with radius R = 1 m at its different height (Fig. 7). The average difference between the results obtained by the numerical method and results obtained by dependence (1) is 1%. This fact indicates that it is possible to determine the buckling stresses by the found dependence with sufficient accuracy.

Fig. 7. Comparison of the numerical calculation results in Ansys software with the results of calculation according to dependence (1) for plate with R = 1 m

5 Accounting of the Plate Support Stiffness In real metal structures the boundary conditions of the plates vary between simply support and rigid support. In flat plate this condition is taken into account by the clamping coefficient [16]. In this work, the influence of plate support is modeled by means of composite I-beam, whose flanges have variable stiffness (Fig. 7). The real support conditionals of wall may be modeled by varying torsional stiffness of flanges. The calculation model has the following support and load conditionals. On the front I-beam face 1 linear bounds are set along the X and Z axes, on the back face 2 – only along the X axis (Fig. 8). The edge 3 of the front face of the lower flange is fixed along all degrees of freedom. A uniform distributed load is applied to the back face 2. The supporting force resulting from fixing of the front face 1 along the Z axis models a uniform distributed load on the face 1.

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According to the plan of the numerical experiment, the geometric parameters presented in Table 2 were varied. 512 models were compiled and calculated. Table 2. Variable geometrical parameters of the plates The geometrical parameter The values of the variable parameters d, m

0.3, 0.6

a, m

1, 2,

t, mm

4, 6, 8, 10, 12

tf , mm

15, 20, 25, 30

R, m

0, 1, 2, 5

Fig. 8. Model for investigation of boundary conditions of the plate

An integrate parameter that determines the stiffness of the support boundaries of the plate is torsional moment of inertia of the flange, determined by Jt = β · b · tf3 ,

(2)

where β is coefficient dependent on the ratio b/tf , β = 0.313 at b/tf = 10, β = 0.333 at b/tf > 10. The influence Jt on buckling stresses for the investigated plate thicknesses at different curvature radii is shown in Fig. 9. The dependence of the influence, on average, is close to linear for flat and curved plates. It can be seen that increasing curvature of the plate the influence of supporting stiffness on relative level of critical stresses σcr is weaker in comparison with flat plate.

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Fig. 9. Influence of the torsional moment of inertia of the flange on buckling stresses

The results were processed in the following sequence. Cases in which the flange loses local buckling before wall, which corresponds to the case of simply support of the plate. The flange with a large thickness can be considered rigid, therefore, the plate edge connected with the flange can be considered rigid supported. On that basis, cases in which a change of flange thickness doesn’t lead to a change of buckling stresses are identified and excluded from the calculation. On the remaining results, the calculation dependence was obtained for calculating buckling stresses taking into account the support.  tw 1.003 σcr = 0.121 · Ks0 ·  d0.134 a d

·

· JT0.02  R 0.793 ,

(3)

d

·E where Ks0 = 3 π1−μ ≈ 760000 MPa. ( 2) Comparison of the calculation results obtained according to dependence (3) with the numerical calculation results in Ansys software is shown in Fig. 10 for plate with R = 1 m. The diagrams for plates with other curvature radii look similar. The largest error of calculated dependence (3) is observed for plate with R = 5 m and t = 0.012 m and is equal 10% in two points. The average error of the dependence (3) is equal 4% in all calculated points. 2

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Fig. 10. Comparison of the numerical calculation results with the results of calculation according to dependence (3) for plate with R = 1 m

6 Conclusion The results of performed investigations have shown the expedience and efficiency of using walls curved on radius along the longitudinal axis in rods and beam working under compression conditions. The walls curved on radius increase the buckling stresses by 2–5 times in comparison with flat walls. To estimate buckling stresses in dependence to geometrical parameters and support conditions of walls the calculated dependencies were obtained, which give good correspondence with numerical investigations.

References 1. SP 16.13330.2017 «SNIP II-23-81». Stal’ny’e konstrukcii, Moscow (2017) 2. GOST 33169-2014. Krany’ gruzopod”emny’e. Metallicheskie konstrukcii. Podtverzhdenie nesushhej sposobnosti. Standartinform, Moscow (2015). 50 p. 3. Sokolov, S.A., Grachev, A.A.: Ustojchivost’ plastiny’ s prodol’ny’m rebrom. Vestnik MGTU im. N.E‘.Baumana. Ser. «Mashinostroenie», № 4, p. 117–127 (2015) 4. Grachev, A.A., Sokolov, S.A.: Vliyanie maly’x otklonenij ot ploskoj formy’ e’lementov tonkostenny’x metallicheskix konstrukcij na ix napryazhennoe sostoyanie. In: Evgrafova, A.N., Popovicha, A.A. (eds.) Sovremennoe mashinostroenie: Nauka i obrazovanie: materialy’ 6-j mezhdunarodnoj nauchno-prakticheskoj konferencii. Izd-vo Politexn, un-ta, SPb (2017). https://doi.org/10.1872/mmf-2017-65. ISSN 2223-0807 5. Sokolov, S., Grachev, A.: Local criterion for strength of elements of steelwork. Mech. Eng. Res. Educ. Int. Rev. Mech. Eng. 12(5), 448–453 (2018) 6. Grachev, A.A., Sokolov, S.A.: Vliyanie neploskix e’lementov na rabotosposobnost’ tonkostenny’x metallicheskix konstrukcij gruzopod”emny’x mashin. Nauchno-texnicheskie vedomosti SPbGPU. 2-2(147)72012. SPb Izd. Politexnicheskogo universiteta, pp. 78–81 (2012)

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7. Lukin, A.O., Aplatov, V.Yu., Chernyshev, D.D.: Sovershenstvovanie konstruktivnogo resheniya balki s gofrirovannoj stenkoj. Vestnik SGASU. Gradostroitel’stvo i arhitektura, № 2 (2016) 8. Bal’zannikov, M.I., Kholopov, I.S., Solov’ev, A.V., Lukin, A.O.: Primenenie stal’nyh balok s gofrirovannoj stenkoj v gidrotekhnicheskih sooruzheniyah. Vestnik MGSU, pp. 34–41 (2013) 9. Shadur, L.A.: Wagons. Konstrukciya, teoriya i raschet. Transport, Moscow (1980). 439 p. 10. Kulkova, N.N.: Osobennosti napryazhenno-deformirovannogo sostoyaniya elementov kranovoj korobchatoj balki pri raschete stenok v zakriticheskoj oblasti. Tr VNIIPTmash (1981) 11. Manzhula, K., Naumov, A., Sokolov, S.: Local buckling of box-shaped beams due to skew bending. IOP Conf. Ser.: Earth Environ. Sci. 337, 1–10 (2019). 012086 12. Naumov, A.V., Manzhula, K.P.: Raschetno-eksperimental’noe issledovanie mestnoj ustojchivosti korobchatyh balok s krivolinejnymi stenkami. Nauchno-tekhnicheskie vedomosti SPbPU. Estestvennye i inzhenernye nauki 25(3), 108–119 (2019) 13. Manzhula, K.P., Naumov, A.V.: Influence of flections radius value to local buckling of boxesshaped beams with non-linear walls. Int. Rev. Mech. Eng. (IREME) 11(5), 326–331 (2017) 14. Manzhula, K.P., Naumov, A.V.: Optimizaciya geometrii i massy korobchatoj balki s krivolinejnymi stenkami pri raschete na mestnuyu ustojchivost’ ot izgibayushchego momenta. Nauchno-tekhnicheskij vestnik Bryanskogo gosudarstvennogo universiteta, №4, pp. 481–487 (2019) 15. LLOYD Rules for Classification and Construction IV Industrial Services. Germanischer Lloyd SE (2011) 16. Alfutov, N.A.: Osnovy rascheta na ustojchivost’ uprugih sistem. Mashinostroenie, Moscow (1978). 312 p.

Interactive Synthesis of Technological Dimensional Schemes Kirill P. Pompeev(B) , Andrey A. Pleshkov, and Viktoriya A. Borbotko ITMO University, Saint Petersburg, Russia [email protected]

Abstract. The paper considers the problem of accuracy assurance during part manufacturing. Ensuring the required accuracy is one of the aspects of achieving the level of quality, implied during product design. It should start even before the actual manufacturing process, during process planning, which is one of the main stages in product life-cycle. In digital manufacturing, while using automatic and semi-automatic equipment, including CNC machines, a multi-operational technological process must be reliable in the context of ensuring its accuracy parameters, i.e. must guarantee in advance the compliance with requirements of dimensional accuracy and surfaces location and orientation which is specified in a part drawing. This can be achieved by carrying out a dimensional accuracy analysis of a technological process at its design stage, which will reduce time and material costs when introduced in production, due to a significant decrease in the probability of obtaining defects, related to precision parameters. When conducting a dimensional accuracy analysis of a technological process, it is necessary to preliminarily create diagrams of linear dimensions and circular runouts. Creating these diagrams and performing the required calculations manually is a rather time-consuming process. This fact as well as the lack of necessary software packages (even though there exist several programs for calculating linear and diametric dimensions and machining allowances, developed by authors of the present paper) today at industrial enterprises engineers do not conduct dimensional accuracy analysis when designing multi-operational manufacturing processes and rely solely on their own experience and the experience of adjusters and CNC machine operators. And this, in turn, holds back the process of manufacturing process automation, which is unacceptable during the transition to digital manufacturing. As a solution to this problem, it is proposed to introduce into the practice of technological process design developed calculation programs and a software tool in development, based on methods of interactive synthesis of linear dimensional and runout diagrams, described in the paper. Keywords: Mechanical engineering · Instrument making · Technological preparation of production · Digital manufacturing · Product life-cycle · Manufacturing process · Product · Part · Operational workpiece · 3D-model · Modeling · Interactive synthesis · Linear dimension diagram · Runouts diagram · Dimensional analysis · Graph theory · Quality · Technological process reliability to ensure accuracy

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 122–135, 2021. https://doi.org/10.1007/978-3-030-62062-2_13

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1 Introduction As long as there exists industrial production, ensuring the quality of manufacturing parts will always remain relevant. The present stage of production development has finally affirmed the following point of view on the quality of manufactured products: a client should purchase defect-free products, which quality is not 90 or 99%, but exactly 100%, and its production should be carried out without fixable, let alone irreparable defects [1]. Quality becomes the main tool for reducing costs, as producing a part correctly at first try is always cheaper than eliminating the obtained defects [2]. The technical quality of products is established and ensured at product life-cycle (PLC) stages like marketing research, product design, technology development and product manufacturing. Moreover, the technological support of parts quality indicators already begins at the design stage [2–6]. Aspects of quality assurance, highlighted in technical literature, are very diverse. Significant attention is given to the problem of ensuring the surface roughness of workpieces during turning [1–3], milling [7, 8], grinding [9], as well as modeling, prediction and optimization of surface roughness expected after processing [9–14]. Another equally important aspect of product (parts) manufacturing quality is to ensure dimensional accuracy [15–17] and requirements for surfaces location and orientation [5, 6]. Problems of improving the quality of processing [18] and its productivity [19, 20] through, among other ways, optimization of cutting parameters [21–24] are also considered. Currently, the industrial production is facing a transition towards digital manufacturing (DM) [25]. Its organization is carried out by introducing a number of concepts, including the concept of Industry 4.0 [26–28]. Under these conditions, a close integration of design and technological process (TP) planning systems [25, 29, 30] is necessary at different life-cycle stages for an effective objects (products) modeling and a production processes simulation [31, 32]. It should also be noted that, as a part of Industry 4.0 concept implementation, the issue of organizing DM as a set of interacting cyberphysical systems (CFS) is being actively considered [28, 33, 34], which, as stated in [34], include an operational workpiece (its 3D model) with the ability to independently determine the next manufacturing resource for its processing in accordance with the current state of a production system. Accordingly, the role and importance of a 3D model of an operational workpiece in the manufacturing process planning and organization is increasing, which is possible if these processes are automated. It can be concluded that during the technological process planning, the entire set of interconnected operational workpieces (their 3D models) should be developed in an automated (automatic) mode [35, 36], taking into account the assurance of the part accuracy parameters without defects [4–6, 37]. To accomplish the latter, it is necessary to minimize or eliminate the influence of the subjective factor by creating knowledge, rules and techniques bases [4–6, 16, 38–40]. In any production, product quality indicators are closely related to the accuracy of manufacturing parts of which it is composed. It is known that the accuracy of a part is understood as its conformity to requirements of the drawing: to its dimensions, geometric shape, requirements for machined surfaces location and orientation and the degree of

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roughness [15]. In the present paper, issues related to ensuring a given dimensional accuracy and required surfaces location and orientation is considered.

2 Methods and Materials 2.1 The Role of Dimensional Accuracy Analysis of a Technological Process in Accuracy Assurance of Parts Manufacturing Process planning is one of the important stages of product life-cycle, at which the possibility of quality assurance of a product (part) manufactured according to the process designed using computer-aided process planning (CAPP) is established and determined. Simultaneously, the effectiveness of computer-aided design (CAD) systems and the quality of decisions depend on built-in principles, methods, rules and algorithms for converting existing technical and technological knowledge [15, 37, 39, 41] into specific technological process for parts manufacturing. Therefore, technologies designed in a CAPP must have the reliability of ensuring a given dimensional accuracy [16] and the requirements for the cross-referenced tolerances (orientation and location of surfaces) [5, 6, 9, 39], which can be characterized by an accuracy margin indicator [5, 15, 16], defined as the ratio of given error of an accuracy parameter to its error expected during the TP implementation. Dimensional accuracy analysis (DAA) of TP allows designing technologies for manufacturing parts that are reliable in terms of accuracy parameters [5, 16]. However, to date, DAA of TP is not carried out in production enterprises due to the process complexity and the lack of appropriate software. When introducing technologies designed under such conditions to production, it cannot guarantee that drawing requirements for dimensional and surface orientation and location accuracy are met. When conducting a DAA of TP, it is necessary to preliminarily draw dimensional diagrams of linear sizes and runouts [6, 16, 39]. It should be noted that drawing such diagrams and performing dimensional calculations based on them manually is a labour intensive process. Currently, programs for automated calculation of linear and diametric dimensions [40, 42] based on techniques described in [4, 16, 40] have been developed, however their introduction in CAPP in practice is constrained by the lack of programs for synthesis of schemes of linear dimensional and requirements for mutual arrangement (RMA). The implementation of interactive synthesis of linear dimensional and runout diagrams is proposed in the interactive mode between user and a PC.

3 Results 3.1 Interactive Synthesis of Linear Dimensional Diagrams Currently, the development of an interactive synthesis program for linear dimensional diagrams and its graphical interface is underway. The program logic is based on the following proposed methodology, consisting of nine stages:

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6. 7. 8. 9.

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loading a preliminary simplified part sketch; numbering planar elements of the part; indicating numbers for intermediate states (allowance) of each planar element; designation of machining operations numbers that should be included in the dimensional diagram; automatic generation of vertical lines corresponding to the final state, and, if intermediate states are present, of each planar element, as well as formation of areas for constructing graphs of design dimensions, misalignments (if present), allowances and technological dimensions; plotting of graph with closing links dimensions (design dimensions, misalignments and allowances) and their automatic designation; generation of a graphs system of technological dimensions for each operation and their automatic designation; identification and display of dimension chains; data transfer to the calculation module.

At the first stage, a user (technologist) loads a simplified (if necessary) sketch of a part in the image format (Fig. 1) into the program. A possible need for simplification of the part sketch is due to the fact that for clarity, it is advisable to spread topologically coincident planar elements of the part in the longitudinal direction of the sketch. Moreover, linear dimensions of chamfers and grooves specified by a design engineer are often satisfied from the finished machined planar elements. In this case, linear dimension chains take the form K i = T j , and therefore, design dimensions of corresponding elements can be omitted from the graph of design sizes in the interactive synthesis of dimensional diagrams. In addition, corresponding technological sizes will be absent in the graph of technological dimensions, and the number of equations in linear dimensional chains system will be reduced. However, in the presented example, these dimensions are taken into account.

Fig. 1. Numbering of planar elements in a simplified part sketch

Then, at the second stage, the user numbers all planar elements of the part on the sketch. To do this, the user, having preliminarily selected the corresponding item on the toolbar, indicates all planar elements (EP) which numbers are immediately displayed on the monitor, while moving along the sketch sequentially from left to right, as shown in

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Fig. 1. Due to the fact that the part sketch is loaded in the image format, the numbering is carried out without reference to the lines of planar elements, since a numbered sketch will be used only for clarity and convenience. At the third stage, the user indicates the number of intermediate states of each planar element that determine the number of allowances needed for processing. In order to achieve this, the following is interactively indicated for each planar element: the surface (plane element) number in the final state, which corresponds to a vertical line with the same number in the linear dimensional diagram being synthesized; the number of its intermediate states; the surface location in its intermediate states (left or right) relative to itself in the final state. During the fourth stage, the user indicates numbers for operations where technological dimensions are satisfied, so that during the dimensional diagram generation horizontal areas are prepared for graphs of technological dimensions to be built. Also during this stage, the engineer specifies the numbers of base surfaces for each operation to display on the diagram as symbols, considering whether the surface belongs to the left or right side of the part (workpiece). After that, by pressing the appropriate button, the program automatically performs the fifth stage, at which the workspace for building corresponding graphs is generated. An example of the obtained result of the third, fourth and fifth stages is presented in Fig. 2.

Fig. 2. The result of the third, fourth and fifth stages of interactive synthesis of a linear dimensional diagram

At the sixth stage, the technologist in interactive mode successively generates dimensions graphs, which are the closing links in the technological linear dimensional chains, in the following sequence: graph of design dimensions G1 = (EPif , K i ); misalignments graph (if present) G2 = (EPih , ei ); allowances graph G3 = (EPih , Z i ). The vertices of graphs are explicit or imaginary planar elements (EPih , or EPif ) in any of their possible states h (f being the final), and the edges represent connections between them: design

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dimensions K i ; misalignments ei ; allowances Z i , where i is the number of an element or bond. The generation of design dimensions or misalignments graph is carried out after clicking the corresponding button on the toolbar. Each design dimension or misalignment is introduced by specifying its left and right borders, i.e. surfaces and/or axes that are connected by this dimension. Since design dimensions are bilateral non-oriented relations, they have arrows on both sides. Numbering and designation of dimensions is carried out automatically and sequentially; however, the user will be able to change numbers at their discretion. The allowance graph generation is carried out similarly, however this one is oriented, unlike the graph of design dimensions (or misalignments). Therefore, the corresponding relations will have a direction from the subsequent to the previous element state, according to which arrows are automatically put, indicating the direction of this relationship. According to the same principle of graph generation, at the seventh stage, for each operation in the area allocated to it on the diagram an oriented graph G4 = (EPih , T i ) of technological dimensions T i (constituent units in technological linear dimensional chains) satisfied on this operation is constructed. Each dimensional relationship is directed to the surface being machined and oriented relative to the other machined surface or the base of an operation, which is indicated by the arrow at the end of this relation. The result is shown as a system of oriented graphs. An example of the program workspace after constructing graphs of design and technological dimensions and allowances, performed at the fifth and sixth stages of the interactive dimensional diagram synthesis, is shown in Fig. 3.

Fig. 3. The program workspace after generating graphs of design and technological dimensions and allowances

After that, at the eighth stage linear technological dimensional diagrams are identified. In order to achieve this, the user indicates a closing link on the diagram (design dimension, misalignment, or allowance) by clicking on its designation. Afterwards, the

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technologist sequentially indicates all links (technological dimensions) of the linear dimensional chain, moving from the left border of the closing link and to the right. First, the user clicks on the technological dimension designation on the diagram, and then determines via selection if its function is decreasing or increasing by pressing the corresponding button on the toolbar. In this case, the equation of the identified dimensional chain is written into a special exchange file in CSV format. For the correct generation of the exchange file, dimensional chains are identified in the following sequence: firstly for design dimensions, then for allowances and finally for misalignments, starting with the first, according to the closing link number, and ending with the last. The file structure is described in [42]. At the final ninth stage, after identifying all dimension chains, the results are exported to the module for calculating linear technological dimensions with the special exchange file generated at the previous step. The module for automated calculation of linear technological dimensions is described in [42]. It provides an automatic (using an exchange file in CSV format) and a manual input of equations of technological linear dimension chains. 3.2 Interactive Synthesis of Runouts Diagrams The calculation of intermediate diametrical dimensions of outer and inner surfaces of revolution is carried out in the developed program according to the method described in [4, 16], which was improved to take into account the presence of finishing operations in the TP for manufacturing bodies of revolution [40]. In accordance with it, the calculation of intermediate diametrical dimensions is carried out based on determining minimum estimated allowances, which consist of a uniform and a non-uniform part. The uniform part of a minimum estimated allowance is made up of roughness (Rzi-1 ) and depth of a surface’s defective layer (hi-1 ) obtained during the previous processing and being eliminated when removing the allowance. The non-uniform part of the allowance to be removed (emax ) is determined by geometric summation of shape errors (curvature, warpage, taper, etc.), errors of machine tool, basing and workpiece fixing [16]. The value of the uniform part of allowance is determined based on reference data [41]. To establish the non-uniform part of allowance being removed during processing of each surface of a body of revolution, it is advisable to apply the method for determining runouts that occur during the TP of a part, with the use of an appropriate runouts diagram [6, 16]. It is convenient to determine the unevenness of allowances and, accordingly, calculate intermediate diametrical dimensions only after determining the possibility of automatic assurance of specified requirements for location and orientation (circular runouts or misalignments) of surfaces of revolution, displayed on the runouts diagram (if available), with (or without) aligning. The rate of allowance unevenness for processing a surface of revolution varies along its diameter from zero to a certain maximum value, which corresponds to the runout vector value of the surface between its neighboring states, before and after processing respectively. This vector is denoted as Bzi on a runouts diagram. Its value is defined as an arithmetic difference between two vectors: a runout of the surface before processing and a runout of the same surface after processing. Since both of these vectors do not change

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their direction, only the last vector has a decrease in its value. The latter is a reflection of the fact that after surface treatment, its axis approaches the machine spindle axis, for example, during turning or grinding. In [16] mentions a well-known fact that the runout value of a surface of revolution originating from its axis displacement is determined as a doubled value of this displacement (B = 2e). Taking this statement into account, to calculate the minimum estimated allowance on the side being removed during processing of the surface of revolution, the emax i will be determined as half of the Bzi value, i.e. emax i = Bzi / 2. Considering the above, a program for interactive synthesis of runouts diagram and its graphical interface is being developed. The program logic is based on the following proposed methodology that consists of eight stages: 1. construction of a simplified cross-section of a part in a CAD system, considering allowances for workpiece processing; 2. numbering of surfaces of revolution; 3. numbering of base surfaces and machining operations that should be included in the runouts diagram; 4. generation of a workspace for runouts diagram construction; 5. construction of runouts graphs appearing as a result of TP operations; 6. calculation of surface runout, which appeared during TP operations, and clarification whether requirements for mutual arrangement are met; 7. calculation of allowances unevenness; 8. transfer of data on allowances unevenness to the calculation module to determine allowances and intermediate diametrical dimensions. When constructing a simplified cross-section of a part in a CAD system at the first stage, chamfers and grooves are removed, if there are no requirements for their mutual orientation and if they are not processed at least twice during different operations. Furthermore, cylindrical steps of the same diameter are spread apart so that they do not coincide topologically (on the sketch) with each other. In addition, when allowances are added to a surface, the sketch is transformed so that these surfaces also do not coincide with each other. In this case, the number of allowances for each surface is one less than the number of its states. The numbering of surfaces of revolution, carried out during the second stage, is performed with respect to its number and state. It is usually a two-digit, and, in some cases, three-digit number, with last indicating the surface condition: zero is final, and a nonzero integer is preliminary. One or two digits before the last one indicate the surface number. Usually, outer surfaces of revolution are numbered first from left to right, and then inner ones in the same order. After determining numbers for machining operations included in the runouts diagram and numbers for surfaces used in these operations (except for obtaining the initial workpiece) as basic, as well as the need of a zone formation for constructing technological requirements for mutual arrangement during the third stage, the workspace for a subsequent runouts diagram construction is automatically generated at the fourth step. The result of the first four stages of interactive synthesis of a runouts diagram is shown in Fig. 4.

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Fig. 4. The result of the first, second, third and fourth stages of interactive synthesis of a runouts diagram

The runouts diagram created interactively at the fifth stage and shown in Fig. 5 as an example, is, in fact, a system of directed graphs and is formed by a sequential construction j of directed graphs of runouts Gn = (E ih , Bi ), appearing during the considered TP j operations. Edges of these graphs are runout vectors, denoted as Bi , where the superscript is the operation number, and the subscript is the surface number, with respect to its state.

Fig. 5. Result of the fifth stage of interactive synthesis of a runouts diagram

The runout diagram is generated in the following order. Firstly, a runouts graph for a workpiece operation, where surfaces predetermined by a technologist get their initial state, is built, i.e., the runout vectors, formed in the original workpiece (stamped or cast), are shown in the form of arcs with arrows at the end at lines. After that, runout graphs for machining operations are constructed in order of their execution. Within an operation, the runout of the base surface is indicated first. Then runouts of remaining surfaces formed in the workpiece are indicated, according to the following condition: runouts of a surface before processing are indicated first, and then runouts of the same surface after processing. Furthermore, for outer surfaces formed

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during an operation, vectors of their runouts are denoted, starting from the surface with the largest diameter, moving to the ones with the smallest diameter, and holes, on the contrary, are indicated from a smallest diameter to the largest. Also plotted on the diagram are vectors Bzi and runouts of tested complexes of mutual arrangement requirements denoted as Bk-l , where k is the number of the dependent surface and l is the number of the base surface in the complex of mutual arrangement requirements. At this stage, it is possible to adjust the length of horizontal lines denoting outer and inner surfaces of rotation in different states for the convenience of perception of the resulting runouts diagram. The calculation of surfaces runouts appearing during TP operations, performed at the sixth stage, is carried out depending on the runout type (surface of original workpiece, base surface, surface being machined, surface after machining, surfaces orientation requirements) according to formulas and expressions presented in [6, 16]. At the same time, the possibility of satisfying mutual arrangement requirements is checked, i.e., the following condition must be met: the calculated value of a verifiable requirement must not exceed its value specified by the drawing. If requirements are met, it is possible to proceed to the seventh stage. If requirements are not met, then a few necessary adjustments are required in the TP, including, for example, a change of base, an increase in base surfaces precision, or a change of type and accuracy class of the fixture used for workpiece basing. Only after it is established that the requirements are met in a given TP, it is possible to proceed to further calculations, i.e., to the seventh stage, where, according to the formula presented in [6, 16], the Bzi and emax i values, used for the following determination of allowances and intermediate diametrical dimensions, are calculated for each allowance. At the final eighth stage, the values of calculated unevenness of allowances emax i are transferred via an exchange file to the module for automated calculation of intermediate diametrical dimensions of surfaces of revolution and allowances for their processing. This module is currently being finalized considering the improved calculation methodology described in [40].

4 Discussion As mentioned earlier, currently, enterprises do not carry out dimensional analysis of TP due to the labor intensity of the process and the lack of appropriate software. When introducing technologies designed without these calculations into manufacturing, accuracy assurance of dimensions and surfaces orientation depends on a technologist, adjuster and operator of a CNC machine tool. Software implementation of methods for interactive synthesis of linear dimensional and runouts diagrams, as well as the introduction of software, developed to automate the dimensional accuracy analysis of TP, into the practice of process planning will ensure that drawing requirements for dimensional accuracy and surfaces orientation and location are met. This will improve the quality of TP design and ultimately reduce time and material costs when introducing TP in enterprises. In addition, it is a significant step towards digital manufacturing, as this way the impact of the human factor is reduced. Initially developed for calculating operational linear and diametrical dimensions and allowances, the proposed software can be used as an expert system for analyzing existing

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TP of parts manufacturing to determine the rationality of choosing machining allowance values, as well as the possibility of automatic accuracy assurance of linear technological dimensions. Apart from that, the use of the developed software significantly reduces estimation time compared to manual calculation. As an example, the program for calculating linear dimensions and allowances, which was used for examination of a current TP for a part manufacturing, made it possible to estimate 47 dimension chains, 25 of which were multi-link, with 3 to 6 links, within 2 h, including data input. On the contrary, manual calculation (with one ten-minute break per hour) took about 14 h (almost 2 shifts). All described above will help to draw the attention of engineers and engineering technologists to the problem of DAA of TP necessity in designing multi-operational technologies.

5 Conclusion To sum up, the results obtained in this study play an important role in modern mechanical engineering and instrumentation technology, since the introduction of the DAA of TP for parts manufacturing in the practice of process planning is one of the steps on the way to digital manufacturing. The combined use of techniques for interactive synthesis of linear dimensional and runouts diagrams together with programs for calculating operational linear dimensions, intermediate diametric dimensions and machining allowances will improve the quality of TP design, i.e. this will allow preparing technologies for manufacturing parts like solids of revolution that ensure the needed accuracy of design dimensions and requirements for mutual arrangement with a certain margin. This will guarantee TP reliability in terms of precision, as well as reduce the cost of its introduction into production by eliminating the need to manufacture pilot or setting batches or adjusting parts, which is usually done to confirm the compliance with the specified accuracy of dimensions and of mutual arrangement requirements. Instead, it will be confirmed by calculation. And only one adjusting part will be enough to correct the cutting mode for different methods of workpiece processing.

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To Calculation of Corones Lens Compensators Bimetallic Pipelines of Hydro Power and Nuclear Power Vladimir A. Pukhliy1(B) , Efim A. Kogan2 , and Sergey T. Miroshnichenko1 1 Sevastopol State University, Streat Universitetskaya 31, Sevastopol 299053, Russia

[email protected] 2 Moscow Polytechnic University, B. Semenovskaya 38, Moscow 107023, Russia

[email protected]

Abstract. The construction of equilibrium systems of two-layer shells of revolution (cylindrical, conical, toroidal) with respect to the calculation of the corrugations of lens compensators for pipelines of hydropower and nuclear energy is given. The solution of boundary value problems for corrugations of lens compensators is based on the application of a modified method of successive approximations. To accelerate the convergence of the solution, the method of telescopic shift of the Lanczos power series is used, in which the original Mac-Loren series are represented through shifted Chebyshev polynomials. Keywords: Hydropower · Nuclear power · Pipelines · Lens compensators · Cylindrical · Conical · Toroidal two-layer shells · A modified method of successive approximations

1 Introduction In hydropower and nuclear power for pipelines with internal cladding of the outer layer, lens compensators are widely used to compensate for temperature expansion, misalignment of pipelines and vibration effects. Usually they are equipped with an internal protective casing and, when calculating their strength, they can be considered as composite shells of rotation [1–4]. Figure 1 shows a general view of the axial single-lens compensator (a) and the axial section of the compensator with the inner protective layer (b). I, II, VIII, IX - cylindrical two-layer shells; III, V, VII - toroidal two-layer shells; IV, VI - conical two-layer shells; A, C, F - arbitrary points on different sections of the corrugation, B, D, E - points of conjugation of the sections). The theory of corrugated shells has long been developed, starting with the works of V.I. Feodosiev [1], L.E. Andreeva [2] and others. The current state of the theory is reflected in the cycle of works by V. Moskvitin [3]. The general equations of thermoelasticity of non-constructive orthotropic two- and three-layer shells in a geometrically nonlinear formulation were obtained in [4] under the assumption that the inner layer is © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 136–149, 2021. https://doi.org/10.1007/978-3-030-62062-2_14

To Calculation of Corones Lens Compensators Bimetallic Pipelines

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Fig. 1. General view of the lens compensator.

deformed in accordance with the Kirchhoff – Love hypothesis, and the outer layer (or two outer layers) are low-hard coatings for which transverse shear strains are significant. They were taken into account in accordance with the Tymoshenko hypothesis. As the coordinate surface, the lower boundary surface of the inner (first) layer of the shell was used. This approach makes it easy to extend the equations to the case of multilayer shells of irregular structure. In this paper, using the approach previously proposed in [4], we construct systems of linear equilibrium equations for two-layer shells of revolution as applied to the calculation of corrugations of lens compensators for hydropower and nuclear power pipelines. For each of the layers, the Kirchhoff - Love hypotheses about a direct undeformable normal are assumed to be valid. Further, the variational method is used to obtain the equations in the forces — moments for non-gentle bilayer cylindrical, conical, toroidal shells and thermoelasticity ratios. An algorithm for the analysis of the stress-strain state (SSS) of an elastic composite system consisting of (4, b) of 4 bi-metal cylindrical shells is described; 3 toroidal shells; 2 conical two-layer shells. We considered 2 types of loading corresponding to axisymmetric deformation and loading corresponding to antisymmetric deformation (Fig. 2): a) the compensator is loaded with uniformly distributed internal pressure q. Boundary contours (sections AA and BB are rigidly fixed) (Fig. 2, a). Then obviously the longitudinal displacements on the left and right boundary contours are equal to zero. b) The left edge of the compensator (BB section is rigidly fixed, and the axial displacement z is specified on the right edge (section AA) (Fig. 2, b). In this case, the longitudinal displacements on the left boundary contour are zero, and on the right boundary they are equal to z. c) The compensator is deformed under the action of the angular and axial displacements specified on both contours (Fig. 2, c). Then the angular displacements on the left edge; uz = − d2ω , vz = −ω; on the right edge uz = d2ω , vz = ω.

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Fig. 2. Compensator deformation schemes for different loading

2 Derivation of Linear Equations of Non-gentle Bilayer Shells We consider shells in a curvilinear orthogonal coordinate system α1 , α2 , z. As the coordinate surface, the lower boundary surface of the inner (first) layer of the shell is used. It is believed that the coordinates α1 and α2 coincide with the lines of the main curvatures of the coordinate surface. The transverse coordinate z coincides with the direction of the external normal to the coordinate surface, hk – is the thickness of the k-th layer (k = 1,2), h = h1 + h2 – is the shell thickness, Ai (α1 , α2 ) – are the Lame coefficients (i = 1,2), κi – are the curvatures of the coordinate lines, are tangential and normal movement of the coordinate surface. The distribution of tangential displacements along the thickness follows a linear law ui (z) = ui + zϑi

(1)

where ϑi – are the angles of rotation of the normal to the coordinate surface ϑi = ki ui −

1 ∂w , (i = 1, 2). Ai ∂αi

The tangential deformation components are defined as: 1 ∂u1 1 ∂A1 1 ∂u2 1 ∂A2 + u2 + k1 w, ε22 = + u1 + k2 w A1 ∂α1 A A ∂α A2 ∂α2 A1 A2 ∂α1  1 2 2   . u1 A2 ∂ u2 A1 ∂ + ω= A2 ∂α2 A1 A1 ∂α1 A2 ε11 =

Parameters for changing curvatures and torsion [6]: κ1 =

1 ∂ϑ1 1 ∂A1 1 ∂ϑ2 1 ∂A2 + ϑ2 , κ2 = + ϑ1 , A1 ∂α1 A1 A2 ∂α2 A2 ∂α2 A1 A2 ∂α1

(2)

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 2    ∂ w 1 ∂u1 1 1 ∂A1 ∂w 1 ∂A2 ∂w 1 ∂A1 + k1 τ =− − − − u1 A1 A2 ∂α1 ∂α2 A1 ∂α2 ∂α1 A2 ∂α1 ∂α2 A2 ∂α2 A1 A2 ∂α2   1 ∂u2 1 ∂A2 (3) + k2 − u2 . A1 ∂α1 A1 A2 ∂α1 It is believed that the material of the shell layers obeys Hooke’s law. The main directions of elasticity of the layers at each point coincide with the directions of the coordinate lines α1 , α2 . It is assumed that the shell is exposed to high temperatures T = T(α1 , α2 , z) unevenly distributed over its thickness and surface. Therefore, the physicomechanical characteristics of the layers Ei(k) – elastic moduli, Gij(k) – shear moduli, α(k) – coefficient of linear thermal expansion are considered temperature-dependent (k = 1, 2). In this case, the stresses in the layers are represented as the sum of the components due to elastic and temperature deformations (k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k) (k)

σ11 = σ110 − σ1T , σ22 = σ220 − σ2T , σ12 = σ120 = G12 ε12 , (k = 1, 2), (k)

σ110 = (k)

σiT =

(k)

E1

(k)

1 − μ2(k) (k)

Ei

2 1 − ν(k)

(ε11 + μ2(k) ε22 ), σ220 =

(k)

E2

1 − μ2(k)

(k)

(k)

(ε22 + μ2(k) ε11 ,

(1 + μ(k) )α (k) ) T (α1 , α2 , z), (i = 1, 2),

(4)

where μ(k) – is the Poisson’s ratio of the k-th layer. Equations of Equilibrium and Boundary Conditions. The equilibrium equations and the natural boundary conditions of two-layer shells are obtained by applying the principle of possible Lagrange displacements

δΠ − δA1 − δA2 = 0.

(5)

Here δΠ − δA1 − δA2 are the variations of the potential energy of shell deformation, the work of the external surface load, reduced to its coordinate surface and external contour forces. The following description is accepted for them: ⎧ ⎡ ⎤⎫ ˚ ⎨ 2 2 2  ⎬  ⎣ σij(k) (z) δεij(k) ⎦ A1 A2 d α1 d α2 dz (6) δΠ = ⎭ ⎩ Vk

k=1

i=1 j=1

δA1 = (p1 δu1 + p2 δu2 + qδw)A1 A2 d α1 d α2

(7)

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 δA2 =

⎡ ⎣

Γ1

+

⎤

2 

(N1j∗ δ uj

+ M1j∗ δϑj ) + Q1∗ δ w⎦A2 d α2

j=1

  2

 (Ni2∗ δ ui + Mi2∗ δϑj ) + Q2∗ δw A1 d α1

(8)

i=1

Γ2

where p1 , p2 , q – are the components of the surface load in the directions of the axes α1 , α2 , z, respectively, Nij∗ , Qi∗ , and Mij∗ – and are the external tangential forces applied to the shell contour, the transverse forces, bending and torques. Using the representation of stresses (4), strains (2), (3) and displacements (1) and introducing the total specific forces and moments in the shell according to the formulas (1)

(2)

(1)

(2)

Ni j0 = Ni j0 + Ni j0 , NiT = NiT + NiT , (1)

(2)

(1)

(2)

Mi j0 = Mi j0 + Mi j0 , MiT = MiT + MiT , (i, j = 1, 2), where (k) Ni j0

 =

(k) σi j0 dz,

(k) NiT

 =

hk

Mi(k) j0



= hk

hk

σi(k) j0 zdz,

Mi(k) T

(k)

σiT dz 

= hk

σi(k) zdz, (i, j = 1, 2; k = 1, 2) T

(9)

To Calculation of Corones Lens Compensators Bimetallic Pipelines

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expression (6) for the potential strain energy can be reduced to

(10)

From the conditions of arbitrariness and independence of the variations of displacements, taking into account expressions (7), (8), we obtain a system of three equations of shell equilibrium in efforts and moments.   ∂(A2 M11 ) ∂(A2 N11 ) ∂A2 1 ∂(A21 N12 ) ∂A2 − N22 + + k1 − M22 ∂α1 ∂α1 A1 ∂α2 ∂α1 ∂α1 1 ∂(k1 A21 M12 ) + A1 A2 p1 = 0, + A1 ∂α2   ∂(A1 N22 ) ∂(A1 M22 ) ∂A1 1 ∂(A22 N12 ) ∂A1 − N11 + + k2 − M11 ∂α2 ∂α2 A2 ∂α1 ∂α2 ∂α2 1 ∂(k2 A22 M12 ) + A1 A2 p2 = 0, A2 ∂α1    1 ∂(A22 M12 ) ∂ ∂A1 M11 + ∂α2 ∂α2 A2 ∂α1 2    ∂(A1 M22 ) ∂ 1 ∂A2 − M22 ∂α2 ∂α1 A1 ∂α1 +

   1 ∂(A2 M11 ) ∂ 1 ∂ − ∂α1 A1 ∂α1 ∂α2 A2    1 ∂(A21 M12 ) ∂ 1 ∂ + ++ 2 ∂α1 A ∂α2 ∂α2 A2 1

−A1 A2 (k1 N11 + k2 N22 ) + A1 A2 q = 0.

(11)

The natural boundary conditions follow from the contour integrals of the variational equation.

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Note that for shells of revolution, the middle surface of which is formed by rotating a plane curve around a straight line lying in the same plane, the basic relations given above are simplified [7], since in this case the Lame A1 coefficient does not depend on the coordinate α2 , therefore A1,2 = 0. ∂u1 ∂A2 1 ∂u2 + k1 w, ε22 = + (A1 A2 )−1 u2 + k2 w ∂α1 A2 ∂α2 ∂α1 ∂A2 1 ∂u1 1 ∂w 1 ∂u2 + − (A1 A2 )−1 u2 , ϑi = ki ui − ω= A1 ∂α1 A2 ∂α2 ∂α1 Ai ∂αi 1 ∂ϑ1 1 ∂ϑ2 1 ∂A2 k1 = , k2 = + ϑ2 A1 ∂α1 A2 ∂α2 A1 A2 ∂α1 1 ∂ϑ2 1 ∂ϑ1 1 ∂A2 κ12 = + − ϑ2 A1 ∂∂1 A2 ∂∂2 A1 A2 ∂∂1 ε11 = A−1 1

(12)

Using formulas (2)–(4), (9), we obtain the thermoelasticity relations connecting the forces and moments with the deformations of the coordinate surface and the curvature change parameters: N11 = B1 ε11 + B2 ε22 + A1 κ11 + A2 κ22 , N22 = B1 ε22 + B2 ε11 + A1 κ22 + A2 κ11 , M11 = A1 ε11 + A2 ε22 + D1 κ11 + D2 κ22 , M22 = A1 ε22 + A2 ε11 + D1 κ22 + D2 κ11 , N12 = B12 ω + A12 κ12 , M12 = B12 ω + D12 κ12 (13) In relations (13), the membrane Bi (i = 1, 2), membrane – bending Ai and bending Di stiffness parameters of the two-layer shell are equal to:   1 1 B1 = E1 (T )dz + E2 (T )dz, 1 − μ21 1 − μ22 h1

B2 =

μ1 1 − μ21

h2



E1 (T )dz + h1

 B12 =

(1) G12 (T )dz +

h1

 A12 =

(1)

1 1 − μ21

1 1 − μ21



G12 (T )zdz +

μ1 A2 = 1 − μ21 D1 =



E1 (T )zdz + h1

h1

h2

(2)

G12 (T )zdz, 1 1 − μ22

μ2 E1 (T )zdz + 1 − μ22



E1 (T )z 2 dz + h1

E2 (T )dz,

(2) G12 (T )dz,

h2







h2

h1

A1 =

μ2 1 − μ22

1 1 − μ22

 E2 (T )zdz, h2



E2 (T )zdz, h2

 E2 (T )z 2 dz,

h2

To Calculation of Corones Lens Compensators Bimetallic Pipelines



(1) G12 (T )z 2 dz

D12 =

 +

h1

μ1 D2 = 1 − μ21

(2) G12 (T )z 2 dz,

h2

 E1 (T )z 2 dz +

h1

143

μ2 1 − μ22

 E2 (T )z 2 dz. h2

The temperature components of the efforts and moments, taking into account the variability of the elastic moduli and the coefficients β1 , β2 of linear thermal expansion of the layers are determined by the formulas: 1 NT = 1 − μ1 1 MT = 1 − μ1

δ1 0

δ1 0

1 E1 β1 Tdz + 1 − μ2

1 E1 β1 Tzdz + 1 − μ2

0 E2 β2 Tdz, −δ2

0 E2 β2 Tzdz. −δ2

From the obtained general equations, equations of various types of shells of revolution follow. Cylindrical Bilayer Shells Taking α1 = x, α2 = s/R, and considering that κ1 = 0, κ2 = 1/R, A1 = 1, A2 = R, expressions for the strain components can be written as w ∂u1 ∂u1 ∂u2 ∂u2 ∂w , ε22 = + , ε12 = + = ω, ϑ1 = − , ∂x ∂s R ∂x ∂s ∂x ∂w ∂ 2w u2 ∂ 2w 1 ∂u2 ϑ2 = − , κ11 = − 2 , κ22 = − 2, R ∂x ∂x R ∂s ∂s ∂ 2w 1 ∂u2 κ12 = −2 . R ∂x ∂x∂s ε11 =

The equilibrium equations of a two-layer cylindrical shell have the form   ∂N11 ∂N12 ∂N22 1 ∂M12 ∂M22 ∂N12 + + p1 = 0, + + + + p2 = 0, ∂x ∂s ∂x ∂s R ∂x ∂s ∂ 2 M22 ∂ 2 M11 ∂ 2 M12 N22 + + q = 0. + 2 − ∂x2 ∂x∂s ∂s2 R

(14)

Equations (14) correspond to those given in [4], [5]. Conical shell For a truncated conical shell (Fig. 3), the Lame and curvature parameters are equal A1 = 1, A2 = b + s · sin γ = r, κ1 = 0, κ2 = cos γ /A2 = cos γ /r

(15)

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Fig. 3. Conical shell of rotation

Taking α1 = s, α2 = θ, the deformation relations for the cone can be represented as ∂u1 ∂A1 ∂us + (A1 A2 )−1 u2 + κ1 w = , ∂s ∂θ ∂s  ∂u2 1 ∂uθ −1 ∂A2 εθθ = A−1 A + κ w = + cos γ · w , + u + sin γ · u ) (A 1 2 1 2 s 2 ∂θ ∂s r ∂θ ∂uθ sin γ A1 ∂  −1  A2 ∂  −1  1 ∂us A u1 + A u2 = + − uθ = ω, εsθ = A2 ∂θ 1 A1 ∂s 2 r ∂θ ∂s r   1 ∂w ∂w 1 ∂w 1 ∂w cos γ · uθ − . (16) ϑs = κ1 u1 − = − , ϑθ = κ2 u2 − = A1 ∂α1 ∂s A2 ∂α2 r ∂θ εss = A−1 1

Therefore, the equilibrium equations of a two-layer truncated conical shell are written as follows: ∂ ∂Nsθ ∂Nss + − sin γ · Nθθ + rps = 0, (sNss ) + b ∂s ∂s ∂θ ∂ cos γ ∂Mθθ ∂Msθ ∂Nθθ ∂Nsθ + sin γ (sNsθ ) + sin γ · Nsθ + b + + cos γ ∂θ ∂s ∂s r ∂θ ∂s sin 2γ Msθ + rpθ = 0, + r 1 ∂ 2 Mθθ ∂Mθθ ∂ 2 Mss ∂ 2 Msθ ∂Mss + − sin γ · r + 2 + 2 sin γ 2 2 ∂s ∂s∂θ r ∂θ ∂s ∂s sin γ ∂Msθ − cos γ · Nθθ + rq = 0. +2 r ∂θ sin γ

(17)

For the truncated conical shell axisymmetrically deformed under the transverse load and the temperature effect, the equilibrium Eqs. (17) are simplified, since displacements, deformations, and specific forces are independent of the circumferential coordinate θ: ∂ ∂Nss − sin γ · Nθθ + rps = 0, (sNss ) + b ∂s ∂s ∂ 2 Mss ∂Mθθ 1 ∂ 2 Mθθ ∂Mss r − sin γ · − cos γ · Nθθ + rq = 0. + + 2 sin γ ∂s2 r ∂θ2 ∂s ∂s sin γ

(18)

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Toroidal Shell For a toroidal shell  of circular shape (Fig. 4), the Lame parameters and curvature are equal [7] A1 = ρ 1 + ρr cos α2 , A2 = r.

Fig. 4. Toroidal shell of circular cross section

From Fig. 4 shows that one of the centers of curvature O1 lies on the axis of symmetry, the other O2 in the center of the section. Therefore, by Meunier’s theorem, r cos θ , and the second main the first principal radius of curvature is R1 = cosR α = ρ +cosα radius of curvature R2 = r. As can be seen from Fig. 4, the lines α = ξ = xr are circles of radius R(θ) obtained by the section of the torus by planes perpendicular to the axis of symmetry (parallel), and the lines θ = sr2 are circles of radius r (meridians). The relationship of the Cartesian rectangular coordinates of an arbitrary point A of the cross section with the curvilinear coordinates of point A: x = R(θ) cos ξ = (ρ + r cos θ) cos ξ = (ρ + r cos θ) cos α, y = R(θ) sin ξ = (ρ + r cos θ) sin ξ = (ρ + r cos θ) sin α, z = r sin θ Therefore, the equilibrium equations of the two-layer toroidal shell are reduced to     ∂Nαα ∂ r ∂ r + (ρ + r cos θ) − 2r sin θ Nαθ + cos α − sin α Mαα ∂α ∂θ ρ + r cos θ ∂α   cos α · sin θ ∂ Mαθ + r(ρ + r cos θ)p1 = 0, + cos α − r ∂θ ρ + r cos θ     ∂Nθθ 1 ∂Mθθ ∂Nαθ 1 ∂Mαθ + +r + + r sin θ · [Nαα − Nθθ (ρ + r cos θ) ∂θ r ∂θ ∂α r ∂α

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1 + (Mαα − Mθθ )] + r(ρ + r cos θ)p2 = 0, r  −1 2   ∂ 2 Mαα r ∂Mθθ r ∂ Mθθ ∂ 2 Mαθ ∂Mαα + − 2 + 2 + sin θ · ρ + r cos θ ∂α2 ∂α∂θ ρ + r cos θ ∂θ2 ∂θ ∂θ + cos θ · (Mαα − Mθθ ) − r cos α · Nαα − (ρ + r cos θ)Nθθ + r(ρ + r cos θ)q = 0. (19) The elastic conjugation conditions for the compensator sections with respect to displacements ansd forces are given, for example, in [8]. For the considered elastic system (Fig. 1b), composed of 4 cylindrical two-layer shells; 3 toroidal two-layer shells; 2 conical two-layer shells of the equation, the relationship between the “global” coordinates (X, Y) of an arbitrary point in the corresponding section and the local coordinates of each shell for different sections of the meridian of the composite system of bimetallic shells of revolution can be written as: – – – –

  for site I: X = − R1 − (r2 − r3 )ctg γ + h +  l + x; Y = R − h1 , 0 ≤ x ≤ l; for site II: X = − R1 − (r2 − r3 )ctg γ + h + x; Y = R − h1 , 0 ≤ x ≤ l1 ; for site VIII: X = R 1 − (r2 − r3 )ctg γ + h + x; Y = R − h1 , 0 ≤ x ≤ l; for site IX: X = − R1 − (r2 − r3 )ctg γ + h + l + x, Y = R − h1 , 0 ≤ x ≤ l1 . Here x is the local coordinate measured from the left edge of the cylindrical shells;

– – – – –

for site III: (A - ppoizvolna toqka na povepxnocti cpa) X = −[R1 − (r2 − r3 ) ctg γ + R2 − (R2 − h1 ) cos θ], Y = r3 − (R2 − h1 ) sin θ, for site VII: X = R1 − (r2 − r3 )ctg γ + R2 − (R2 − h1 ) cos θ, Y = r3 − (R2 − h1 ) sin θ, 2 −r3 for site IV: XC  = −[R1 −(r2 −r3 )ctg γ+s·cos γ], YC  = r3 +s·sin γ, 0 ≤ s ≤ rsin γ ; r2 −r3 for site VI: XC = R1 − (r2 − r3 )ctg γ + s · cos γ, YC = r3 + s · sin γ, 0 ≤ s ≤ sin γ .

For Sects. 4, 6, C and are arbitrary points on the generatrix of the cones, spaced a distance s from the smaller base of the cone (see Fig. 4); For Sect. 5: (A is an arbitrary point on the surface of the junction of the two-layer torus V)  π π XA = −(R1 + h1 ) sin θ, YA = (R1 + h1 ) cos θ, − ≤ θ ≤ 2 2

3 The Solution of Boundary Value Problems for Calculating the Stress - Strain State of an Elastic Structure To solve the stated boundary value problems and analyze the stress - strain states of an elastic structure, we apply the method of expanding unknown functions into trigonometric series in the circumferential coordinate θ. The solution of the resulting systems of ordinary differential equations with variable coefficients is constructed using the modified method of successive approximations [9–11].

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Previously, systems of differential equations are represented in the normal Cauchy form:  dXm Bν,m Xν + fm , m = 1, 2, . . . , s. = dξ s

(20)

ν=1

Variable Bν,m coefficients and free fm terms are represented through shifted Chebyshev polynomials: fm =

q 

fm,r (dr · r!)−1

r=0

r 

ak Tk* (ξ ), Bν,m =

q 

bν,m,r dr−1

r=0

k=0

r 

ak Tk* (ξ ). (21)

k=0

Here q is the degree of the interpolation polynomial; ak - coefficients of expansion of ξ r in a series of polynomials of Chebyshev Tk∗ (ξ ). In expressions (21) dr = 1 for r = 0 and dr = 22r−1 for the remaining r. The general solution of the system of Eqs. (20) has the form [9, 10]:   s ∞   Cμ d0−1 a0 T0∗ (ξ ) δ + Xm,μ,n Xm = μ=1

+

n=1

q 

∞   −1  tm,j,0 dj+1 (j + 1)! ak Tk∗ (ξ ) + Xm,n ,

j=0

k=0

j+1

(22)

n=2

where tm,j,0 = fm,r for j = r; μ is the number of the fundamental function; Cμ - integration constant. In the solution, δ = 1 if m = μ and δ = 0 for the remaining μ. The first approximation Xm,μ,1 is obtained by substituting the zero approximation: s  m = Bν,m Xν . d0−1 a0 T0∗ (ξ ) δ in the right-hand side of the system dX dξ ν=1

Subsequent approximations are carried out according to the formulas: Xm,μ,n =

β 

tm,μ,n,j



n+j−1  −1 dn+j−1 (n + j − 1)! × ak Tk∗ (ξ );

j=1

Xm,n =

β 

k=0

  −1 tm,n,j dn+j−1 (n + j − 1)! ×× ak Tk∗ (ξ ) , n+j−1

j=1

(23)

k=0

where β = n (q + 3) – 2. The systems of fundamental functions (23) are uniformly converging series, and the coefficients tm,μ,n,j and tm,n,j are determined through the coefficients of the previous approximation using the recurrence formulas: tm,μ,n,j =

q s   ν=1 r=0

bν,m,r tν,μ,n−1,j−r (n + j − 1)−1 ×

r  γ =0

(n + j − 1 − γ ),

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V. A. Pukhliy et al.

tm,n,j =

q s   ν=1 r=0

bν,m,r tν,n−1,j−r (n + j)−1 ×

r 

(n + j − γ ).

γ =0

The constants Cμ included in the general solution (22) are found from the boundary conditions. Here, to accelerate the convergence of the solution, the method of telescopic shift of the Lanczos power series is used [12]. The idea of the method is that the McLoren series at our disposal is telescopically shifted to a much shorter series, without losing accuracy. For this, the possibility of representing any power series in terms of shifted Chebyshev polynomials on the interval [0,1] is used. This approach was tested on a number of applied problems and described in the works of V.A. Puchlia [10, 11]. Example Let us consider the stress-strain state of the lens compensator, which is an elastic system of cylindrical, conical, toroidal bimetallic shells of revolution (Fig. 1) under 3 load cases (Fig. 2). The geometric parameters of the lens compensator are as follows: D = 100 sm; d = 76 sm; h = 0,5 sm; l1 = 1,4 sm; h1 = 1,2 sm; R = 3,6 sm; r 1 = 46 sm; r 2 = 47 sm; r 3 = 40,2 sm; r 4 = 40 sm; z1 = 0; z2 = 3,4 sm; z3 = 3,0 sm; z4 = 6,1 sm; μ = 0,3. The thickness of the anti-corrosion surfacing is 0.4 cm and is made of stainless austenitic steel grade 08X19H10G2B. Figure 5 shows the results of stress calculations, while in Fig. 5, a - for the 1st, Fig. 5, b - for the 2nd, Fig. 5, c - for the 3rd type of loading.

Fig. 5. Plots of meridional (to the left of the axis of symmetry of the corrugation) and circumferential (to the right of the axis of symmetry) stresses on the outer surface.

The calculation results show that the meridional stresses are determinative from the point of view of calculating the corrugations for strength.

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4 Conclusions A variational method is used to obtain a system of linear equilibrium equations for two-layer shells of revolution as applied to the calculation of corrugations of lens compensators for pipelines of hydropower and nuclear power. Equations are given in the efforts - moments for non-gentle bilayer cylindrical, conical, toroidal shells and the ratio of thermoelasticity. An algorithm for the analysis of the stress-strain state (SSS) of an elastic composite system consisting of bimetallic cylindrical, toroidal conical shells under the action of the above 3 types of loading is described. To solve the boundary value problem for the corrugations of lens compensators, a modified method of successive approximations was applied. To accelerate the convergence of the solution, the method of telescopic shift of the Lanczos power series is used, in which the original Mac-Loren series are represented through shifted Chebyshev polynomials. Acknowledgement. The study was financially supported by the Russian Federal Property Fund and the city of Sevastopol in the framework of project No. 18-48-920002.

References 1. Feodosiev, V.I.: Elastic elements of precision instrumentation, Oborongiz (1949), 344 p 2. Andreeva, L.E.: Elastic Elements of Devices, Mashgiz (1962) 3. Moskvitin, G.V.: Low cycle strength of compensating elements of pipelines with screw and ring corrugations. - Abstract of the dissertation (2002) 4. Kogan, E.A., Lopanitsyn, E.A.: Oscillations of structurally-orthotropic laminated shells under thermal force loading. In: Proceedings of the RAS. Solid Mechanics, no. 3, pp. 117–132 5. Timoshenko, S.P., Voinovsky-Krieger, S.: Plates and shells, Fizmatgiz (1963), 636 p 6. Filin, A.P.: Elements of the theory of shells. 2nd, add. and reslave, Stroyizdat, Leningrad. Branch (1975), 256 p 7. Karmishin A.V., Lyaskovets V.A., Myachenkov V.I., Frolov A.N.: Statics and dynamics of thin-walled shell structures, Mechanical Engineering (1975), 376 p 8. Strength, stability, fluctuations. Handbook in Three Volumes. Birger, I.A., Panovko, Ya.G. (eds.) Mechanical Engineering (1968) 446 p 9. Pukhliy, V.A.: The solution of initial-boundary value problems of mathematical physics by the modified method of successive approximations. Review of Applied and Industrial Mathematics, 24(1), 37–48 (2017) 10. Pukhliy, V.A.: An analytical method for solving boundary value problems of shell theory. In: Proceedings of the XIII All-Union. Conference on the Theory of Plates and Shells. Part IV Publishing House of TPI, Tallinn, pp. 101–107 (1983) 11. Pukhliy, V.A., Kogan, E.A., Pukhliy, K.V.: The Karman boundary value problem for the elastic bending of bimetallic hydropower pipelines. Power Plants and Technologies, 5(3), 25–35 (2019) 12. Lantsosh, K.: Practical Methods of Applied Analysis, Fizmatgiz (1961), 524 p

Thermoelasticity of Flexes Bimetallic Pipelines Atomic Energy Vladimir A. Pukhliy1(B) , Efim A. Kogan2 , and Sergey T. Miroshnichenko1 1 Sevastopol State University, Street Universitetskaya 31, Sevastopol 299053, Russia

[email protected] 2 Moscow Polytechnic University, B.Semenovskaya 38, Moscow 107023, Russia

[email protected]

Abstract. The issues of thermoelasticity of bends of bimetallic pipelines of nuclear energy are considered taking into account ovalization of the cross section. The dependence of the physicomechanical characteristics of the material of pipelines on temperature and on exposure to radiation is taken into account. An algorithm for the analytical solution of the Karman boundary value problem for elastic bending based on the modified method of successive approximations is presented. Keywords: Bimetallic pipelines · Toroidal shells · Bends · Thermoelasticity · Radiation exposure · Method of successive approximations

1 Introduction The main circulating pipelines of the VVER-1000 reactor are designed to organize the circulation of the coolant to the reactor and from the reactor to the PGV-1000 steam generator through 4 loops (Fig. 1). The coolant of the first circuit is pure water with boric acid dissolved in it (H3 BO3 ). Boric acid with a sufficiently high concentration (16 g H3 BO3 /1 kg H2 O) is used to compensate for the excessive fission ability of fresh nuclear fuel. Water enters the reactor at a pressure of P = 16 MPa with a temperature of T = 280 °C. In the reactor core, water is heated to T = 315 °C and then sent to the PGV-1000 steam generator, where it transfers heat to the primary coolant. The water of the primary circuit during the operation of the reactor acquires a high induced radioactivity, even if there is no excess violation of the tightness of the TVEL shells, since there are almost always impurities in the water (sludge - products of corrosion, erosion, salt, etc.), which are activated in core. DU-850 mm coolant circulation pipelines are made of low-alloy carbon steel of pearlitic class 10GN2MFA with cladding of the inner surface of pipelines with 08Kh18N10G2B stainless steel. Thus, NPP pipelines operate under complex loading conditions, primarily due to external mechanical loads, aggressive media, high-temperature heating [1, 2], and radiation exposure [3]. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 150–162, 2021. https://doi.org/10.1007/978-3-030-62062-2_15

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Fig. 1. Scheme of the main circulation pipelines of the VVER-1000 reactor installation: 1 reactor; 2 - steam generator PGV-1000; 3 - the main circulating pump GCN-195M; 4- main circulation pipelines DU-850.

In the developed reactors of the future, water at supercritical parameters, liquid helium, as well as liquid metal coolants, liquid-salt coolants are used as a coolant. The temperature of the coolants in these reactors reaches a value of the order of T = 1000 °C. As a result, temperature stresses in combination with mechanical stresses from force actions and radiation exposure can cause structural failure. Therefore, when making more accurate calculations of the stress-strain state and strength of the power elements of the reactor structure and the resource of centrifugal pumps, it is necessary to take into account the change in the mechanical and thermophysical characteristics of the material: elastic modulus E and linear expansion coefficient α and radiation exposure of the material. from coordinates and temperature In a previous work of the authors [4], the problem of the stress-strain state of elastic bends of bimetallic pipelines of hydropower under the influence of internal pressure was considered (the Karman problem). In the present work, the approach developed by the authors earlier extends to the problem of thermoelasticity of bends of bimetallic pipelines of nuclear energy taking into account the effect of radiation.

2 Geometric Parameters of Bending of a Bimetallic Pipeline Curved sections of pipelines (the so-called pipe bends) are the most damaged elements of the piping systems of thermal and nuclear power plants. The longitudinal forces arising during bending of a pipe with a curved axis lead to flattening of the cross section (ovalization of the pipe) [5]. A pipe in the bending zone can be considered as a toroidal bimetallic shell of circular cross section (see Fig. 2). We introduce a curvilinear coordinate system ξ, θ, ζ. The coordinate −h/2 ≤ ζ ≤ h/2 is measured normal to the median surface. As can be seen from Fig. 2, the lines ξ = xr = α1 are circles of radius R(θ) obtained by the section of the torus by planes perpendicular to the axis of symmetry (parallel), and the lines θ = sr2 = α2 are circles of radius r (meridians).

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Fig. 2. Toroidal shell of circular cross section

The lengths of arcs counted along the coordinate lines are equal

Using the relationship of the rectangular Cartesian coordinates of an arbitrary point A of the cross section with the curvilinear coordinates of point A: x = R(θ) cos ξ = (ρ + r cos θ) cos ξ = (ρ + r cos α2 ) cos α1 , y = R(θ) sin ξ = (ρ + r cos θ) sin ξ = (ρ + r cos α2 ) sin α1 , z = r sin θ = r sin α2

(1)

you can get the necessary geometric parameters of the bend of the bimetallic pipeline: The first quadratic shape of the junction surface of the toroidal shell, due to the orthogonality of the introduced curvilinear system of toroidal coordinates, will be equal to I = ds2 = a11 d ξ 2 + 2a12 d ξ d θ + a22 d θ2 = A21 d α12 + A22 d α22 . In this case, the Lame parameters, determined in the general case by the formula          ∂ r¯  ∂x 2 ∂y 2 ∂z 2 = Ai =  + + , ∂qi  ∂q1 ∂q2 ∂q3 where r¯ = r¯ (q1 , q2 , q3 ) = r¯ (ξ, θ, ζ ) is the radius vector of an arbitrary point A, taking into account formulas (1), will be equal   r A1 = ρ 1 + cos α2 , A2 = r (2) ρ

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From Fig. 2 shows that one of the centers of curvature O1 lies on the axis of symmetry, the other O2 in the center of the section. Therefore, by Meunier’s theorem, the first principal radius of curvature is R1 =

ρ + r cos θ R = , cos α cos α

(3)

and the second main radius of curvature R2 = r.

(4)

It is easy to verify that the obtained expressions (2)–(4) satisfy the Peterson – Codazzi – Gauss relations [6]:     1 ∂A2 1 ∂A1 ∂ A1 ∂ A2 = = , , ∂ξ R2 R1 ∂ξ ∂θ R1 R2 ∂θ      1 ∂ 1 ∂A2 ∂ 1 ∂A1 1 + . =− R1 R2 A1 A2 ∂ξ A1 ∂ξ ∂θ A2 ∂θ The radii of curvature R1 , R2 determine the angles between the faces of the shell element in planes orthogonal to the median surface. In the plane tangent to the middle surface, the angles between the faces of the element will be determined by the values ds1 ds2 R13 and R23 , where are R13 , R23 – the radii of curvature equal for a toroidal shell with a circular meridional section [6]. R13 =

R , R23 = ∞. sin α

Knowing the main radii of curvature, we can determine the second quadratic form

the coefficients of which characterize the normal curvature of the coordinate lines, and b12 is the surface torsion parameter. In the accepted notation, taking into account relations (1), they are determined by the formulas [7].   ∂ 2 r ∂r 1 ∂r r cos α b11 = √ · · = ρ 1 = 2 cos α2 , a11 a22 ∂α12 ∂α1 ∂α2 ρ b12 = √

∂ 2 r ∂ 2 r ∂r ∂r ∂r 1 ∂r · · = 0, b22 = √ · · = r. 2 a11 a22 ∂α1 ∂α2 ∂α1 ∂α2 a11 a22 ∂α2 ∂α1 ∂α2 1

3 Stress-Strain State of Toroidal Bimetallic Shells The stress-strain state of the toroidal bimetallic shells in the bending zone can be described by the equations in the forces - moments [8]   T2 ∂A2 T1 ∂A1 S2 S1 Q1 + − A1 A2 + − − q1 = 0, ∂α1 ∂α2 R23 R13 R1

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  T1 ∂A1 T2 ∂A2 S1 S2 Q2 + + A1 A2 + + + q2 = 0, ∂α2 ∂α1 R13 R23 R2   ∂A2 Q1 T1 ∂A1 Q2 T2 + − A1 A2 + − qz = 0, ∂α1 ∂α2 R1 R2   ∂A2 M1 M2 ∂A1 H2 H1 + − A1 A2 + + Q1 = 0, ∂α1 ∂α2 R23 R13   ∂A1 M2 M1 ∂A2 H1 H2 + + A1 A2 + − Q2 = 0, ∂α2 ∂α1 R13 R23 Here αi (i = 1, 2) - coordinate lines, Ti , Si , Qi (i = 1, 2) - internal forces: Mi , Hi (i = 1, 2) - bending and torques, b12 - Lame parameters. The elasticity relations connecting the forces and moments with the relative deformations of the junction surface and the curvature change parameters are determined by the formulas: T1 = B1 ε1 + B2 ε2 − C1 κ1 − C2 κ2 − NT , T2 = B2 ε1 + B1 ε2 − C2 κ1 − C1 κ2 − NT , M1 = C1 ε1 + C2 ε2 − D1 κ1 − D2 κ2 − MT , M2 = C2 ε1 + C1 ε2 − D2 κ1 − D1 κ2 − MT .

(5)

In this case, the temperature components of the forces and moments, taking into account the dependence of the elastic moduli and the coefficients β1 , β2 of linear thermal expansion of the layers on temperature, are found by the formulas: 1 NT = 1 − μ1 1 MT = 1 − μ1

δ1

1 E1 β1 Tdz + 1 − μ2

0

δ1 0

1 E1 β1 Tzdz + 1 − μ2

0 E2 β2 Tdz, −δ2

0 E2 β2 Tzdz. −δ2

In relations (5), now the membrane Bi (i = 1, 2), membrane - bending Ci and Di bending stiffness parameters of the two-layer shell are equal to: B1 =

1

 δ 1

1 − μ21 0

E1 dz +

⎛

0

1 1 − μ22

E2 dz B2 = −δ2

1⎜ 1 C1 = ⎝ 2 1 − μ21 ⎛ C2 =

1 ⎜ μ1 ⎝ 2 1 − μ21

δ1 0

1 − μ21

1 E1 zdz − 1 − μ22

δ1 E1 zdz − 0

μ1

μ2 1 − μ22

δ1 E1 dz + 0

0 −δ2

0 −δ2

μ2 1 − μ22

⎞ ⎟ E2 zdz ⎠, ⎞ ⎟ E2 zdz ⎠,

0 E2 dz, −δ2

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1 1 D1 = 3 1 − μ1

D2 =

1 μ1 3 1 − μ21

δ1

1 1 E1 z dz + 3 1 − μ2

0

2

0

δ1 E1 z 2 dz + 0

1 μ2 3 1 − μ22

155

E2 z 2 dz, −δ2

0 E2 z 2 dz. −δ2

3.1 Accounting for the Effects of Radiation As is known, structural materials under the action of irradiation undergo structural transformations, accompanied by numerous effects, which result in additional volumetric deformation QI, and the elastic and plastic characteristics of the material change. The degree of change in properties and the number of defects in a metal during irradiation depend on the total particle flux, irradiation temperature, and metal recrystallization temperature. Irradiation causes an increase in yield strength and strength, a decrease in the plasticity resource, and an increase in the critical temperature of the transition from a brittle to a viscous state. Important radiation effects also include gas and vacancy swelling of materials, accompanied by significant dimensional changes, warping, cracking and destruction of products. The radiation resistance of structural materials is mainly increased by alloying and regulating the microstructure (its grinding). Of all types of irradiation, due to the high penetrating power of neutral particles (neutrons and gamma rays), neutron irradiation, which causes bulk damage to the material, has the strongest influence. To assess the effect of neutron irradiation on the stress-strain state of structural elements, we consider a homogeneous isotropic body occupying a half-space z ≥ 0. If neutrons with the same average energy and intensity ϕ0 [neutron/(m2 · s)] are emitted at the body boundary (z = 0) parallel to the z axis, then the intensity of the neutron flux reaching the plane z = const is determined as follows [3]: ϕ(z) = ϕ0 · e−μz

(6)

The quantity μ is called the macroscopic effective cross section. For any chemical element, you can write: μ = σ n0 = σ

A0 ρ , s m−1 . A

Here, σ – is the effective cross section assigned to one core, n0 – is the number of cores in 1 cm3 ; A0 – is the Avogadro number; A – is the atomic weight; ρ – is the density. In the stationary case, taking into account (6), by the time moment t, the integral flow passes through section z: I (z) = ϕ0 te−μz .

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Thus, we can assume that the change in the volume of a substance is directly proportional to the integral flow I(z): QI = BI (z) Here B – is a constant coefficient obtained from the experiment. Note that the value of I0 = φ0 t gives the total neutron flux per unit surface area of the body. In nuclear reactors, ϕ0 is of the order of 1017 ÷ 1018 neutrons/(m2 • s), and I 0 reaches values of 1023 ÷ 1027 neutrons/m2 , while the volumetric strain QI reaches 0,1. Therefore, depending on the energy of neutrons and the irradiated material, the value of B can be of the order of 10−28 –10−24 m2 /neutron. It should be emphasized that the dependence of the elastic modulus E, yield strength σT , strength, and the entire tensile diagram on the integral flux I 0 of various energies was studied experimentally after irradiation of the samples in nuclear reactors. It is noted that the elastic modulus E upon irradiation increases to 5%. The yield strengths and strength of materials also increase with radiation exposure. Thus, in terms of its effect on elastic structures, radiation exposure is basically the opposite of thermal exposure, which reduces the yield strength of the material and other characteristics. For the analytical description of radiation hardening of the yield strength of the material, the following formula is used:

Here σT , σT0 – is the yield strength of irradiated and non-irradiated materials; A, ξ are material constants determined from experiments. Thus, when solving the boundary-value problem of calculating thermoelastic aggregates of nuclear energy, the known temperature dependences of the elastic modulus E can be used, in which radiation hardening of the physicomechanical characteristics of the material should also be taken into account. 3.2 Accounting for the Deformation of the Contour of the Cross Section of the Pipe Bend Since the deformation of cross sections in the bending region is accompanied by a significant decrease in stiffness, then when analyzing the ground stress state when bending curved pipes and open section rods, we can go to the equations of the semi-momentless theory of cylindrical shells V.Z. Vlasov [9] generalized to the case of flexible shells of double curvature [10, 11]. ∂S ∂T1 + + rq1 = 0, ∂ξ ∂θ

∂T2 T1 ∂S r + + + Q2 + rq2 = 0, ∂ξ ∂θ R13 R2

r r ∂Q2 − rqu = 0, T1 + T2 − R1 R2 ∂θ

∂H ∂M1 + − rQ1 = 0, ∂ξ ∂θ

∂M2 − rQ2 = 0. ∂θ

(7)

Equations (7) are equations of linear theory. They do not take into account the effect of pipe bending deformation on the arrangement of internal forces. They also do not allow taking into account the influence of the internal pressure p0 on the stiffness of the

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pipe. Therefore, it is necessary to take into account the deformation of the contour of the cross section of the pipe bend. Next, we compose the equilibrium conditions for the shell element, taking into account the change in its shape during the deformation of the pipe. For this, it is necessary to supplement Eqs. (7) by replacing the radii of curvature of the normal sections R1 and R2 and the radius R13 with their values after deformation R∗1 , R∗2 , R∗13 , which reflects the bending of the shell element. In addition, it is necessary to take into account the torsion strain of the element characterized by the parameter τ. Provided that A1 , A2 ≈ const we obtain: τ=

∂ϑ2 ∂ϑ1 = . A2 ∂α2 A1 ∂α1

After making additions that take into account the change in the shape of the pipe during bending and torsion, Eqs. (7) for a shell element in a deformed state are rewritten in the form: ∂S ∂T1 + + rQ2 τ + rq1 = 0, ∂ξ ∂θ

r r ∂Q2 + 2rSτ − rqz = 0, T1 + ∗ T2 − ∗ R1 R2 ∂θ

∂T2 T1 ∂S r + + ∗ + ∗ Q2 + rq2 = 0, ∂ξ ∂θ R13 R2 ∂H ∂M1 + − rQ1 = 0, ∂ξ ∂θ

(8)

∂M2 − rQ2 = 0 ∂θ

Here the quantities Q1 and S1 = S2 = S in the first three equations are discarded. Equations (8) are nonlinear, since they include the product of efforts on the curvature and torsion of the deformed coordinate surface Q2 τ , Sτ . Unknown shell deformation parameters contain quantities R∗1 , R∗2 , R∗13 . Equations (8) can be simplified by discarding terms that are non-linear with respect to unknown functions, but at the same time, while retaining terms that are non-linear with respect to the load parameters: bending moment M 0 and transverse pressure p0 (physical nonlinearity). The main part of the geometric nonlinearity of bending of the curvilinear part (bending) of the pipe is a consequence of a change in the Karman effect as a result of a change in the curvature of the axis. As a result of the initial deformation, the cross section ofthe pipe remains circular R02 = Rr = r, and the curvature of the axis varies from 1 ρ to 1 ρ 0 , but remains constant along the pipe.   1 M cos θ 1 + cos θ. (9) = = ρ0 ρ EI R01 We consider the deformation components in two parts: the first part is determined under the assumption that the cross sections are not deformed, and the second part is composed of the components resulting from the deformation of the cross section contour. Then, the curvature ε0 of the longitudinal axis of the pipe bend after application of the load, calculated without taking into account the deformation of the contour, (we call it the original one): ε0 = æ r cos θ

(10)

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can be calculated taking into account (9) by the formula æ =

M 1 1 . − = 0 ρ ρ EI

(11)

The total elongation of the longitudinal fibers ε1 is determined as the sum of the elongations: the original ε0 and the elongation caused by the deformation of the cross sections ε1 = ε0 +

r ∂u + Wy . ∂ξ ρ0

(12)

In expression (12), the relative elongation of the longitudinal fibers caused by their movement in the plane of curvature of the pipe axis (Karman effect) is taken into account by the term ρr0 Wy , where Wy = w cos θ − v sin θ − Wy0 ,

(13)

where Wy0 – is the displacement of the pipe section as a rigid whole. Hereinafter, all displacement components (u, v, w, Wy ) describe only a part of the pipe deformation associated with a change in the shape of the cross sections. Assuming, in accordance with the assumptions of the semi-momentless theory, that the relative elongation in the circumferential direction and the shift of the median surface are small in comparison with the corresponding derivatives, the relations between deformations and displacements can be written in the form usual for the linear theory of shells: ∂v + w = 0; ∂θ æ1 = −

∂u ∂v + = 0; ∂θ ∂ξ

ϑ = −ϑ2 =

∂w − v. ∂θ

1 ∂ϑ 1 ∂ 2w 1 ∂ϑ ; τ =− . ; æ2 = − 2 r ∂ξ r ∂θ r ∂ξ

(14) (15)

We exclude the forces N 2 , Q2 , S from the equilibrium Eqs. (8) and take into account that for a given load (moment M and internal pressure of intensity p) the components q1 , q2 and q have the value: q1 = q2 = 0; qn = p0 . As a result, we obtain an equation containing only 2 unknown quantities N 1 and M 2 :       ∂ 2 R∗2 ∂ 2 M2 ∂M2 ∂ 2 R∗2 ∂ 2 N1 ∂ − τ + + N 1 ∂ξ 2 ∂ξ ∂θ ∂θ 2 R∗1 ∂θ 2 r 2 ∂θ 2     2 ∂ r 1 ∂ M2 ∂2  ∗  ∂ − − − R p = 0. sin θ · N (16) 1 ∂θ ρ 0 ∂θ R∗2 ∂θ ∂θ 2 2 Equation (16) includes the radii of curvature and torsion of the deformed middle surface of the pipe, which are related to the initial curvature by the following relationships:   ∂ϑ r cos θ 1 ∂ 2w 1 1 1 1− ; 0 sin θ = R013 ≈ R∗13 . = − ; = (17) R∗1 ρ0 r ∂ξ 2 R∗2 r ∂θ ρ

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Further solving the problem of displacements, we use the relation between deformations and displacements (11)–(15) and the elasticity relations. Substituting these relations and expressions (17) into Eq. (16), we obtain a linear equation in the displacements of the following form:   r ∂u r ∂2 ∂ 3u r ∂ 2 ∂u cos θ − 0 sin θ + 0 2 (w cos θ − v sin θ) + 0 2 3 ∂ξ ∂ξ ρ ∂θ ρ ∂ξ ρ ∂ξ   2    2 2  ∂ ∂ 1 r 1 r 2 θ − v sin 2θ − 2θ w cos w sin 2θ − v sin + 2 ∂θ 2 ρ0 ∂θ2 ρ0        3 2 2 2 2 ∂ϑ ∂ ∂ ∂ ϑ ∂ w r ∂ − kν2 3 ε ϑ sin θ + cos θ − 1 − ε + 0 0 ∂θ ∂θ ∂θ2 ∂θ 2 ∂ξ 2 ρ 0 ∂θ2 r ∂ r ∂2 p ∂ 3ϑ − 0 = 0. (ε0 sin θ) + 0 2 (ε0 cos θ ) − r Eh ∂θ3 ρ ∂θ ρ ∂θ

(18)

Here ε0 – is the initial deformation defined by expression (10);  kν – is the  dimensionless parameter of the pipe wall thickness: kν = h cν r; cv = 12(1 − ν 2 ). Equation (18) contains 4 unknown functions: displacements u, v, w and rotation angle ϑ. By attaching to it 3 Eqs. (14), connecting u, v, w and ϑ, we obtain a closed system of equations in displacements. The arbitrary integration constants obtained by integrating the resulting system of 4 partial differential equations are found from the boundary conditions of the problem. 3.3 The Solution of the Boundary Value Problem To solve the boundary value problem, we apply the method of expanding unknown functions into trigonometric series in the circumferential coordinate θ. In this case, for each expansion term, the periodicity conditions in the coordinate θ are satisfied. The boundary conditions at the ends of the pipe are satisfied due to variable expansion coefficients, which are functions of the longitudinal coordinate ξ. Imagine the radial component of the displacement w in the form of a series:  fn (ξ ) cos nθ , n = 1, 2, . . . , n = 1, 2, . . . (19) w= n

Here fn (ξ ) are the longitudinal coordinate functions ξ to be determined. The main components of the movement and the angle of rotation are written as follows: u=− ϑ =−

 1 f (ξ ) cos nθ ; 2 n n n

 n2 − 1 n

n

fn (ξ ) sin nθ ;

v=−

1 n

Wy =

n

fn (ξ ) sin nθ, n = 1, 2, . . . ;

  n+2 n−2 1 + fn−1 cos nθ. fn+1 2 n n+1 n−1

(20)

Here, W y – are the displacement components describing the part of the pipe deformation associated with the change in the shape of the cross sections (Karman effect) are determined by formula (13).

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Substituting the decompositions of the components of displacement u, v, w and rotation ϑ (19)–(20) into the equilibrium Eq. (18), we obtain an infinite system of ordinary differential equations of the fourth order (each) with respect to unknown functions , where :   ⎧ 3 M0 IV ⎪ ⎨ f1 (ξ0 ) − 4 μ + 2 f2 (ξ0 ) = 0; II (ξ ) + f IV (ξ ) + a f (ξ ) + a II an,n−2 fn−2 (ξ0 ) + an,n−1 fn−1 0 0 n n 0 n,n+1 fn+1 (ξ0 ) n ⎪ ⎩ +an,n+2 fn+2 (ξ0 ) = Pδ2n.

Indexes n will take thefollowing values: 1 for n = 2; n = 2, 3, 4,…; δ2n = 0 for n = 2. Variable ξ0 will vary within√the following limits: L . −l0 ≤ ξ0 ≤ l0 , where l0 = kν 2r Series (19)–(20) have good convergence. Therefore, to obtain good results for μ < 10, it suffices to store three terms in them, not counting n = 1, i.e. n = 2, 3, 4. The truncated system of ordinary differential equations, in the general case with variable coefficients, can be written as: ⎧ IV II ⎪ ⎨ f2 + a2 f2 + a23 f3 + a24 f4 = P; (21) f3IV + a32 f2II + a3 f3 + a34 f4II = 0; ⎪ ⎩ IV f4 + a42 f2 + a43 f3II + a4 f4 = 0. The system of Eqs. (21) is a system of ordinary differential equations of the 12th order. In the present work, an analytical approach based on the use of a modified method of successive approximations by V. A. Pukhliy [12, 13] is used to integrate the system of Eqs. (21). 3.4 Application of the Modified Method of Successive Approximations First, we represent the system of differential Eq. (21) of the 12th order in the normal Cauchy form:  dXm = Bν,m Xν + fm , m = 1, 2, . . . , s. dξ s

(22)

ν=1

In accordance with the modified method of successive approximations [12, 13], variable coefficients and free terms are represented through shifted Chebyshev polynomials: Bν,m =

q  r=0

bν,m,r dr−1

r  k=0

ak Tk∗ (ξ ), fm =

q  r=0

fm,r (dr · r!)−1

r 

ak Tk∗ (ξ ).

(23)

k=0

Here q – is the degree of the interpolation polynomial; ak – are the coefficients of the expansion of ξ r in a series in Chebyshev polynomials Tk∗ (ξ ). In expressions (23), d r = 1 for r = 0 and dr = 22r−1 for the remaining r.

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The general solution of the system of Eqs. (22) has the form [12, 13]: Xm =

s  μ=1

 Cμ d0−1 a0 T0∗ (ξ ) δ +

∞  n=1

 Xm,μ,n +

q 

∞    −1 tm,j,0 dj+1 (j + 1)! ak Tk∗ (ξ ) + Xm,n j+1

j=0

k=0

n=2

(24) where tm,j,0 = fm,r for j = r; μ – is the number of the fundamental function; Cμ integration constants. In solution (24), δ = 1 if m = μ and δ = 0 for the remaining μ. The first approximation Xm,μ,1 is obtained by substituting the zeroth approximation: d0−1 a0 T0∗ (ξ ) δ into the right side of the homogeneous system:  dXm Bν,m Xν . = dξ s

ν=1

Subsequent approximations are carried out according to the formulas: Xm,μ,n =

β 

 tm,μ,n,j dn+j−1 (n + j − 1)!

−1

j=1

Xm,n =

β 



n+j−1

ak Tk∗ (ξ );

k=0

 tm,nj dn+j−1 (n + j − 1)!

−1

j=1



n+j−1

ak Tk∗ (ξ ) ,

(25)

k=0

where β = n (q + 3) – 2. Note that the systems of fundamental functions (25) are uniformly converging series, and the coefficients tm,μ,n,j and tm,n,j are determined through the coefficients of the previous approximation using the recurrence formulas:

tm,μ,n,j =

q s   ν=1 r=0

bν,m,r tν,μ,n−1,j−r (n + j − 1)−1

r !

(n + j − 1 − γ ).

γ =0

The constants Cμ included in the general solution (24) are found from the boundary conditions.

4 Conclusion A complete system of geometric and deformation relationships that determine the differential surface geometry of a junction of bimetallic toroidal shells, as well as physical relationships for the forces and moments of such shells, taking into account the dependence of physical and mechanical characteristics on temperature and radiation exposure, is presented.

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An algorithm for studying the stress-strain state of bends of bimetallic toroidal shells as applied to piping systems of nuclear reactors is described. The possibility of applying an analytical approach to calculating the strength and rigidity of pipe bends based on the use of the modified method of successive approximations is shown. The study was financially supported by the Russian Federal Property Fund and the city of Sevastopol in the framework of project No. 18-48-920002.

References 1. Getman, A.F.: Resource of operation of vessels and pipelines of nuclear power plants. In: Hetman, A.F. (ed.) Energoatomizdat (2000), 427 p 2. Arkadov, G.V.: Reliability of equipment and pipelines of nuclear power plants and optimization of their life cycle (probabilistic methods). In: Arkadov, G.V., Getman, A.F., Rodionov, A.N. (eds.) Energoatomizdat (2010). 424 p 3. Platonov, P.A.: The effect of irradiation on the structure and properties of metals. In: Platonov, P.A. (ed.) Mechanical Engineering (1971), 402 p 4. Pukhliy, V.A., Kogan, E.A., Pukhliy, K.V.: The Karman boundary value problem for the elastic bending of bimetallic hydropower pipelines. - Power plants and technologies, vol. 5, no. 3, pp. 25–35 (2019) 5. Axelrad, E.L.: Calculation of pipelines. In: Axelrad, E.L., Ilyin, V.P. (eds.) Engineering (1970), 240 p 6. Volmir, A.S.: Stability of deformable systems. In: Volmir, A.S. (ed.) Nauka (1967), 984 p 7. Rashevsky, P.K.: Differential geometry course. In: Rashevsky, P.K. (ed.) Gostekhizdat (1950), 360 p 8. Pukhliy, V.A.: On a boundary value problem for elastic bending of pipelines of hydraulic structures and nuclear power plants. In: Pukhliy, V.A., Nagolyuk, L.O., Kutovoi, L.V., Pichugova, L.N., A.V. Chuklin, V.P. Galenina (eds.) Theory of Mechanisms and Machines, vol. 12, no. 2(24), pp. 3–15 (2014) 9. Vlasov, V.Z.: The general theory of shells and its application in technology. In: Vlasov, V.Z. (ed.) Gostekhizdat (1949). 784 p 10. Axelrad, E.L.: Thin-walled curved rods and pipes. In: Axelrad, E.L. (ed.) Sat. “Studies in structural mechanics.” Proceedings of the LIIZhT, 249, 147–168 (1966) 11. Axelrad, E.L.: Semi-momentless theory of flexible shells. In: Axelrad, E.L. (ed.) Proceedings of the X All-Union Conference on the Theory of Shells and Plates, September 22–29, 1975, Tbilisi: Metsniereba Publishing House, pp. 487–497 (1975) 12. Pukhliy, V.A.: The method of analytical solution of two-dimensional boundary value problems for systems of elliptic equations. J. Comput. Matem. Mate. Phys. 18(5), 1275–1282 (1978) 13. Pukhliy, V.A.: About one approach to solving boundary value problems of mathematical physics. Differential equations, 15(11), 2039–2043 (1979)

Vibration Active of Machines with Elastic Transmission Mechanism Yuri A. Semenov(B) and Nadezhda S. Semenova Peter the Great St.Petersburg Polytechnic University, St.Petersburg, Russia [email protected]

Abstract. The authors consider some issues of machine dynamics, taking into account the elasticity of the transmission mechanism links. Such mechanisms are based on gear drives, which even under the constant external load have variable forces acting on their teeth, i.e. they are a source of vibration activity of the machine. The paper analyzes the causes of internal vibration activity of a spur gear train: variable gearing stiffness, kinematic errors of gears. Keywords: Variable gearing stiffness · Kinematic errors of gears · Dynamic errors

1 Introduction One of the most vibroactive mechanisms of machines are gear trains. They have a wide range of variable forces created by a large number of perturbing factors. First of all, this involves external dynamic loads. Thus, a spur gear is affected by relatively low-frequency external perturbations coming from the motor and the actuator with frequencies rv (r = 1, 2, . . . , with v as the frequency of actuating link rotation), as well as by perturbations with gear and wheel rotating frequencies, i.e. ω1 and ω2 . In spur gears (and primarily in gears with straight teeth) there is a periodic change in tooth stiffness in the engagement phase. This is due to the fact that a different number of teeth is involved in torque transmission (at any given moment of time, either one or two pairs of teeth can be in contact at the same time, depending on the position of contact points on the engagement line). Thus, the process of teeth pairing will be accompanied by a redistribution of load between the teeth at the boundaries of the engagement phases from the one-pair contact to the double-pair contact, which can cause forced and parametric vibrations. Typical for spur gears are forces associated with kinematic errors of gears: tooth profile errors, base pitch errors, assembly errors, etc. The teeth of gears, due to inaccuracy in their manufacture and elastic deformations, come into contact not at the engagement line, but at some off-design point, as a result of which there will be no common normal between the contacting profiles, i.e. a hit will occur.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 163–172, 2021. https://doi.org/10.1007/978-3-030-62062-2_16

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2 External Dynamic Loads The design model of a machine with rigid links does not allow us, in principle, to determine deformation of real links and structural elements of the kinematic pairs. However, in modern machines, these deformations and the associated vibrations (elastic vibrations) of mechanisms are of paramount importance. They can significantly affect the accuracy of mechanisms and cause significant dynamic loads in the links and their joints. To study them, more complex dynamic machine models that take into account the elasticity of the links have to be used. In principle, all the links of a machine, the elements of all kinematic pairs are elastic: shafts, gears, bearings, etc. Obviously, a complete account of the elasticity of all these links and kinematic pairs will make the model so complicated that the dynamic analysis will become practically impossible. Therefore, not all links are considered elastic, but only those whose elasticity can significantly affect the results of the dynamic analysis are singled out. The experience in machine design [1–4] shows that taking into account torsional deformations of shafts and deformations of gear teeth is of greatest importance in such systems. [5] describes the dynamic model of a gearbox with external loads, presented in Fig. 1. Here, the motor rotor 0 drives the gears 1 and 2 of the single-stage gearbox and the actuating link 3. The sections between links 0 and 1, 1 and 2, 2 and 3 are considered elastic; c01 , c12 , c23 are their rigidity; b01 , b12 , b23 are resistance coefficients; J0 , J1 , .J2 , J3 are inertia moments of the links relative to their own rotation axes; M 0 and M c are the driving moment and the moment of resistance. In this case, the bending compliance of the shafts and the compliance of the supports were not taken into account. The system under consideration has four degrees of freedom; as the generalized coordinates we have taken the absolute angles of rotation of the motor rotor q0 , the gears q1 − q2 and the actuating link q3 . It is more convenient to bring these coordinates to the axis of the motor rotor by introducing new generalized coordinates: ϕ0 = q0 ; ϕ1 = q1 ; ϕ2 = i12 q2 ; ϕ3 = i12 q3 ,

Fig. 1. The dynamic model of a machine unit

(1)

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with i12 the gear ratio. In this case, the deformation of the shafts and the deformation of the gears, adduced to the drive wheels, were defined as follows: −1 (ϕ3 − ϕ2 ), θ01 = ϕ1 − ϕ0 ; θ12 = rb2 q2 − rb1 q1 = rb1 (ϕ2 − ϕ1 ); θ23 = i12

(2)

with rb1 , rb2 as the radii of the base circles. To compose the Lagrange equations of the second kind, the kinetic and potential energies, as well as the dissipative function, were determined. Kinetic energy of the system   3 3 3 ϕ˙ s 2 1 1 1 2 Js q˙ s = Js = Js∗ ϕ˙ s2 , T= 2 2 i0,s 2 s=0

s=0

(3)

s=0

where −2 −2 J2 ; J3∗ = i12 J3 J0∗ = J0 ; J1∗ = J1 ; J2∗ = i12

(4)

are the inertia moments of the transmission mechanism links, adduced to the rotation axis of the motor rotor. Potential energy 

1 1 cs−1,s θ2s−1,s = cs∗ (ϕs − ϕs−1 )2 . = 2 2 3

3

s=1

s=1

(5)

with −2 2 c12 ; c3∗ = i12 c23 . c1∗ = c01 ; c2∗ = rb1

(6)

1 1 2 bs−1,s θ˙s−1,s = bs∗ (ϕ˙ s − ϕ˙ s−1 )2 , 2 2

(7)

Dissipative function Φ=

3

3

s=1

s=1

with −2 2 b12 ; b3∗ = i12 b23 . b1∗ = b01 ; b2∗ = rb1

(8)

External forces acting in a mechanical system are the driving moment M 0 applied to the rotor of the motor and the moment of resistance M c applied to the actuator. Composing the Lagrange equations in the form   ∂T d ∂T ∂ ∂ − =− − + Ms , (9) dt ∂ ϕ˙ s ∂ϕs ∂ϕs ∂ ϕ˙ s we obtain (s = 0, 1, 2, . . . , 3):

⎫ J0 ϕ¨ 0 + b1 (ϕ˙ 0 −ϕ˙ 1 ) + c1 (ϕ0 − ϕ1 ) =0 , ⎪ ⎪ ⎬ J1 ϕ¨ 1 + b1 (ϕ˙ 1 −ϕ˙ 0 ) + b2 (ϕ˙ 1 −ϕ˙ 2 ) +1 (ϕ1 − ϕ0 ) +2 (ϕ1 − ϕ2 ) = 0, J2 ϕ¨ 2 + b2 (ϕ˙ 2 −ϕ˙ 1 ) + b3 (ϕ˙ 2 −ϕ˙ 3 ) + c2 (ϕ2 −ϕ1 ) + c3 (ϕ2 − ϕ3 ) = 0, ⎪ ⎪ ⎭ J3 ϕ¨ 3 + b3 (ϕ˙ 3 −ϕ˙ 2 ) + c3 (ϕ3 −ϕ2 ) = M3 .

(10)

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The resulting differential Eqs. (10) describe the motion in the transmission mechanism, as in a system of absolutely rigid bodies connected in series with each other by non-inertial elastic-dissipative elements. Figure 2 shows a four-mass vibration system corresponding to the system of differential Eqs. (10).

Fig. 2. The reduced model of a free chain system

In the above work, another form of writing the equations of a mechanical system motion was given, useful for the cases where the law of motion of the motor rotor is known. From the equations of motion (10) we select the equation corresponding to s = 0 and turn to the deformation coordinates characterizing the displacement of the masses relative to the motor rotor θs = ϕs − ϕ0 (t).

(11)

In this case, the equations of motion of the system can be written as ⎫ ⎪ J1 θ¨ 1 + b1 θ˙ 1 + b2 (θ˙ 1 − θ˙ 2 ) + c1 θ1 + c2 (θ1 − θ2 ) = − J1 ϕ¨ 0 (t), ⎬ J2 θ¨ 2 + b2 (θ˙ 2 − θ˙ 1 ) + b3 (θ˙ 2 − θ˙ 3 ) + c2 (θ2 − θ1 ) + c3 (θ2 − θ3 ) = − J2 ϕ¨ 0 (t), ⎪ ⎭ J3 θ¨ 3 + b3 (θ˙ 3 − θ˙ 2 ) + c3 (θ3 − θ2 ) = − J3 ϕ¨ 0 (t) + M3 ,

(12)

J0 ϕ¨ 0 − b1 θ˙1 − c1 θ1 = M0 .

(13)

Figure 3 shows a vibratory system with a fixed left end, for which the equations of forced vibrations caused by the applied active and transportable inertia forces coincide with Eqs. (12)–(13).

Fig. 3. The reduced model of a system with a fixed end

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Analyzing the obtained formulae for dynamic errors, we should emphasize that the most dangerous for the system is the main resonance that occurs when the perturbation frequency v coincides with the first natural frequency k1 . For relatively low-speed machines it is possible to fulfill the condition v g10 .

(25)

If condition (25) is violated, the teeth of the gear and the wheel will open, and formula (23) obtained above will no longer characterize the forced vibrations in the original system. The limiting state of the system at which the teeth of the mating wheels open is determined by the condition M2 = 0. Then the equation of the gear motion is: J1 q¨ 1 = c01 (q0 − q1 ).

(26)

Substituting into Eq. (26) the value q1 from formula (22) and omitting the terms characterizing natural vibrations, we obtain

Vibration Active of Machines with Elastic Transmission Mechanism

−J1

2 ξ2 ω2 c12 g10 rb1 1 0

c1 − J1 ξ21 ω02

 sin ξ1 (ω0 t − ) = M0 + ω0 t −

2 c12 g10 rb1

c1 − J1 ξ21 ω02

171

 sin ξ1 (ω0 t − ) − ω0 t c01 .

2 and transformations we find After substitution c1 = c01 + c12 rb1

g10 sin ξ1 (ω0 t − ) = M0 . 2 )−1 + (c − J ξ2 ω2 )−1 (c12 rb1 01 1 1 0 This formula can be considered as the equation that allows determining the opening time t of the teeth:

 M0 1 1 sin ξ1 (ω0 t − ) = + . 2 g10 c12 rb1 c01 − J1 ξ21 ω02 For the resulting equation to have real roots, it is necessary    M0  1 1  −   ≤ 1. 2 g10  c12 rb1 J1 ξ21 ω02 − c01 

(27)

Inequation (27) is a condition for opening the teeth [9].

5 Conclusion The paper discusses the causes of internal vibration activity of a spur gear [8–15]: variable gearing stiffness, kinematic errors of gears. Dynamic errors and loads in a gear drive are determined. In spur gears in the absence of friction forces, the possibility of parametric resonance is considered. It is shown that at low rotational speeds, mating teeth can open only at a large ratio g10 / 3 , i.e. either with a significant amplitude of the gearing error, or with a very small static load (in a lightly loaded transmission). From condition (25), the admissible value of the kinematic error amplitude is obtained.

References 1. Kolovsky, M.Z., Evgrafov, A.N., Semenov, Yu. A., Slousch, A.V.: Advanced Theory of Mechanisms and Machines. Springer, Heidelberg (2000). 394 p. (rus.) 2. Kolovsky, M.Z.: Dynamics of machines. Mashinostroenie (1989). 263 p. (rus.) 3. Veits, V.L., Kolovsky, M.Z., Kochura, A.E.: Dynamics of controlled machine aggregates. Nauka, Moscow (1984). 352 p. (rus.) 4. Vulfson, I.: Dynamics of cyclic machines. Springer, Heidelberg (2015). 390 p. 5. Semenov, Yu.A., Semenova, N.S.: Determination of Dynamic Errors in Machines with Elastic Links, p 163–174. Springer, Heidelberg (2019) 6. Semenov, Y.A., Semenova, N.S., Egorova, O.V.: Dynamic mesh forces in accounting of the time variable mesh stiffness of a gear train. Rev. Mech. Eng. 12(9), 736–741 (2018)

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7. Livshits G. A. Dynamics of gear transmission in connection with the accuracy of engagement.//Trudy TSNIITMASH. 1963. (rus.) 8. Kolovsky, M.Z., Chernikov, A.V.: On the influence of kinematic accuracy of gear transmission on the vibration activity of the machine unit.//Problems of mechanical engineering and reliability (1995). № 4. (rus.) 9. Andrienko, P.A., Karazin, V.I., Kozlikin, D.P., Khlebosolov, I.O.: About implementation harmonic impact of the resonance method. In: Lecture Notes in Mechanical Engineering, pp. 83–90 (2019) 10. Evgrafov, A.N., Karazin, V.I., Petrov, G.N. Analysis of the self-braking effect of linkage mechanisms. In: Lecture Notes in Mechanical Engineering, pp. 119–127 (2019) 11. Advances in Mechanical Engineering: Selected Contributions from the Conference “Modern Engineering: Science and Education”, Saint Petersburg, Russia, June 2014. Evgrafov, A. 2016 Lecture Notes in Mechanical Engineering (2016) 12. Evgrafov, A.N., Karazin, V.I., Khisamov, A.V.: Research of high-level control system for centrifuge engine. Int. Rev. Mech. Eng. 12(5), 400–404 (2018) 13. Semenov, Yu.A., Semyonova, N.S.: Theory of mechanisms and machines in examples and problems. Part 2. Saint Petersburg, Polytechnic University. UN-TA, 2016 -282 p. (rus.) 14. Abramov, B.M.: Vibrations of straight-toothed gears. Kharkiv /KSU 1968. 175 p. (rus.) 15. Vibration in technology. Handbook in six volumes. M.: mechanical engineering (1981). (rus.)

Method for Calculating and Selection the Optimal Profile of Clearance in Pairs of Cylinder-Piston Group of Piston Engine Alexander Yu. Shabanov, Anatolii A. Sidorov(B) , Pavel N. Brodnev, Oleg V. Abysov, and Victor V. Rumyantsev Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia [email protected]

Abstract. The authors present an analysis of the work of the cylinder-piston group parts and a method for determining the clearance values in the parts of the cylinder-piston group (CPG) of a diesel engine. The method is developed on the basis of calculating the power of mechanical losses in the cylinder-piston group of a diesel engine while maintaining the oil burning loss within the permissible limits for operation without scuffing of the cylinder-piston group. This ensures a guaranteed clearance in the loaded zone of the “piston-cylinder sleeve” friction pair, taking into account the heat-stressed and deformed state of the piston and engine cylinder over the entire range of operating power conditions of the engine. Keywords: Cylinder-piston group parts · Diesel engine · Power of mechanical losses · Oil burning losses · Friction losses · Wear · Without scuffing the cylinder operation · Profile of working clearances in the CPG

1 Introduction Friction units of diesel engines are always subject to dynamic effects due to the nature of its operation. The most loaded are the parts of the cylinder-piston group (CPG), which are affected by mechanical and thermal loads when internal combustion engines operate. Under the influence of thermal loads, deformations and temperature stresses are brought about in the CPG parts. The parts change their dimensions, which leads to changes in the clearance in the interfaces and, accordingly, to a change in the thickness of the lubricant layer. When the load is dynamically changing, the lubricant layer shows its own elasticity, which depends on the frequency of exposure. Dynamic perturbation and temperature changes cause additional internal friction in the lubricant layer, which changes the viscosity and the ability of the lubricant layer to accept the load. This accelerates the thermomechanical destruction of the lubricant, which also depends on the thickness (volume) of the lubricant layer [1–7]. In determining the optimum profile of the clearance in the mating of piston the diesel engine is necessary to consider all these factors and especially the power conditions of © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 173–184, 2021. https://doi.org/10.1007/978-3-030-62062-2_17

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the engine where possible scuffing of cylinder bore, according to its application and operating conditions. Then it is necessary to simulate the temperature and deformed state of the piston and cylinder in the selected power conditions of the engine. Finally, solve the problem of forming lubricant layers in the area of the side surface of the piston and piston rings of the engine, taking into account the real profile of the working clearance in the CPG. It is obvious that the power conditions of the engine can not be clearly defined, the task is variable, and the design development involves choosing the best profile that meets the conditions stated above. The correct estimation of the stress-strain state of the CPG parts, taking into account their temperature state, is a prerequisite for reliable operation of the diesel engine in operation at the design stage. Calculation of the Clearance Profile in the Coupling of the Cylinder Piston Group of a Diesel Engine Choosing the values and profiles of clearance in the CPG of a diesel engine is a responsible and quite complex task and involves solving two problems. The first task is to ensure no-scuffing working of the cylinder-piston group, that is, there are clearances in the loaded zone of the “piston-cylinder sleeve” friction pair in the entire range of operating power conditions. This problem is solved by ensuring a guaranteed clearance across the entire lateral working surface of the piston in any engine operating power conditions from idling when the engine is warmed up to the nominal. The second task is to reduce the power of mechanical losses in the cylinder-piston group of diesel engines while maintaining the loss of oil to carbon monoxide within acceptable limits. To solve this problem, we select the oval-barrel profile of the piston, the size and location of the contact spot of the piston skirt, which guarantees the best working conditions of the “piston-cylinder” friction pair, i.e., minimal power loss on friction. The solution of the set tasks must be made taking into account the dependence of geometrically complex thermal and mechanical deformations of the piston and cylinder on the engine operating power conditions. The choice of values and profiles of clearances in the CPG can be solved in the course of experimental finishing works. But this is extremely problematic due to the inability to provide a combination of the entire intended range of engine operating power conditions and experienced versions of pistons in the process of debugging, long duration, high cost and the lack of, in the end, guarantees of obtaining a positive result for specific operating conditions of the engine. In this case, the most preferable method is to profile the working clearances in the CPG using mathematical modeling of friction processes, taking into account the calculated heatstressed and deformed state of the piston and cylinder of the engine. At the same time, the need for final experimental verification of the decision still remains, but the number of variants of the CPG design that are submitted to full-scale tests is significantly reduced.

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The Solution of this Task Consists of the Following Stages – simulation of the temperature and deformed state of the piston and cylinder in the selected power conditions of engine operation, which can be used for scuffing of the bore in the CPG; – solving the problem of forming lubricant layers in the area of the side surface of the piston and piston rings of the engine, taking into account the real profile of the working clearance in the CPG. The problem is solved in a multi-variant power conditions with the subsequent selection of the best profile that meets the conditions stated above. To solve the problem, we introduce the following concepts: – “cold” profile of the working surface of the piston-the size of the mounting clearances formed at the stage of manufacturing the part. This profile does not depend on the engine operating power conditions; – “hot” profile of the working surface of the piston-the clearances values taking into account the thermomechanical deformations of the piston. This profile is variable depending on the temperature deformations of the piston, and therefore it depends on the operating power conditions of the engine. The relationship between the clearance values of the “cold” and “hot” profiles is determined by an obvious relationship: h (z) = c (z) + tm (z), where h (z) is the clearance value of the “hot” profile in the design power conditions; c (z) - the clearance value of the “cold” profile; tm (z) - the value of radial thermomechanical deformations of the piston in the design power conditions; z - coordinate of the calculated point in the height of the lateral surface of the piston. To illustrate the proposed method of selecting the working clearance profile in the optimization CPG, a forced marine diesel engine was used. The operating base speed range of the crankshaft is 1200, 1600, 2000 and 2400 rpm. A sketch of the design of the basic version of the piston with an oval-barrel profile is shown in Fig. 1. Profile on the head part is conical, on the loaded surface of the skirt is of the “one-sided barrel” type. The cylinder profile is a regular circular cylinder. The calculation of thermomechanical deformations of the piston and cylinder of the diesel engine is performed in three-dimensional formulation by the finite element method. In our work, we used the modeling block of the SOLIDWORKS complex. The boundary conditions of thermal and mechanical loading were set based on calculations using the methods developed by the authors and described in detail in the works [1, 2, 8, 9].

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Fig. 1. Sketch of the side surface profile of the basic version of the piston of the studied diesel engine

2 Methods and Materials Calculations of the temperature and stress-strain state of the piston and cylinder of the diesel engine were performed on the selected reference power conditions. The results of radial deformations of the piston in the nominal operating power conditions are shown in Figs. 2 and 3. Based on the results of these calculations, the “hot” profiles of the lateral surface of the piston were determined and analyzed (Figs. 4 and 5). For further analysis of the quality of the selected piston profile, information on radial thermomechanical deformations of the cylinder was used. The results of calculating these deformations in the selected power conditions are shown in Fig. 6. It is obvious that thermomechanical deformations of the working surface of the cylinder in our case are qualitatively similar in all design power conditions and contribute to an increase in the “piston-cylinder” clearance in all zones of the friction pair. The totality of the information obtained allows us to draw the following conclusions about the quality of the choice of the “cold” clearance profile in the CPG of the basic version of the piston.

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Fig. 2. Radial deformations of the piston in the nominal operating power conditions, cross section perpendicular to the axis of the piston pin

Fig. 3. Radial deformations of the piston in the nominal operating power conditions, cross-section along the axis of the piston pin

Taking into account the working deformations of the cylinder sleeve, the clearance values along the piston skirt are clearly overstated. This is one of the reasons for the increased oil consumption on carbon monoxide observed for the basic version of the diesel engine. In addition, they cause an increased intensity of transverse movement of the piston, which, with a small thickness of the steel cylinder liner and without an intermediate support shoulder, negatively affects the resource and acoustic characteristics of the engine. Also unfavorable is the nature of the distribution of clearances in the head of the piston. The absence of a smoothed “barrel-shaped” surface when the piston is skewed within the clearance leads to contact with the cylinder surface and the need to increase

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Fig. 4. Calculation of the “hot” profile of the lateral surface of the basic version of the piston in the nominal operating power conditions, the cross section is perpendicular to the axis of the piston pin

Fig. 5. Calculation of the “hot” profile of the lateral surface of the basic version of the piston in the nominal operating power conditions, cross-section along the axis of the piston pin

the “cold” clearances in this part. This also does not provide optimal sealing conditions for the combustion chamber and increases the oil consumption for carbon monoxide.

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Fig. 6. Distribution of radial displacements of the working surface of the diesel cylinder sleeve

At the same time, the contact spot area in the “piston-sleeve” pair is clearly overstated, the hydrodynamically optimal profile of the working surface is poorly expressed, which leads to an increase in friction losses in the CPG and a decrease in the bearing capacity of the piston skirt.

3 Results Comparing Variants of the Profiles of the Pistons For further comparative studies, four variants of oval-barrel profiles of the modified version of the piston were used, differing in the size and location of the “contact spot” of the “piston-cylinder” interface. In all versions of these pistons, the clearance values are reduced. In contrast to the basic profile of the “one-sided barrel” type, all modified profiles are of the “two-sided barrel” type with different degrees of completeness. In this case, the position of the contact zone is consistent with the position of the piston pin axis to reduce the amplitude of the lateral swing of the piston axis when it moves within the clearance. One of the modified versions of the profile, which later showed the best result, is shown in Fig. 7.

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Fig. 7. Sketch of the side surface profile of the modified version of the piston of the studied diesel engine

The results of calculating the “hot” profile of this variant of the piston at the nominal engine operating power conditions are shown in Fig. 8 and 9.

Fig. 8. Calculation of the “hot” profile of the lateral surface of the modified version of the piston in the nominal operating power conditions, the cross section is perpendicular to the axis of the piston pin

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Fig. 9. Calculation of the “hot” profile of the lateral surface of the modified version of the piston in the nominal operating power conditions, cross-section along the axis of the piston pin

4 Discussion As follows from the results obtained, for the developed “hot” profile, the hydrodynamic quality of the working surface of the piston skirt has been improved, the position of the contact spot has approached the piston swing axis, which helps to reduce the amplitude of the swing of the piston axis relative to the cylinder axis. The size of the friction zone has decreased, which should have the effect of reducing friction losses in the CPG. To assess the degree of influence of the studied profiles of the side surface of the piston, a simulation of the lubrication and friction processes in the diesel CPG was performed for all versions of the piston in comparison with the basic one. Modeling of friction processes in the CPG was performed using the software and methodological complex “Tribology of the CPG”, developed by the authors of the article [8, 9]. As a base for comparison, the initial version of the piston with four rings was adopted (Fig. 1). The proposed profile options were calculated for a modified version of the piston with three rings. Calculations were performed for four values of the speed of rotation of the crankshaft (at n = 2400, 2000, 1600, 1200 rpm) of a marine diesel engine working on the ship’s propeller. The results of calculating the power of mechanical losses are summarized in Tables 1 and 2. Analysis of the results shows that the use of a modified version of the CPG provides a significant reduction in the power of friction losses. The combined effect of replacing the set of piston rings and the profile of the piston skirt allows you to reduce the friction power in the CPG compared to the basic version by 14…17%, depending on the engine operating power conditions. It should be noted that the contribution to this increase in

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Table 1. Power balance of mechanical losses of a marine diesel engine, the basic version of the diesel piston Power conditions, n, rpm

Power of mechanical losses, kW/cylinder

Total power of mechanical losses, kW/cylinder

First compression ring

Second compression ring

First oil ring

Second oil ring

Trunk

2400

4,90

3,09

2,41

2,31

3,56

16,27

2000

3,92

2,52

1,93

1,85

2,83

13,05

1600

2,91

1,98

1,43

1,38

2,10

9,80

1200

2,05

1,42

0,95

0,92

1,44

6,78

Table 2. Power balance of mechanical losses of a marine diesel engine, a modified version of the diesel piston Power conditions, Power of mechanical losses, kW/cylinder Total power of n, rpm First compression Second Oil ring Trunk mechanical losses, ring compression ring kW/cylinder 2400

4,89

3,24

2,72

2,65

13,49

2000

3,99

2,69

2,09

2,07

10,84

1600

2,93

2,07

1,54

1,60

8,14

1200

2,13

1,58

0,99

1,10

5,80

engine efficiency is made by changing the design of the piston rings and the profile of the lateral surface of the piston skirt. Experimental verification of the proposed version of the CPG design generally confirmed the conclusions made on the basis of the above-mentioned computational study.

5 Conclusion Reducing friction losses in the global aspect is very important. For example, 50… 55% of the power generated by a piston engine is spent on overcoming friction and wear of materials (in a well-established modern automobile engine-20… 25% of energy). In the process of operation, these losses increase. Their level largely determines the efficiency of the engine, as well as material and energy costs. According to the UN, 30% of all energy produced in the world is used to overcome friction forces, and 20% of the materials produced in the world are used to restore machines after wear. Therefore, this problem is studied by many specialists [10–21], which confirms its relevance.

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Prospects for research on this issue should be aimed at further improving the methods of computational modeling of friction pairs, taking into account the operating power conditions of the piston engine, the use of advanced coatings with a low coefficient of friction, the development of new low-viscosity lubricants and additives. Implications 1. Thus, the developed method allows at the design stage to predict the efficiency of CPG in minimizing the power of mechanical losses while ensuring without scuffing of the cylinder bore at the operation piston group, thereby increasing the reliability and efficiency of the engine as a whole. 2. It should be noted that the methods used in the calculations are completely universal and can be used in the design or modernization of the CPG of any four-stroke piston engine.

References 1. Shabanov, A.Yu., Galyshev, Yu.V., Sidorov, A.A.: Anti-friction composition as a means to restore worn engine performance. Int. Rev. Mech. Eng. 11(5), 305–310 (2017) 2. Shabanov, A.Yu., Galyshev, Yu.V., Zaitsev, A.B., Sidorov, A.A.: Pilot study of influence of high-temperature viscosity of engine oil on technical and economic indicators of a high-speed piston internal combustion engine. Mod. Mech. Eng. Sci. Educ. 6, 804–814 (2017) 3. Putintsev, S.V., Anikin, S.A., Ageev, A.G.: Application of the triboadaptivity principle for profiling the piston skirt of a high-speed diesel engine. Izvestiya Vuzov 5(662) (2015) 4. Vellandi, V., Namani, P., Bagavathy, S., Chalumuru, M.: Design and Development of an UltraLow Friction and High Power-Density Diesel for the Indian Market. SAE Technical Paper 2020-01-0834 (2020). https://doi.org/10.4271/2020-01-0834 5. Chichinadze, A.V., Berliner, E.M., Braun, E.D., Bushe, N.A., Buianovskii, I.A., Gekker, F.R., Goriacheva, I.G., Grib, V.V., Demkin, N.B., Dobychin, M.N., Evdokimov, Iu.A., Zakharov, S.M., Kershenbaum, V.Ia., Luzhnov, Iu.M., Mamkhegov, M.A., Mikhin, N.M., Romanova, A.T.: Friction, Wear, and Lubrication (Tribology and Triboengineering). Mashinostroenie, Moscow (2003). 575 p. (in Russia) 6. Kolovsky, M.Z., Evgrafov, A.N., Slousch, A.V., Semenov, Yu.A.: Theory of Mechanisms and Machines: Textbook for High Schools. Publishing House “Academia”, Moscow (2013). 560 p. (in Russian) 7. Yakunin, R.V.: Methodological foundations of optimizing the profile of the piston skirt of the internal combustion engine to reduce mechanical losses. Dissertation, Moscow, NAMI (2019) 8. Petrichenko, R.M., Petrichenko, M.R., Kanishchev, A.B., Shabanov, A.Yu.: Friction and heat transfer in piston rings of internal combustion engines Edited by RM Petrichenko. LSU Publishing House, Leningrad (1990) 9. Shabanov, A.Yu.: Work of the ring seal of high-speed diesel. Dissertation, Leningrad, LPI them. MI Kalinina (1985) 10. Kennedy, M., Hoppe, S., Esser, J.: Piston ring coating reduces gasoline engine friction. MTZ Worldw 73(5), 40–43 (2012) 11. Rozhdestvensky, Yu.V., Gusev, A.M.: Radial profiling of the guide part of the internal combustion engine piston. Vestnik SUSU 11, 78–84 (2006)

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12. Belogub, A.V.: Specification of force loading parameters for calculating the profile of the piston skirt. Bull. Engine Constr. 1, 59–63 (2015) 13. Evgrafov, A.N., Karazin, V.I., Khisamov, A.V.: Research of high-level control system for centrifuge engine. Int. Rev. Mech. Eng. 12(5), 400–404 (2018) 14. Meng, Z., Ahling, S., Tian, T.: Study of the Effects of Oil Supply and Piston Skirt Profile on Lubrication Performance in Power Cylinder Systems. SAE Technical Paper 2019-01-2364 (2019) 15. Fang, C., Meng, X., Xie, Y.A.: Piston tribodynamic model with deterministic consideration of skirt surface grooves. Tribol. Int. 110, 232–251 (2017) 16. Ter-Mkrtichyan, G.G., Yakunin, R.V.: Improvement of methods for optimal profiling of the piston skirt in order to reduce mechanical losses. Proc. NAMI 262, 184–186 (2015) 17. Holmberg, K., Andersson, P., Nylund, N., Mäkelä, K., Erdemir, A.: Global energy consumption due to friction in trucks and buses. Tribol. Int. 78, 94–114 (2014). https://doi.org/10. 1016/j.triboint.2014.05.004 18. Lu, Y., Li, S., Wang, P., Liu, C.S., Zhang, Y., Müller, N.: The analysis of secondary motion and lubrication performance of piston considering the piston skirt profile 2018, 27 (2018). Article ID 3240469. https://doi.org/10.1155/2018/3240469 19. Joseph, S., Johnson, J., Umate, M., Vipin, T.: A Study of Friction Behavior of a Single Cylinder Gasoline Engine. SAE Technical Paper 2018-32-0078 (2018). https://doi.org/10.4271/201832-0078 20. Liu, C., Lu, Y.-J., Zhang, Y.-F., Li, S., Müller, N.: Numerical study on the lubrication performance of compression ring-cylinder liner system with spherical dimples. PLoS One 12(7), e0181574 (2017). https://doi.org/10.1371/journal.pone.0181574 21. Evgrafov, A.N., Petrov, G.N.: Computer simulation of mechanisms. Lectures Notes in Mechanical Engineering, pp. 45–56. Springer International Publishing, Switzerland (2017). https://doi.org/10.1007/978-3-319-53363-6_6

Oscillations of Double Mathematical Pendulum with Noncollinear Joints Alexey S. Smirnov1,2(B) and Boris A. Smolnikov1,2 1 Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia

[email protected] 2 Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences,

St. Petersburg, Russia

Abstract. This paper considers the dynamics of a double mathematical pendulum whose cylindrical joints axes are not collinear to each other and form a certain acute angle between them. Such system is the simplest scheme of spatial two-link manipulator and can also be a component of various multi-link robotic devices. Basic geometric and kinematic characteristics of a double pendulum are given, which are necessary for deriving the equations of its motion. Small oscillations of the system near the lower equilibrium position are studied. As a result, formulas for the natural frequencies and modes of small oscillations are obtained. The dependence of frequencies and modes on the angle between the pendulum joints is revealed using a qualitative and graphical analysis of found expressions. The obtained results are both of theoretical interest and important practical value. Keywords: Spatial double pendulum · Noncollinear joints · Free oscillations · Linear model · Natural frequencies and modes of small oscillations

1 Introduction A lot of attention is given to issues of pendulum systems dynamics and control of their movement quite recently, as evidenced by numerous publications [1–8]. The double pendulum and its various modifications are of particular interest among pendulum structures, and they are very popular objects of research in the area of oscillating systems with several degrees of freedom, and also in the related science disciplines. This is due to their important practical applications in the design of various manipulators to use them as elements of complex robotic structures and also in the modeling of biodynamic systems. Main directions of studying the double pendulum in recent decades can be distinguished: creation and analysis of computer models with visualization of dynamic behavior [9–11], chaotic behavior analysis [12–18], controlled motions analysis [19– 21], dynamics and stabilization of inverted double pendulum [22–25], dynamics with vibrating suspension point [26–29], construction of optimal movement modes [30–32]. However, only the flat double pendulum is usually considered in existing works, when the axes of both joints are collinear. Nevertheless, the joint axes make up a certain © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 185–193, 2021. https://doi.org/10.1007/978-3-030-62062-2_18

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angle in many real manipulators and pedipulators [33, 34]. As a result, such a double pendulum becomes spatial and therefore its movements get the more interesting character, encompassing many different configurations. The dynamic behavior of such constructions has not yet been studied in detail in the existing literature, and at the same time such systems have important practical and promising significance which ensures the relevance of this study.

2 Design Scheme of Spatial Double Pendulum We turn to a study of the design scheme of double pendulum consisting of two pivotally connected identical mathematical pendulums of length l and with end loads of mass m. We assume that axes of both cylindrical joints are horizontal and form an angle α between themselves in their main equilibrium position, when both links lie on the same vertical, and joint angles of their rotation are equal to zero, so that these axes are not collinear to each other (Fig. 1). As result, the geometry and kinematics of this system are much more complicated than in the case of a simple flat double pendulum. The Fig. 2 shows the deflected position of the considered double pendulum. We consider the articulated rotation angles θ1 i θ2 as the generalized coordinates. For the convenience of further actions, we introduce a rectangular Cartesian coordinate system xyz with the fixed orthogonal basis ijk connected, so that in the state of main equilibrium the first link is directed along the unit vector i. In addition, we introduce a movable orthogonal basis i j  k  connected with the first link, where unit vector i is directed along the axis of this link from the input (first) joint to the output (second) joint rotated by an angle α relatively to the input joint.

Fig. 1. Equilibrium position

Fig. 2. Deflected position

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3 Deriving of Dynamic Equations of the System We proceed to the calculation of kinetic and potential energies of the system. It is clear that energies of the first load are easily determined, since its speed is v1 = l θ˙ 1 and the vertical coordinate is given by expression x1 = l cos θ1 . Main difficulty to derive the equations of this system is to find the speed and vertical coordinate of the second end load which makes a complex spatial motion determined by the angles θ1 , θ2 and α. The simplest way to find the velocity of this load is a coordinate method in which it is necessary to determine the Cartesian coordinates of this load in a fixed basis and then find the projections of velocity on the corresponding coordinate axes. With this purpose, we obtain the expression for radius vector r 2 of the second load. For this we first find the vectors l 1 i l 2 which have the following length and direction of links in the basis i j  k  with simple representations:   l 1 = li , l 2 = l cos θ2 i + cos α sin θ2 j − sin α sin θ2 k  . Summarizing these vectors, we obtain the desired vector r 2 :   r 2 = l 1 + l 2 = l (1 + cos θ2 )i + cos α sin θ2 j  − sin α sin θ2 k  .

(1)

(2)

Considering the obvious relations between the orts of the bases i j  k  and ijk i = cos θ1 i + sin θ1 j, j  = − sin θ1 i + cos θ1 j, k  = k,

(3)

we get the expression for r 2 in motionless basis ijk: r 2 = l[((1 + cos θ2 ) cos θ1 − cos α sin θ2 sin θ1 ) i

 + ((1 + cos θ2 ) sin θ1 + cos α sin θ2 cos θ1 )j − sin α sin θ2 k .

(4)

From this expression, Cartesian coordinates of the second load in the xyz coordinate system can be found: ⎧ ⎨ x2 = l[(1 + cos θ2 ) cos θ1 − cos α sin θ2 sin θ1 ] (5) y = l[(1 + cos θ2 ) sin θ1 + cos α sin θ2 cos θ1 ] . ⎩ 2 z2 = −l sin α sin θ2 Differentiating these expressions with respect to time, one can find the projection of the velocity of second load on coordinate axes and determine velocity square:  2 v22 = x˙ 22 + y˙ 22 + z˙22 = l 2 (1 + cos2 α + 2 cos θ2 + sin α cos2 θ2 )θ˙ 21  +2 cos α(1 + cos θ2 )θ˙ 1 θ˙ 2 + θ˙ 22 . Therefore, kinetic energy of the system is determined by expression  2 1 1 T = m(v12 + v22 ) = ml 2 (2 + cos2 α + 2 cos θ2 + sin α cos2 θ2 )θ˙ 21 2 2

(6)

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 1 T ˙ +2 cos α(1 + cos θ2 )θ˙ 1 θ˙ 2 + θ˙ 22 = θ˙ A(θ)θ, 2

(7)

where θ = [θ1 , θ2 ]T is the column of generalized coordinates, and kinetic energy matrix A(θ) called the matrix of inertial coefficients is symmetric (AT = A), and it has the form

 2 + cos2 α + 2 cos θ2 + sin2 α cos2 θ2 cos α(1 + cos θ2 ) A(θ) = ml 2 . (8) 1 cos α(1 + cos θ2 ) Potential energy of this system is determined by expression (θ) = mg(3l − x1 − x2 ) = mgl[3 − (2 + cos θ2 ) cos θ1 + cos α sin θ2 sin θ1 ], (9) where an additive constant is added for convenience, so that  = 0 at equilibrium position θ1 = 0, θ2 = 0. Substituting expressions (7) and (9) into Lagrange equations of the second kind in matrix form ∂ d ∂T ∂T =− , − dt ∂ θ˙ ∂θ ∂θ

(10)

we obtain, after series of transformations, the equations of motion of spatial double pendulum in the well-known matrix form ˙ + C(θ) = 0, A(θ)θ¨ + B(θ, θ)

(11)

˙ and C(θ) are where the matrix A(θ) has the representation (8), and the columns B(θ, θ) determined by expressions



− 2(1 + sin2 α cos θ2 )θ˙ 1 + cos αθ˙ 2 θ˙ 2 2 ˙ B(θ, θ) =ml sin θ2 , (1 + sin2 α cos θ2 )θ˙ 21

 (2 + cos θ2 ) sin θ1 + cos α sin θ2 cos θ1 C(θ) =mgl . (12) sin θ2 cos θ1 + cos α cos θ2 sin θ1 The nonlinear mathematical model (11) together with (8) and (12) allows us to formulate and solve many problems on the dynamic behavior of a spatial double pendulum.

4 Analysis of System’s Small Oscillations In this article, we restrict ourselves to analyze the linear model of spatial double pendulum and to find out how angle α affects the frequencies and modes of its free oscillations. Linearizing Eq. (11) for this purpose, we arrive to the linear matrix equation A0 θ¨ + C0 θ = 0,

(13)

where the constant symmetric matrices of inertial A0 and quasi-elastic C0 coefficients have the forms



  5 2 cos α 3 cos α 2 A0 = ml , C0 = mgl . (14) 2 cos α 1 cos α 1

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We will seek a solution of Eq. (13) in the form of single-frequency harmonic oscillations θ =  cos(k0 t + δ0 ),

(15)

where  is the unknown column of amplitudes, k0 is the natural oscillation frequency, δ0 is the initial phase. Substituting solution (15) into Eq. (13), we obtain the matrix algebraic equation (C0 − k02 A0 ) = 0.

(16)

This equation has a nontrivial solution only when determinant turns to zero: det(C0 − k02 A0 ) = 0.

(17)

Revealing this determinant taking into account representations (14), it is easy to obtain a biquadratic equation for the desired frequency. This equation in the dimensionless form is (1 + 4 sin2 α)p04 − 4(1 + sin2 α)p02 + 2 + sin2 α = 0,

(18)

where the notation is introduced: p0 = k0 /k is the dimensionless natural frequency of the spatial double pendulum, and k = g/l is the frequency of small oscillations of the mathematical pendulum of length l. Solving Eq. (18) with respect to p0 , we find the dimensionless frequencies:   2(1 + sin2 α) ± 2 − sin2 α ps0 = , s = 1, 2. (19) 1 + 4 sin2 α Natural frequencies ks0 are obtained by multiplying the dimensionless frequencies by factor k. We emphasize that the lower sign corresponds to the first frequency (for s = 1), and the upper sign corresponds to the second frequency (for s = 2). Index “0” for frequencies shows that obtained frequencies correspond to the linear model. Now we proceed to the analysis of dependence (19). It is easy to see that for the flat double pendulum to which the value α = 0 corresponds, we obtain widely known results from [19, 21]:   √ √ (20) p10 = 2 − 2 ≈ 0.7654, p20 = 2 + 2 ≈ 1.8478. In another particular case, when α = π/2, the double pendulum can be called “orthogonal”, and its frequencies according to (19) will be  3 ≈ 0.7746, p20 = 1. p10 = (21) 5 In order to reveal qualitatively the influence of the angle between the joints on the natural frequencies, we obtain approximate formulas for these frequencies near the

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boundary values α = 0 and α = π/2. Expanding (19) in the Taylor series in α in the locality of point α = 0 and holding only two first terms, we find     √ √   √ √ 5 2−7 2 5 2+7 2 p10 = 2 − 2 1 + (22) α , p20 = 2 + 2 1 − α . 8 8 It can be seen that coefficient at α2 in the bracket for the frequency p10 is 0.0089, while for the frequency p20 it is (−1.7589). Therefore, as α increases from 0, the first frequency increases very slightly, while the second frequency begins to decrease significantly. Similarly, approximate formulas can also be obtained near α = π/2 by expanding expressions (19) in the power series of a small parameter ε = π/2 − α, keeping as before only two terms:    1 2 3 1 1 − ε , p20 = 1 + ε2 . (23) p10 = 5 60 4 It can be seen from this, that with decreasing α from π/2, the first frequency decreases only slightly, while the second frequency increases to a greater extent. Curves of the dependence of dimensionless frequencies ps0 on the angle α are presented in Fig. 3. It is seen clearly from them that the first frequency is almost independent of α, because its highest value (at α = π/2) differs from the lowest one (at α = 0) by only 1.2%. The second frequency depends on α much more significantly.

Fig. 3. Oscillation frequencies

Fig. 4. Oscillation modes

We proceed to finding the oscillation modes corresponding to the found oscillation frequencies ps0 . They are determined by the ratio between the components of the column

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 = [Θ1 , 2 ]T , i.e. the ratio between the oscillation amplitudes 1 and 2 of the rotation angles θ1 i θ2 in the pendulum joints. We characterize the oscillation mode by the relation βs0 = 2 /1 . This value can be found from any of the two scalar equations corresponding to the matrix Eq. (16), for example, the second, because these equations become linearly dependent after substituting frequencies in them due to the condition (17). As a result, after mentioned transformations we obtain  3 ± 2 2 − sin2 α  cos α. (24) βs0 = − cos 2α ± 2 − sin2 α Let us study these dependencies. In the case of flat double pendulum, when α = 0, we get √ √ β10 = 2 − 1 ≈ 0.4142, β20 = − 2 − 1 ≈ −2.4142. (25) In the case of orthogonal double pendulum at α = π/2, we find β10 = 0, β20 = −∞,

(26)

and here we have indeterminacy 0/0 in the limit at α → π/2 which is easily revealed. Formulas (26) mean that oscillations in both degrees of freedom in this case are independent. This fact also is followed from expressions (14) for the matrices A0 and C0 , when each of them becomes diagonal at α = π/2. Finally, we also obtain approximating formulas for the ratio of oscillation amplitudes near the boundary values α = 0 and α = π/2. Expanding (24) in the Taylor series in α in the locality of point α = 0 and holding only two first terms, we find   √  √  2 2 α . (27) βs0 = −1 ∓ 2 1 ± 4 To find approximating formulas near the point α = π/2 we set again ε = π/2 − α, and after that we obtain ε 2 (28) β10 = , β20 = − . 2 ε Curves of the dependence of values βs0 on the angle α are presented in Fig. 4, and they clearly demonstrate the change in oscillation modes with increasing angle between the joint axes.

5 Conclusion Summarizing the research results, we can conclude that the spatial double pendulum has richer properties than conventional two-link flat one. The obtained formulas represent only the first stage of the study of this structure and therefore they are only primary results. It is possible further to solve a number of more complex practical problems based on these results. We note the following main promising areas of subsequent research: the study of nonlinear oscillations of this system, the analysis of its controlled movements, the construction of rational modes of its functioning and the selection of their optimal parameters.

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24. Reshmin, S.A.: Decomposition method in the problem of controlling an inverted double pendulum with the use of one control moment. J. Comput. Syst. Sci. Int. 44(6), 861–877 (2005) 25. Lupina, T.A.: Evaluation of the sustainability of the vertical equilibrium of the inverted double pendulum with visco-elastic elements. Vodniy Transp. 3(15), 67–73 (2012) 26. Bulanchuk, P.O.: Controlling the motion of a double pendulum by vibration the suspension point. Vestnik of Lobachevsky University of Nizhni Novgorod 4(5), 2041–2042 (2011) 27. Bulanchuk, P.O., Petrov, A.G.: Suspension point vibration parameters for a given equilibrium of a double mathematical pendulum. Mech. Solids 48(4), 380–387 (2013) 28. Kholostova, O.V.: On the motions of a double pendulum with vibrating suspension point. Mech. Solids 44(2), 184–197 (2009) 29. Kholostova, O.V.: Problems of solid body dynamics with vibrating suspension, p. 308. IKI, Izhevsk (2016) 30. Formalskii, A.M.: Motion control of unstable objects. Fizmatlit, Moscow (2014). 232 p 31. Leontev, V.A., Smirnov, A.S., Smolnikov, B.A.: Optimal damping of two-link manipulator oscillations. Robot. Tech. Cybern. 2(19), 52–59 (2018) 32. Lavrovskii, E.K., Formalskii, A.M.: The optimal control synthesis of the swinging and damping of a double pendulum. J. Appl. Math. Mech. 65(2), 219–227 (2001) 33. Smirnov, A.S., Smolnikov, B.A.: Static analysis of the two-links manipulator with noncollinear joints. Robot. Tech. Cybern. 4(17), 47–51 (2017) 34. Leontev, V.A., Smirnov, A.S., Smolnikov, B.A.: Dynamic analysis of the two-links manipulator with noncollinear joints. Robot. Tech. Cybern. 1(18), 56–60 (2018)

The Method for Determining the Risks of the Transport of Dangerous Goods Dmitry R. Stakhin(B) and Dmitry G. Plotnikov Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia [email protected]

Abstract. The paper deals with the organization of transportation of dangerous goods. Difficulties in ensuring the safety conditions for the transport of dangerous goods were noted, including due to the lack of appropriate controls and methods for determining the dangerous zones of the transport route. The article presents the existing methods for determining risks in the transport of dangerous goods. In this paper, a new method has been developed and proposed for determining the risks of transporting not only dangerous goods, but also other types of cargo transportation. Keywords: Analysis · Transportation of dangerous goods · Risks during transportation · Probability and damage of the event

1 Introduction According to the UN, the share of dangerous goods in the global cargo turnover is constantly growing and currently reaches almost half of the cargo turnover [1]. The transport of dangerous goods has a significant potential for emergency risks. There is an urgent need for measures to reduce this potential to the level of residual risks acceptable to society and the state [2, 3]. In addition, freight transportation has a serious impact on the environment, in this connection it is necessary to correctly choose a route and evaluate possible alternative modes of transport [4, 5]. Existing methods for identifying risks: 1. Methods of a qualitative risk assessment: a) b) c) d)

Check List Method; PSA (Process safety analysis); FMEA (Failure mode and effects analysis); HAZOP (Hazard and operability studies).

2. Methods of quantitative risk assessment: [6, 7] a) FTA (Fault tree analysis); © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 194–203, 2021. https://doi.org/10.1007/978-3-030-62062-2_19

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b) ETA (Event tree analysis); c) CCA (Cause and consequence analysis); d) HEA (Human error analysis). The results of the risk analysis are used to make informed decisions regarding the choice of safety of the transportation technology [8, 9]. The complexity of the risk analysis of the transport system is mainly due to the fact that at any point on the route and during loading/unloading an emergency and the same event can occur can lead to completely different consequences. Moreover, risk analysis can be defined as the process of solving many problems: identifying hazards in the system, analyzing the probabilities of the distribution of hazards in space, frequency analysis, dividing by reason, and analyzing the consequences [10–12]. A common problem with existing methods is that when creating risk models, the possible causal relationship of events is not taken into account [13, 14]. Risk analysis and risk assessment are closely linked. Risk assessment is traditionally understood as the process of determining the risk value of the analyzed dangers for a person, material values and the environment. When assessing the risk of the technical process of transporting dangerous goods, the emphasis should be on the analysis of risks to the life and health of people, taking into account those involved in the technical process and those who accidentally fall into the danger zone [15, 16]. As a result of the analysis, data should be obtained on the frequency of risks and their consequences. A significant drawback of existing risk assessment methods is that the probability of an event occurring is considered separately from the possible consequences. In this case, it is often impossible to obtain objective information about the degree of danger of a cargo, “separated” from a route and technology [17]. Some risks arising from the loading/unloading and transportation of dangerous goods: – – – – – – – – – – – – – –

Inconsistency of packaging and labeling of goods; failure of fastening mechanisms; errors in the preparatory work; breakage of fastening; failure of loading and unloading equipment; shortcomings in the design of the car and equipment; failures due to defects and wear; violation of traffic rules and railway rules; dangerous external influences on the cargo; unsatisfactory road conditions; poor quality maintenance and TR; erroneous operator exposure; poor organization of labor; unsatisfactory weather conditions. The main goal is to determine the total risk during the transport of dangerous goods.

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2 Methods and Materials Due to the fact that it is impossible to take into account a causal relationship with existing methods of risk analysis, a technique has been developed that is devoid of this drawback. The technique is being implemented in stages. 1.

Determination of the most significant negative events associated with the transport of dangerous goods. Negative events - lead to negative consequences 2. The negative consequences that arise as a result of the transport of dangerous goods are determined. Negative consequences - any consequences that lead to an increase in the cost of transportation of goods. 3. The probability with which negative consequences arise is determined. It is revealed on the basis of expert assessment. Experts are invited to determine the probability of risk and the magnitude of the consequences in a separate group from 1 to 5. The average value for each group is used. 4. The amount of damage that occurs as a result of the negative impact on the transport of dangerous goods is indicated. The amount of damage is given as a percentage of the amount of damage in monetary terms. It is revealed on the basis of expert assessment. Experts are invited to determine the probability of risk and the magnitude of the consequences in a separate group from 1 to 5. The average value for each group is used. 5. The risk that arises as a result of a negative event is calculated. It is defined as the product of the probability of an event P and the amount of damage D when this event occurs. R=P·D 6. This dependence is presented on the graph. The abscissa axis is the damage from the event, the ordinate axis is the probability of this event. 7. These risks are divided into managed and non-managed. Based on this, the total risk of all events is calculated. The calculation is made using graph theory methods. For this, it is necessary to construct a graph of connectivity. 8. A graph of criteria connectivity is constructed. A connected graph is a graph containing exactly one connected component. This means that there is at least one path between any pair of vertices of this graph [18]. For further calculation, two restrictions must be introduced: a) The graph of connectivity should be oriented. In order to follow the sequence of events. A directed graph is a graph whose edges are assigned a direction [18]. Directional edges are also referred to as arcs, and in some sources simply as edges. b) The graph must not contain cycles. In order to carry out the calculation correctly (Fig. 1). 9. The total risk is calculated. The general formula for determining the total risk, consisting of two dependent events: R1,2 = R1 + R2 − P1 · P2 · (D1 + D2 ), where R1 and R2 are the risks of individual events;

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Fig. 1. An example of a directed graph of connectivity.

P1 and P2 are the probabilities of these events; D1 and D2 - damage from these events. If event 2 is a consequence of event 1, therefore, these two events are joint. It follows that they occurred simultaneously. As a result, the total risk will be considered as the sum of the risks of two events. However, due to the fact that the events occurred simultaneously, they have a common (joint) part, which must be subtracted from this amount. If there are several groups of negative events, then the final risk is determined by the sum of the risks of each group: R = ΣRi 10. Each event is assigned a probability group (Table 1). This group divides all the probabilities into separate blocks. Table 1. Probability groups. №

Probability of the event

Interval, %

Group 0 Event Excluded

0

Group 1 Extremely low probability

0–10

Group 2 Low probability

10–20

Group 3 Average probability

20–30

Group 4 High probability

30–45

Group 5 Extremely high probability 45 and more

11. Corrective measures that need to be taken to prevent negative events and reduce the risk of these consequences are determined. The purpose of corrective measures is to transfer the event into a group with a lower probability. The cost of these events is indicated. 12. The effect of the corrective measure is determined as the difference between the cost of carrying out these measures and the magnitude of the risk associated with a negative event.

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It should be noted that not every preventive action is beneficial in terms of application. A preventive action is not necessary if this leads to higher costs compared with the risk of a negative event. 13. The events to which the adjustment is applicable are transferred to a lower probability group of events and they are assigned the average probability of an event from this group: – From groups 4 and 5 to group 1; – From groups 3, 2, 1 to group 0. 14. The total risk with the changed probabilities is recounted. 15. The result of the calculation is analyzed.

3 Results Consider the application of this technique in the transport of dangerous goods. The results of paragraphs 1, 2, 3, 4 and 5 are summarized in Table 2. Table 2. Negative events and their data. №

Name of a negative event

Consequences

Probability, %

Damage amount

1

Deficiencies in the design of the car and equipment

:Incorrect delivery conditions : Damage to the cargo

10

20

2

2

Failures due to wear defects, etc.

: Incorrect delivery conditions :Damage to cargo : Incorrect delivery time

15

60

9

3

Unsatisfactory road conditions

: Damage to the vehicle : Damage to the cargo : Violation of the delivery conditions

40

40

16

Risk

A graphical representation of the probability of a negative event (paragraph 6) is shown in Fig. 2. Under item 8, we construct an oriented graph of connected events. We unite events in a dependent group.

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Fig. 2. Dependence of the probability of an event on damage.

Event Group: 1-2-3. (Figure 3), the data are summarized in Table 3.

Fig. 3. Criteria connectivity graph. Table 3. Column data 1–2–3. № P

D R

1

0.1

20

2

2

0.15 60

9

3

0.4

40 16

The total risk is calculated according to paragraph 9: R = R1 + R2 + R3 −P1 :P2 :(D1 + D2 ) − P1 :P3 :(D1 + D3 ) − P2 :P3 :(D2 + D3 ) − P1 :P2 :P3 :(D1 + D2 + D3 ) = 2 + 9 + 16 − 0.1:0.15:(20 + 60) − 0.1:0.4:(20 + 40):0.15:0.4:(60 + 40) :0.1:0.15:0.4:(20 + 60 + 40) = 16.68. We introduce corrective measures and analyze their need (Table 4). Points 10, 11, 12 and 13. The adjusted values are considered in Fig. 4 and Table 5. We determine the negative events in the group after adjustment. We calculate the total risk with the corrected data. According to paragraph 14. R = R1 + R3-P1 · P3 · (D1 + D3) = 2 + 2 − 0.1 · 0.05 · (20 + 40) = 3.7.

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4 Discussion Currently, all the methods used to assess risk are poorly used in the transport and logistics sectors. In this regard, a methodology was developed and proposed to determine the risks in the transportation of not only dangerous goods, but also other types of goods. The

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Fig. 4. Dependence adjusted graph connectivity. Table 5. Corrected column data 1-2-3 № P

D R

1

0.1

20 2

3

0.05 40 2

main aspect that distinguishes this technique is the consideration of a causal relationship between negative events [18]. This accounting is necessary due to the fact that events can occur simultaneously and depend on certain conditions [19]. This suggests that it is incorrect to consider such events separately, they should be dependent on each other, and this dependence should be calculated. The determining circumstance of risk reduction is the use of corrective measures that will be aimed at minimizing the likelihood of negative events during the transportation of dangerous goods [20–23].

5 Conclusion The article illustrates the methodology taking into account the causal relationship. Currently, such methods are gaining popularity [24]. As a result of the methodology: 1.

The probabilities of the event and the damage from this event that occur during loading/unloading and transportation of dangerous goods are listed; 2. The risk arising from the loading/unloading and transportation of dangerous goods has been calculated; 3. Constructed graphs of connected events; 4. Defined probability groups; 5. Demonstrated dependencies of the probability of the event; 6. The total risk of the event is calculated. 7. Preventive measures have been introduced to reduce the likelihood of a negative event; 8. The adjusted probabilities of negative events are determined; 9. Recalculated the total risk of the event; 10. The difference in risks was identified before and after the introduction of corrective measures. Thanks to the implementation of this methodology, the organization will be able to more easily and more accurately calculate the risks arising from the transport of

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dangerous goods, as well as determine the need for the implementation of corrective measures. From these calculations, we can conclude that the magnitude of the risk is reduced. This decrease in the risk of the probability of a negative event is associated with the fact that when calculating the risk of each group, the total (related) related part of the negative event is subtracted, which makes this calculation more correct compared to the usual summation of the risks of negative events.

References 1. Akimov, V.A., Sokolov, Y. I.: Risks of transportation of dangerous goods (2011) 2. Martynyuk, I. V.: Selection of optimal routes for transportation of dangerous goods based on the results of risk assessment of traffic safety violations and damages from them. Bull. Rostov State Univ. Railway Transp. 3, pp. 103–106 (2006) 3. Kondratov, S.V., Novikov, A.N., Tryashtsin, A.P.: Analysis and risk assessment in the transport of dangerous goods. World Transp. Technol. Mach. 1, 87 (2016) 4. Stakhin, D., Goncharov, K.: Features of the choice of power unit for electric tricycle. In: E3S Web of Conferences. – EDP Sciences, vol. 140, p. 02010 (2010). (rus.) 5. Minnikhanov, R.N.: Transportation of dangerous goods on public roads. New Technol. Mater. Equipment Russ. Aerosp. Ind. 368–372 (2018) 6. Baranov, Y. N., Tryashtsin, A.P.: Analysis and risk assessment in the transport of dangerous goods by road in the agro-industrial complex. Bull. Agrarian Sci. 26(5) (2010) 7. Caliendo, C., De Guglielmo, M.L.: Quantitative risk analysis on the transport of dangerous goods through a bi-directional road tunnel. Risk Anal. 37(1), 116–129 (2009) 8. Sokolov, Y.I.: Questions of safety of transportation of dangerous goods. Probl.Risk Anal. 6(1), 38–74 (2009) 9. Kazantzi, V., Kazantzis, N., Gerogiannis, V.C.: Risk informed optimization of a hazardous material multi-periodic transportation model. J. Loss Prev. Process Ind. 24(6), 767–773 (2012) 10. Li, S., Meng, Q., Qu, X.: An overview of maritime waterway quantitative risk assessment models. Risk Anal. Int. J. 32(3), 496–512 (2012) 11. Łukasik, Z., Ku´smi´nska-Fijałkowska, A., Kozyra, J.: Transport of dangerous goods by road ´ aska (2017) from a European aspect. Zeszyty Naukowe. Transport/Politechnika Sl˛ 12. Janno, J., Koppel, O.: Human factor as the main operational risk in dangerous goods transportation chain. Bus. Logistics Mod. Manag. 17, 63–78 (2017) 13. Van Raemdonck, K., Macharis, C., Mairesse, O.: Risk analysis system for the transport of hazardous materials. J. Saf. Res 45, 55–63 (2013) 14. Ebrahimi, H., Tadic, M.: Optimization of dangerous goods transport in urban zone . In: Decision Making: Applications, vol. 1, no. 2, pp. 131–152 (2018) 15. Isaeva, E.I., Kurlov, A.S., Shubin, A.A.: Accounting of risks during transportation of dangerous goods. In: Current Issues of Organization of Road Transport and Traffic Safety (rus.) 16. Harari, F.: Graph Theory. URSS, Moscow (2003). 300 p 17. Semenov, Y.A., Semenova, N.S.: Determination of dynamic errors in machines with elastic links. In: Evgrafov, A. (ed.) Advances in Mechanical Engineering. LNME, pp. 163–174. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39500-1_17 18. Chekanin, V.A., Chekanin, A.V.: New effective data structure for multidimensional optimization orthogonal packing problems. In: Evgrafov, A. (ed.) Advances in Mechanical Engineering. LNME, pp. 87–92. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-295 79-4_9

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19. Cassini, P.: Road transportation of dangerous goods: quantitative risk assessment and route comparison. J. Hazard. Mater. 61(1-3), 133–138 (1995) 20. Vamanu, B.I., Gheorghe, A.V., Katina, P.F.: Critical Infrastructures: Risk and Vulnerability Assessment in Transportation of Dangerous Goods. TSRRQ, vol. 31. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30931-6 21. Gheorghe, A.V. et al.: Advanced spatial modelling for risk analysis of transportation dangerous goods In: Probabilistic safety assessment and management, pp. 2499–2504. Springer, London (2004).https://doi.org/10.1007/978-0-85729-410-4_401 22. Zhi-guo, C.Z.C., Hong, T.: Risk assessment on road transportation system for dangerous goods. Ind. Saf. Environ. Prot. 6 (2007) 23. Eliseev, K.V.: Contact forces between wheels and railway determining in dynamic analysis. numerical simulation. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 61–70. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72929-9_8

The Method of Exponential Differential Inequality in the Estimation of Solutions of Nonlinear Systems in the Vicinity of the Zero of the State Space G. I. Melnikov1 , V. G. Melnikov1,2 , N. A. Dudarenko1 , and V. V. Talapov1(B) 1 ITMO University, Saint Petersburg, Russia

[email protected] 2 Saint-Petersburg Mining University, Saint Petersburg, Russia

Abstract. We consider a system of differential equations in the Cauchy form, containing homogeneous linear and cubic forms with respect to phase variables and small perturbing functions. We consider the system as a polynomial dynamical system under constant disturbances. We propose a method for constructing a nonlinear exponential differential inequality for a quadratic Lyapunov function and a square inequality. By integrating the exponential differential inequality, we obtain an estimate of the Lyapunov function from a function of time and initial conditions and an estimate of the phase variables. Keywords: Nonlinear systems · Homogeneous cubic and linear forms · Stability analysis · Lyapunov function · Differential comparison equations · Quadratic differential inequality · Exponential differential inequality · Functional estimates of solutions

1 Introduction This study is considering a nonlinear dynamical system which represents by a system of differential Cauchy equations. It is contains linear homogeneous forms regarding to phase variables, homogeneous cubic forms, and functions of constantly acting perturbations. To study linear systems N.N. Krasovsky [1] and other authors developed a method of linear differential inequalities with definitely positive Lyapunov functions [2–9] for the nonlinear systems [10, 11]. Analytical or numerical integration of linear and nonlinear differential inequalities constructed for definite positive functions leads to estimates of the phase coordinates with functions of time and initial conditions. The estimates are sufficient, and they can be refined by constructing and integrating more precise differential inequalities. Study [8] was proposed a theorem on the stability of motion, based on various nonlinear differential inequalities for Lyapunov functions. To date, the method of differential inequalities in the theory of stability has received significant development in the next work [2–4], etc. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 204–211, 2021. https://doi.org/10.1007/978-3-030-62062-2_20

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In this paper, we propose estimates for the practical stability of transient processes on a basis of quadrature-integrable differential inequalities containing Lyapunov functions and exponential functions. The inequalities are analytically integrated and lead to quantitative and qualitative assessments of transients. These estimates characterize the stability of dynamical systems with nonlinear polynomial characteristics This method of quantitative and qualitative transient processes assessments can applying to generalized equations of perturbed motion containing nonlinear functions in the form of extent series.

2 Formulation of the Problem Consider a polynomial system of differential Cauchy equations satisfying the conditions for existence and continuity of solutions and having the form x˙ = Ax + X (x) + F(t, x)

(1)

Here x – a column-vector of phase variables, X(x) – column-vector of the homogeneous polynomials of third and fifth degrees with column coefficients, F(t, x) – columnvector of small permanent perturbations, – matrix of constant coefficients of the system linear part x = [x1 , . . . , xn ]T , A = [aij ]n1 , F(t, x) = [F1 (t, x), . . . , Fn (t, x)]T , X = [X1 (x), . . . , Xn (x)]T Xs =

5  v1 +v2 +...+vn =3

pvs 1 ...vn x1v1 . . . xnvn =

5 

pv xv , pv = [pv1 , . . . , pvn ]T

(2)

(3)

v=3

Dynamical system (1)–(3) is specified in a small vicinity zero phase space: ¯ = {x : |xs | ≤ hs , s = 1, . . . , n], x0 ∈ D ¯0 ⊂ D D

(4)

We assume that all own eigenvalues λ1 , . . . , λn of matrix A have negative real parts, less than some negative constant α: λs = αs + iβs , αs−1 ≤ αs < α < 0, s = 1, . . . , n

(5)

We have a spectrum of the eigenvalues of matrix A S = {λs = αs + iβs , αs ≤ α < 0, s = 1, . . . , n}

(6)

in which connection λ1 = α1 + iβ1 denotes the root with largest real part. To system (1) we associate the unperturbed system with the variables ξ , and also a linear system with variable x˜ ξ˙ = Aξ + X (ξ )

(7)

x˙˜ = A˜x

(8)

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3 Methods and Materials 3.1 Formation of Quadratic and Exponential Differential Inequalities Relatively to the Lyapunov Function Let the Lyapunov function be chosen for system (1) V (x) in the form of a definite positive homogeneous quadratic form with a constant square matrix L V =

1 T x Lx, L = [lij ]n1 , 2

(9)

In accordance with the Lyapunov method, it is chosen on the basis of definitely negative quadratic form W (2) (x) = xT Bx, B < 0.

(10)

Introduce the domain G(h) in a vicinity of zero given by an inequality of the form V ≤ h2 ¯ ¯ G(h) = {V (x1 . . . xn ), h : V (x1 . . . xn ) ≤ h2 } ⊂ D,

(11)

where h – assignable positive constant. ¯ ¯ and in the domain of G(h) in the domain of D, function V satisfies the differential equation V˙ =

n  ∂V j=1

∂xj

x˙ j =

∂V ∂V ∂V ∂V ∂V x˙ = T (Ax + X (x) + F(t)), T = [ ,..., ] ∂xT ∂x ∂x ∂x1 ∂xn

(12)

Equation (12) contains three components: a homogeneous quadratic polynomial in phase variables qs , a homogeneous fourth degree polynomial and a linear polynomial with variable coefficients: V˙ = W (2) (x) + W (4) (x) + W (1) (x, t)

(13)

At 

W (2) (x) =

v1 +...+vn =2

W (4) (x) =

 v=4

pv(2) xv1 . . . xnvn = 1 ,...,vn 1

pv(4) xv =



pv(2) xv ;

(14)

v=2

∂V ∂V X (x), W (1) (x, t) = T F(x) T ∂x ∂x

Definitely negative quadratic form W (2) (x) can be majorized with a definite negative form like (−aV ) at a > 0 according to the A.D. Gorbunov [7, 10] method. Linear small ¯ function W (1) with small perturbations are estimated in the domain D(4)  with small function of time f (t), or a small constant using a condition of the form xj  ≤ cj , j = 1, . . . , n. The homogeneous fourth degree form W (4) (x) is estimated by the fourth degree form (bV 2 ) with a positive constant b.

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Substituting these estimates into differential Eq. (13), we obtain a nonlinear quadratic differential inequality for a definite positive function V V˙ ≤ −aV + bV 2 + f (t), at V ≤ h, a > 0, b > 0, f (t) ≥ 0,

(15)

with a differential comparison equation of the Ricatti type u˙ = −au + bu2 + f (t), at u ≥ 0, a > 0, b > 0, f (t) ≥ 0.

(16)

Inequality (15) and Eq. (16) cannot be integrated by quadratures in general form. If one particular solution is known u = y1 , then they integrate after changing variables V = V1 + y1 , u = y + y1 . We associate the quadratic inequality (15) with an exponential differential inequality integrable by quadratures. Quadratic function with two constants a, b in inequality (15) approximated by an exponential function with two constants α, k, while the error approximations ε(t) we include in the small perturbation function f (t). We get −2aV + bV 2 + f (t) = α exp(−kV ) + f1 (t) at f1 (t) = f (t) + ε(t)

(17)

Constants α, k, defined by expansion of the exponent and comparison with the left side of equality (17) have the form k = b/2a, α = 4a2 /b

(18)

Substituting expression (17) into the right side of inequality (15), we obtain the nonlinear exponential differential inequality V˙ ≤ αe−kV + f1 (t) at f1 (t) = f (t) + ε(t)

(19)

From the estimate of a definite positive function V follow the estimates of absolute values of generalized coordinates for the set of solutions with the initial conditions V (x0 ) = V0 .

4 Results and Discussion 4.1 Example Consider the system of differential Cauchy equations containing a homogeneous cubic form and the function with a small permanent perturbation. |q1 | ≤ h, |q2 | ≤ h, x˙ 1 = −2x1 + 0, 1x1 (x12 + x22 ) x˙ 2 = −x2 + 0, 2x2 (x12 + x22 ) + f (t) |f (t)| ≤ ε

(20)

Where ε - positive constant defining the domain in which the system’s motion is considered, x0 = [0.2; 0]. For function from system (20), we obtain the differential inequality: V˙ ≤ (−2V + 0, 4V 2 ) + f (t) at V < h2 , |f (t)| < ε

(21)

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We approximate the quadratic function by an exponential function with a small error, in which we include the function f (t), we get 1 −2V + 0, 4V 2 ≈ a(e−kV − 1) ≈ a(−kV + k 2 V 2 ) 2 From here ak = 2, ak 2 /2 = 0.4, or finally k = 0.8, a = 2.5. So, with a small error, we have −2V + 0.4V 2 ≈ 2.5(e0.8V − 1)

(22)

Differential inequality (21) takes the form V˙ ≤ 2.5(e−0.8V − 1) + f1 at |f1 | ≤ ε

(23)

where the error in equality (22) is included in the function f1 . Multiplying term by exponential inequality (23) on the 0.8e0.8V , we get 0.8V˙ e0.8V − 0.8(f1 − 2.5)e0.8V ≤ 2.5

(24)

U˙ + 0.8(2.5 − f1 )U ≤ 2.5 at e0.8V

(25)

d (λ(t, U )) ≤ 2.5λ(t) dt

(26)

Or

finally get

Where t U = exp(0.8V ), λ(t) = exp

0.8(2.5 − f1 )dt

(27)

0

Let us perform term-by-term integration of inequality (26) over the time interval [0, t], we get t λ(t)U − U0 ≤ 2.5

λ(t)dt at U = exp(0.8V0 ) 0

We obtain an upper bound for the positive function 1 (U0 + 2.5 U ≤ λ(t)

t

t at U = exp(0.8V ), λ(t) = exp (2 − 0.8f1 (t))dt 0

λ(t)dt) 0

(28)

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Let us logarithm inequality (28) term by term, we obtain t (0.8V ≤ ln[exp(0.8V0 + 2.5

λ(t)dt] − ln(λ(t)) 0

or finally t V ≤ 1.25 ln(exp(0.8V0 + 2.5

λ(t)dt) − 1.25 ln(λ(t))

(29)

0

Simulation results for a given system (20) of the Lyapunov function V, her assessment V, as well as the graph of the difference of these functions (errors) are presented for cases of absence of disturbance (Fig. 1) and small permanent indignation (Fig. 2).

Fig. 1. Results of modeling the Lyapunov function (solid line) and its estimates (dash-dotted line), error graph (dashed line)

Fig. 2. Results of modeling the Lyapunov function (solid line) and its estimates (dash-dotted line), error graph (dotted line)

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These estimates are correct from the moment until that time, when the right side of inequality (29) reaches the boundary of the domain, or is true on the interval, if the right side of inequality (29) does not reach this boundary.

5 Conclusion A nonlinear dynamic system is considered under the conditions of constantly acting small force disturbances. A method for constructing and integrating a nonlinear exponential differential inequality for a definitely positive Lyapunov function, integrable by quadratures, replacing a quadratic differential inequality, which is not analytically integrable in general form, is proposed. As a result of integrating the differential inequality, functional upper estimates of the transient processes of the dynamic system are determined. These inequalities complement and expand the capabilities of the method of nonlinear differential inequalities [8, 12, 13]. Along with modern methods of structural synthesis and kinematic analysis of mechanical systems [14–16] the proposed method can be effectively used for complex analysis nonlinear mechanical systems. The given example shows the possibilities of using the method of exponential differential inequalities in studying the practical stability of systems with one degree of freedom motion, as well as systems with several degrees of freedom [17, 18]. Acknowledgments. The reported study was funded by the Russian Foundation for Basic Research (project 19-38-90290), and by the Government of Russian Federation (Grant 08-08).

References 1. Krasovskii, N.: Problems of control and stabilization in dynamical systems. J. Math. Sci. 100, 2458–2469 (2000) 2. Vassilyev, S., Yadykin, I., Iskakov, A., Kataev, D., Grobovoy, A., Kiryanova, N.: Participation factors and sub-Gramians in the selective modal analysis of electric power systems. IFACPapersOnLine 50, 14806–14811 (2017) 3. Vassilyev, S., Kosov, A.: Common and multiple Lyapunov functions in stability analysis of nonlinear switched systems. In: AIP Conference Proceedings, vol. 1493, pp. 1066–1073 (2012) 4. Martynyuk, A.A., Martynyuk-Chernienko, Y.A.: Analysis of the set of trajectories of nonlinear dynamics: stability and boundedness of motions. Differ. Equ. 49, 20–31 (2013) 5. Melnikov, V.G.: Chebyshev economization in Poincare-Dulac transformations of nonlinear systems. Nonlinear Anal. Theory Methods Appl. 63(5–7), e1351–e1355 (2005). https://doi. org/10.1016/j.na.2005.01.080 6. Rabysh, Y., Grigoriev, V.V., Bystrov, S.V., Sporyagin, A.: Application of qualitative exponential instability conditions for dynamic processes estimation. Sci. Tech. J. Inf. Technol. Mech. Opt. 12(1), 36–40 (2012). (in Russian) 7. Gorbunov, A.D.: Some questions of the qualitative theory of ordinary linear homogeneous differential equations with variable coefficients. Sci. Notes Moscow State Univ. Math. 165(4), 39–78 (1954). (in Russian) 8. Melnikov, G.I.: Some questions of the direct Lyapunov method. Rep. Acad. Sci. 110(3), 326–329 (1956). (in Russian)

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9. Melnikov, V.G.: Chebyshev economization in transformations of nonlinear systems with polynomial structure. In: International Conference on Systems – Proceedings, vol. 1, pp. 301–303 (2010) 10. Melnikov, G.I.: Dynamics of nonlinear mechanical and electromechanical systems. Leningrad, Machine Engineering, 200 p. (1975). (in Russian) 11. Bellman, R.: On the Poincare-Lyapunov theorem. Nonlinear Anal. 4, 297–300 (1980) 12. Krasovskii, N.N.: Control of a dynamic system. In: Ural Scientific Center of the Academy of Sciences USSR (Sverdlovsk), 199 p. (1985). (in Russian) 13. Ivanov, S.E., Melnikov, V.G.: On the equation of fourth order with quadratic nonlinearity. Int. J. Math. Anal. 9(53–56), 2659–2666 (2015). https://doi.org/10.12988/ijma.2015.510249 14. Zlatanov, V.: Stationary oscillation in two-mass machine aggregate with universal-joint drive. In: Evgrafov, A. (ed.) Advances in Mechanical Engineering. LNME, pp. 105–113. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29579-4_11 15. Vulfson, I.I.: Some peculiarities of electric drive impact on the dynamics of cyclic machines. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 163–176. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72929-9_17 16. Aleksandrov, A., Tikhonov, A.: Electrodynamic stabilization of earth-orbiting satellites in equatorial orbits. Cosm. Res. 50(4), 313–318 (2012) 17. Dyashkin-Titov, V.V., Zhoga, V.V., Nesmiyanov, I.A., Vorob’eva, N.S.: Dynamics of the manipulator parallel-serial structure. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 33–43. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-729 29-9_5 18. Melnikov, G.I., Dudarenko, N.A., Malykh, K.S., Ivanova, L.N., Melnikov, V.G.: Mathematical models of nonlinear oscillations of mechanical systems with several degrees of freedom. Nonlinear Dyn. Syst. Theory 17(4), 369–375 (2017)

The Application of the Tubular Sorbing Elements Based on the Composition Powders of Zeolites in Adsorption Cryogeno-Vacuum Pumps Aleksey V. Gotsiridze, Pavel A. Kuznetsov, Alexandra O. Prostorova, and Valeriy P. Tretyakov(B) Peter the Great St.Petersburg Polytechnic University, St. Petersburg, Russia [email protected]

Abstract. The article considers the issues of increasing the efficiency of cryogenic vacuum pumps to create a deep vacuum used in production processes during chemical and physical experiments. It is noted that the use of zeolites as a sorbent material, in comparison with activated carbons, provides a more efficient absorption of hydrogen and helium. It is proposed to increase the heat sink using compositions of zeolite and metal powders with high thermal conductivity, pressed onto the surface of the tubes. It is shown that the best results are achieved when using the composition of zeolite - copper powder. It is proposed to use the process of elastostatic pressing for the manufacture of coaxial-cylindrical samples. The original design of the pump with coaxial-cylindrical samples with the pressing of the sorbent composite layer is presented. The results of practical tests of the pump are presented. The prospects of using the developed adsorption cryogenicvacuum pumps with sorbent elements based on zeolite and copper powders are considered. This research work was supported by the Academic Excellence Project 5–100 proposed by Peter the Great St. Petersburg Polytechnic University. Keywords: Cryogenic vacuum technology · Adsorption · Sorbent elements · Zeolites · Composite materials · Powder materials · Elastostatic pressing

1 Introduction Cryogenic vacuum pumps are technological devices for obtaining the deep vacuum required for the implementation of various chemical and physical processes, including the production of new materials, chemical reactions under special conditions, vacuum spraying, and so on. Cryogenic vacuum pumps operate on the principle of adsorption of gas molecules by substances with high sorbing properties. Activated carbons, silica gels and zeolites are most often used as sorbing materials [1, 2]. To maintain the sorption properties at a high level, the sorbent material is cooled by removing heat from the adsorbent layer through the metal surface with which it is in contact with the refrigerant - various liquefied gases, for example, nitrogen. Silica gels and zeolites, unlike coals, do © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 212–218, 2021. https://doi.org/10.1007/978-3-030-62062-2_21

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not form fine dust and do not contaminate vacuum plants. In addition, zeolites absorb hydrogen and helium more efficiently than other adsorbent materials. Therefore, they are a promising group of adsorbents for obtaining high vacuum in cryogenic plants. The efficiency of the cryogenic vacuum pump is largely determined by the type of sorbing system [3]. The trend in the design of adsorption cryogenic-vacuum pumps are the use of thinner layers of adsorbent, an increase in the area of the pumping surface and an increase in the thermal conductivity of sorbent compositions. At the same time, in order to create optimal working conditions for the adsorbent, its compaction and pressing on the surface of the panel or tube are required. Can be used to increase the heat sink the composition of zeolite and metal powders with high thermal conductivity. The optimal content of zeolite and metal powder in the composition ensures the location of the adsorbent particles in the pores of the metal matrix, which reduces dust formation when using the material and increases heat dissipation. The sintering is necessary to increase the strength of thin pressed layers and to carry out non-contact radiation heating.

2 Methods and Materials The main properties of sorbent materials are their sorption characteristics. Figure 1 shows a summary graph that allows to compare the adsorption properties of different sorbent materials. The adsorption curves shown in this figure corresponding to the maximum level of adsorption properties of this material were measured at a temperature of liquid nitrogen after degassing of 673 K. The nitrogen was used as an adsorbate.

Fig. 1. Adsorption isotherms of composite sorbent materials

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Curve 1 in Fig. 1 illustrates the sorption properties of sorbent elements obtained from a mixture based on copper powder with 16% zeolite powder CaE-T. The adsorption capacity of sorbent elements obtained from a mixture based on aluminum powder of the ASD grade with 10% zeolite is slightly lower (curve 2). Pure metal sorbent elements obtained from the aluminum-magnesium alloy powder PAM-5 grade, which underwent preliminary chemical treatment, have similar properties (curve 3). In the low vacuum region, the adsorption capacity of these sorbing series has comparable values. However, upon transition to the region of lower pressures, the difference in the amount of gas absorbed by the unit mass of the sorbent material differs by almost an order of magnitude. The sorbing elements obtained from PAM-4 grade aluminum-magnesium alloy powder that have undergone chemical treatment after the molding operation have even lower adsorption capacity (curve 4). At a pressure of 10−1 … 10−2 mm Hg the adsorption capacity of this material is 5–10 times lower than the capacity of the PAM-5 powder. In the region of higher vacuum, the adsorption isotherms of these materials approach each other. More efficient, technological and promising in comparison with flat panels are sorbing elements of coaxial-cylindrical shape. The modern development of powder metallurgy technologies allows the use of various methods of pressing of the powder materials [4–7]. The methods of isostatic pressing are most effective for pressing powders onto cylindrical surfaces [8–10]. The tubular sorbent elements are much more technological in the assembly and operation of the pump, as well as more effective in heat transfer. Two types of tubular sorbent elements are possible - with the external location of the layer of sorbent material when the refrigerant flows inside the pipeline, and with the internal - when liquid gas washes the outer surface of the pipe, and the evacuated volume is connected with the internal cavity of the sorbing element. The manufacture of such sorbent elements can be carried out in two ways. The first method consists in molding hollow cylindrical products from powder sorbent material, then mechanically securing them or gluing them on an internal or external surface, a section of a pipeline or sleeve made of compact metal. Another method involves pressing onto a corresponding surface of a cooled metal substrate of a sorbent material in a powder state [11, 12]. The second method of manufacturing sorbent elements is more technologically advanced, because it allows to combine two operations - molding the sorbent mass into a porous body and its attachment to the substrate. With this method, the sorbent layer is fixed without the use of an adhesive, the presence of which imposes a limitation on the temperature regime of the sorbent material. It also provides reliable mechanical and thermal contact between the sorbent layer and the cooled substrate, since the contacting surface of the sorbent layer repeats the relief features of the surface of the substrate. This leads to an increase in the contact area between the adsorbent layer and the substrate and, as a result, contributes to a more intense heat removal. The contact area of the sorbent layer and the substrate can be increased by preliminary applying a relief on its surface. For applying a sorbent mass to a cylindrical substrate, two main schemes are most suitable: compression (for the manufacture of sorbent elements with an external location

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of the sorbent layer) and distribution (for the case when a layer of sorbent material is applied to the inner surface of the substrate). Modern technologies of isostatic pressing of powders make it possible to press powder layers from 2 to 5 mm thick both on the inner and outer surfaces of the bushings, depending on the specific design of the pump. In this work the one of the most effective isostatic pressing method was used in which polyurethane is applied as a mobile medium [13–15]. The scheme of elastostatic pressing on the outer surface of the sleeve (according to the crimping scheme) is shown in Fig. 2.

Fig. 2. Schematic diagram of elastostatic pressing of the sorbent layer on the outer surface of the sleeve

The powder material 4 is compacted and pressed onto the surface of the substrate 7 by axially compressing the elastic forming element 5 enclosed in the container 3 by the punch 1. The axial movement of the powder is limited by the upper 2 and lower 8 covers. The rod 6 provides centering of the elastic forming element. The sorbent tubular elements proposed in this work allow creating fundamentally new designs of cryosorption pumps [16]. The Fig. 3 presents one of the possible schemes of an adsorption cryogenic high-vacuum pump. A vessel 2 with refrigerant 3, concentrically located sleeves 4 and 5, on the surface of which adsorbent 6 is pressed, is placed inside the housing 1. For applying the sorbing mass to the cylindrical sleeves, compression (for the manufacture of sorbent elements with an external arrangement of the sorbent layer) and distribution (for the case when a layer of sorbent material applied to the inner surface of the substrate) are used in the manufacture of sleeve 4. The adsorbent was pressed on both sides. To reduce heat fluxes, copper screens 7 are used on the surface of the adsorbent. The housing is equipped with an outlet pipe 8 to which an auxiliary high-vacuum volume pump can be connected. In this case, the pump in question will be used as a cooled adsorption trap. The adsorbent is heated in the regeneration process using the heater 9.

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Fig. 3. Schematic diagram of the adsorption high-vacuum cryogenic pump with tubular sorbent elements.

3 Results To compare the technical characteristics of vacuum pumps, the time was determined (see Fig. 4), necessary to cool down to a temperature of liquid nitrogen the granules filled with SKT-3 granulated coal (curve I) and the composite sorbent material CaE-T PMS-2 (curve 2).

Fig. 4. Dependence of the temperature of a layer of granulated carbon SKT-3 (curve I) and composite sorbent material (curve 2) versus the cooling time.

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The studies have shown that at a pressure below 10−4 mm Hg the use of composite sorbent material provides 5-10 times higher sorption capacity in comparison with granulated coal and significantly reduce the volume occupied by the sorbent material and the dimensions of the cryosorption pump. As a result of practical tests, it was found that the use of coaxial-cylindrical sorption elements in vacuum pumps provides a pumping speed of up to 100 m3 /s. In this case, the time for the adsorption cryogenic-vacuum pump to reach the operating mode was 80–120 min. Operation of the pump with tubular composite sorbent elements showed: high mechanical strength of the sorbent layer and its reliable attachment to the metal substrate, absence of dust in the pumped-out volume and high technical characteristics (maximum residual pressure 5 × 10−6 mm Hg).

4 Discussion An adsorption high-vacuum cryogenic pump with tubular sorbing elements allows oilfree pumping of devices for spectral studies of elements in technical objects. An important advantage of composite sorbent elements is the absence of adsorption dust that occurs when using traditional sorbent materials. The adsorption dust affects, for example, the deterioration of the characteristics of the optical elements or the violation of the specified friction conditions on the mating surfaces of the test elements. In addition, this dust, saturated with the gas depressurization of the object, is subsequently a powerful source of gas evolution, leading to an increase in the maximum achievable pressure in the chamber. The results are of practical importance and serve as the basis for the continuation of new research.

5 Conclusion The studies showed that the use of composite sorbent elements instead of the traditionally used active carbons can significantly increase the productivity of technological vacuum equipment by reducing the pump operating time by about 10 times. Such pumps can be effectively used in large-volume cryogenic-vacuum installations, for example, in vacuum spectrophotometers, and provide oil-free pumping of devices for spectral studies, vacuum deposition plants, as well as for the production of ultrapure metals and alloys. This research work was supported by the Academic Excellence Project 5–100 proposed by Peter the Great St. Petersburg Polytechnic University.

References 1. Glazkov, A.A., Saksagansky, G.L.: Vacuum of Electrophysical Installations and Complexes, p. 184. Energoatomizdat, Moscow (1985) 2. Shumyatsky, Y.I: Industrial adsorption processes, p. 183. Moscow: Kolos (2009) 3. Hoffman, D., Singh, B., Thomas III, J. (eds) Handbook of Vacuum Engineering and Technology, p. 736. TECHNOSPHERE (2011)

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4. Tsemenko, V.N., Ganin, S.V., Phuc, D.V.: Research and simulation of the deformation process of dispersion-hardened powder in a capsule. Mater. Phys. Mech. 25(1), 68–76 (2016) 5. Rudskoy, A.I., Tsemenko, V.N., Ganin, S.V.: A study of compaction and deformation of a powder composite material of the ‘aluminum–rare earth elements’ system. Met. Sci. Heat Treat. 56, 542–547 (2015). https://doi.org/10.1007/s11041-015-9796-3 6. Allison, P., Grewal, H., Hammi, Y., Hammi, Y., Brown, H., Whittington, W., Horstemeyer, M.: Plasticity and fracture modeling. experimental study of a porous metal under various strain rates, temperatures, and stress states. J. Eng. Mater. Technol. 135, 041008 (2013). https://doi. org/10.1115/1.4025292 7. Oslanec, P., Skrobian, M.: Pecularities of gas analysis in Al and Mg powders. Powder Metall. Prog. 17(1), 65–71 (2017) 8. Timokhova, M.I.: Industrial technology for the automated production of grinding balls by quasi-isostatic pressing. Refract. Ind. Ceram 52(6), 389–392 (2012). https://doi.org/10.1007/ s11148-012-9438-x 9. Timokhova, M.I.: Advantages of quasi- isostatic pressing for powder materials. Refract. Ind. Ceram 53(3), 147–150 (2012). https://doi.org/10.1007/s11148-012-9483-5 10. Jonsén, P., Häggblad, H.A., Gustafsson, G.: Modelling the non-linear elastic behaviour and fracture of metal powder compacts. Powder Technol. 284(3), 496–503 (2015). https://doi.org/ 10.1016/j.powtec.2015.07.031 11. Timokhova, M.I.: The method of quasi-isostatic pressing of ceramic and refractory products. New Refract. 11, 18–22 (2018) 12. Timokhova, M.I.: Some features of the method of quasi-isostatic pressing of ceramic and refractory products. New Refract. 3, 17–20 (2019) 13. Kuznetsov, R.V., Kuznetsov, P.A., Karachevtsev I.D.: Elastostatic pressing of sintered working inserts with a gradient structure for bimetal plain bearings. In: Evgrafova, A.N., Popovich, A.A. (eds.) Modern Engineering: Science and Education: Materials of the 8th International Scientific and Practical Conference, pp. 663–773. Saint Petersburg Publishing House of Polytechnic University (2019) 14. Kuznetsov, R.A., Kuznetsov, P.A.: A new way of manufacturing bimetal products on the basis of the technology of casting with crystallization under pressure. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 119–127. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39500-1_13 15. Kuznetsov, P.A., Prostorova, A.O.: Isostatic pressing of coaxial cylindrical sorption elements for cryogenic vacuum pumps, no. 5, pp. 3–4. Kholodilnaya Tekhnika (2018) 16. Gotsiridze, A.V., Kuznetsov, P.A., Prostorova A.O.: Adsorption cryogenic vacuum pumps with coaxial cylindrical sorption elements, no. 4, pp. 3–4. Kholodilnaya Tekhnika (2018)

Research of Air Suspension of Shock Machine Mikhail N. Polishchuck, Arkadii N. Popov, Alexey K. Vasiliev(B) , and Dmitrii V. Reshetov Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia [email protected], [email protected]

Abstract. The mathematical and computer models of the shock testing machine air suspension are considered. Estimates are obtained for the influence of the main characteristics and parameters of the suspension elements: pressure, piston area and the stroke of pneumatic cylinders, the properties of shock absorbers and the inertia block mass on the parameters of reproducible shock pulses and the dynamic load onto the foundation. Keywords: Shock testing machine · Air suspension · Shock absorbers · Computer simulation

1 Introduction A significant part of products during their life cycle is repeatedly subjected to various kinds of mechanical influences. Such influences can disable the product or significantly worsen its performance [1, 2]. To confirm the reliability of products and their quality, diagnostic tests with mechanical testing equipment, such as centrifuges [3, 4], oscillation stands [5–8], shock machines [9–11], etc. are performed. Shock machines perform single and/or multiple strikes. Single shock machines are characterized by peak values, by a shape and duration of the reproduced acceleration pulses, while multiple shock machines, in addition, by the frequency of the shocks repetition. At the moment, shock machines with air suspension are widely used [12–14]. They can perform a number of functions: – allow to reduce the mass of inertia of the shock machines unit; – reduce the impact of shock loads onto the foundation; – protect shop equipment from high-frequency oscillations that occur when reproducing pulses with high shock accelerations. Air suspension is now widely used in vehicles: trucks and motor vehicles, passenger rail cars, as well as in technological equipment. It is a mandatory structural element of electrodynamic oscillation stands. However, the criteria for shock machine air suspension synthesis differ from those used in vehicles and in traditional technological equipment.

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2 Problem Statement When designing an air suspension, it is important to know how the characteristics of pneumatic cylinders (or pillows), shock absorbers, the inertia block mass affect the parameters of a shock machine, as well as the requirements for the machine foundation. These issues are not studied and not covered well enough in the technical and scientific literature. We can only note papers [15–20]. The purpose of this article is to obtain general estimates of shock machine air suspension characteristics and to show the direction of further research in this area. As an example, SMU500-3 multiple shock machine (Fig. 1) developed at The Peter the Great Saint Petersburg Polytechnic University is to be reviewed.

Fig. 1. Structural diagram of SMU500-3 multiple shock machine: 1 – tested product; 2 – accelerometer; 3 – shock table; 4 – shock pulse generator (programmer); 5 – inertia block (seismic mass); 6 – guides; 7 – drive pneumatic cylinder; 8 – suspension pneumatic cylinders; 9 – receivers; 10 – shock absorbers

Shock Table 3, fixed in guides 6 and raised to its upper position by drive pneumatic cylinder 7, then with tested product 1 installed onto it falls down to shock pulse generator (programmer) 4. Kinetic energy of the table is converted into energy of the generator deformation and kinetic energy of inertia block (IB) 5. IB kinetic energy, in its turn, transfers into potential energy of the air compressed in suspension pneumatic cylinders 8, and oil heating due to throttling in shock absorbers 10. Rigidity of suspension cylinders 8 in a wide range allows adjusting receivers 9 connected to them. The air suspension consists of two blocks: a block of shock absorbers and a block of pneumatic cylinders connected to the receivers. Shock absorbers absorb mechanical energy of a shock, turning it into heat.

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3 Design Model In order to select the characteristics of an air suspension, let us consider its simplified design model. In its initial state (Fig. 2), IB is in an equilibrium position, the total weight of the shock table with the tested product and IB is balanced by the force acting from the side of suspension pneumatic cylinders

Fig. 2. Design diagram of air suspension in its initial position (before shock): 1 – shock table; 2 – inertia block; 3 – suspension pneumatic cylinders; 4 – shock absorbers; 5 – workshop floor (foundation)

p0 S = (M + m)g,

(1)

where p0 is initial pressure in the piston cavity of the suspension pneumatic cylinders; S is an effective area of the pistons; M is IB mass; m is a mass of the table with the product; g is free fall acceleration. When shocked, IB moves down (Fig. 3) M s¨ = p(s)S − Mg + Fa (˙s) − ma,

(2)

where s is IB coordinate; p(s) is current pressure in the piston cavity;Fa (˙s) is shock absorbers resistance force; a is the table acceleration.

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Fig. 3. Design diagram of air suspension in shock mode

In this case, the processes speed is relatively high, so pressure changing is fairly well described by the adiabatic law: pV γ = const [21]. Considering the equations for an initial volume of air V 0 in the piston cavity of the pneumatic cylinder and its value V during movement V0 = Vr + SL, V0 = Vr + S(L + s),

(3)

where V r is a total volume of receivers; L is an initial position of the piston, we will get  γ Vr + SL . (4) p = p0 Vr + S(L + s) Most analytical methods for the shock absorber’s resistance force determining are based on the assumption that this force is related to the velocity by a linear relation [22], which does not correspond to a physical model of the shock absorber described, for example, in [23, 24]. In case of air suspension, it’s better to use “force-velocity”

Fig. 4. “Force-velocity” characteristic of KYB444119 series shock absorber

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experimental characteristic, which is provided for in the product catalogues of shock absorbers manufacturers. Figure 4 shows the characteristics for gas-oil shock absorbers of KYB444119 series [25], which are used in SMU500.3 machine. For this type of shock absorber the resistance force can be approximated by the following equation:  √ β1 s˙ n, s˙ ≥ 0, β1 = 1800 , (5) Fa = Fa (˙s) = √ −β2 −˙sn, s˙ < 0, β2 = 650 where n is a number of shock absorbers. After the shock, IB first moves down by inertia, and then returns to its original position under the action of the suspension pneumatic cylinders (Fig. 5)

Fig. 5. Design diagram of air suspension when returning to original position after shock

M s¨ = p(s)S − Mg − Fa (˙s).

(6)

4 Research Results Based on design diagrams, a simulation model of air suspension was obtained in MATLAB Simulink software environment. Using this model, the following dependencies have been studied: – influence of IB mass on the peak shock acceleration; – influence of IB and shock absorbers masses on static and dynamic components of the load onto the foundation; – influence of IB and shock absorbers masses on damping time for oscillations caused by a shock;

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– influence of suspension rigidness on a shock pulse and IB oscillations. The main purpose of IB is to reduce negative influence of a shock due to IB deadweight. It is a traditional method of shock machines designing in case of air suspension absence. In case of air suspension, kinetic energy of the falling table is transferred to kinetic energy of IB, and then to potential energy of compressed air and work on oil throttling in the shock absorbers. For design reasons, IB mass and dimensions shall be reduced, provided that shock acceleration and frequency of shocks required for testing products will not be distorted. Figure 6 shows a graph of acceleration obtained in a simulation at the mass of shock table with the tested product of 40 kg and different values of IB mass. The analysis shows that if IB mass is by an order more than the mass of the shock table with the tested product, peak acceleration A and reproduced shock pulse duration τ practically do not change when IB mass changes. Thus, in one of the modes of SMU5003 shock machine operation, at IB mass M = 800 kg, we get A = 640 m/s2 , τ = 10.2 ms (Fig. 6, a), at M = 400 kg: A = 580 m/s2 , τ = 10 ms. If masses of IB and the shock table

Fig. 6. Results of shock pulse simulation at different IB masses: a - at IB mass M = 800 kg; b at IB mass M = 80 kg

Fig. 7. Dependence of maximum IB movement on its mass at shock

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do not differ significantly, the changes become more significant. For example, if M = 80 kg, m = 40 kg, we have: A = 360 m/s2 , τ = 9 ms (Fig. 6, b). A similar situation occurs, when IB is moved. Figure 7 shows the dependence of IB movement on its mass obtained in the simulation with reproduction of shock acceleration with a peak value of 640 m/s2 and duration of 10.2 ms. IB movement increasing in case of multiple shock machines entails a problem related to maintaining the stability of the peak shock acceleration within the desired range and ensuring the frequency of shock pulses. For single shock machines, IB mass reducing, while keeping the specified shock pulse, can be compensated by increasing the shock table velocity prior to a shock, as IB movement increasing here is not critical. Damping increasing in shock absorbers leads to a decrease in dynamic load onto the foundation, which is obvious. However, the design of present-day shock absorbers is so that their characteristics depend discretely on the stroke, besides, throttling at small and large strokes differs significantly. At the stroke decreasing, shock absorbers start operating as rigid springs, so there is a certain limited number of absorbers, exceeding which does not make sense and even negatively affects the shock machines operation. This effect is most pronounced when reproducing large peak accelerations and there is no IB movement caused by a large number of shock absorbers, which in this case simply stop functioning. The suspension rigidness changes non-linearly when IB is moved due to pressure changes in the piston cavity of the suspension pneumatic cylinders in accordance with (4). To ensure condition (1), initial pressure in the piston cavity and the total area of the pistons are to be related values; only one of them is independent value. Its selection is based on the task formulation: for example, the piston diameter can be justified by the machine design features or conditions of drives unification within shock machines line of the test equipment manufacturer. On the other hand, pressure applied to the suspension pneumatic cylinders is always limited by the supply line pressure. In workshop pneumatic networks, pressure usually does not exceed 1 MPa, standard pneumatic elements of pneumatic automation equipment manufacturers are designed for such pressure. For small IB movements, the equation for pressure and elastic force can be linearized, and we will get F ≈ (M + m)g − cs,

(7)

where rigidness c is defined by the following equation c=

(M + m)gγ . L + Vr /S

(8)

Thus, the suspension rigidness can be changed by adjusting (selecting) the receiver volume. Transient time, during which IB returns to its initial stationary state after a shock is stopped, is related to a number of shock absorbers by a non-linear relation (Fig. 8). In the “Underdamping area”, transient time decreases as a number of shock absorbers increases. Herewith, the transient process is of oscillatory type. In the “Overdamping area”, an increase in a number of shock absorbers leads to a delay of the transient

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Fig. 8. Graph of dependence of transient process time on number of shock absorbers at different table accelerations

process (with an aperiodic type of transient process). At critical damping, there are no oscillations, and the transient time is minimal. In addition to impact on the peak shock acceleration, IB mass also determines mass and size characteristics of shock machines, as well as static and dynamic load onto the workshop floor, where the equipment is installed. Mass and size characteristics are traditionally directly related to the cost of equipment, while static and dynamic loads are initial data for calculating the technological machine foundations according to building codes. It is impossible to build a workshop or locate a shock machine in an existing building not knowing these loads. Studies of IB mass influence on the loads onto the workshop floor are shown in Fig. 9, 10. All the calculations have been performed for SMU500-3 shock stand (see Fig. 1) with the following parameters: – – – –

m = 40 - mass of the table, kg; τ = 0.010 - duration of a shock pulse, s; S = 0,0316 - area of the suspension cylinders pistons, m2 ; L = 0.075 - stroke of the suspension pneumatic cylinder pistons, m.

Dependence of the maximum (during shock) load R onto the foundation on IB mass M at an optimal number of shock absorbers (minimum of maximum load) is built for three different values of the peak acceleration (Fig. 9). The maximum load onto the foundation consists of two components: reaction from shock absorbers F a and reaction of air suspension cylinders F. Since with IB mass growth, at a given impact (shock pulse of the table), it accelerates more slowly, its velocity and movement, including their maximum values, decrease. Since F a is proportional to the velocity degree, this component of the load onto the foundation decreases with mass increasing. Influence of air suspension cylinders on the maximum load is more complex. On the one hand, F is proportional to the movement, and, therefore, with IB mass increasing it shall decrease. On the other hand, initial pressure in the pneumatic

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Fig. 9. Dependence of maximum load onto foundation R on inertia block mass M and peak shock acceleration A

cylinder (and, therefore, current pressure) increases with mass growth, since p0 = (M + m)·g/S, so F can grow with mass growth. So we see conflicting trends, which explains a “saddle” on the graph and the possible optimal load, which is achieved at different numbers of shock absorbers. Dependence of the maximum dynamic load Rd onto the foundation on the inertia block mass M is built for three different values of the peak acceleration A (Fig. 10). Points on the graph shown in Fig. 10 and in Fig. 9 correspond to the optimal numbers of absorbers for each particular IB mass. Optimum – at the minimum of the maximum dynamic load Rd .

Fig. 10. Dependence of maximum dynamic load Rd onto foundation on inertia block mass M and peak shock acceleration A

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The authors have evaluated the influence of SMU500-3 multiple shock machine design parameters changes on dynamic and static loads onto the foundation. The study of the main relations that determine effectiveness of shock loads damping, which occurs when reproducing a shock pulse at typical shock accelerations for a case of a single shock reproducing, has been conducted. The studies have been performed at IB mass changing relative to the design value of 800 kg, at increasing the mass of the table with the tested product, and the peak shock accelerations within the range from 50 to 300 g. A part of the study results are presented in Table 1, where designations correspond to those introduced earlier. Besides, Rsh = ma, characterizing the peak value of the force in a shock pulse, has been added. Table 1. Assessment of structural factors influence on load onto foundation M, kg

M, kg

600

80

A(g)

N

50

4

R, H

Rd , N

Rd /R, %

Rsh „ N

Rd /Rsh , %

9,892

3,221

33

39,240

8

800

4

11,680

3,047

26

8

1000

4

13,538

2,943

22

8

75

4

11,409

2,825

24

36,786

8

100

5

12,801

3,972

31

49,050

8

125

7

14,295

5,221

36

61,313

9

800

It has been found that 25% change in IB mass in the direction of its decrease or increase leads to changing a share of the dynamic component in the total load onto the foundation within the range of 4…8%. Changing the mass of the shock table with the product relative to the value of 100 kg by 25% in one direction or another also leads to changing a share of the dynamic component in the total load onto the foundation within the range of 4…8%. An important assessment characteristic of shock machines operation is a ratio of the dynamic load onto the foundation Rd to the force of the table shock when reproducing shock acceleration Rsh . During the simulation, it has been found that within the entire range of IB mass changes and changes in the mass of the table with the product, changes in the peak shock acceleration from 50 to 300 g, i.e. 6 times, Rd /Rsh ratio has increased only twice (in percentage terms). An important performance indicator of operation of stands with air suspension is an assessment of a share of dynamic load in the load onto the foundation when the peak shock acceleration changes. This component of the load changes with the mass of the table with a product of 75 kg three times from 24 to 77%.

5 Conclusion A suspension plays an important role in present-day pneumatic shock machines. It allows to reduce the dynamic influence onto the foundation and workshop equipment, as well as to reduce IB mass. Since the mathematical model of the processes occurring in shock

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machines is non-linear, the study of the suspension operation and optimal choice of its parameters is a complex multi-factor problem that should be solved using computer modeling. The research results presented in this paper are obtained in relation to SMU500-3 shock machine. In the future, they can be extended and generalized for a wide type of shock machines, taking into account the study of influence on phases of the table and IB movement of detailed models of generators, shock absorbers, guides, pneumatic equipment and IB itself, which is a multi-layer structure made of steel plates.

References 1. Kashyap, M,P., Sharadkumar, P.P.: Dynamic characterization of Shock Table. In: MultiDisciplinary Sustainable Engineering: Current and Future Trends, pp. 71–75 (2016). https:// doi.org/10.1201/b20013-12 2. Ievlev, P.V.: Influence of applied shock parameters on the dynamic characteristics of the third level electronic modules. Bull. Voronezh State Tech. Univ. 13, 53–60 (2017) 3. Evgrafov, A.N., Karazin, V.I., Kozlikin, D.P., et al.: Centrifuges for Variable Accelerations Generation. Int. Rev. Mech. Eng. 11(5), 280–285 (2017) 4. Popov, A.N., Polishchuck, M.N., Pulenets, N.Y.: Test centrifuge arrangement analysis. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 139–151. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11981-2_13 5. Harris, C.M., Piersol, A.G.: Harris’ Shock and Vibration Handbook, 5th edn. McGraw-Hill Professional, USA (2002) 6. Menyhardt, K., Nagy, R., Maruta, R.S.: Design of modular vibration testing equipment. Appl. Mech. Mater. 801, 333–337 (2015). https://doi.org/10.4028/www.scientific.net/AMM. 801.333 7. Fu, Y., Li, B.: Design of vibration test bench for electric locomotive bogie. In: Liu, B., Jia, L., Qin, Y., Liu, Z., Diao, L., An, M. (eds.) EITRT 2019. LNEE, vol. 640, pp. 611–623. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-2914-6_58 8. Andrienko, P.A., Karazin, V.I., Khlebosolov, I.O.: Bench tests of vibroacoustic effects. Lect. Notes Mech. Eng. 11(6), 11–17 (2017) 9. Lalanne, C.: Mechanical Shock. ISTE Ltd and John Wiley, London (2009) 10. Babuška, V., Sisemore, C.: The Science and Engineering of Mechanical Shock. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-12103-7_9 11. Popov, A.N., Polishchuck, M.N.: Testing Machines. Publishing House Ural Worker, Yekaterinburg (2017) 12. Lee, S.J.: Development and analysis of air spring model. Int. J. Automot. Technol. 11, 471–479 (2010). https://doi.org/10.1007/s12239-010-0058-5 13. Sun, T.C., Luo, A.C.J., Hamidzadeh, H.R.: Dynamic response and optimization for suspension system with non-linear viscous damping. Inst. Mech. Eng. Part K J. Multi-body Dymanics 214, 181–187 (2000) 14. Quaglia, G., Sorli, M.: Air suspension dimensionless analysis and design procedure. Veh. Syst. Dyn. 35(6), 443–475 (2016) 15. Rastegar, J.: A new class of high-G and longduration shock testing machines. Smart Structures and NDE for Industry 4.0. (2018) https://doi.org/10.1117/12.2296345 16. Popov, A.N., Polishchuck, M.N., Pulenec, N.E., et al.: Adjustment logic for pneumatic shock mashines. Mod. Mech. Eng. Sci. Edu. 6, 503–514 (2017) 17. Popov, A.N., Polishchuck, M.N., Vasiliev, A.K.: Method for reproducing shock acceleration in mechanical testing. Int. Rev. Mech. Eng. (I.R.E.M.E.) 14(2), 105–110 (2020). https://doi. org/10.15866/IREME.V14I2.18204

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18. Popov, A.N., Vasilev, A.K.: Pneumatic shock machine simulation. In: Proceedings of International Scientific and Practical Conference Science Week at SPbPU, pp. 12–15 (2015) 19. Gerenshtein, A.V., Kastryulina, N.S.: Methods of mathematical modeling of pneumatic systems. In: Achievements of Science - Agricultural Production, UGAUA, pp. 30–35 (2016) 20. Chernyshev, A.V., Belova, O.V., Krutikov, A.A., et al.: Numerical study of unsteady thermal processes in elements of pneumatic systems. Radio Eng. 4, 44–50 (2007) 21. Dobromirov, V.N., Gusev, E.N., Karunin, M.A., et al.: Shock absorbers. Design. Calculation. Tests M.: MSTU MAMI, p. 184 (2006) 22. Novikov, V.V., Ryabov, I.M., Chernyshov, K.V., et al.: Hydraulic shock absorber with variable resistance. Bull. Sci. Edu. 1(6), 7–11 (2018) 23. Kireev, A.V., Kozhemyaka, N.M., Burdugov, A.S.: Test bench trials of the electromagnetic regenerative shock absorber. Int. J. Appl. Eng. Res. 12, 6354–6359 (2017) 24. Guan, D., Jing, X., Shen, H., et al.: Test and simulation the failure characteristics of twin tube shock absorber. Mech. Syst. Sig. Proc. 122, 707–719 (2019). https://doi.org/10.1016/j.ymssp. 2018.12.052 25. KYB company (2020) https://www.kyb.com. Accessed 04 June 2020

Optimization of Parameters of Cyclic Machines When Crossing Resonance Zones Iosif I. Vulfson(B) Saint Petersburg State University of Industrial Technologies and Design, Bolshaya Morskaya Str. 18, 191186 Saint Petersburg, Russia [email protected]

Abstract. It analyzes a number of ways to reduce the excitation of vibrations when crossing the resonance zones of cyclic machines with constant and variable positions of the center of mass. For parametric resonances, in order to maintain dynamic stability, methods are proposed for determining threshold values of dissipative characteristics and the duration of the intersection zone. The results can be used to increase the productivity of machines, the accuracy of reproduction of the programmed movement of the working bodies, the implementation of technological requirements, as well as the ergonomic working conditions of the human operator. Keywords: Center of mass · Resonance · Parametric oscillations · Dynamic stability

1 Pseudo Resonances The terms “pseudo-resonance” or “false” resonance were proposed by Ya. Panovko as applied to forced oscillations during fast crossing of frequency ranges in the resonance zone in order to prevent the oscillation amplitude from reaching its maximum values [1]. Let us explain this with a simple example of a linear oscillatory system with one degree of freedom described by the differential equation aq¨ + b˙q + cq = F0 cos ωt.

(1)

Having divided all the terms of Eq. (1) by a, we write it in the following form: q¨ + 2n˙q + k 2 q = W0 cos(ωt − γ),

(2)



   where 2n = b/a; k 2 = c/a; W0 = F0 a; γ =arctg 2n k 2 − ω2 ; (0 ≤ γ ≤ π). First, consider the case when there is no resistance force. The solution consists of solving a homogeneous equation and a particular solution of an inhomogeneous equation: q = C1 cos kt + C2 sin kt + Y (t). © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 231–246, 2021. https://doi.org/10.1007/978-3-030-62062-2_23

(3)

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We are given the initial conditions q(0) = q0 , q˙ (0) = q˙ 0 . Then, therefore, q0 = C1 + Y (0); q˙ 0 = C2 k + Y˙ (0), and q˙ 0 Y˙ (0) sin kt − Y (0) cos kt − sin kt + Y (t) . q = q0 cos kt + 

  k

  k

3 1

(4)

2

Group of terms 1 describes free oscillations, whose frequency is equal to the natural frequency, and the amplitude depends on the initial conditions. The group of terms 2 defines the so-called accompanying vibrations, the frequency of which is also equal to the natural frequency, but unlike free vibrations, the amplitude here does not depend on the initial conditions, but is determined by the violation of the continuity of the particular solution Y and its derivatives [2]. Term 3 corresponds to forced oscillations, which depend both on the parameters of the driving force and on the parameters of the oscillatory system. When considering the behavior of the system in the vicinity of resonance (ω = k), one can, without loss of generality, exclude free oscillations from consideration by accepting. In this case, according to formula (4), we obtain the uncertainty, for the disclosure of which we will present in the following form: q=

W0 t sin[0, 5(k − ω)t] sin[0, 5(k + ω)t]. k + ω 0, 5(k − ω)t

(5)

When ω → k, q = [W0 t/(2k)] sin(kt). It follows that in the absence of resistance forces, the resonance amplitude grows according to a linear law, therefore, with a fast crossing of the resonance zone, emergency consequences can be avoided, i.e. there is a so-called pseudo-resonance. This mode is usually implemented during acceleration of the car and to a lesser extent - during coasting. When linear resistance is taken into account, the amplitude and phase of oscillations are determined as A= 

W0 (k 2

− ω 2 )2

+ 4n2 ω2

; γ = arctg

2nω . − ω2

k2

(6)

At resonance; q = [W0 /(2nω)] sin(ωt); Amax = W0 /(2nω); γ = π/2. Note that periodic oscillations with natural frequencies are usually considered a sign of a resonant state of a system. In particular, the experimental studies indicate the existence of cases where such oscillations occur in nonresonant zones, i.e. with absence ω = k. This is confirmed by the oscillograms of the oscillations of the loop-forming organs of the warp knitting machines shown in Fig. 1 with a significant difference between the natural frequencies and the excitation frequencies [2–4]. The main reason for this effect is a violation of the continuity of geometric characteristics, i.e. position functions and its derivatives: (ϕ),  (ϕ),  (ϕ),  (ϕ).

Optimization of Parameters of Cyclic Machines When Crossing Resonance Zones

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Fig. 1. Oscillograms with non-resonant periodic excitation

When analyzing the amplitude (AFC) and phase-frequency (PFC) characteristics in engineering calculations, the dimensionless form of dependencies is preferable (6). For this purpose, we introduce the dynamic coefficient æ, which is equal to the ratio  of the amplitude of the forced oscillations A to the so-called static amplitude ACT = F0 c. (The static amplitude is the deformation of the system under the influence of the amplitude value of the driving force applied under static conditions.) According to (6) æ=

1 A = , 2 ACT (1 − z )2 + 4z 2 δ2

(7)

where z = ω/k is the coefficient of frequency detuning, equal to the ratio of the frequency of the driving force to the natural frequency; δ = n/k = ϑ/(2π) is damping coefficient; ϑ – logarithmic decrement. The function æ(z), which is the dimensionless form of the frequency response, corresponds to the curve shown in Fig. 2. When z = 0, we have æ = 1. With growth, the function æ first increases, reaching a maximum in the vicinity of the resonance. The value of æ corresponds to the minimum of the radical expression of the denominator of formula (7), when √ at large values, which is fundamentally æmax = æ(z∗ ) = 1/(2δ 1 − δ2 ). However, √ possible during installation. Here z∗ = 1 − δ2 . In the zone z > z∗ the function æ asymptotically tends to zero. Excluding resistance δ = 0, (8) æ(z) = 1/ 1 − z 2 An analysis of dependences (7) disclosed that for small dissipative factors play a decisive role only directly in the resonance zone; for and the results practically coincide; √ therefore, in nonreasoming zones, they often use a simpler dependence (8). For z > 2 we have æ(z) < 1. This means that the amplitude of the forced oscillations is smaller than the static amplitude realized in the vicinity. That often used for the purpose of vibration isolation of oscillatory systems (Fig. 3). With kinematic perturbation, any point or section of the system receives forced motion according to a given law. So, for example, for the sprung crew model (Fig. 4,

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Fig. 2. Graph AFC

Fig. 3. Graph PFC

a), the source of disturbances is the road profile, and for the cam follower (Fig. 4, b), the law of motion determined by the cam profile. In both examples, the input section of the oscillatory system moves according to the position function x(ϕ), where ϕ = ωt. In this case, the movement of the driven mass consists of the portable motion x and the relative q - caused by the excited oscillations. The corresponding differential equation in the absence of other perturbations has the form

Fig. 4. Typical dynamic models

m¨q + b˙q + cq = −m¨x(t).

(9)

Using normal (main) coordinates, the technique described in this and the next sections can be easily extended to a system with a finite number of degrees of freedom. The most natural way to determine the solution of differential Eq. (2), at first glance, is to integrate it using numerical methods before reaching the steady state. However, this way leads to a significant accumulated error and increased complexity of the calculation. Therefore, we will use a more accurate and economical method of constructing a closed solution form. Consider the behavior of the system described by differential Eq. (9) during an arbitrary period of oscillations over period 0 < t < τ. In this case, the solution

Optimization of Parameters of Cyclic Machines When Crossing Resonance Zones

235

is described by the dependence, which includes the initial conditions q(0) = q0 and q˙ (0) = q˙ 0 [2, 3]. However, in this case, the initial conditions are not known to us, since the considered period is preceded by an unlimited number of cycles when the vibrational mode is established. To determine we use the conditions for the periodicity of forced oscillations:

q(q0 , q˙ 0 , t = 0) = q(q0 , q˙ 0 , τ); (10) q˙ (q0 , q˙ 0 , t = 0) = q˙ (q0 , q˙ 0 , τ ). Now we have a system of two equations with two unknowns, and we can use the dependence that is valid for the steady state at an arbitrary oscillation period [2]. Repeating the integration procedure with these initial conditions, we find the final solution q(t), that determines the dynamic error. Thus, the initial conditions corresponding to the steady-state oscillatory regime were found: q0 = C1 ; q˙ 0 = C2 k − C1 n = k(C2 − C1 δ) ≈ kC2 (δ = n/k  1),

(11)

where C1 ; C2 are the integration constants. Repeating the integration procedure with these initial conditions, we find the final solution that determines the dynamic error. The amplitude of the accompanying oscillations at the beginning of the cycle is   2 2 D0 = C1 + C2 = μ [Y 0 (τ) − x0 ]2 + k −2 [Y˙ 0 (τ) − ˙x0 ], (12) where μ is the accumulation coefficient of perturbations, defined as 1 . μ= √ −ϑ N cos 2π N + e−2ϑ N 1 − 2e

(13)

The set of curves μ(N , ϑ), where ϑ is the logarithmic decrement, is shown in Fig. 5. The coefficient can be either more than unity amplification of oscillations, or less (weakening of oscillations). The maximum value lies in the neighborhood of integers, and the minimum value when - an odd number. From formula (13) it follows μ+ = [1 − exp(−ϑN )]−1 ; μ− = [1 + exp(−ϑN )]−1 .

(14)

The value μ+ corresponds to the coincidence of the phase of previously excited oscillations and oscillations in the considered cycle; when μ− these fluctuations are in antiphase. We emphasize an important feature of this method related to the possibility of a rational combination of numerical and analytical methods: numerical integration here defines only a particular solution over a limited period of time; the conditions corresponding to reaching the steady state are identified analytically. The latter significantly affects the accuracy of the solution. Of course, with relatively simple excitation functions, obtaining a closed form of a solution can be carried out without resorting to numerical methods, directly using the analytical form of a particular solution. The use of the numerical-analytical method is especially advisable in those cases when there are continuity discontinuities in the geometric transfer functions or external forces, which is

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Fig. 5. Graphs of the coefficients of accumulation of perturbations

usually characteristic of cam mechanisms. With a more complex form of description of the coercive force, it is advisable to use the numerical-analytical method, which consists of the following steps: – numerical integration of the initial differential equation under zero initial conditions (for example, by the Runge-Kutta method). In this case, we find and; – determination of constants and initial conditions by the formula (11) corresponding to the steady state; – repeated numerical integration with the obtained initial conditions. In this method, numerical integration found only certain intermediate functions calculated over a limited period of time equal to τ, while the steady state is determined analytically (t → ∞) using the periodicity conditions. The latter has a significant effect on increasing the accuracy of the solution and reducing the complexity of the calculation.

2 Forced Vibrations with a Quasi-constant Amplitude-Frequency Characteristic We illustrate the features of oscillatory systems of this class using the dynamic model shown in Fig. 6 for cyclic mechanisms (links 1, 2, 3) mounted on a movable platform 0 with an elastic suspension [2, 3, 5]. Between the output link 3 and the housing is installed a spring, which provides power circuit of the kinematic circuit. In the future, in dynamic analysis, we will consider only one mechanism with reduced inertial and elastic-dissipative characteristics, since duplication of mechanisms is associated only with balancing the horizontal components of dynamic reactions and does not play a fundamental role from the standpoint of the problem under consideration.

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237

Fig. 6. Scheme of the mechanism with an abnormal characteristic

We accept the following conventions: mi - masses; ci - stiffness factors; ψi - scattering coefficients; ϕ, ω - the angle of rotation and the angular velocity of link 1; c1 r = ω2∗ (mD R + m2 r)—coordinates of links 0 and 3. On the crank 1 in the General case can be additionally installed weight with masse mD . Suppose that in the absence of this mass the crank is balanced, i.e. its center of masse S coincides with the point O. Then, when additional load is installed, the center of mass of this link will move to the point D. We take as a generalized coordinate of m2 , and for zero reference – the equilibrium position of the system at. In this case, the kinetic and potential energy are described as follows: T = 0, 5[(m0 + m1 )˙q2 + mD (˙q + Rω cos ωt)2 + m2 (˙q + rω cos ωt)2 ]; V = 0, 5[c0 q2 + c1 (q + r sin ωt)2 ].

(15)

Here m2 is the sum of the masses of units 2 and 3; R = OD; r = OB. After substituting (1) into the second-order Lagrange equation, taking into account the equivalent linear resistance force, we obtain m q¨ + β˙q + (c0 + c1 )q = −[c1 r − ω2 (mD R + m2 r)] sin ωt,

(16)

where m = mi ; β = (c0 + c1 )ψ∗ /(2πω) - reduced coefficient of linear resistance; ψ∗ - reduced scattering coefficient. If we accept z = ω/p, then there is a dynamic unloading at which at a certain frequency ω the kinematic excitation is fully compensated the restoring force in the elastic element. However, in this case, we face a different task - to carry out oscillations of the platform with an amplitude that does not depend (or weakly depends) on the change of ω∗ .

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According to (16), the amplitude of the forced oscillations is determined as c1 r − ω2 (mD R + m2 r)  A= , (17) (c0 + c1 ) (1 − z 2 )2 + 4δ2 z 4  √ where z = ω/p; p = [(c0 + c1 )/m ](1 − δ2 ) ≈ (c0 + c1 )/m – natural frequency; δ = ψ∗ /(4π) is dissipative coefficient. For a clearer physical idea of the problem under study, we first consider a special case, in which (mD = 0) and dissipative forces (δ = 0). Then formula (17) takes the form 1 − ω2 /k 2 , (18) A=r (1 + ζ) 1 − ω2 /p2 √ where k = c1 /m2 ; ζ = c1 /c2 .   The requirement dA/d ω = 0 leads to c1 c0 = m2 (m1 + m0 ), k = p and A = r/(1 + ζ) = const. The result obtained indicates a non-trivial situation when the “amplitude-frequency characteristic” does not depend on the frequency of the disturbance. It is easy to verify that the natural frequency of the system in this case is equal to the natural frequencies of two subsystems, that obtained by breaking the kinematic connection in the hinges B (see Fig. 6). This means that with a “hard” connection of both subsystems the natural frequency remains unchanged. Next, we return to the consideration of the general case corresponding to the original model. Then c1 r − ω2 (m2 r + mD R) rk 2 [1 − ω2 (1 + ρμD )/k 2 ]   = , (19) A= (c0 + c1 ) (1 − z 2 )2 + 4δ2 z 4 p2 (1 + ζ) (1 − z 2 )2 + 4δ2 z 4 where ρ =R/r, μD = mD /m2 . If the condition is satisfied k 2 /(1 + ρμD ) = p2 , we obtain on the basis of (19): r 1 − z 2  A= . (20) (1 + ζ) (1 − z 2 )2 + 4δ2 z 4 Taking into account that, p2 = k 2 (1 + ζ)/μ where, μ = 1 + μ0 + μ1 + μD , we have an additional equation of coupling between dimensionless parameters: ζ = μ / (1 + ρμD ) − 1 > 0.

(21)

Note that under the given additional condition, the dimensionless natural frequency p˜ = p/k depends only on the product ρμD . In dimensionless form, the final dependence describing the amplitude-frequency characteristic has the form a = a∗ κ(z, δ), where a = A/r, a∗ = (1 + ζ)−1 = (1 + ρμD )/μ - amplitudes of forced oscillations of the platform and static amplitudes; κ(z, δ) - dynamic factor, defined as  (22) κ = 1/ 1 + 4δ2 z 4 /(1 − z 2 )2 .

Optimization of Parameters of Cyclic Machines When Crossing Resonance Zones

239

Next, we determine the oscillations of the output link 3 y2 /r = sin ω t − a sin(ω t − γ)    where γ = arctg 2δ z (1 − z)2 . We can show that the amplitude of link 3 is  b = max (y2 /r) = (1 − a cos γ)2 + a2 sin2 γ

(23)

(24)

Figure 7 shows the family of quasi-constant amplitude-frequency characteristics obtained on the basis of (21)–(24) for platform 1 and –b(z) for slave link 3. When plotting the following initial data were adopted: μ0 = 2; μD = 0.5; δ = 0.03. Double indexation of the curves corresponds to a − ρ, b − ρ (numbers correspond to the numerical value of the parameter ρ = R/r).

Fig. 7. Frequency response family

Analysis of the graphs reveals the following characteristics of the frequency response: • With the exception of a narrow frequency range in the vicinity of the resonance zone (z ≈ 1), the amplitudes of the forced oscillations platforms and slave practically keep constant value. • At resonance (z = 1), the platform amplitude is virtually zero (antiresonance), and takes the value for the slave link is b = 1 (A2 = max(y2 ) = r), which corresponds to resonance. For a more detailed study of the behavior of the system directly in the resonance zone in Fig. 8, the corresponding frequency range is highlighted. If ρ = 0, ρ = 1, we have a < b. When ρ = 3 the amplitude of the platform exceeds the amplitude of link 3. The boundary case corresponds an a = b. Taking in the absence of dissipation we have:

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a = b = 0.5; γ = 0 , π. In addition to the modes shown in Fig. 7 the AFC is shown for ρ∗ = 1.5. When ρ < ρ∗ the analysis of the graphs shows, that during the phase of oscillations and the platform and the output link, at the entrance to the resonance zone it remains unchanged (curves a − 0, b = 0), and, conversely, they are reversed (curves a − 3, b − 3). At ρ = ρ∗ and certain distance from the resonance zone a ≈ b ≈ a∗ , as one would expect, this mode is illustrated in Fig. 8, b (a – curve 1; b – curve 2). Note that a similar situation occurs during dynamic damping, when at a certain frequency the reaction from the damper to the vibration protection object “balances” the external disturbance.

Fig. 8. Frequency response in the resonant zone

Analysis of the graphs shows that during the phase of oscillations of the platform and the output link at the entrance to the resonance zone it remains unchanged (curves), and vice versa, they are applied to the opposite (curves). At a certain distance from the resonance zone, as one would expect, this mode is illustrated in Fig. 8, b (a - curve 1; b curve 2). Note that a similar situation occurs during dynamic damping, when at a certain frequency the reaction from the damper to the vibration protection object “balances” the external disturbance. Figure 8 shows the graphs obtained by computer simulation at (for the rest of the data see above non-resonant zone; Fig. 9, a) the vibration amplitudes correspond to the values obtained above see Fig. 5, b, and the oscillation phases of units 0 and 3 are shifted by. In the resonance zone (), the amplitude of the platform tends to zero, and the amplitude of the driven link tends to unity (Fig. 6). Non-resonant zone; Fig. 9, a) the vibration amplitudes correspond to the values obtained above (see Fig. 5, b), and the oscillation phases of units 0 and 3 are shifted by. In the resonance zone (), the amplitude of the platform tends to zero, and the amplitude of the driven link tends to unity (Fig. 6). It should be noted that when crossing the resonance zone, the system sometimes does not have time to fully reach the steady state (see Fig. 6, c), however, a sharp decrease in the amplitude of the platform oscillations in the resonance zone is manifested quite clearly.

Optimization of Parameters of Cyclic Machines When Crossing Resonance Zones

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Fig. 9. Oscillations in a non-resonant and resonant zone: 1 − y1 (t); 2 − y2 (t).

3 Dynamic Characteristics with Variable Parameter Values Below we use the well-known method of transition to “dimensionless time”, where is the angular velocity. In this case, the transformation to the “dimensionless frequencies” is realized. With sufficient distance from the parametric excitations, the function is shown in Fig. 10.

Fig. 10. Graphs q(ϕ): 1. The average value k¯i = ki /ω; 2. max k¯i ; 3. min k¯i

This case is characteristic of acceleration and deceleration of the engine, as well as during the operating mode when the position of the center of mass of the system changes. Analysis of the graph clearly shows that with the “instantaneous” intersection of the resonance zone we have. With an increase in q, the length of stay in the resonance zone increases. Thus, here again we are faced with a typical “false” resonance, the elimination of which is possible with the rapid crossing of this zone. The above results are valid with a constant center of mass of the machine unit. However, in the oscillatory systems of many machine assemblies, center of mass displacements occur, caused by programmed movements of the working bodies and manufactured products, as well as drive mechanisms, lifting and transport functions, and other factors. This is the source of rheonomic connections, which leads to non-stationary frequency characteristics of the drive of the machine, to the appearance of dynamic effects associated with

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the excitation of parametric oscillations and to the possibility of violation of the conditions of dynamic stability [2, 5–11]. The use of the above method of false resonance in this case, at first glance, raises some doubts, since the amplitude during parametric resonance, in contrast to resonance during forced oscillations, increases very intensively exponentially. A way out of this situation is fundamentally possible if we exclude the possibility of crossing the frequency range of the main parametric resonance. However, as analysis shows, when the characteristics of the electric motor and a number of other factors are taken into account, the decrease in the vibration activity of the system is usually effectively implemented outside the lower frequency, which requires the intersection of this resonance zone. Another way is to increase the level of dissipation of the system. It can be shown that in order to exclude the possibility of excitation of the main parametric resonance, the condition where the ripple depth and scattering coefficient [3, 5, 7] must be met. Where in; here, where is the average value; Is the frequency of the main parametric resonance. As the basic object of the dynamic model, we take a rigid element mounted on two elastic supports (Fig. 11).

Fig. 11. Dynamic model

The disadvantages of the adopted option are large energy losses due to overcoming dissipative forces while suppressing the parametric resonance. For greater certainty, with further qualitative dynamic analysis, we specify the functions according to the following dependencies: L1 (ϕ) = h(0, 5 + λ + σ cos ϕ); L2 (ϕ) = h(0, 5 − λ − σ cos ϕ); (λ + σ < 0, 5)

(25)

The parameters and correspond to the constant and variable components of the diation of the center of mass coordinate from the middle position. At σ = 0 L1 = h(0, 5 + λ) = a; L2 = h(0, 5 − λ) = b

(26)

Then the problem coincides with the case of a fixed motionless position of the center of mass considered by Ya.G. Panovko in relation to vehicle vibrations [1]. Note that the parameter L is independent of ϕ: L = L1 (ϕ) + L2 (ϕ) = h. In machines with cyclic mechanisms, a slow change in parameters often dominates. In practice, this often leads to the false conclusion that the influence of non-stationary

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bonds can be neglected. However, in modern technological machines there are frequent cases in which there is a need for multi-frequency movement of the working body, when the given inertial and elastic characteristics contain not only low-frequency, but also highfrequency components [10, 11]. A similar situation also arises when using vibrational linearization in order to reduce the effective values of the Coulomb friction forces. The noted factors create a rheonomic connection and a source of parametric excitation. Practical methods for determining the regions of dynamic instability are usually based either on various modifications of the small parameter method and other asymptotic methods, or on numerical methods. The procedure for solving such problems is far from elementary, especially if we have in mind not standard simplified models with one degree of freedom, but real oscillatory systems of modern machines and mechanisms in which parametric excitation has a multicomponent structure. Below, the problem is solved on the basis of the conditional oscillator method, which avoids a number of difficulties that arise in this case when using traditional solution methods [6]. Above, only the slow components of the center of mass movement were taken into account. However, as shown in [3, 10, 11], the influence of higher frequencies cannot be neglected when studying parametric excitation. Therefore, along with formulas (4) below, similar dependencies of the following form will be used: L∗1 = L1 + r cos jϕ; L∗2 = L2 + r cos jϕ,

(27)

where the number j corresponds to the number of the high-frequency harmonic taken into account (Fig. 12). Here and below, an asterisk with parameters indicates

Fig. 12. Graphs (i = 1; 2)

Fig. 13. Graphs, J , J ∗

L∗i (ϕ) the use of functions that meets the influence of high-frequency harmonics. Figure 13 shows graphs of moments of inertia for a special case, when each of the radii of inertia is respectively equal to the geometric mean between the quantities and. For definiteness, in the numerical evaluation of the influence of the variability of the parameters, we take, N/m, Nm. As generalized coordinates, we take the vertical displacement of the center of mass and the angle of rotation of the swinging object with the moment of inertia or. The studied dynamic model is described by the following system of differential equations:

mq1 + 2δ∗1 mk1∗ q1 + (c1 + c2 )q1 + (c1 L∗1 − c2 L∗2 )q2 = Q1 ; (28) ∗2 J ∗ q2 + 2δ∗2 J ∗ k2∗ q2 + (c1 L∗1 − c2 L∗2 )q1 + (c1 L∗2 1 + c2 L2 )q2 = Q2 .

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Here Q1 , Q2 are the generalized forces; k ∗1 (ϕ), k2∗ (ϕ) are the values of natural frequencies (Fig. 14); ( ) = d /d ϕ; δ∗i = ϑi∗ /(2π) - damping coefficient (Fig. 15); ϑi∗ logarithmic decrement.

Fig. 14. Natural frequencies 1. k1 ; 2. – k2 ; 3. Fig. 15. Dissipation coefficient: s - resonance – k1∗ ; 4. – k2∗ number

When analyzing parametric excitation and using “false resonance”, one may doubt the effectiveness of this method due to the intensive growth of amplitudes, requiring an almost instantaneous transition through the resonance zone. However, the graphs in Fig. 16 indicate a relatively long delay preceding the manifestation of dynamic instability. Even more clearly, this effect is shown in Fig. 17 on the graphs of the envelopes of the resonant amplitudes and on the phase plane.

Fig. 16. Parametric resonance at high frequency excitation

Fig. 17. To the analysis of the delay effect of the dynamic instability mode.

As analysis has shown, the cause of this effect is a significant decrease in the moments of inertia in this zone (see Fig. 13), which led to an increase in the “natural” frequencies

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and to the intersection of instability regions. In Fig. 18 shows 3 regions of parametric resonance and a graph corresponding to the main parametric resonance with frequency. The graph also shows the critical value of the ripple depth depending on the level of dissipation, below which the system maintains dynamic stability. For the adopted model, the critical value of the damping coefficient is:

Fig. 18. Dynamic instability zones

4 Conclusion The article solves the problem of optimal intersection of resonance zones and analysis of the dependence of vibration activity on the displacements of the center of mass of the machine unit. An important result of the study is the elimination of violations of the conditions of dynamic stability in the frequency range of parametric resonances.

References 1. 2. 3. 4.

Pankovko, Ya.G.: Mechanics of a Deformable Solid. Nauka, Moscow (1985). (in Russian) Vulfson, I.I.: Dynamics of Machines. Vibration. Yurait, Moscow (2017). (in Russian) Vulfson, I.I.: Dynamics of Cyclic Machines. Springer, Berlin (2013) Vulfson, I.I., Teterina, V.V.: An experimental study of the forces arising in the drive rods of the loop-forming organs of the warp knitting machine OV-7, no. 5, pp. S.28–34. TSNIITEI “Mechanical Engineering for Light Industry”, Moscow (1968). (in Russian) 5. Vulfson, I.I., Kolovsky, M.Z.: Nonlinear Problems of the Dynamics of Machines. Mechanical Engineering, Leningrad (1968). (in Russian)

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6. Vulfson, I.I.: Parametric synthesis of vibrational systems of mechanisms with anomalous characteristics. In: Evgrafov, A.N. (ed.) Theory of Mechanisms and Machines, no. 2, pp. S40–49 (2003). (in Russian) 7. Vulfson, I.I.: Oscillations of systems with time-dependent parameters. Appl. Math. Mech. 33(2), 331–337 (1969). (in Russian) 8. Bolotin, V.V.: Dynamic stability of Elastic Systems. Gostekhteoretizdat, Moscow (1956). (in Russian) 9. Vulfson, I.I.: On the problem of reducing the vibration activity of machine drives when taking into account the dynamic characteristics of the electric motor. Problems of Mechanical Engineering and Machine Reliability, no. 4, pp. 12–19 (2017) 10. Ganiev, R.F., Kononenko, V.O.: Vibrations of Solids. Nauka, Moscow (1976). (in Russian) 11. Vulfson, I.I.: The conditions of dynamic stability combined with slow and fast parameter changes. Bull. Sci. Technol. Dev. 91(3), 3–13 (2015)

Author Index

A Abyshev, Oman A., 9 Abysov, Oleg V., 173 Aksenov, L. B., 20 B Bahrami, Mohammad Reza, 30 Belogur, Valentina P., 38 Borbotko, Viktoriya A., 122 Brodnev, Pavel N., 173 C Chekanin, Alexander V., 52 Chekanin, Vladislav A., 52 D Dudarenko, N. A., 204 E Evgrafov, Alexander N., 60, 71 Evgrafov, Sergey A., 71 Ezhova, Natalja V., 1 F Filippenko, George V., 80 G Gotsiridze, Aleksey V., 212

Kogan, Efim A., 136, 150 Korendyasev, Georgy K., 92 Kunkin, S. N., 20 Kuzmichev, Ivan Sergeevich, 102 Kuznetsov, Pavel A., 212 L Lagunova, Marina V., 1 M Mäkiö, Juho, 9 Manzhula, Konstantin P., 112 Melnikov, G. I., 204 Melnikov, V. G., 204 Miroshnichenko, Sergey T., 136, 150 P Petrov, Gennady N., 71 Pleshkov, Andrey A., 122 Plotnikov, Dmitry G., 194 Polishchuck, Mikhail N., 219 Pompeev, Kirill P., 122 Popov, Arkadii N., 219 Potapov, N. M., 20 Prostorova, Alexandra O., 212 Pukhliy, Vladimir A., 136, 150

I Ivanova, Liubov A., 1

R Radkevich, Mihail Mihailovich, 102 Reshetov, Dmitrii V., 219 Rumyantsev, Victor V., 173

K Karazin, Vladimir I., 60 Khlebosolov, Igor O., 60

S Salamandra, Konstantin B., 92 Semenov, Yuri A., 163

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Evgrafov (Ed.): MMESE 2020, LNME, pp. 247–248, 2021. https://doi.org/10.1007/978-3-030-62062-2

248 Semenova, Nadezhda S., 163 Shabanov, Alexander Yu., 173 Sidorov, Anatolii A., 173 Smirnov, Alexey S., 185 Smolnikov, Boris A., 185 Stakhin, Dmitry R., 194

T Talapov, V. V., 204 Tereshin, Valery A., 60 Tretyakov, Valeriy P., 212

Author Index V Valiulina, Anastasia A., 112 Vasiliev, Alexey K., 219 Vulfson, Iosif I., 231 Y Yablochnikov, Eugeny I., 9 Z Zhao, Wen, 38 Zhavner, Milana V., 38 Zhavner, Victor L., 38 Zinovieva, Tatiana V., 80