Dynamic Stability of Columns under Nonconservative Forces: Theory and Experiment [1st ed.] 978-3-030-00571-9, 978-3-030-00572-6

Table of contents :
Front Matter ....Pages i-xiii
Fundamentals (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 1-12
Columns under Conservative Forces (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 13-23
Columns under a Follower Force (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 25-36
Columns with Damping (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 37-48
Energy Consideration on the Role of Damping (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 49-69
Cantilevered Pipes Conveying Fluid (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 71-86
Cantilevered Pipes with a Mechanical Element (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 87-102
Columns under a Follower Force with a Constant Line of Action (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 103-113
Generalized Reut’s Column (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 115-128
Columns under a Rocket-Based Follower Force (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 129-143
Columns under a Rocket-Based Follower Force and with a Lumped Mass (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 145-153
Columns under a Rocket-Based Subtangential Follower Force (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 155-165
Pinned-Pinned Columns under a Pulsating Axial Force (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 167-181
Parametric Resonances of Columns (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 183-197
Parametric Resonances of Columns with Damping (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 199-206
Columns under a Pulsating Reut Force (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 207-217
Remarks about Approaches to the Dynamic Stability of Structures (Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama)....Pages 219-221
Back Matter ....Pages 223-236

Citation preview

Solid Mechanics and Its Applications

Yoshihiko Sugiyama Mikael A. Langthjem Kazuo Katayama

Dynamic Stability of Columns under Nonconservative Forces Theory and Experiment

Solid Mechanics and Its Applications Volume 255

Series editors J. R. Barber, Department of Mechanical Engineering, University of Michigan–Ann Arbor, Ann Arbor, MI, USA Anders Klarbring, Mechanical Engineering, Linköping University, Linköping, Sweden Founding editor Graham M. L. Gladwell, University of Waterloo, Waterloo, ON, Canada

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. Springer and Professors Barber and Klarbring welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Nathalie Jacobs, Publishing Editor, Springer (Dordrecht), e-mail: [email protected] Indexed by SCOPUS, Ei Compendex, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink.

More information about this series at http://www.springer.com/series/6557

Yoshihiko Sugiyama Mikael A. Langthjem Kazuo Katayama •

Dynamic Stability of Columns under Nonconservative Forces Theory and Experiment

123

Yoshihiko Sugiyama Small Spacecraft Systems Research Center, Osaka Prefecture University Sakai, Osaka, Japan

Kazuo Katayama Daicel Corporation Tatsuno, Hyogo, Japan

Mikael A. Langthjem Department of Mechanical Systems Engineering Yamagata University Yonezawa, Japan

Additional material to this book can be downloaded from http://extras.springer.com. ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid Mechanics and Its Applications ISBN 978-3-030-00571-9 ISBN 978-3-030-00572-6 (eBook) https://doi.org/10.1007/978-3-030-00572-6 Library of Congress Control Number: 2018954614 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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, express 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 lightweightedness of structures is one of the indexes of an advanced civilization. All structures, including transport vehicles, houses, buildings, commodities, etc., have become lighter with every passing year, while the demand on them to maintain higher integrity and strength has increased. Lightweight structures have continually drawn the attention of structural engineers over the years. It has been noted that lightweightedness may lead to structural failure (loss of structural stability), as it may cause reduction of structural stiffness. The study of structural stability problems has thus become more important than ever. Recent demands for higher speed and energy efficiency in modern transportation vehicles have brought to light stability problems under complex applied forces. Under these circumstances, studies of the stability problems of structures have progressed considerably and have become paramount in the pursuit of progress. Forces on structures are classified into two categories: conservative forces and nonconservative forces. This book deals mainly with structural stability problems relating to nonconservative forces. Forces attributable to active energy sources are prone to be nonconservative. The stability of structures accommodated with energy sources should generally be placed in the category of nonconservative stability problems. Structures under nonconservative forces may lose their stability dynamically. These problems should thus be investigated dynamically. The book comprises 17 chapters. The first 3 chapters describe the fundamentals of the dynamic stability of columns. Chapter 1 gives the definitions of beams and columns, the concept of stability, and basic knowledge about experiments related to the dynamic stability of columns. Chapter 2 discusses the stability of columns under conservative forces from the viewpoint of dynamic stability. Chapter 3 introduces the cantilevered column subjected to a follower (nonconservative) force, the so-called Beck’s column, with discussions of its basic characteristics. Chapters 4 and 5 discuss the effect of damping on the stability of cantilevered columns under a follower force. As an example of a realistic mechanical system with a follower load,

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the stability of cantilevered pipes conveying fluid is discussed, dealing with both theory and experiment, in Chaps. 6 and 7. Another type of column under a nonconservative force, with a constant line of action, the so-called Reut’s column, is introduced in Chap. 8. Experimental verification of Reut’s column is presented in Chap. 9. Chapters 10, 11, and 12 give experimental verifications of the dynamic stability of columns subjected to end rocket thrust, resembling a follower force. Chapters 13, 14, 15, and 16 discuss the dynamic stability of columns under pulsating axial forces. Simple resonances, and combination resonances of sum and difference type, are dealt with through theory, simulation, and experiment. An experimental approach to combination resonances of difference type is described in Chap. 16. Remarks about approaches to the dynamic stability of structures are presented in Chap. 17. Suggested exercises are provided in the Appendix A. Appendix B presents a movie of the dynamic responses of aluminium test columns under a rocket-based follower force, as described in Chaps. 10, 11 and 12. Appendix C presents a movie of flutter motion of an acrylic test column under a rocket-based follower force. The movies are available in the Supplementary Materials (online). The aim of the book is to give a realistic picture of nonconservative problems of structural stability. Among the many types of nonconservative problem for structures, the book mainly discusses the dynamic stability of cantilevered columns subjected to nonconservative (follower) forces within the framework of linear stability theory. Follower forces are defined as forces (vectors) that depend on the deformation of structures and are not derivable from a potential field. Rocket and jet thrusts acting on aerospace structures are typical examples. The book also intends to discuss the physical background of the topic. In addition to the theoretical aspects of the topic, simulations and experiments with columns under nonconservative forces are emphasized. The book is designed as a textbook for an advanced course in structural dynamic stability for senior-level undergraduate and master-level graduate students. Textbooks typically start with the general theory of the topic and then move on to concrete examples. This book starts with simple examples and then moves on to more complex examples with related theories. The book is written with the intention of striking a good balance between phenomenological observations of the dynamic stability of columns and their theoretical background. The initial author was inspired by his mentors, Prof. T. Sekiya in Sakai (where he was active), and the late Profs. V. V. Bolotin in Moscow, R. E. D. Bishop in London, and H. Leipholz in Waterloo. He has been encouraged by many friends, including Profs. T. Iwatsubo in Kobe, M. P. Païdoussis in Montreal, I. Elishakoff in Florida, A. P. Seyranian in Moscow, P. Pedersen in Lyngby, and N. Olhoff in Aalborg, and his academic brothers in aerospace structural engineering, Profs. P. K. Datta in Kharagpur and C.-D. Kong in Gwangju. In the course of studies on the present topic, many former students and researchers at the Sugiyama laboratory, the Department of Mechanical Engineering at Tottori University, and the Department of Aerospace Engineering at Osaka Prefecture University, were engaged in the

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present topic through analytical, computational, and experimental research projects. Their efforts are acknowledged sincerely herewith. The authors acknowledge the conversations and collaborations with all of our colleagues, who joined in the research activities on the related topic. Finally, the authors do hope that this book may present readers with a broad and sound perspective on the dynamic stability of columns under nonconservative forces, and that the book may open a window for students and young engineers, including Ph.D. candidates and researchers, to study the dynamic stability of structures in general. Sakai, Japan Yonezawa, Japan Tatsuno, Japan

Yoshihiko Sugiyama Mikael A. Langthjem Kazuo Katayama

Contents

1

Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Beam and Column . . . . . . . . . . . . . . . . . 1.2 Stability and Stability Criteria . . . . . . . . . 1.3 Experiments with Columns . . . . . . . . . . . 1.4 Preliminary Tests . . . . . . . . . . . . . . . . . . 1.4.1 Deflection Test . . . . . . . . . . . . . . 1.4.2 Vibration Test . . . . . . . . . . . . . . 1.4.3 Decaying Pendulum Motion Test 1.4.4 Decaying Beam Motion Test . . . . 1.5 Influence of Support Conditions . . . . . . . . 1.6 Nonconservative Forces . . . . . . . . . . . . . . 1.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Columns under Conservative Forces . 2.1 Cantilevered Columns . . . . . . . . 2.2 Pinned-Pinned Columns . . . . . . 2.3 Standing Cantilevered Columns . 2.4 Discussion . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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Columns under a Follower Force . . . . . . . 3.1 Beck’s Column . . . . . . . . . . . . . . . . . 3.2 Vibrations of Beck’s Column . . . . . . 3.3 Stability in a Finite Time Interval . . . 3.4 Character of Beck’s Column . . . . . . . 3.5 Nonconservative Nature of a Follower 3.6 Discussion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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Columns with Damping . . . . . . . . . . . . . . . . . . . . 4.1 Cantilevered Columns with Damping . . . . . . 4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . 4.3 Beck’s Column with Damping Introduced . . 4.3.1 Internal Damping Only . . . . . . . . . . 4.3.2 External Damping Only . . . . . . . . . 4.3.3 Both Internal and External Damping 4.4 Pflüger’s Column with Internal Damping . . . 4.5 Dynamic Responses . . . . . . . . . . . . . . . . . . 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Energy Consideration on the Role of Damping . . . . . . . . . . . . 5.1 Energy Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Equation of Motion and Stability Analysis . . . . . . . . . . . . 5.3 Energy Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Energy Balance Equations . . . . . . . . . . . . . . . . . 5.3.2 Energy Balance at the Critical Force . . . . . . . . . . 5.3.3 Discretized Energy Equations . . . . . . . . . . . . . . . 5.4 Flutter Configurations and Phase Angle Functions . . . . . . 5.5 Energy Balance with Small Internal Damping . . . . . . . . . . 5.6 Energy Balance with Both Internal and External Damping 5.7 Energy Growth Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Introduction of Small Internal Damping at the Undamped Flutter Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Cantilevered Pipes Conveying Fluid . . . . . . 6.1 Basic Equations of Motion . . . . . . . . . 6.2 Finite Element Formulation . . . . . . . . . 6.3 Eigenvalue Branches Related to Flutter 6.4 Flutter Configurations . . . . . . . . . . . . . 6.5 Effect of Internal Damping . . . . . . . . . 6.6 Discussion . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Cantilevered Pipes with a Mechanical Element . . . 7.1 Pipes with an Elastic Spring . . . . . . . . . . . . . 7.2 Pipes with a Lumped Mass . . . . . . . . . . . . . . 7.3 Pipes with a Damper . . . . . . . . . . . . . . . . . . . 7.4 Coefficient of Damping of a Dashpot Damper 7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Columns under a Follower Force with a Constant Line of Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Reut’s Column . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Stability Analysis of a Generalized Reut’s Column . 8.3 Approximate Solution by the Galerkin Method . . . . 8.4 Non-Self-Adjointness of Boundary Value Problems 8.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Generalized Reut’s Column . . . . . . . 9.1 Stability Analysis . . . . . . . . . . 9.2 Realization of Reut Force . . . . 9.3 Experimental Setup . . . . . . . . . 9.4 Experimental Results . . . . . . . . 9.5 Reut’s Column with a Damper 9.6 Discussion . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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10 Columns under a Rocket-Based Follower Force 10.1 Equation of Motion and Stability Analysis . 10.2 Rocket Motors . . . . . . . . . . . . . . . . . . . . . 10.3 Test Columns . . . . . . . . . . . . . . . . . . . . . . 10.4 Preliminary Tests . . . . . . . . . . . . . . . . . . . 10.4.1 Test for Bending Stiffness . . . . . . . 10.4.2 Test for Damping Coefficients . . . . 10.4.3 Buckling Test . . . . . . . . . . . . . . . . 10.5 Flutter Test . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Outline of the Test . . . . . . . . . . . . 10.5.2 Test Runs . . . . . . . . . . . . . . . . . . 10.5.3 Test Results . . . . . . . . . . . . . . . . . 10.5.4 Effect of the Size of the Motor . . . 10.5.5 Stability in a Finite Time Interval . 10.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Columns under a Rocket-Based Follower Force and with a Lumped Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Finite Element Formulation and Stability Analysis . . . . . . . 11.2 Rocket Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Estimate of the Effect of a Lumped Mass on the Flutter Limit 11.4 Flutter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Columns under a Rocket-Based Subtangential Follower Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Mathematical Model and Finite Element Formulation 12.2 Rocket Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Test Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Stability Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Experiment with Columns under a Rocket-Based Subtangential Follower Force . . . . . . . . . . . . . . . . . . 12.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Pinned-Pinned Columns under a Pulsating Axial Force 13.1 The Mathieu Equation . . . . . . . . . . . . . . . . . . . . . . 13.2 Stability of the Solution to the Mathieu Equation . . 13.3 Pinned-Pinned Columns . . . . . . . . . . . . . . . . . . . . 13.4 Vibrations in the Vicinity of Upper and Lower Boundaries I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Vibrations in the Vicinity of Upper and Lower Boundaries II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Effect of a Phase Angle in Excitation . . . . . . . . . . . 13.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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176 178 180 180

14 Parametric Resonances of Columns . . . . . . . . . . . . . . . . 14.1 Mathieu-Hill Equations . . . . . . . . . . . . . . . . . . . . . . 14.2 Hsu’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Coupled Mathieu Equation of Columns . . . . . . . . . . 14.4 Hsu’s Resonance Conditions . . . . . . . . . . . . . . . . . . 14.5 Estimate of the Principal Regions of Resonances . . . 14.6 Experiment with Columns Having Clamped-Clamped and Clamped-Pinned Ends . . . . . . . . . . . . . . . . . . . . 14.7 Columns under a Pulsating Follower Force . . . . . . . . 14.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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183 183 184 185 187 188

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189 192 196 197

15 Parametric Resonances of Columns with Damping . 15.1 Approaches to Mathieu-Hill Equations . . . . . . . 15.2 Hsu’s Approach to Coupled Hill Equations . . . 15.3 Effect of Damping . . . . . . . . . . . . . . . . . . . . . 15.4 Second-Order Approximation . . . . . . . . . . . . . 15.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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199 199 200 202 203 205 205

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Contents

16 Columns under a Pulsating Reut Force . . . . . . . . . . . . . . . . 16.1 Columns under a Pulsating Generalized Reut Force . . . 16.2 Finite Difference Formulation and Stability Analysis . . . 16.3 Experiment with Columns under a Pulsating Reut Force 16.3.1 Experimental System . . . . . . . . . . . . . . . . . . . 16.3.2 System for a Pulsating Reut Force . . . . . . . . . . 16.3.3 Test Pieces . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.4 Attachment . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.5 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . 16.3.6 Regions of Parametric Resonances . . . . . . . . . 16.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

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207 207 209 211 211 211 212 213 213 214 215 217

17 Remarks about Approaches to the Dynamic Stability of Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Appendix A: Suggested Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Appendix B: Movie of Test Runs in Chapters 10, 11 and 12 . . . . . . . . . 227 Appendix C: Flutter Motion of a Damped Column under a Rocket-Based Follower Force . . . . . . . . . . . . . . . . . . . . . . 229 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Chapter 1

Fundamentals

The word “column” usually refers to some slender vertical structure. “Column” in this book is used as an engineering term, and means a slender and straight structural member under compression. Here, the definition of a column and some basic concepts related to the dynamic stability of columns are presented.

1.1

Beam and Column

In the terminology of structural mechanics, if a bar is subjected to bending (and shearing), then it is called a beam. If a bar is subjected to axial compression, it is called a column. It is assumed, if not otherwise defined, that beams and columns are initially uniform, straight and elastic. Normally, it is assumed that the longitudinal inertia and rotatory inertia of columns are negligible. Mathematically, a column is an elastic line, accommodated with some mechanical properties distributed along the line. These are bending stiffness/rigidity, mass, damping, and axial compressive stress. In this book, it is assumed that stresses over the cross-section of a column are elastic during bending. It is also assumed that the deflection of columns is small enough for the relation between deflection and bending moment to obey Euler-Bernoulli beam theory. Columns of this sort are slender and may be called long columns. Figure 1.1a shows a simple structure composed of two vertical bars and one horizontal bar. This structure was designed as a portable apparatus for an Euler buckling demonstration in class for a course in structural mechanics [1]. The two vertical bars support the horizontal bar and resist compression due to its weight. The horizontal bar is a device for loading, and at the same time, a weight for load. The bar resists bending due to its own weight, and possibly another weight to be added to increase the load. It is noted that the vertical bars are clamped at the lower ends on the base block, while their upper tip ends were knife-edged and support the upper bar at V-notches on each side of the bar. The tip ends of the vertical bars are © Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_1

1

2

1 Fundamentals

(a) Columns before buckling

(b) Buckled columns

Fig. 1.1 Simple structure with two columns and one beam [1]

free to rotate and move laterally, to realize free ends. The horizontal bar is called a beam, while the two vertical bars are referred to as columns. As an example, let us consider a truss structure. A truss is composed of bars linked together by hinges. Some bars are subjected to axial compression, others to tension. The compression members are columns, while the tension members are called strings. Columns may lose their stability suddenly through buckling (static instability), as shown in Fig. 1.1b. Structural engineers should be aware of the possibility of buckling when dealing with columns. Many books have been published on the buckling of columns and structures [2–7], and somewhat fewer on the dynamic stability of structures [8–13]. This book deals with the dynamic stability of columns, which has a variety of problems of interest.

1.2

Stability and Stability Criteria

The concept of stability is ubiquitous in our social life: we can talk about the stability of the society, a population, and economic and political situations, as well as physical and mechanical systems. By stability or instability, we mean whether a small disturbance leads to a small change in behavior as time passes, or not. If the behavior changes only a little under a given small disturbance, the system is stable; if not, the system is unstable. Many stability criteria have been developed to define the stability of systems. A mild stability criterion requires that the disturbed behavior stay close to the initial state as time passes, while a stronger criterion defines that the behavior is stable only if the disturbed behavior returns exactly to the initial state as time goes to infinity. Many different criteria have been applied in practice, depending on the properties of the systems.

1.2 Stability and Stability Criteria

3

As a mathematical discussion on stability criteria, let us focus on the stability of the motion of columns. In structural dynamics, it is normally assumed that the initial state of a column is stable. Stability theory of columns under applied forces is concerned with the stability of solutions to the equations of motion. Let us consider a cantilevered column of length L subjected to a compressive force P (Fig. 4.1). The equation of motion is given by m

5 @2y @y @2y @4y  @ y þ E þ C I þ P þ EI ¼ 0; @t2 @t @t@x4 @x2 @x4

ð1:1Þ

where y is the deflection at position x and time t, m is the mass of the column per unit length, E is Young’s modulus, and I the cross-sectional area moment of inertia. The product EI is the bending stiffness. It is assumed that the material of the column is viscoelastic and obeys the Kelvin-Voigt constitutive law r ¼ Ee þ E e_ ;

ð1:2Þ

where r is the cross-sectional stress, e is the strain, e_ is its time derivative, and E  is the coefficient of internal damping due to internal friction of the material (the modulus of viscosity). It is also assumed that the column is subjected to external damping due to friction between the column and the surrounding medium. The coefficient of external damping is denoted by C. The boundary conditions are @yð0; tÞ ¼ 0; @t @ 2 yðL; tÞ @ 3 yðL; tÞ EI þ E I ¼ 0; 2 @x @t@x2 yð0; tÞ ¼ 0;

EI

@ 3 yðL; tÞ @ 4 yðL; tÞ þ E I ¼ 0: 3 @x @t@x3

ð1:3Þ

By applying a discretization method, such as the finite difference method (FDM), the Galerkin (Bubnov-Galerkin) method, or the finite element method (FEM), to the equation of motion of the continuous column, we obtain, in the simplest cases, a system of differential equations of the form n d 2 yi dyi X þ þ c kij yj ¼ 0; i dt2 dt j¼1

ði ¼ 1; 2; . . .; nÞ;

ð1:4Þ

where yi ¼ yi ðtÞ denotes generalized coordinates. Let yðtÞ ¼ fy1 ðtÞ; y2 ðtÞ; . . .; yn ðtÞgT : Lyapunov’s stability criterion can be formulated as follows: the solution yðtÞ is stable in the sense of Lyapunov if kyðt0 Þk\e implies that kyðtÞk\dðeÞ for all t0 \t\1;

ð1:5Þ

4

1 Fundamentals

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kyðtÞk ¼ jy1 j2 þ jy2 j2    þ jyn j2 is the Euclidean norm of yðtÞ. e and d are (arbitrarily) small positive numbers. If the condition (1.5) is not satisfied, the solution yðtÞ is said to be unstable. A stronger form of stability is asymptotic stability. The solution yðtÞ is asymptotically stable if kyðt0 Þk\e implies that kyðtÞk ! 0 for t ! 1:

ð1:6Þ

When the stability of a linearized system is considered, the initial conditions are not of significance regarding stability; only the effect of perturbation of force parameters is of practical significance. Considering a small periodic vibration with frequency x ðx [ 0Þ about the stable state of the column, we can reduce the set of differential equations to an pffiffiffiffiffiffiffi  eigenvalue problem with complex eigenvalues of the form k ¼ a þ ix i ¼ 1 . Let us assume that the generalized coordinates can be written in the form yi ðtÞ ¼ hi ekt ¼ hi eat cosðxt þ uÞ:

ð1:7Þ

If this is the case, the asymptotic stability based on the positive parts of the characteristic roots (eigenvalues) is expressed in the following form: the vibrations are stable if all a\0:

ð1:8Þ

The vibrations are unstable if at least one a [ 0. This means that the limit of stability is given by a ¼ 0. When a [ 0 and x [ 0, the straight configuration of the column loses its stability in the form of vibrations with increasing amplitude. This instability is termed flutter, or dynamic instability. When a [ 0 and x ¼ 0, the straight configuration loses its stability in the form of monotonically (exponentially) increasing amplitude. This type of instability is called divergence, or static instability. The different behaviors of the motion are sketched in Fig. 1.2.

Fig. 1.2 Stability of motion

1.2 Stability and Stability Criteria

5

In the laboratory (i.e., in experiments) and in simulations of the dynamics of columns, it is not easy to confirm whether small vibrations return exactly to the initial stable state after infinite time. We may thus have to rely on a relaxed stability criterion defined for a finite time interval. A relaxed version of Lyapunov stability based on the disturbed response in a finite time interval, practical for laboratory experiments, can be expressed as follows: the vibrations are stable if kyk\jyi j

for t0 \t\t0 þ tf ;

for all i:

ð1:9Þ

If condition (1.9) is not satisfied, the vibrations are unstable. The specified threshold ys may be defined as k times the initial deflection, or as a possible limit deflection that guarantees the validity of the assumption of small deflections. The finite time interval tf may be taken as n times the number of periods of vibrations during an experiment or a simulation. A relaxed asymptotic stability criterion, based on a measure of the rate of amplitude growth, may be expressed as follows: the vibrations are stable if, for all a, 0\a\ara for t0 \t\t0 þ tra :

ð1:10Þ

If condition (1.10) is not satisfied, the vibrations are unstable. The specified growth rate ara should be chosen to be as small as possible, while the finite time interval tra should be taken to be as long as possible, to obtain a trustworthy stability bound.

1.3

Experiments with Columns

When conducting experiments with columns, one starts designing test columns, determining their shapes and dimensions, and selecting the material. When the material is selected, the modulus of elasticity (Young’s modulus) E can be found in data books for engineering materials. The modulus of elasticity in the data sheets has been obtained by material testing in tension, not in compression. However, it is normally assumed that the modulus of elasticity of engineering materials takes the same values in compression as in tension. For calculating the bending stiffness EI, one has to determine the cross-section of the column. Consider, for example, the case of a rectangular section having breadth b and thickness t. Then, the moment of inertia I is given by I ¼ bt3 =12. As b and t can be measured directly, it is easy to evaluate the bending stiffness EIc . When one wants to compare the calculated deflection values with test results, say, the buckling load or the flutter limit, it is advisable to conduct preliminary tests of the bending stiffness of the test columns before starting on the main test. The bending stiffness can be obtained by a static deflection test and a free vibration test to give, respectively, static and dynamic values of EI, say EIs and EId . Normally, the values of the bending stiffness/rigidity

6

1 Fundamentals

obtained by the two tests are slightly different, sometimes lower, sometimes higher than the calculated (ideal) value EIc , by several per cent. In the case of a cantilevered column, it is normally lower. When we are concerned with the dynamics of columns, theoretically and experimentally, it is vital to know the magnitude of damping. Damping due to friction in the material of the column is referred to here as internal (structural/ material) damping, while damping due to friction between the column and the surrounding medium is referred to as external damping. At the very least, the order of magnitude of the damping should be confirmed in preliminary tests, to be described in the following.

1.4 1.4.1

Preliminary Tests Deflection Test

As the first step in the preliminary tests, a deflection test should be conducted, to obtain the experimental bending stiffness of the test column. Figure 1.3 shows a sketch of the deflection test. A corresponding experimental setup is shown in Fig. 1.4. With the notations shown in Fig. 1.3, the static bending stiffness EIs is given by the formula

Fig. 1.3 Schematic of a deflection test

Fig. 1.4 Setup for a deflection test

1.4 Preliminary Tests

7

Fig. 1.5 Setup for a vibration test

  a3 3L W 1 : EIs ¼ d 6 a

1.4.2

ð1:11Þ

Vibration Test

Nonconservative systems may lose their stability by flutter (dynamic instability). It is thus desirable to check the dynamic bending stiffness with a vibration test. The dynamic stiffness of the test beam can be obtained by measuring the natural frequencies. Normally, only the first mode is considered in the test. An example of an experimental setup for the vibration test is shown in Fig. 1.5. The test beam is excited by a shaker and the dynamic deflection is recorded by, for example, a non-contact type deflection sensor. The first natural (resonant) frequency x1 (Hz) is read on the recorder. The dynamic bending stiffness is then given by the expression EId ¼

4p2 mL4 x21 ; a41

ð1:12Þ

where a1 is the first eigenvalue of the characteristic equation of a cantilevered beam (a1 ¼ 1:875) and m is the mass per unit length of the beam.

1.4.3

Decaying Pendulum Motion Test

For measuring the coefficient of external damping, it will be assumed that the test beam is a rigid bar. Let us consider a pendulum model of the bar, pinned at the top

8

1 Fundamentals

Fig. 1.6 Schematic of a pendulum motion test in air (h_ is the time derivative of the angle h)

end and hung down vertically, as shown in Fig. 1.6. The equation of motion of the pendulum is given by 1 3 d 2 h 1 3 dh 1 mL 2 þ CL þ mgL2 h ¼ 0; 3 dt 3 dt 2

ð1:13Þ

where h is the angle of the declined bar to the vertical line and g is the acceleration due to gravity. Equation (1.13) can be written as d2h dh þ x2 h ¼ 0; þ 2fx dt2 dt

ð1:14Þ

with C 2fx ¼ ; m

 x¼

 3g 1=2 ; 2L

ð1:15Þ

where f is the damping ratio and x the frequency of the angular vibration. With the relation of the damping ratio f and the logarithmic decrement d (measured value), f ¼ d=2p, the coefficient of external damping is given by the expression   dm 6g 1=2 C¼ : 2p L

ð1:16Þ

1.4 Preliminary Tests

1.4.4

9

Decaying Beam Motion Test

Starting with Eq. (1.1) with P ¼ 0, and applying Galerkin’s method with the first eigenfunction of free vibration of a cantilevered beam, one obtains the reduced equation of motion  df1 d 2 f1  þ x21 ¼ 0; þ j þ cx21 2 dt dt

ð1:17Þ

  C E a1 EI 1=2 : j ¼ ; c ¼ ; x1 ¼ 2 m E L m

ð1:18Þ

where

The damping ratio f is expressed in the form 2fx1 ¼ j þ cx21 :

ð1:19Þ

If one can find the logarithmic decrement d with the decaying beam motion test, one obtains the modulus of viscosity (the coefficient of internal damping) E  in the form c¼

E d C ¼  : px1 mx21 E

ð1:20Þ

Figure 1.7 shows a schematic of a decaying beam motion test. Decaying free oscillations of a test beam, under internal and external damping, can be picked up, for example, by a non-contact type deflection sensor to find the damping ratio f and the first natural frequency x1 .

Fig. 1.7 Schematic of a decaying beam motion test

10

1 Fundamentals

Fig. 1.8 Design of clamped end. ① Clamping bed, ② Test piece, ③ Clamping pad, ④ Washer, ⑤ Bolt, ⑥ Milled cavities

5

4 3

6

2 1

1.5

Influence of Support Conditions

It is noted that the bending stiffness of a cantilevered column mounted in a test rig may possibly not coincide with the calculated ideal value, as it is very difficult to realize a perfect/ideal clamped end. The bending of a column in a test rig may be influenced slightly by the condition of the clamped end. To realize a possibly better clamping, it is advised that milled cavities are made, as shown in Fig. 1.8. The cavities may in effect ensure that the edge of the clamping presses firmer than otherwise. In comparing the calculated and experimental values of the bending stiffness, it is preferable to adopt the value obtained by the preliminary tests for the calculated/theoretical prediction. The boundary conditions for cantilevered columns are the most reproducible and the most easily realizable conditions among all when we are concerned with experiments regarding the dynamics of columns. For example, let us imagine that we wish to realize pinned (hinged, simply supported) end conditions. We may realize this with bearings (see Fig. 14.5b), or other devices, for zero moment. In the case of the dynamics of a column having a pinned end, the effect of the bearings on the dynamics of the beams/columns is complicated, and the dynamics will be affected considerably by it. In short, the pinned end, as described earlier in connection with the portable buckling set in Fig. 1.1a, is not reliable in a dynamical test. It is noted here that all of the tests and experiments described in this book were conducted upon metric engineering units with a force unit of kgf (kilogram-force). The unit of kgf can be converted into the SI force unit of N (Newton) by multiplication with the gravity acceleration g ¼ 9:81 m=s2 .

1.6 Nonconservative Forces

1.6

11

Nonconservative Forces

Forces applied to structures are classified into two categories: conservative and nonconservative forces. A conservative force is defined as a force having the property that the work done by it is path-independent, and is thus dependent only on positions. If the force is conservative, it may be called a potential force. Gravity is a typical conservative force. If a beam is elastic and the forces acting on it are conservative, the stability of this type of column can be discussed within the regime of conservation of mechanical energy, and thus within the framework of the theory of elastic buckling [2–7]. If a force has no property as defined for conservative ones, then it is nonconservative. It is highlighted in this text that there are very many active nonconservative forces attributable to energy sources and flows. These are, for example, jet thrusts, fluid dynamic forces, time-dependent forces, etc. These forces are prone to cause dynamic instability of structures, as structures under such forces may gain energy from them during disturbed motion. Stability problems of structures related to fluid dynamic forces are categorized as fluid-structure interaction problems [14–16], while the problems with time-varying forces are discussed in the field of parametric resonances of dynamical systems [17]. Jet and rocket thrusts, typical in the field of aerospace structural engineering, are forces that are attributable to fluid flow, and they may be nonconservative when they act on a structure with a free end. These forces are dependent on the deformation of the structure, and typically follow the deformation. They are called “follower forces”. In the case of the stability of a column, a follower force can be nonconservative only when the column is cantilevered, not when the column is supported at both ends. The stability of columns under such forces should be treated dynamically, that is, within the framework of dynamic stability [17–20].

1.7

Discussion

This chapter has discussed the definitions of beam and column, and their stability criteria, including a relaxed version of Lyapunov stability based on the disturbed response in a finite time interval, which is practical for laboratory experiments. As damping plays an important role in the dynamic stability of columns, the basic procedures of the decaying pendulum motion test (for the coefficient of internal damping) and the decaying beam motion test (for the coefficient of external damping) have been introduced. These fundamentals regarding experiments on the dynamics of columns have been emphasized to encourage experiment-friendly students, scientists and engineers. The stability problems of nonconservative structural systems were first compiled in book form in 1963 (English version) by Bolotin, in his classical monograph entitled “Nonconservative Problems of the Theory of Elastic Stability” [18]. The

12

1 Fundamentals

classification of forces and structural systems was discussed by Ziegler in 1968, in his book entitled “Principles of Structural Stability” [19]. Recent progress in regard to nonconservative stability problems in modern physics has been thoroughly discussed by Kirillov in his comprehensive monograph [20], in 2013. The present book deals with the dynamic stability of cantilevered columns subjected to nonconservative forces, theoretically and experimentally. The book aims to bridge the gap between the classical books by Bolotin and Ziegler and the recent monograph by Kirillov.

References 1. Sugiyama, Y. (1984). Euler buckling demonstration. Experimental Techniques (Educational Techniques), 8(8), 27–28. 2. Timoshenko, S. P., & Gere, J. M. (1961). Theory of elastic stability. New York: McGraw-Hill. 3. Gerard, G. (1962). Introduction to structural stability theory. New York: McGraw-Hill. 4. Croll, J. G. A., & Walker, A. C. (1972). Elements of structural stability. London: Macmillan. 5. Kirby, P. A., & Nethercot, D. A. (1979). Design for structural stability. London: Granada Publishing. 6. Singer, J., Arbocz, J., & Weller, T. (1998). Buckling experiments, experimental methods in buckling of thin-walled structures, basic concepts, columns, beams, and plates (Vol. 1). New York: Wiley. 7. Elishakoff, I., Li, Y., & Starnes, J. H., Jr. (2001). Non-classical problems in the theory of elastic stability. Cambridge: Cambridge University Press. 8. Simitses, G. J., & Hodges, D. H. (2006). Fundamentals of structural stability. New York: Butterworth-Heinemann. 9. Panovko, Y. G., & Gubanova, I. I. (1965). Stability and oscillations of elastic systems (Translation from the Russian text in 1964. Foreword by W. Flügge), New York: Consultants Bureau. 10. Leipholz, H. (1970). Stability theory—An introduction to the stability of dynamic systems and rigid bodies. New York: Academic Press. 11. Simitses, G. J. (1990). Dynamic stability of suddenly loaded structures. Berlin: Springer. 12. Hodges, D. H., & Pierce, G. A. (2002). Introduction to structural dynamics and aeroelasticity. Cambridge: Cambridge University Press. 13. Xie, W.-C. (2006). Dynamic stability of structures. Cambridge: Cambridge University Press. 14. Blevins, R. D. (1977). Flow-induced vibration. New York: Van Nostrand Reinhold Company. 15. Païdousis, M. P. (1998). Fluid-structure interactions, slender structures and axial flow (Vol. 1). London: Academic Press. 16. Païdousis, M. P. (2004). Fluid-structure interactions, slender structures and axial flow (Vol. 2). London: Elsevier-Academic Press. 17. Bolotin, V. V. (1964). The dynamic stability of elastic systems. San Francisco: Holden-Day Inc. 18. Bolotin, V. V. (1963). Nonconservative problems of the theory of elastic stability. New York: Pergamon Press. 19. Ziegler, H. (1968). Principles of structural stability. Massachusetts: Balisdell Publishing Co. 20. Kirillov, O. N. (2013). Nonconservative stability problems of modern physics. Berlin: Walter de Gruyter.

Chapter 2

Columns under Conservative Forces

Leonard Euler (1707–1783) was arguably the foremost mathematical scientist of the 18th century. Euler and Daniel Bernoulli established the basic theory of beams and columns. The structural stability of columns under a compressive (conservative) force was first studied by Euler in 1744 [1, 2]. This chapter discusses columns from the viewpoint of dynamic stability.

2.1

Cantilevered Columns

Euler obtained the expression for the buckling load of a uniform elastic column with one end clamped and the other free, as shown in Fig. 2.1 [1–3]. The cantilevered column is referred to as Euler’s column. Buckling is a structural engineering term used to express sudden lateral failure of a structure due to compressive loading. Buckling may also be termed loss of static stability, static instability, or bifurcation of the static equilibrium. A vertical force is assumed to be applied to the column at its free end. The force is a dead load. The buckling load P
is well known from textbooks on the strength of materials and is given by the formula [1–3]: P
¼

p2 EI : 4 L2

ð2:1Þ

It is noted that instability—also static instability—is essentially a dynamic (that is, a time-dependent) process. Let us consider the dynamics of Euler’s column

© Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_2

13

14

2 Columns under Conservative Forces

Fig. 2.1 Euler’s column

through an electronic analog computer-based experiment (analog experiment) [4]. It is assumed that the column is subjected to a vertical force at its free end during its bending motion. The equation of motion is given by m


 @2y @2y @2 @2y þ P þ EI ¼ 0: @t2 @x2 @x2 @x2

ð2:2Þ

This equation can be expressed in the following set of equations: m

@2y @2 ¼  ½Q; @t2 @x2

ð2:3Þ

where Q ¼ M þ Py; M ¼ EI

@2y : @x2

ð2:4Þ

To discretize the equation of motion, we rely here on the finite difference method (FDM). Let us divide the column into N cell elements of length Dx ¼ L=N and take the collocation point to be at the center of each element, as shown in Fig. 2.2.

2.1 Cantilevered Columns

15

Fig. 2.2 Column divided into N cell elements

Central finite differences give the following approximate formulae of the differential expressions: dyi 1 yi þ 1  yi yi  yi1  1  þ ðyi þ 1  yi1 Þ; ¼ dx 2 " Dx Dx # 2Dx     d 2 yi 1 dy dy 1   ¼ 2 ðyi þ 1  2yi þ yi1 Þ: Dx dx i þ 1 dx i1 Dx dx2 2

ð2:5Þ

2

Applying these expressions in the partial differential equation (2.3), we obtain a system of ordinary differential equations d2 y 1 ¼ ðQi þ 1  2Qi þ Qi1 Þ; dt2 mDx2

i ¼ 1; 2; . . .; N:

After two time integrations, we obtain ZZ 1 ðQi þ 1  2Qi þ Qi1 Þdtdt þ C1 t þ C2 ; yi ¼  mDx2
 EI Dx2 Pyi ; i ¼ 1; 2; . . .; N: Qi ¼ 2 yi þ 1  2yi þ yi1 þ Dx EI

ð2:6Þ

ð2:7Þ

The corresponding boundary conditions are given by: For the clamped end: y ¼ @y @x ¼ 0, at x ¼ 0, y1 ¼ y1 ¼ 0 at the station 1=2:

ð2:8Þ

For the free end: M ¼ 0; @Q @x ¼ 0, at x ¼ L, MN þ MN þ 1 ¼ 0; QN þ 1  QN ¼ 0; at the station N þ 1=2:

ð2:9Þ

Figure 2.3 shows the electronic circuits for the relaxed differential equations for Euler’s column, with N ¼ 6. The circuits are composed of integrators, summing amplifiers, and multipliers, together with potentiometers. The electronic circuit is said to be equivalent to the mechanical system, as the governing equations are identical. The reduced dynamical systems have five degrees of freedom, as one end

16

2 Columns under Conservative Forces

Fig. 2.3 Analog computer circuits for Euler’s columns

(a) Circuits

(b) Symbols and functions

is fixed, and so the deflection of element at the end vanishes. As for the accuracy of the 6-element-FDM approximation, the difference between the approximate and the exact value of the first natural frequency is about 0.5%. For the second natural

2.1 Cantilevered Columns

(a)

(d)

17

(b)

(c)

(e)

(f)

Fig. 2.4 Records of vibrations observed in the analog computer-based experiment of Euler’s column [4]

frequency, the difference is about 2.0%. The electronic circuit was realized in the Hitachi ALS-45 analog computer. Figure 2.4 shows records of the vibrations observed in the analog experiment (simulation) of Euler’s column with increasing loading parameters, where P
is the Euler buckling load. It is seen that the amplitude of the harmonic vibration is constant as long as the load is less than the critical value, and that the first eigenfrequency decreases with increasing load parameter. Finally, as the frequency vanishes, the motion becomes divergent. It is interesting to watch the dynamic behavior shown in Fig. 2.4e, as it appears to be non-harmonic. It is noted that the effective elastic restoring force is vanishing just at the vicinity of the (sub-) critical value, and so the dynamics of the column is sensitive to small disturbances. In the case of the present simulation with an old-fashioned analog computer, small noises in electronic components act as disturbances for the dynamics, just as “in the real world”. If one is to conduct a numerical simulation with a modern computer, one will likely miss such a non-harmonic dynamic response in the vicinity of the critical load. The frequency was read from the record of the vibrations. The relation between the first eigenfrequency x and the load P is shown in Fig. 2.5, where x
1 is the first natural frequency of a cantilevered beam. The straight line suggests that mechanical energy is conserved. Figures 2.4 and 2.5 indicate that the loss of structural stability in Euler’s column is attributable to a loss of the elastic restoring force. Let us now discuss the realizability of the free end of Euler’s column. No method for realizing the free end of Euler’s column has been devised for studying its dynamic stability experimentally, to the best of the authors’ knowledge. In order to conduct an experiment on the dynamics of Euler’s column, we have to design a modified Euler’s column, having a rigid body at its free end as a loading device, and take the mass of the loading device into account. Simple equipment for studying the dynamics of Euler’s column with a tip rigid body was demonstrated, for example, by Pippard in his book [5].

18

2 Columns under Conservative Forces

Fig. 2.5 Relation between the first eigenfrequency and the load [4]

2.2

Pinned-Pinned Columns

Now let us consider a column pinned (simply supported) at both ends, as shown in Fig. 2.6. This column is a popular model for discussing the buckling of structures [2, 3]. The equation of motion is given by m

@2y @2y @4y þ P 2 þ EI 4 ¼ 0: 2 @t @x @x

ð2:10Þ

The boundary conditions are yð0; tÞ ¼ 0;

@ 2 yð0; tÞ ¼ 0; @x2

Fig. 2.6 Pinned-pinned column

yðL; tÞ ¼ 0;

@ 2 yðL; tÞ ¼ 0: @x2

ð2:11Þ

2.2 Pinned-Pinned Columns

19

The general solution to this boundary-value problem, Eqs. (2.10) and (2.11), is given by yðx; tÞ ¼ Aeixt sin

npx ; L

n ¼ 1; 2; . . .

ð2:12Þ

Now let us consider the dynamics of the fundamental (first) mode, that is, the first natural (eigen-) frequency x1 , for n ¼ 1: Substitution of Eq. (2.12) into Eq. (2.10) results in the relation x21 P þ ¼ 1; 2 x01 P
1

ð2:13Þ

where x01 is the first eigen frequency when P ¼ 0 and P
1 is the first buckling load. These parameters are given by x201 ¼

p4 EI ; L4 m

P
1 ¼

p2 EI : L2

ð2:14Þ

The relation (2.13) suggests conservation of mechanical energy. The first term corresponds to the kinetic energy, while the second term corresponds to the potential energy. Equation (2.13) can be written in the form x1 ¼ x
1

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 1 : P
1

ð2:15Þ

This equation shows that the eigenfrequency x1 is a real number, and that it decreases as the compressive load P increases, for P\P
1 . When P ¼ P
1 , x1 vanishes. When P [ P
1 , the frequency x1 becomes imaginary, x1 ¼ ir, where r is a positive real number. Equation (2.12) then shows that motion of the column is exponentially divergent. Now let us discuss the case in which x1 ¼ 0. If x1 [ 0, the period of vibration with cyclic frequency x1 is given by T ¼ 2p=x1 . But if x1 ¼ 0, T becomes infinite. This implies that, if the column is forced to move by a small dynamic disturbance, it will return to the initial position only after infinite time. In reality, once it moves out, it does not return. This type of instability phenomenon is called “divergence” . Divergence is defined as a monotonically increasing (non-vibrating) motion of a mechanical system, departing from its initial equilibrium state, in the regime of small motion. It is noted that the motion of the column at the critical load is dynamically unstable. However, if the column is treated within the framework of static stability, in which only static disturbances are considered, it is defined as neutrally stable at the buckling load.

20

2.3

2 Columns under Conservative Forces

Standing Cantilevered Columns

The stability of elastic columns subjected to a distributed vertical force, due to self-weight, is one of the classical topics in the theory of elastic stability [2, 3]. Now let us discuss the dynamics of a long uniform column under a distributed axial force, as shown in Fig. 2.7. The equation of motion of an elastic column subjected to a distributed axial force q per unit length is given by m

@2y @4y @2y @y þ EI þ q ð L  x Þ  q ¼ 0: 2 4 2 @t @x @x @x

ð2:16Þ

The boundary conditions are yð0; tÞ ¼ 0;

@yð0; tÞ ¼ 0; @x

EI

@ 2 yðL; tÞ ¼ 0; @x2

EI

@ 3 yðL; tÞ ¼ 0: @x3

ð2:17Þ

The following nondimensional parameters are defined: g¼

qL3 ; EI

n¼

mx2 L4 qx2 L4 ¼ ; EI gEI

ð2:18Þ

where g is the acceleration due to gravity. Now let us consider an experiment with a vertical cantilever. The vibration and stability of flexible vertical bars can be observed easily by holding a long rubber cylinder vertically upward and making its length longer and longer, step by step. Eventually, the bar will not be able to keep its straight form and will bend.

Fig. 2.7 Standing cantilevered column under self-weight

2.3 Standing Cantilevered Columns

21

This simple experiment is satisfactory for obtaining a qualitative understanding of the stability of a vertical cantilever. However, the quantitative experimental validation is not so simple, as straight, uniform test specimens are needed. In this section, we will discuss a test of a vertical cantilever. Two long (straight) cylindrical brass bars, which were treated with hot-stretching so as to maintain their straightness, were selected as test specimens. They had a nominal diameter of 3.0 mm, and their maximum length was 2000 mm. The measured values of self-weight per unit length q and bending stiffness EI are shown in Table 2.1. A sketch of the test-setup is shown in Fig. 2.8. The motion of the specimen was restricted to one plane by thin cotton strings. A specimen 100 mm shorter than the predicted divergence length was mounted vertically upward in a support block, with the lower end clamped. The specimen was checked and adjusted to keep its configuration straight and vertical by controlling three screw legs on the support block. After these careful preparations, the length of the specimen was increased in small steps up to the critical length giving divergence. The experimental divergence length was found to be Ldes ¼ 1:88  103 mm, while the theoretical prediction was Ldt ¼ 1:87  103 mm. It is noted that this experimental length Ldes was obtained by the static method. At the critical length, the specimen could not maintain its vertically straight configuration. The straight and divergent configurations observed in the test are shown in Fig. 2.9a, b. Figure 2.9c shows the first mode of divergence predicted by theory. As mentioned earlier, the test specimen was loosely constrained by two cotton strings. The purpose of these strings was to prevent large deflection of the test specimen, even at divergence. The specimen was thus not subjected to plastic deformation. Table 2.1 Measured values of q and EI

Fig. 2.8 Schematic of the test setup

Test specimen No. 1 Test specimen No. 2

q (N/mm)

EI (N mm2)

5.91  10−4 5.90  10−4

4.94  105 4.91  105

22

2 Columns under Conservative Forces

(a) Stable column

(b) Buckled column

(c) Bucked configuration by theory

Fig. 2.9 Configurations of standing columns [6] (Some vertical strings in the photographs are the vertical reference lines)

Fig. 2.10 Relations between dimensionless load parameter η and frequency parameter n [6]

Now let us proceed to the second phase of the test, to find the relation between the length and the frequency of the vertically standing column. The length of the specimen was shortened in appropriate steps. The mean period of transverse oscillation was measured by a stopwatch at different lengths of the specimen. The theoretical and experimental relations between the load parameter η and the frequency parameter n, defined in Eq. (2.18), are plotted in Fig. 2.10. The solid line in Fig. 2.10 was obtained by a numerical method, as described in reference [6]. The line is almost straight, but not exactly. For engineering purposes, it is reasonable to

2.3 Standing Cantilevered Columns

23

assume that it is straight. The extrapolation of the test results cuts the axis n ¼ 0: The intersection predicts the experimental divergence limit by the dynamic method, Lded ¼ 1:88  103 mm.

2.4

Discussion

This chapter has discussed three classical buckling problems of columns from the viewpoint of dynamic stability. The nature of the stability of columns under conservative forces can be understood correctly only if they are discussed within the context of the time domain. It has been demonstrated that the columns are unstable in divergence at the buckling load, which can be obtained through the static method.

References 1. Timoshenko, S. P. (1953). History of strength of materials. New York: McGraw-Hill. (1983, Dover Publications, Inc. New York). 2. Timoshenko, S. P., & Gere J. M. (1961). Theory of elastic stability. New York: McGraw-Hill. (2009, Dover Publications, Inc., New York) 3. Pflüger, A. (1964). Stabiliätsprobleme der Elastostatik. Berlin: Springer. 4. Sugiyama, Y., Fujiwara, N., & Sekiya, T. (1969). Studies on nonconservative problems of instability of columns by means of analog computer. In Proceedings of the 18th Japan National Congress for Applied Mechanics, 1968 (pp. 113–126). Tokyo: The University of Tokyo Press. 5. Pippard, A. B. (1985). Response and stability. Cambridge: Cambridge University Press. 6. Sugiyama, Y., Ashida, K., & Kawagoe, H. (1978). Buckling of long columns under their own weight. Bulletin of the Japan Society of Mechanical Engineers, 21(158), 1228–1235.

Chapter 3

Columns under a Follower Force

An elastic cantilevered beam subjected to a follower force, the so-called Beck’s column, is an ideal/classical structural model in the theory of nonconservative stability problems [1]. The stability of columns associated with follower forces is a relatively new topic in the field of structural stability, at least in comparison with the part dealing with conservative forces. This chapter discusses the basic aspects of a column under a follower force, its positive and negative features.

3.1

Beck’s Column

Let us consider a cantilevered column as shown in Fig. 3.1. It is assumed that the column is subjected to a follower force, having a line of action that is tangential to the tip end of the column. The nonconservativeness of a force can be verified by showing that the force does a different amount of work when taking different paths. The nonconservative nature of a follower force will be discussed in Sect. 3.4. The column is referred to as Beck’s column, named after Max Beck who evaluated first the critical force in 1952 by carrying out a dynamic stability analysis [1–3]. The equation of motion of the column is given by m

@2y @2y @4y þ P 2 þ EI 4 ¼ 0: 2 @t @x @x

ð3:1Þ

The boundary conditions are yð0; tÞ ¼ 0; @ 2 yðL; tÞ EI ¼ 0; @x2

@yð0; tÞ ¼ 0; @x @ 3 yðL; tÞ EI ¼ 0: @x3

© Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_3

ð3:2Þ

25

26

3 Columns under a Follower Force

Fig. 3.1 Beck’s column

The solution to Eq. (3.1) is written as yðx; tÞ ¼ Y ð xÞeixt :

ð3:3Þ

Now let us introduce the nondimensional quantities x z¼ ; L

p¼

PL2 ; EI

n¼

mx2 L4 : EI

ð3:4Þ

Inserting Eq. (3.3) into Eqs. (3.1) and (3.2), with the notations Eq. (3.4), we obtain the ordinary differential equation nY þ p

d2 Y d4 Y þ 4 ¼ 0; 2 dz dz

ð3:5Þ

and the boundary conditions Y ð0Þ ¼ 0;

dY ð0Þ ¼ 0; dz

d2 Y ð 1Þ ¼ 0; dz2

d3 Y ð 1Þ ¼ 0: dz3

ð3:6Þ

The general solution to the ordinary differential equation (3.5) is given by Y ðzÞ ¼ C1 sin r1 z þ C2 cos r1 z þ C3 sinh r2 z þ C4 cosh r2 z;

ð3:7Þ

where r12 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p þ p2 þ 4n ; 2

r22 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p þ p2 þ 4n : 2

ð3:8Þ

3.1 Beck’s Column

27

The condition for the solution Eq. (3.7) to satisfy the boundary conditions Eq. (3.6), with the condition for the existence of a nontrivial solution, yields the characteristic equation for the eigenvalues ðp; nÞ detðp; nÞ ¼ p2 þ 2n þ p

pffiffiffi n sin r1 sinh r2 þ 2n cos r1 cosh r2 ¼ 0:

ð3:9Þ

Figure 3.2a reproduces the eigenvalue curve in Beck’s original paper, in which the first two eigenvalues are evaluated. The first six eigenvalues ni (i ¼ 1; 2; . . .; 6) are shown in Fig. 3.2b. Here n0i denotes the ith eigenvalue of the unloaded cantilever. With an increasing force parameter p, the first eigenvalue n1 increases, while the second n2 decreases, and finally, the two coincide at the critical force, gc ¼ 20:05. This value was first obtained numerically by Beck. Although there are other multiple coincidence points between the third and fourth, the fifth and sixth, and so on (see Fig. 3.2b), it is the lowest coincidence point that gives the fundamental flutter point. The column loses its stability by flutter in two-mode-coupling. The flutter limit P is given by P ¼ 20:05

EI : L2

ð3:10Þ

It is noted that the eigenvalues ðp; nÞ are plotted in Fig. 3.2 only when the nondimensional frequency n takes a positive real value, when p  p . When p [ p , the nondimensional frequency n becomes complex, and thus the frequency x becomes complex as well and takes the form x ¼ a  ib (with positive a and b). Expression (3.3) is then written as yðx; tÞ ¼ Yð xÞebt eiat :

ð3:11Þ

Expression (3.11) represents a vibration with an increasing amplitude and thus dynamic instability when p [ p , as one of the characteristic roots has a positive

Fig. 3.2 Eigenvalue curves for Beck’s column. a Eigen-value curve by Beck [2], b behavior of the first six eigenvalues [4]

28

3 Columns under a Follower Force

real part. In this section we simply followed a classical method as applied by Beck. However, to understand strictly the dynamic stability of Beck’s column, we shall start with the solution to Eq. (3.1) in the form yðx; tÞ ¼ Yð xÞekt :

ð3:12Þ

In this case, the eigenvalues will be plotted in the complex plane, k ¼ ReðkÞ  iImðkÞ ¼ r  ix, that is, the so-called Argand diagram.

3.2

Vibrations of Beck’s Column

Now let us discuss the dynamic behavior of Beck’s column in an analog computer-based experiment [5]. The basic procedure is the same as described in Chap. 2, except for the circuit for the free end, which is slightly different from the case for Euler’s column. Figure 3.3 shows the record of vibrations observed in the analog computer-based experiment (simulation) of Beck’s column. The vibrations in the case of Beck’s column with an increasing load are quite different from those in the case of Euler’s column. The clear harmonic motion at P = 0 is modified as the load parameter increases. Beating takes place at a subcritical state (Fig. 3.3e). Finally, the amplitude of the vibration becomes very large in flutter motion. Figure 3.3f shows that Beck’s column gains additional kinetic energy once the load exceeds the flutter limit. The surplus kinetic energy is attributable to the work done by the applied follower force. This is the clearest evidence that the force does work and is

Fig. 3.3 Vibrations observed in the analog experiment for Beck’s column [5]

3.2 Vibrations of Beck’s Column

29

Fig. 3.4 Relation between the eigenfrequencies and the load [5]

nonconservative. The frequencies read on the record are plotted in Fig. 3.4 and compared with Beck’s result, where x1 and x2 are the first and second eigenfrequency, respectively. Figures 3.3 and 3.4 demonstrate how the column loses its stability by internal resonance due to coincidence of the first and second eigenfrequencies.

3.3

Stability in a Finite Time Interval

Although the concept of asymptotic stability is defined in terms of infinite time, it is impossible to treat stability behavior in an infinite time in practice, say, in simulations and experiments. The following definition of instability, in relation to stability in a finite time interval [6], is applied in the present simulation: the vibrations of the column are unstable if the deflection of the disturbed column exceeds ten times the initial disturbance (deflection) during the operation of the analog computer. This definition can be justified, firstly because it means instability from an engineering point of view. Secondly, even if we adopt a stronger criterion (threshold) than the ten-times criterion, for example, the hundred-times criterion, the critical value (in experiment and simulation) would not be improved considerably in practice. Figure 3.5 shows a record of motions of Beck’s column in the vicinity of the critical load. It is seen that the difference of the critical value by the ten-times and the value by the hundred-times criteria is very small, just a 0.3% increase in the case of the latter criterion.

30

3 Columns under a Follower Force

Fig. 3.5 Stability in a finite time interval

3.4

Character of Beck’s Column

After Beck’s paper, the theory of structural stability of nonconservative systems was developed extensively, with a large number of papers on Beck’s column and related topics. The progress of nonconservative stability problems of structures under follower forces has been complied in books [6–8] and discussed in review papers [9–12]. However, Beck’s result, as well as other related results, have not always been appreciated by structural engineers. Some have a negative idea of the concept of follower forces [13], while some are more positive [14, 15]. How realistic is Beck’s column, the elastic cantilever with a follower force? At present, it is recognized that Beck’s column is not realistic in the sense that no mechanism has been proposed by means of which this column-force system can be reproduced in the laboratory. Beck’s column suggests only one thing: if a tangential follower force is applied to an elastic cantilever, the column does not buckle, but rather loses stability by flutter in two-mode-coupling. Beck simply demonstrated the necessity of applying a dynamic criterion to determine the critical load of the column. Beck’s column incorporated the area of structural stability as a branch of the general stability theory of dynamical systems. Another weak point of Beck’s column, it may be said, is that internal (material) damping is neglected. When one deals with the dynamics of structures, it is optional to take damping into account. Beck’s column may lack physical significance in the area of the dynamic stability of structures and it may not be qualified as a physically significant structural model. Then is Beck’s column worthless for structural dynamists to deal with? Let us recall the concept of an ideal fluid, in which viscosity is neglected, in the theory of fluid dynamics. Knowing that elementary courses in fluid dynamics usually begin with an ideal fluid, not with real fluid with viscosity, together with a clear statement of the limitation of the ideal fluid, we may refer to Beck’s column as an ideal column in the theory of nonconservative stability problems. The column can be justified only when its limitations are described clearly. It is noted that Beck wrote in his paper [2, in German]; “The result found here is, as yet, valid only for an ideal, damping-free column”. Ziegler [16] has shown that, “in the case of nonconservative systems, even the smallest damping can, in certain circumstances, lower the critical load significantly. By the present problem, it is however not easy

3.4 Character of Beck’s Column

31

to find a physically flawless formulation (model) for the damping which does not complicate the mathematical problem significantly.” As internal (structural) damping is neglected in Beck’s column, the column is conservative when the applied follower force is below the critical value. This can be understood easily, as the dynamics of the column has its solution in the form of real harmonic functions when the force is below the flutter limit. This fact implies that there is no energy input and output (for more detailed discussions, refer to Sect. 5.4). Ziegler [7] was the first to explain the essential features of the elastic cantilever subjected to a tangential follower force system known as Beck’s column. Despite the presence of Beck’s short paper published in 1952 [2], Ziegler, in his longer paper of 1952 [16], applied the dynamic stability criterion to a double pendulum loaded by a tangential follower force and thoroughly discussed the basic characteristics of nonconservative stability problems. At present, no one knows how Ziegler and Beck imagined a tangential follower force. However, knowing that sequential photographs of body-bending flutter due to an end rocket thrust, as shown in Fig. 3.6, was published openly in 1956 [17], it is probable that they might have imagined the force as the end thrust in rocket-propelled flight vehicles and missiles. The dynamic stability of a free-free rocket-propelled flying beam was investigated by Feodosiev [18] (the beam is referred to as Feodosiev’s beam), Beal [19], Wu [20, 21] and Wu et al. [22]. The relation between Beck’s column and a free-free rocket-propelled beam was demonstrated by Ohshima and Sugiyama [23], as shown in Fig. 3.7. The effect of follower forces on the aeroelastic stability of flexible structures was investigated by Chae [24]. An experimental approach to a

Fig. 3.6 Sequential photographs of the bending flutter of a free-free flying beam propelled by end rocket thrust [17] (reproduced by kind permission of The Royal Aeronautical Society’s Aeronautical Journal)

32

3 Columns under a Follower Force

Fig. 3.7 Relation between a free-free rocket-propelled beam and Beck’s column [23]

rocket-propelled flying beam is evidently not an easy task. Theoretical and experimental approaches to modified versions of Beck’s column may give a qualitative perspective of the dynamics and stability of the rocket-propelled flying beam.

3.5

Nonconservative Nature of a Follower Force

Following Bolotin’s thought experiment [1], let us discuss the nonconservative nature of a follower force by considering a cantilevered column subjected to such a force, as shown in Fig. 3.8. It is assumed that the force follows the deflection of the column such that its direction remains tangential to the tip deflection of the column. The angle of the tangent to the vertical is denoted by u. It is assumed here again that the column is elastic and is dealt with in the regime of linearized beam theory. To prove the nonconservative nature of a follower force it is necessary to show that the work done by the force is path-dependent. Figure 3.8a shows the initial state of a column under a follower force, while Fig. 3.8b shows the deformed state and Fig. 3.8c the final state. Now let us discuss the work done by the applied force P on a path taken from the initial/starting point O to the final/end point A. Figure 3.9 shows two different paths from the initial position O to the final one A. It is seen that there are many possible paths from point O to point A. The work done by the force is given by WP ¼ Wx þ Wy ;

ð3:13Þ

where WP is the total work done by the applied force P, Wx is the work done by the x component of the force, Px ¼ P cos u  P, and Wy the work done by the y component, Py ¼ P sin u  Pu.

3.5 Nonconservative Nature of a Follower Force

33

Fig. 3.8 Column under a tangential follower force

Fig. 3.9 Paths taken by the force

If one can find at least two arbitrary paths on which the work done by the force depends, this is sufficient to show that the force has no potential, and so is nonconservative. Now let us consider the two paths in Fig. 6.9: Path I : Point O to B ðu ¼ 0Þ and then B to A: Path II : Point O to C ðy ¼ 0Þ and then C to A ðu ¼ uA Þ:

34

3 Columns under a Follower Force

We need now to recall the displacement of point A in the x direction, which is given by 1 d¼ 2

ZL  2 dy dx: dx

ð3:14Þ

O

It is noted that the displacement y and the slope u are small quantities of first order, while the displacement in the x direction d is a small quantity of second order. This is the reason why d is not shown in Fig. 3.8. Thus, one obtains Wx ¼ Pd; ZyA Wy ¼ P

udy:

ð3:15Þ

o

It is understood that Wx is the work done by the vertical (conservative) component of the force P, while Wy is the work by the lateral (nonconservative) component, as the former does not depend on the path taken, while the latter does. If the path takes the route from O to A via B, then the work Wy is negative, and if it goes on O to A via C, the work is zero. Let us consider another path: Path III: Point O to D, D to E ðu ¼ uA Þ; and E to B to A ðy ¼ yA Þ: It is found that Wy is positive. It has thus been shown that a follower force possesses a nonconservative nature. Now let us imagine a quasi-static motion of the column. The present simple thought experiment suggests that if the angle u and the tip deflection y are in phase, then the force does negative work, and therefore the force stabilizes the column, while if they are out of phase, the force does positive work, and thus it destabilizes the column. Imagine that the column moves, in one way, from O to A in the first mode, then the column is stabilized by the force, while in the second mode, it may be destabilized. This suggests that the column subjected to a follower force would not lose its stability in the first mode. One more suggestion obtained by this thought experiment. When the vibration configuration of the column takes place in a fixed mode and is symmetrical around the undeformed x-axis, then the work done by the force vanishes in one cycle of oscillation. To break this symmetry of the vibration configuration, that is, to restore the nonconservative nature of a follower force, it is necessary to introduce damping to the column. The role of damping in the stability of Beck’s column will be discussed in detail in Chaps. 4 and 5.

3.6 Discussion

3.6

35

Discussion

It is worth noting that a follower force is produced not only by jet or rocket thrust, but also by dry friction. The squeal generated in rotating disks, such as that caused by automobile disk brakes, is another type of dynamic instability due to a follower force. The force is caused by dry friction between the rotating disk and the brake pad. The onset of brake squeal is equivalent to the passage through the stability limit in Beck’s column [25–28] . The concept of friction-induced follower forces is thus of importance in the design of brake systems for transportation vehicles. Pre-twisted cantilevered beams subjected to a distributed follower force are another engineering application, found in the case of a drill bit during the drilling process [29].

References 1. Bolotin, V. V. (1963). Nonconservative problems of the theory of elastic stability. New York: Pergamon Press. 2. Beck, M. (1952). Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes. Zeitschrift für Angewandte Mathematik und Physik, 3, 225–228. 3. Timoshenko, S. P., & Gere, J. M. (1961). Theory of elastic stability. New York: McGraw-Hill (New York: Dover Publications, Inc., 2009). 4. Sugiyama, Y., Katayama, T., & Sekiya, T. (1970). Studies on nonconservative problems of instability of columns by difference method. In Proceedings of the 19th Japan National Congress for Applied Mechanics 1969 (pp. 23–31). Tokyo: The University of Tokyo Press. 5. Sugiyama, Y., Fujiwara, N., & Sekiya, T. (1969). Studies on nonconservative problems of instability of columns by means of analog computer. In Proceedings of the 18th Japan National Congress for Applied Mechanics 1968 (pp. 113–126). Tokyo: The University of Tokyo Press. 6. Leipholz, H. (1970). Stability theory. New York: Academic Press. 7. Ziegler, H. (1968). Principles of structural stability. Massachusetts: Blaisdell Publishing Co. 8. Leipholz, H. (1980). Stability of elastic systems. Alphen aan den Rijn: Sijthoff & Noordhoff. 9. Langthjem, M. A., & Sugiyama, Y. (2000). Dynamic stability of columns subjected to follower loads: A survey. Journal of Sound and Vibration, 238(5), 809–851. 10. Elishakoff, I. (2005). Controversy associated with the so-called follower forces. Applied Mechanics Review, 58, 117–142. 11. Datta, P. K. (2011). Aeroelastic behaviour of aerospace structural elements with follower force: A review. International Journal of Aeronautical and Space Sciences, Korean Society of Aeronautical and Space Sciences, 12(2), 134–148. 12. Venkateswara Rao, G., & Singh, G. (2001). Revisit to the stability of a uniform cantilever column subjected to Euler and Beck loads—Effect of realistic follower forces. Indian Journal of Engineering and Materials Sciences, 8, 123–128. 13. Koiter, W. T. (1996). Unrealistic follower forces. Journal of Sound and Vibration, 194(4), 636–638. 14. Sugiyama, Y., Langthjem, M. A., & Ryu, B.-J. (1999). Realistic follower forces. Journal of Sound and Vibration, 225(4), 779–782. 15. Sugiyama, Y., Ryu, S.-U., & Langthjem, M. A. (2002). Beck’s column as the ugly duckling. Journal of Sound and Vibration, 254(2), 407–410. 16. Ziegler, H. (1952). Die Stabilitätskrierien der Elastomechnik. Ingeniur-Archiv, 20, 45–56.

36

3 Columns under a Follower Force

17. Farrar, D. J. (1956). Structures. The Aeronautical Journal, the Royal Aeronautical Society, 60, 712–720. 18. Feodosiev, V. I. (1965). On a problem of stability. Prikladnaya Mathematica i Mekhanika, 29, 391–392. (in Russian). 19. Beal, T. R. (1965). Dynamic stability of a flexible missile under the constant and pulsating thrust. AIAA Journal, 3, 486–494. 20. Wu, J. J. (1975). On the stability of a beam under axial thrust subjected to directional control. Journal of Sound and Vibration, 42, 45–52. 21. Wu, J. J. (1976). On missile stability. Journal of Sound and Vibration, 49, 141–147. 22. Wu, L., Xie, C., & Yang, C. (2012). Aeroelastic stability of a slender missile with constant thrust. Procedia Engineering, 31, 128–135. 23. Ohshima, T., & Sugiyama, Y. (2003). Dynamic stability of free-free beam subjected to end rocket thrust and carrying a heavy payload at its nose. In Proceedings of 2003 International Conference on Physics and Control (PhysCon 2003) (Vol. 4, pp. 1115–1120). St. Petersburg. 24. Chae, S. (2004). Effect of follower forces on aeroelastic stability of flexible structures, (Doctoral Thesis). School of Aerospace Engineering, Georgia Institute of Technology, ProQuest Information and Learning Company, Ann Arbor. 25. Nishiwaki, M. (1993). Generalized theory of brake noise. Proceedings of the Institution of Mechanical Engineers, 207, 195–202. 26. Mottershead, J. E., & Chan, S. N. (1995). Flutter instability of circular discs with frictional follower loads. Journal of Vibration and Acoustics, 117, 161–163. 27. Bigoni, D., & Noselli, G. (2011). Experimental evidence of flutter and divergence instabilities induced by dry friction. Journal of the Mechanics and Physics of Solids, 59(10), 2208–2226. 28. Kurnik, W., Przybylowicz, P. M., & Bogacz, W. (2018). Bigoni and Noseli’s experiment: Is it evidence for flutter in the Ziegler column? Archive of Applied Mechanics, 88, 203–213. 29. Fazelzadeh, S. A., Karimi-Nobandegani, A., & Mardanpour, P. (2017). Dynamic stability of pretwisted cantilever beams subjected to distributed follower force. AIAA Journal, 55(5), 955–964.

Chapter 4

Columns with Damping

Ziegler discovered in 1952 that the introduction of damping in an elastic system under a follower force may have a destabilizing effect [1]. This caused a great deal of interest among structural dynamists, interest that has continued to date [2–16]. This chapter discusses the effect of internal damping on the flutter limit of a cantilevered column under a follower force. In place of Beck’s column, we consider Pflüger’s column, which has a tip mass [7, 8]. This chapter deals with a Pflüger’s column with internal (Kelvin-Voigt type) and external damping.

4.1

Cantilevered Columns with Damping

Let us consider a column as shown in Fig. 4.1. It is assumed that the cantilevered column has a concentrated mass M at its free end. This type of column, with no damping, was first considered by Pflüger and is referred to as Pflüger’s column. The equation of motion for the column shown in Fig. 4.1 is given by m

@2y @y @5y @2y @4y þ E I þC þ P 2 þ EI 4 ¼ 0: 2 4 @t @t @t@x @x @x

ð4:1Þ

The boundary conditions are @yð0; tÞ ¼ 0; @x @ 3 yðL; tÞ @ 2 yðL; tÞ E I þ EI ¼ 0; @t@x2 @x2 yð0; tÞ ¼ 0;

E I

@ 4 yðL; tÞ @ 3 yðL; tÞ @ 2 yðL; tÞ þ EI ¼ M : @t@x3 @x3 @t2 ð4:2Þ

Equations (4.1) and (4.2) can be brought into nondimensional forms by making use of the nondimensional quantities © Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_4

37

38

4 Columns with Damping

Fig. 4.1 Pflüger’s column with internal and external damping

x n¼ ; L j¼

y g¼ ; L CL2

ðmEI Þ1=2

;

 1 t EI 2 s¼ 2 ; L m  1 E  EI 2 c¼ 2 : EL m

p¼

PL2 ; EI

l¼

M ; mL

ð4:3Þ

The nondimensional equation of motion is written as @2g @g @5g @2g @4g þc þj þ p 2 þ 4 ¼ 0; 4 2 @s @s @s@n @n @n

ð4:4Þ

while the boundary conditions become @gð0; sÞ ¼ 0; @n @ 3 gð1; sÞ @ 2 gð1; sÞ c þ ¼ 0; @s@n2 @n2 gð0; sÞ ¼ 0;

4.2

@ 4 gð1; sÞ @ 3 gð1; sÞ @ 2 gð1; sÞ c þ ¼ l : @s2 @s@n3 @n3

ð4:5Þ

Stability Analysis

The dynamical system (4.4) and (4.5) can be converted into a set of ordinary differential equations by applying the standard finite element method. The characteristic equation is ultimately obtained in matrix form [5]: Mvss þ Cvs þ Kv ¼ 0;

ð4:6Þ

where M is a symmetric mass matrix, C a symmetric damping matrix, and K ¼ Ks þ Kl , where Ks is a symmetric stiffness matrix and Kl is a non-symmetric load matrix. The displacement field v is assumed in the form

4.2 Stability Analysis

39

vðsÞ ¼ xeks :

ð4:7Þ

With Eq. (4.7), the characteristic equation (4.6) can be written as the standard eigenvalue problem kz ¼ Az; with

  x z¼ ; x_

 0 I ; A¼ M1 K M1 C

ð4:8Þ



ð4:9Þ

where x_ ¼ @x=@s, and 0 and I are the zero matrix and the unit matrix, respectively. The characteristic root k ¼ r  ix can be obtained by using, for example, the QR-algorithm. Within the concept of asymptotic stability, the stability of the dynamical system is determined by the sign of the real part of k, which is r. If r [ 0, the system is unstable, while if r\0, the system is stable. The flutter limit is given by the loading parameter p at which r ¼ 0 with x 6¼ 0, as sketched in Fig. 1.2.

4.3

Beck’s Column with Damping Introduced

Now let us consider Beck’s column with internal and external damping (with no tip mass: l ¼ 0). This section reproduces and reviews some well-known basic results which are, however, essential for further discussions, especially in the subsequent Chap. 5, on the role of damping in stability analysis of a damped Beck’s column.

Fig. 4.2 Critical flutter load and frequency when only internal damping is introduced (j ¼ 0)

40

4.3.1

4 Columns with Damping

Internal Damping Only

Figure 4.2 shows the critical (flutter) load pcr and the corresponding flutter
fre6 quency xcr as functions of the internal damping parameter c 10  c  0:1 , with no external damping ðj ¼ 0Þ. It is seen that both the flutter load and frequency remain practically constant by small c-value—over four decades, 106  c.0:01. Within this range, pcr  10:94 and xcr  5:40. It is noted that in the critical load for an undamped column, that is, Beck’s column, pcr ¼ p ¼ 20:05, with the corresponding flutter frequency x ¼ 11:02. The jump down in pcr from 20.05 to 10.94 when vanishingly small (but non-zero) internal damping is introduced is known as the destabilizing effect (paradox) of small damping.

4.3.2

External Damping Only

The corresponding results when only external damping is introduced are given in Fig. 4.3. It is seen that this type of damping alone has a stabilizing effect. This can be proved mathematically [9]. It is finally noted that introduction of small external damping to Beck’s column does not lead to a jump in flutter load, as with small internal damping.

4.3.3

Both Internal and External Damping

The dependence of the flutter load on internal damping becomes more complicated when external damping is included as well. This is illustrated by Fig. 4.4, which shows the critical load pcr and the critical frequency xcr as functions of the internal damping parameter c for two cases with different values of the external damping

Fig. 4.3 Critical flutter load and frequency when only external damping is introduced (c ¼ 0)

4.3 Beck’s Column with Damping Introduced

41

(a)

(b) Fig. 4.4 Critical flutter force and frequency when both internal and external damping are introduced

parameter j, j ¼ 0:01 in part (a) and j ¼ 0:1 in part (b). In the first case, a logarithmic scale has been used for the c-axis, as the values of pcr fall off rapidly for a small value of c. Increasing the amount of internal damping has, in general, a destabilizing effect for small values of c, in the sense that @pcr =@c\0, and a stabilizing one for larger values, in the sense that @pcr =@c [ 0. The value of c at which the change in its effect occurs depends on the value of j, that is, the mutual balance between j and c is of importance. This section suggests that the consideration of both internal and external damping is of importance (and complicated) when one conducts experiments as well as stability analyses involving follower forces. However, from now on in this chapter, we shall discuss the effect of internal damping on the stability of cantilevered columns under a tangential follower force in greater detail.

42

4.4

4 Columns with Damping

Pflüger’s Column with Internal Damping

Figure 4.5 shows the Argand diagrams (root locus diagrams) with the first and second eigenvalue branches for Pflüger’s column with the nondimensional concentrated mass l ¼ 1:0. It is noted that the horizontal axis is the real part of the eigenvalue k, while the vertical one is the corresponding imaginary part. The flutter limit is determined in the Argand diagrams as the smallest value at which one of the branches crosses the imaginary axis. It is seen from Fig. 4.5a that Pflüger’s column, and thus also Beck’s column, loses its stability by flutter in two-mode-coupling, while Fig. 4.5b, c show that the damped column does so by flutter in a single mode. The flutter limits obtained by considering damping, p , and those by neglecting damping, p , are shown in Fig. 4.6.

(a)

=0

(b)

= 0. 001

(c)

= 0. 01

Fig. 4.5 Argand diagram of Pflüger’s column without and with internal damping [10]

Fig. 4.6 Stability map of Pflüger’s column when damping is neglected and introduced [10]

4.4 Pflüger’s Column with Internal Damping

43

The stability map in Fig. 4.6 shows that the flutter boundary obtained by taking account of small damping is approximately half of that obtained by neglecting damping. It is worth noting the behavior of the flutter-prone (critical) eigenvalue branch in relation to the possibly destabilizing effect of damping. The critical eigenvalue branch runs just in the vicinity of, and parallel with, the imaginary axis, keeping a small rate of amplitude growth, as shown in Fig. 4.5b.

4.5

Dynamic Responses

Now let us consider the dynamic response of Pflüger’s column with small internal damping (external damping is neglected) when an external disturbance is applied to the column. Here, a unit impulse to the tip end is assumed as the disturbance. The system of linear differential equations for the column is now modified to the form Mvss þ Cvs þ Kv ¼ f ðsÞ;

ð4:10Þ

where f ðsÞ is the excitation vector. The dynamic response of the system (4.10) can be obtained through a method proposed by Fawzy and Bishop [11], and later generalized by Newland [12]. The impulse responses of the undamped case of Pflüger’s column are shown in Fig. 4.7, in which the three loading parameters in the vicinity of the flutter limit are selected to demonstrate the sub-critical, critical and super-critical responses of the column. It is noted that flutter of Pflüger’s column, and also of Beck’s column, is two-mode-coupling flutter. The subcritical behavior is a beating vibration between the first and second modes. The dynamic behavior of a damped Pflüger’s column in the vicinity of the flutter limit is shown in Fig. 4.8, when a small internal damping c ¼ 0:001 is taken into account ðexternal damping j ¼ 0Þ. Figure 4.8a shows the neutral steady-state motion at the critical flutter value p ¼ p ¼ 7:91, when the real part of the characteristic exponent r ¼ ReðkÞ ¼ 0. Figure 4.8b shows the behavior when the loading parameter p exceeds the critical value, p ¼ 9:37 [ p , when r ¼ 103 . Here, r is referred to as the growth rate index. It is noted that the behavior is unstable in the sense of asymptotic stability, as r is positive, although very small. The amplitude of the column can become infinite as time goes to infinity. In reality, say, in the case of a physical experiment or an analog/numerical experiment, we observe the behavior only within a finite time interval. Behavior of the kind shown in Fig. 4.8b may be judged as stable by the concept of stability in a finite time interval. This fact suggests a gap between the theoretical and practical/experimental stability criteria. Figure 4.9 shows two different responses of the column ðl ¼ 1:0Þ at the same loading parameter p ¼ 12:0 [ p but with two different magnitudes of damping, c ¼ 0:001 and c ¼ 0:01, respectively. Figure 4.10 shows the Argand diagram for the damped column with a different magnitude of damping, when p ¼ 12:0.

44

4 Columns with Damping

(a)

(b)

(c) Fig. 4.7 Dynamic responses of Pflüger’s column ðl ¼ 1:0; c ¼ 0; j ¼ 0Þ [10]

The diagram shows that r ¼ 0:0054 for c ¼ 0:001 and r ¼ 0:0532 for c ¼ 0:01, when p ¼ 12:0. Figures 4.9 and 4.10 show that the column with c ¼ 0:01 has a larger growth rate than that with c ¼ 0:001. This fact indicates another (qualitative) destabilizing effect of damping. Now let us discuss the stability of Pflüger’s column by a relaxed concept of stability, stability in a finite time interval, based on the growth rate index r . This type of relaxed stability criterion was originally suggested and discussed in 1965 by Herrmann and Jong [2]. They proposed a relaxed criterion based on a measure of the rate of amplitude growth during one period of oscillation. Here, let us discuss a slightly modified stability concept. This concept proposes the following: if the real part of the eigenvalue is positive, but smaller than a prescribed value, the column is said to be stable in the prescribed in a finite time interval: stable if r  r for s  s ; unstable if not. Figure 4.11 shows the flutter bounds determined by the different stability criteria when small damping is considered. The flutter limit is determined by the relaxed stability criterion in a finite time interval for five specified growth rate indices: r ¼ 0; 103 ; 102 ; 101 and 5  101 . It can be seen in Fig. 4.11 that the concept

4.5 Dynamic Responses

45

Fig. 4.8 Dynamic responses of a damped Pflüger’s column for l ¼ 1:0 and c ¼ 0:001 [10]

of relaxed stability can bridge the gap between the flutter prediction by including damping, and the prediction by neglecting damping.

4.6

Discussion

The mathematical aspects of the destabilizing effect of damping have been discussed thoroughly by Kirillov, who also discussed the same effect in rotor dynamics [15, 16]. Bigoni et al. [17] discussed flutter onset in nonconservative systems by

46

4 Columns with Damping

(a)

(b) Fig. 4.9 Responses of the column at p ¼ 12:0 ðl ¼ 1:0Þ [10] Fig. 4.10 Argand diagram showing the relation between growth rate and follower force magnitude [10]

4.6 Discussion

47

Fig. 4.11 Stability predictions for Pflüger’s column ðl ¼ 1:0Þ in terms of the relaxed concept of stability within a finite time interval

dealing with Pflüger’s column with two dissipation mechanisms: the air resistance and the internal damping.

References 1. Ziegler, H. (1952). Die Stabilitätkrierien der Elastomechanik. Ingenieur-Archiv, 20, 49–56. 2. Herrmann, G., & Jong, I.-C. (1965). On the destabilizing effect of damping in nonconservative elastic systems. Journal of Applied Mechanics, 32(3), 592–597. 3. Ziegler, H. (1968). Principles of structural stability. Waltham: Blaisdell Publishing Co. 4. Bolotin, V. V., & Zhinzher, N. J. (1969). Effects of damping on stability of elastic systems subjected to nonconservative forces. International Journal of Solid and Structures, 5, 965– 989. 5. Kirillov, O. N., & Seyranian, A. P. (2005). The effect of small internal and external damping on the stability of distributed nonconservative systems. Journal of Applied Mathematics and Mechanics, 69(4), 529–552. 6. Tommasini, M., Kirillov, O. N., Misseroni, D., & Bigoni, D. (2016). The destabilizing effect of external damping: Singular flutter boundary for the Pflüger column with vanishing external dissipation. Journal of the Mechanics and Physics of Solids, 91, 204–215. 7. Pflüger, A. (1955). Zur Stabilität des tangential gedrückten Stabes. Zeitschrift für Angewandte Mathematik und Mechanik, 51(4), 191. 8. Sugiyama, Y., Kashima, K., & Kawagoe, H. (1976). On an unduly simplified model in the non-conservative problems of elastic stability. Journal of Sound and Vibration, 45(2), 237– 247. 9. Pedersen, P. (1984). Sensitivity analysis for non-self-adjoint systems. In Komkov, V. (Ed.), Sensitivity of Functionals with Applications to Engineering Sciences (pp. 119–130). Berlin: Springer.

48

4 Columns with Damping

10. Ryu, S.-U., & Sugiyama, Y. (2003). Computational dynamics approach to the effect of damping on stability of a cantilevered column subjected to a follower force. Computers & Structures, 81, 265–271. 11. Fawzy, I., & Bishop, R. E. D. (1976). On the dynamics of linear nonconservative systems. In Proceedings of the Royal Society of London, A, 352, 25–40. 12. Newland, D. E. (1989). Mechanical vibration analysis and computation (pp. 226–257). New York: Academic Press. 13. D’Annibale, F., Ferretti, M., & Luongo, A. (2016). Improving the linear stability of the Beck’s beam by added dashpots. International Journal of Mechanical Sciences, 110, 151– 159. 14. Zamani, V., Kharazmi, E., & Mukherjee, R. (2015). Asymmetric post-flutter oscillations of a cantilever due to a dynamic follower force. Journal of Sound and Vibration, 340, 253–266. 15. Kirillov, O. N., & Verhulst, F. (2010). Paradoxes of dissipation-induced destabilization or who open Whitney’s umbrella? Zeitschrift für Angewandte Mathematik und Mechanik, 90(6), 462–488. 16. Kirillov, O. N. (2013). Nonconservative stability problems of modern physics. Berlin: De Gruyter. 17. Bigoni, D., Misseroni, D., Tommasini, M., Kirillov, O. (2018). Detecting singular weak-dissipation limit for flutter onset in reversible systems. Physical Review, E, 97, 023003.

Chapter 5

Energy Consideration on the Role of Damping

In Sect. 4.3 of the previous chapter, it has been shown that introduction of small internal damping to Beck’s column leads to a considerable reduction in the flutter limit, from p
¼ 20:05 (for the undamped case) to pcr ¼ 10:94 (for the damped case). This effect is referred to as the destabilizing effect of small damping. This chapter presents an energy-based discussion on the role of internal damping in the dynamics of Beck’s column with damping.

5.1

Energy Considerations

The concept of energy plays a prominent role in mechanics. Lagrange’s equations of motion and Hamilton’s principle, which are both based on variational principles, or the calculus of variations [1, 2], start out from energy expressions to give exact equations of motions. Energy and energy-like quantities are also the basic ingredients in approximation methods such as Rayleigh’s method, the Rayleigh-Ritz method, and the Galerkin method [1]. And most importantly, energy expressions are the starting point for modern finite element formulations [3, 4], as well as for most sensitivity analysis approaches [5]. In addition to forming the basis of exact and approximative computation methods, considerations of energy balance have also contributed to obtaining a deeper understanding of stability problems, particularly stability in fluid-structure interaction problems. This road of progress was pioneered by Benjamin in his famous two-part paper on the dynamics of stability of articulated pipes conveying fluid [6, 7]. Based on an energy balance equation, Benjamin explained qualitatively how the pipes have to move in a dragging motion (similar to a swimming eel, see, e.g., Refs. [8, 9]) in order for the fluid flow to do positive work on the pipes, and hence to generate flutter. Studies on the stability properties of more general fluid-structure interaction problems by Benjamin [10, 11], Crighton and Oswell [12], and Landahl [13] were also based on energy consideration. © Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_5

49

50

5 Energy Consideration on the Role of Damping

Herrmann and Nemat-Nasser [14] discussed the stability of Ziegler’s pendulum model and brought forward the idea that internal damping has a dual role in the dynamics of mechanical systems, as a mechanism for dissipating energy out of the system and as a mechanism for modifying the vibrations, and that the latter mechanism is responsible for the destabilizing effect of damping. Païdousssis and Deksnis [15] discussed the relation between Ziegler’s pendulum and Beck’s column, in connection with articulated pipes conveying fluid. Energy considerations of Beck’s column have previously been reported in Refs. [14, 16–18]. These articles all employ a Galerkin expansion of the flutter configurations. Semler et al. [19] considered a follower-force-loaded continuous structure within the framework of a Galerkin expansion, namely the cantilevered fluid-conveying pipe. For this problem, Semler et al. found that the phase angles between the normal modes (in terms of the Jordan normal form) play an important role in the stability characteristics. However, the Galerkin-based approach implies highly complex energy consideration, even with just three modes. The present chapter offers an energy-based discussion on the role of damping in the dynamics and stability of Beck’s column with damping. Instead of the Galerkin representation, we will consider here a non-discretized ‘actual flutter mode’ representation, with a continuously varying phase angle, as in the related studies in Refs. [20–22]. This gives significantly simpler energy expressions and provides, through this, a means for obtaining additional information. A direct comparison with the results of Semler et al. is thus difficult. Both approaches show, however, that phase angles are of major importance. But the main emphasis of this chapter is on the effect of internal damping on the dynamic properties of a column under a follower force.

5.2

Equation of Motion and Stability Analysis

The equation of motion for Beck’s column with damping is written in the nondimensional form @2g @g @5g @2g @4g þc þj þ p 2 þ 4 ¼ 0; 4 2 @s @s @s@n @n @n

ð5:1Þ

while the boundary conditions are given by @gð0; sÞ ¼ 0; @n @ 3 gð1; sÞ @ 2 gð1; sÞ c þ ¼ 0; @s@n2 @n2 gð0; sÞ ¼ 0;

@ 4 gð1; sÞ @ 3 gð1; sÞ c þ ¼ 0; @s@n3 @n3

as given in Eqs. (4.4) and (4.5) with l ¼ 0.

ð5:2Þ

5.2 Equation of Motion and Stability Analysis

51

Writing the displacement function gðn; sÞ in the form gðn; sÞ ¼ ^gðnÞexpðksÞ;

k ¼ a þ ix;

i¼

pffiffiffiffiffiffiffi 1;

ð5:3Þ

with Eq. (5.3), Eqs. (5.1) and (5.2) can be written as the eigenvalue problem   0000 000 00 k2 ^g þ k j^g þ c^g þ ^g þ p^g ¼ 0; ^gð0Þ ¼ 0; 00

ð5:4Þ

0

^g ð0Þ ¼ 0;

ðkc þ 1Þ^g ð1Þ ¼ 0;

00

ðkc þ 1Þ^ g ð1Þ ¼ 0;

where a dash indicates differentiation with respect to n. Eigenvalue pairs (k, p) can be determined numerically without discretizing (5.4). However, for ease of stability analysis, the problem is discretized by applying the finite element method [3]. Equation (5.4) is then transformed into the matrix eigenvalue problem 

 k2 M þ kðjM þ cSÞ þ ðS  pLc þ pLnc Þ a ¼ 0;

ð5:5Þ

where M is the mass matrix, S is the stiffness matrix, Lc and Lnc are the conservative and nonconservative load matrices, respectively, and a ¼ aR þ iaI is the complex eigenvector, with aR being the real part and aI the imaginary part. With the column divided into N elements, the matrices in Eq. (5.5) are of size 2N  2N, as each node has two degrees of freedom. The numerical results to follow have been obtained using 20 finite elements. The flutter force, denoted by pcr , is defined as the smallest value of p for which r ¼ 0 (the flutter force for the undamped case will be denoted by p
). The flutter frequency corresponding to pcr is denoted by xcr . The flutter force and frequency of Beck’s column with damping are reproduced in Fig. 5.1, as shown earlier in Fig. 4.2, to help with the discussions in the following sections.

Fig. 5.1 Effect of internal damping on the flutter force and frequency of Beck’s column with damping (reproduced from Fig. 4.2)

52

5 Energy Consideration on the Role of Damping

5.3

Energy Expressions

5.3.1

Energy Balance Equations

Multiplication of Eq. (5.1) by the lateral velocity of the column, @g=@s, followed by integration over the length ð0  n  1Þ, yields a power (rate of work) balance equation. When the boundary conditions in Eq. (5.2) are incorporated via integration by parts we obtain the power equation in the form d dWnc dWd þ : ðT þ V  W c Þ ¼ dt ds ds

ð5:6Þ

The left side of Eq. (5.6) represents the rate of increase of mechanical (kinetic and potential) energy. The individual terms, the kinetic energy T, the elastic energy V and the work done by the conservative component of the force W, are given by 1 T¼ 2

Z1  2 Z1  2 2 Z1  2 @g 1 @ g 1 @g dn; V ¼ dn; W ¼ p dn: @s 2 2 @n @n2 0

0

ð5:7Þ

0

The terms on the right side of Eq. (5.6) represent the sources responsible for the energy increase. The power delivered by the nonconservative component of the force is

dWnc @g @g ¼ p ; @s @n n¼1 ds

ð5:8Þ

and the power ‘delivered’ (that is ‘minus’ the power dissipated) by the damping force is  3 2 # Z 1 "  2 dWd @g @ g ¼ j þc dn: ð5:9Þ @s ds @s@n2 0

5.3.2

Energy Balance at the Critical Force

Following Benjamin [6], the motion of the column over a time interval s1  s  s2 will be considered by assuming that the shape of the column at the time s2 coincides with that at s1 . The increase in mechanical energy is given by

5.3 Energy Expressions

53

 3 2 # Zs2 Zs2 Z1 "  2 @g @g @g @ g DE ¼  p ds  j þc dnds; @s @n n¼1 @s @s@n2 s1

s1

ð5:10Þ

0

where the first term on the right side is the work done by nonconservative component of the applied force DWnc , and the second term is the work done by damping DWd . Harmonic vibrations, characterized by DE ¼ 0, exist exactly at the critical force pcr , where the real part of the eigenvalue k is zero, that is, a ¼ 0, and x ¼ xcr . These vibrations can be expressed as gðn; sÞ ¼ AðnÞ cosðxcr s þ /ðnÞÞ;

ð5:11Þ

where AðnÞ is the amplitude function and /ðnÞ is the phase angle function. Inserting Eq. (5.11) into Eq. (5.10), and carrying out the time integration with s1 ¼ 0 and s2 ¼ 2p=xcr for one period of oscillation, gives (with the aid of Mathematica [23]) DE ¼ DWnc þ DWd ¼ DWnc þ DWde þ DWdi

Z1 2 d/ ¼ pcr pAð1Þ xcr jp AðnÞ2 dn dn n¼1 0 2 3   2 !2  Z1  2 2 2 dA d/ d / d A d/ 5dn ¼ 0: þA 2 þ A  xcr cp 4 2 dn dn dn dn dn2

ð5:12Þ

0

In this expression, the work done by the damping force DWd has been separated into the part done by external damping and the part done by internal damping, denoted by DWde and DWdi ; respectively. The representation in Eq. (5.11) of the flutter configuration can be contrasted with the Galerkin form gðn; sÞ ¼

J X

 Aj ðnÞ cos xcr s þ /j ;

ð5:13Þ

j¼1

used in Refs. [14, 17, 19, 24]. The differences between the constant phase angles /j for the J individual modes may provide valuable information (e.g., refer to Ref. [19]), however, the equation corresponding to Eq. (5.9) becomes too complicated to be useful (see again [19]).

54

5.3.3

5 Energy Consideration on the Role of Damping

Discretized Energy Equations

The discrete (matrix-form) versions of the terms in (5.12) are given by (see, e.g., Ref. [25]) 1 DWnc ¼  2

2p=x Z cr

Re

h

kaT eks




i pcr Lnc aeks ds;

0

1 DWd ¼  2

2p=x Z cr

Re

h

kaT e

 ks


i

ð5:14Þ

kðjM þ cSÞaeks ds;

0

where a superscript T means transposition, and a superscript asterisk (
) means complex conjugation. Evaluation and separation of DWd into external and internal parts (as in Eq. (5.12)) gives  DWnc ¼ pcr p aTR Lnc aI  aTI Lnc aR ;  DWde ¼ xcr jp aTI MaI þ aTR MaR ;  DWdi ¼ xcr cp aTI SaI þ aTR SaR :

ð5:15Þ

These expressions are applied in the numerical examples to follow. The phase angle / at element number e is evaluated as /e ¼ arctanðaIe =aRe Þ, where aIe and aRe are 0 the ð2e  1Þ th element in aI and aR , respectively. The phase angle gradients d/e =dn are evaluated by finite differences.

5.4

Flutter Configurations and Phase Angle Functions

Flutter configurations at pcr and the corresponding phase angle functions /ðnÞ are shown in Fig. 5.2 for two different amounts of internal damping c ðj ¼ 0Þ. As the dissipative work DWd is negative definite, Eq. (5.12) shows that the relation

d/ dn

n¼1

\0

ð5:16Þ

must hold to satisfy the zero energy balance, DE ¼ DWnc þ DWd ¼ 0. The plots of the phase angle function /ðnÞ verify this, as shown in Fig. 5.2. When the force parameter p is increased beyond the flutter level pcr , DE ¼ DWnc þ DWd will become larger than zero, and unstable oscillations will set in. Figure 5.3 illustrates the energy balance in a force parameter range around the critical value pcr ¼ 10:94, with damping parameter c ¼ 1:0  106 ðj ¼ 0Þ. The evolution of the gradient ½@/=@nn¼1 with increasing p is shown in Fig. 5.4. It is seen that the gradient decreases rapidly as the undamped critical force level

5.4 Flutter Configurations and Phase Angle Functions

55

(a)

(b) Fig. 5.2 Flutter configurations gðn; sÞ (arbitrary scale) and phase angles /ðnÞ (in degree) (j ¼ 0) [22]

pcr ¼ p
¼ 20:05 is approached. This is consistent with, and may explain, the fact that, with small damping, the flutter within the force range pcr \p\p
is ‘mild’, while ‘violent’ flutter sets in for p [ p
[26–28]. It is remarked that the wavenumber kðnÞ of the elastic wave travelling along the beam is defined as minus the gradient of the phase angle [29], that is, k ð nÞ ¼ 

d/ : dn

ð5:17Þ

The work done by the follower force thus varies as DWnc   pcr

@/ @n

 pcr kð1Þ: n¼1

ð5:18Þ

56

5 Energy Consideration on the Role of Damping

Fig. 5.3 Work balance around the critical force (c ¼ 1:0  106 ; j ¼ 0) [22]

Fig. 5.4 Phase angle gradient at the free end [22]

This implies that introduction of internal damping into Beck’s column causes a progressive wave (travelling wave), and that the nodal point moves continuously along the undeformed axis, at the speed v ð nÞ ¼

x : k ð nÞ

ð5:19Þ

More importantly, Eq. (5.18) suggests that the follower force does work on the column during each period of oscillation only if the column vibrates with a progressive wave component. This is analogous to the flutter instability of cantilevered pipes conveying fluid, in which the Coriolis force acts as a velocity-dependent

5.4 Flutter Configurations and Phase Angle Functions

57

mechanism (localized damping mechanism at the free end), in addition to the internal and external damping mechanism considered here [21, 30, 31]. Now let us discuss the effect of damping on the flutter configurations. Figure 5.5 shows three examples of flutter configurations and phase angles for different cases of damping: for the undamped case (c ¼ j ¼ 0) in Fig. 5.5a, for a case of external damping only (c ¼ 0; j ¼ 0:1) in Fig. 5.5b, and for a case with both external and internal damping ðc ¼ 2  104 ; j ¼ 0:1Þ in Fig. 5.5c. In the undamped case, flutter occurs through coalescence of the first and second eigensolution (k, p). For this reason, the flutter configuration is principally a linear combination of the first and second eigenmode, although small components of the third and higher modes are also present. The position of the nodal point is fixed, that is, the vibrations do not have any travelling wave character. When the part to the left of the nodal point moves up, the part to the right moves down, and vice versa. Hence, there is a phase-shift of 180° across the nodal point. As there is no energy dissipation, the follower force cannot produce any energy either. Accordingly, ½@/=@nn¼1 ¼ 0, in agreement with Eq. (5.18). That is to say, the follower force of Beck’s column is “conservative”, in the sense that there is no energy input from it when it is subcritical. It is truly nonconservative only when the follower force parameter exceeds the critical value. The discretized expression for DWnc (Eq. (5.15)) provides an alternative interpretation of the same thing: the follower force can do work only if the eigenvector a is complex- and it becomes complex only when p [ p
. When external damping alone is introduced (Fig. 5.5b), the flutter configurations keep their basic character, but the stepped phase angle function is smoothed out.  Smoothing of the 180 step implies that the nodal point will move slowly between n 0:4 and n 0:5, as the wavenumber kðnÞ is large there (cf. Eq. (5.19)). When internal damping is included as well (j [ 0; c [ 0; Fig. 5.5c), thinking in terms of a Galerkin modal expansion, the second eigenmode will be damped more than the first, and the flutter vibrations will, with increasing internal damping, be dominated more and more by the first eigenmode. This is because the Kelvin-Voigt damping model considered here damps the higher modes with higher intensity, the intensity factor being proportional to the squared natural frequency of the mode [32]. External (viscous) damping, on the other hand, damps each eigenmode with the same intensity. As the second eigenmode is suppressed, the phase angle function /ðnÞ will, with increasing internal damping, begin to resemble the ones shown in Fig. 5.2.

5.5

Energy Balance with Small Internal Damping

This section discusses the balance between the work done at the critical force by the follower force, DWnc , and the work done by the internal damping, DWdi , in more detail, for the case of small internal damping, 0\c 1 (and no external damping, j = 0).

58

5 Energy Consideration on the Role of Damping

(a) Undamped case;

(b) With external damping only;

(c) With internal and external damping; Fig. 5.5 Flutter configurations and phase angles for different cases of damping [22]

5.5 Energy Balance with Small Internal Damping

59

Fig. 5.6 Work done by the follower force and by small internal damping [22]

Figure 5.6 shows DWnc and DWdi as functions of the small damping parameter c. It is noted that DWnc is equal to DWdi at the flutter limit. The figure shows that these terms grow linearly with increasingly small c. This means that DWnc =c constant and DWdi =c constant. Referring to Fig. 5.1, it can be seen that pcr and xcr are approximately constant for small internal damping, 0\c 1. Equation (5.18) then gives the simple expression

@/ @n

n¼1

=c constant:

ð5:20Þ

This is verified by Fig. 5.7, which shows that the relation holds within the range of c for which 0\c  0:01: For a sufficiently small damping parameter, Eq. (5.20) can thus be written in the form

d/ ¼ c þ c; dn n¼1

ð5:21Þ

where c+ is a positive constant. Substitution of this expression into Eq. (5.18) gives DWnc  pcr c þ c:

ð5:22Þ

This expression shows that the work done by the follower force is directly proportional to the damping parameter c: Inserting Eq. (5.18) into Eq. (5.12) gives, with j ¼ 0,

60

5 Energy Consideration on the Role of Damping

(a) Phase angle gradient

(b) Phase angle gradient divided by the damping parameter γ

Fig. 5.7 Phase angle gradient for small internal damping [22]

Z 1 " pcr ¼ c þ xcr

  2 )# 2 ( 2  2  d/ A dA d / d   d/ þA 2 þ A dn; 2 2 dn dn dn dn dn

ð5:23Þ

0

 ðnÞ ¼ AðnÞ=Að1Þ. For vanishingly small c, d/=dn and d 2 /=dn2 will also where A 2   @ A=@n,  : Then, Eq. (5.23) can be be very small in comparison with A; and @ 2 A=@n approximated by pcr c þ xcr

Z1  2  2 d A dn: dn2

ð5:24Þ

0

It is interesting to note that Eqs. (5.23) and (5.24) are independent of the magnitude of damping as a consequence of Eq. (5.22). This verifies the well-known result, as shown in Fig. 4.2 (and in Fig. 5.1), that the effect of vanishingly small (internal) damping is independent of the actual magnitude of the coefficient of damping.

5.6

Energy Balance with Both Internal and External Damping

The behavior of pcr with varying internal damping c when external damping j is included was discussed in Sect. 4.3.3 (see Fig. 4.4). It can be noted that it is the mutual balance between c and j, not just their absolute magnitudes, that governs

5.6 Energy Balance with Both Internal and External Damping

(a) Work balance

61

(b) Phase angle gradient

Fig. 5.8 Work balance and phase angle function (j ¼ 0:1) [22]

the magnitude of the critical force pcr . This is more easily understandable in light of the energy equations, as DWnc and DWd ð¼ DWdi þ DWde Þ are now both functions of c and j. The balance between the work done by the follower force, DWnc , and that done by the damping, DWd , as a function of internal damping is shown in Fig. 5.8a for the case of the external damping parameter j ¼ 0:1. The figure should be seen together with Fig. 4.4. For a small value of c, it is seen that the gradients of the work curves are almost zero, yet Fig. 4.4 shows that pcr decreases rapidly here. The explanation for this can be found in the behavior of the phase gradient function ½d/ðnÞ=dnn¼1 , which is shown in Fig. 5.8b. It can be seen that ½d/ðnÞ=dnn¼1 is decreasing in a similar way to Fig. 5.7a. The critical force pcr can then, according to Eq. (5.18), decrease even when DWnc is constant. With external damping included, the phase gradient function ½d/ðnÞ=dnn¼1 is still approximately a linear function of c, as seen in Fig. 5.8b. The destabilizing effect of small internal damping illustrated in Fig. 5.1 (and also in Fig. 4.4) will be considered in the following. Figure 5.9 shows a small portion of Fig. 5.8a for j ¼ 0:1 and 0  c  0.005, drawn with a linear scale of c. The work done by the damping is separated into internal and external parts. It is seen in Fig. 5.9 that introduction and increase of internal damping implies naturally an increase in its work, DWdi . In the case considered in Fig. 5.9, however, the increase of internal damping implies that the work done by external damping DWde decreases, and at a larger rate than DWdi increases. The drop in DWdi is mainly due to the reduction of the flutter frequency (see Fig. 4.4b, right) and the reduction of the second eigenmode-component (in the sense of a Galerkin expansion) (Fig. 5.5). As a consequence, the work done by the follower force DWnc must decrease to satisfy the energy balance. As ½d/=dnn¼1

62

5 Energy Consideration on the Role of Damping

Fig. 5.9 Work done during one period with small internal damping [22]

decreases, the critical force pcr will necessarily have to decrease as well. By a further increase in the internal damping, the drop in DWde starts to ‘flatten out’, and for c > 0.001, DWnc actually starts to increase. But the steep gradient of ½d/=dnn¼1 implies that pcr necessarily must continue to decrease. In this way, the destabilizing effect of damping continues until c 0:03 (see Fig. 4.4b).

5.7

Energy Growth Rate

It is interesting to consider the energy rate during passage of the flutter limit, expressed by the quotient v ¼ DWnc =DWdi ;

ð5:25Þ

which may also be used as a stability criterion. The vibrations are stable if v\1; critical if v ¼ 1; unstable if v [ 1: Using Eq. (5.12), the energy growth rate ½@v=@pp¼pcr can be evaluated as

ð5:26Þ

5.7 Energy Growth Rate

63

Fig. 5.10 Energy rate v ¼ DWnc =DWdi (‘Beta’ means j and ‘Gamma’ means c in the figure) [22]

2  , Z1  @v 1 d/ 4j A2 dn ¼ Að1Þ2 @pp¼pcr xcr dn n¼1 0 8 9 3  2 !2 =  Z 1 : 3 B4 A44

8 9 > =

0 : ¼ 0 > ; ; > : > 0 9 > =

The condition for a non-trivial solution leads to the characteristic equation

ð10:9Þ

132

10

Columns under a Rocket-Based Follower Force

  Aij ða; l; m; j; c; p; kÞ ¼ 0;

ði; j ¼ 1; 2; 3; 4Þ;

ð10:10Þ

for the known parameters a; l; m; j; c and the loading parameter p. The complex characteristic roots are expressed in the form k ¼ r þ ix;

ð10:11Þ

where r is the rate of amplitude growth and x is the nondimensional eigenfrequency. The condition r ¼ 0 ðx [ 0Þ yields the flutter limit.

10.2

Rocket Motors

Small-sized solid rocket motors were manufactured by Daicel Co. (Harima Plant, Tatsuno-shi, Japan). The motors consisted of a solid propellant and hardware. The propellant was specially mixed for the present experiments to realize the aimed thrust level. The hardware, consisting of a motor case, forward and aft caps, a nozzle, and an ignitor, were conventional for use in repeated combustion tests of the propellant. The hardware was originally designed to be strong enough for reusability and safety in the combustion tests, and it is thus massive and heavy. The mass of the hardware is 13.73 kg, while the mass of the propellant is 0.90 kg. Three motors were tested on the test bench in order to calibrate their thrust. A typical thrust curve of a rocket motor is shown in Fig. 10.3. The average thrust of the motor was approximately 40 kgf (392 N), while the burning time was about 4 s. Although the thrust curve was not flat but rather linearly increasing during burning, it was assumed for the stability estimate that the thrust was constant and equal to the average value of 40 kgf (392 N). The measured Fig. 10.3 Thrust curve [1–3]

10.2

Rocket Motors

133

Fig. 10.4 Stability chart based on Pflüger’s column [1–3] (a = 0, l = 29.2, m = 0, c = 0.001, and j = 0)

data of the rocket motors are as follows: the average mass of the motor M = 14.18 kg, distance LR = 0.20 m, and rotatory inertia J = 0.120 kg m2.

10.3

Test Columns

In consideration of the nominal thrust of 40 kgf (392 N), the test columns were made of aluminium bars with a cross-section of 6.0 mm  30.0 mm and mass per unit length m = 0.481 kg/m. As the thrust (load) is assumed to be constant, the length of a column is the variable for loss of stability (flutter), and the flutter limit is given as the critical length. For determination of the range of lengths, the stability chart for Pflüger’s column, as discussed in Chap. 4, was applied, taking small (assumed) damping into account. The nondimensional mass of the rocket motor l = 29.2, and so 1/l = 0.0343, for the standard length of L = 1000 mm. A preliminary flutter estimate based on Pflüger’s column yields a stability chart as shown in Fig. 10.4. The chart suggests the length range 800–1400 mm. The chart, with the test runs, shows how to narrow-in the critical length from both sides.

10.4

Preliminary Tests

10.4.1 Test for Bending Stiffness A static deflection test revealed that the (static) bending stiffness EIs ¼ 3:37  106 kgf mm2 33  106 N mm2 , while a vibration test gave the   (dynamic) bending stiffness EId ¼ 3:47  106 kgf mm2 34  106 N mm2 . The average value of the two measured results was adopted as the nominal bending the test columns with the value EI ¼ 3:42  106 kgf mm2 stiffness of  6 2 33:6  10 N mm . This value is used hereafter in estimating the performance of the test columns.

134

10

Columns under a Rocket-Based Follower Force

Table 10.1 Measured data of test columns Dimensions of cross section (mm)

Length L (mm)

Mass per unit length m (kg/m)

Bending stiffness EI (kgf m2)

Nondimensional coefficient of damping Internal c External j

6.0  30.0

800– 1400

0.481

3.42  106

4.94  10−4

1.19  10−3

10.4.2 Test for Damping Coefficients A damped pendulum test gave the nondimensional coefficient of external damping as j ¼ 1:1  103 . A decaying beam motion test gave the nondimensional modulus of viscosity as c ¼ 5:4  104 . The measured physical data of the test columns are summarized in Table 10.1. It is noted that the nondimensional quantities defined in Eq. (10.3) vary with the length of the column.

10.4.3 Buckling Test It is of interest to conduct a buckling test of the test column under a conservative loading, prior to the flutter test. The main aim of the buckling test is to check the measured bending stiffness of the test columns. A schematic of the buckling problem for a cantilevered column is shown in Fig. 10.5. A corresponding setup with two vertical columns and an upper bar, as discussed in Chap. 1, is shown in Fig. 10.6. The length of the two columns was L ¼ 1000 mm. The load Q ¼ 2P was applied on the upper plate. The deflection of the test column was read by a microscope. The experimental results in terms of deflection are plotted in Fig. 10.7a. The same data are represented in Southwell plots [4], as shown in Fig. 10.7b. The slope of the curve gives the buckling load as Qex ¼ 16:1 kgf ð158 NÞ. This corresponds to Pex ¼ 8:1 kgf ð79:1 NÞ for the single column, while the corresponding theoretical estimate of the buckling load, by taking account of the average bending stiffness, is Pte ¼ 8:4 kgf ð82:5 NÞ.

10.5

Flutter Test

10.5.1 Outline of the Test Although it has been suggested that a follower force may be realized using a rocket motor, the explosives control law has prohibited the handling of rocket motors

10.5

Flutter Test

Fig. 10.5 Buckling of a cantilevered column

Fig. 10.6 Setup for the buckling test [1]

135

136

10

Columns under a Rocket-Based Follower Force

(a) Load-deflection curve

(b) Southwell plots

Fig. 10.7 Experimental results of buckling test (Q ¼ 2P) [1]

(pyrotechnic devices) in non-licensed laboratories. The present tests with rocket motors were carried out in the licensed rocket motor functional test facility in the Harima Plant, Daicel Co. (Tatsuno, Japan), with the help of licensed staff members. As the motors were specially prepared for the present flutter tests, they were costly, and only seven motors were supplied. The “flutter length” of the test column was determined by narrowing-in the critical length from both sides, as outlined in the stability chart of Fig. 10.4. A schematic of the setup for the flutter test is shown in Fig. 10.8. A bird’s eye view of the setup is shown in Fig. 10.9. One end of the column was attached to a clamp (with a milled cavity, as explained in Sect. 1.5), as shown in Fig. 10.10. The rocket motor was mounted onto the column at its tip end and suspended from the ceiling by a thin wire of about 3 m length. The motion of the column was recorded by a motor-driven camera and a video camera, which were mounted in the ceiling. The axial strain was recorded by strain gauges (at both sides of the column) to calibrate and confirm the thrust. A target disk for a deflection sensor was placed on one side of the column.

10.5

Flutter Test

137

Fig. 10.8 Schematic of the setup for the flutter test. ① Clamp, ② Test column, ③ Rocket motor, ④ Thin wire from the ceiling, ⑤ Strain gauge for axial strain, ⑥ Deflection sensor with a target disk, ⑦ Motor-driven camera, ⑧ Video camera

Fig. 10.9 Setup for the flutter test [1]

10.5.2 Test Runs Run No. 1: The first test run was done by using the longest column of length L ¼ 1400 mm. Just after ignition of the motor, the test column lost its stability by divergence, swaying immediately to one side. Run No. 2: The second run was conducted by using the shortest column of length L ¼ 800 mm. The column was stable. It is recalled that the thrust load was

138

10

Columns under a Rocket-Based Follower Force

Fig. 10.10 Clamped end [1]

40 kgf, while the (experimental) buckling load of the column of length L ¼ 1000 mm was 8.1 kgf. The buckling load of a column of length L ¼ 800 mm is estimated to be about 13 kgf. This run thus confirms that the column does not lose its stability by buckling, under a load three times the estimated buckling load. It was confirmed that the critical flutter length is located within the designed range of the test columns. Run No. 3: The third run was done using a column with an average length between the longest and the shortest, L ¼ 1100 mm. It was observed that the column lost its stability by flutter. Run No. 4: Knowing that the column with the length of L ¼ 1100 mm was unstable, one with a length of L ¼ 1000 mm was selected for the fourth run. It was confirmed that the column was stable. It was estimated that the critical length for the flutter may be within the range 1000\Lcr \1100 mm. Run Nos. 5–7: The last three runs were conducted to search for the critical length that initiates flutter. The test results were as follows: Run No: 5 : L ¼ 1050 mm; unstable by flutter: Run No:6 : L ¼ 1025 mm; unstable by flutter: Run No: 7 : L ¼ 1000 mm; unstable by flutter: The actual thrust acting upon the test column was calibrated by checking the recorded axial strain. The test results are plotted in Fig. 10.4. Here, open circles mean that the test column was stable, while by solid circles, the column was unstable.

10.5

Flutter Test

139

The present flutter test was the first experimental trial test with a rocket-based follower force with a limited number of motors. The test results look rather qualitative. However, the test could show experimental evidence that cantilevered columns under a follower force lose their stability by flutter.

10.5.3 Test Results Photographs of the disturbed motion under a rocket thrust were recorded every quarter of a second by a motor-driven camera installed in the ceiling. Figure 10.11 shows sequential pictures of the flutter motion observed in Run No. 7. The test column used in Run No. 7, damaged by bending into a shape resembling the second eigenmode, is shown in Fig. 10.12. Figure 10.13 shows the axial thrust (obtained by the recorded axial strain) and the dynamic deflection in Run No. 7.

Fig. 10.11 Flutter motion in Run No. 7 (top view) [1–3]

140

10

Columns under a Rocket-Based Follower Force

Fig. 10.12 Damaged column after flutter-type instability in Run No. 7 [1]

Fig. 10.13 Axial thrust and dynamic deflection in Run No. 7 [1–3]

Figures 10.11, 10.12 and 10.13 show experimental evidence that the cantilevered column loses its stability by flutter.

10.5.4 Effect of the Size of the Motor The first estimate of the flutter force, as shown in Fig. 10.4, was made by assuming the motor to be a mass point, while the real motor was of course a solid body of a

10.5

Flutter Test

141

Fig. 10.14 Flutter estimates by considering the size of the motor [1–3]

finite size. Theoretical flutter estimates, obtained by taking the size of the motor into account, are shown in Fig. 10.14, together with the experimental flutter points. The thick solid lines are for the estimate according to Pflüger’s column model. The dashed lines show the predictions by taking the distance LR ¼ 0:20 m and the mass M ¼ 14:2 kg into account. The thin solid lines show the flutter estimate by accounting for the rotatory inertia J ¼ 0:120 kg m2 , in addition to LR and M. The internal damping was also considered by adopting the measured (simplified) value of c ¼ 5:0  104 . It is found in Fig. 10.14 that the experimental “flutter length” may come close to the theoretical estimate computed by considering the motor as a solid body and by neglecting damping.

10.5.5 Stability in a Finite Time Interval The theoretical flutter estimates are based on the concept of asymptotic stability in infinite time. But in the present experiment the force due to the rocket motors could be applied to the test columns only for a short time; thus it suffices to rely on the stability in a finite time interval (refer to Sect. 1.2). The stability criterion for a finite time interval states that the column is

142

10

and

Columns under a Rocket-Based Follower Force

stable if jgj\njgi j for 0  s  sf ; unstable if jgj  njgi j for 0  s  sf ;

ð10:12Þ

where gi is the deflection at s ¼ 0 and n is a specified amplification factor. This factor is given by   n ¼ exp r sf ;

ð10:13Þ

where r is the critical exponent (the rate of amplitude growth), as discussed in Chap. 1. Let us take n ¼ 10. With the burnout time of the motor tf ¼ 4:0 s, the dimensionless finite time is given by sf ¼ 33:4. The critical exponent is found to be r ¼ 0:069. The stability criterion for the present experiment, based on the rate of amplitude growth, can be given in the form: the column is stable and unstable

if r\0:065for 0  s  33:4; otherwise

The Argand diagram for the test columns is shown in Fig. 10.15. Here, the critical exponent r is drawn with a dotted line. The open circles are the critical points, while the solid circles are representative points of the nondimensional thrust. The concept of asymptotic stability says that the left-hand side of the ðr; xÞ plane is stable and the right side of the plane is unstable. The flutter load by neglecting damping is found to be p ¼ 12:60. When damping is taken into account, the first eigenvalue branch (locus) starts in the left-hand side of the plane and crosses the imaginary axis (r ¼ 0) at p ¼ 5:65. If we neglect damping and apply the stability in a finite time interval, with the critical exponent r ¼ 0:065, the nondimensional thrust parameter pf , obtained by considering internal damping, is found to be pf ¼ 12:6. This means that the stability limit in a finite time interval, and with damping included, agrees with the flutter limit by neglecting damping, at least in the case of the present experiment. This verifies the experimental flutter limit.

10.6

Discussion

This chapter has shown that a cantilevered column under a follower force loses its stability by flutter. It has been shown that a follower force can be devised in the form of the jet thrust of a solid rocket motor mounted on the column at its tip. It has been found that the experimental flutter limit of the test column, with the concept of stability in a finite time interval, agrees with the flutter limit obtained by neglecting damping. This may verify the flutter analysis with damping neglected as the upper bound of the flutter limit of Beck’s column and its modified versions. A movie of

10.6

Discussion

143

Fig. 10.15 Argand diagram for the test columns [1–3]

the dynamic responses of the test columns, as described in this chapter, is available in the Supplementary Materials (online) in Appendix B. A movie of flutter motion of an acrylic test column, damped in comparison with aluminium columns, is available in the Supplementary Materials (online) in Appendix C.

References 1. Katayama, K. (1997). Experimental verification of the effect of rocket thrust on the dynamic stability of cantilevered columns. Doctoral Thesis, Department of Aerospace Engineering, Osaka Prefecture University. 2. Sugiyama, Y., Katayama, K., Kinoi, S. (1990). Experiment of flutter of cantilevered columns subjected to a rocket thrust, In AIAA-90-0948-CP, AIAA/ASME/ASCE/ASC 31st Structures, Structural Dynamics and Materials (SDM) Conference, Long Beach, California (pp. 1893– 1898). 3. Sugiyama, Y., Katayama, K., & Kinoi, S. (1995). Flutter of cantilevered column under rocket thrust. Journal of Aerospace Engineering, 8, 9–15. 4. Southwell, R. V. (1932). On the analysis of experimental observations in problems of elastic stability. Proceedings of the Royal Society of London, A, 135, 601–616.

Chapter 11

Columns under a Rocket-Based Follower Force and with a Lumped Mass

The experiment described in Chap. 10 demonstrated that the cantilevered column under a rocket thrust lose its stability by flutter. This chapter will take a similar but more quantitative experimental approach to the column subjected to a rocket thrust and equipped with a lumped mass. The stability analysis is carried out by applying the finite element method to discuss the effect of a rigid body on the flutter bound.

11.1

Finite Element Formulation and Stability Analysis

The stability of uniform columns under a follower force has been discussed in a large number of papers. However, actual slender rockets and missiles are equipped with discrete parts, such as payloads, tanks, engines, etc., so the mass distribution of these slender structures is no longer uniform. The effect of three concentrated masses on the stability of cantilevered columns has been studied by Kounadis and Katsikadelis [1]. The maximum controlled follower force acting on a free-free beam carrying a concentrated mass has been discussed by Park and Mote Jr [2]. This chapter is going to discuss the effect of a lumped mass on the flutter limit of a cantilevered column subjected to a rocket thrust. The aim of the experiment in this chapter is, first, to obtain more quantitative experimental verification of the effect of a follower force, and second, to discuss the effect of a lumped mass on the flutter limit of the columns under a follower thrust. Let us consider a cantilevered column having a lumped mass and a rocket motor at its tip, as shown in Fig. 11.1. Figure 11.2 shows a mathematical model of this system. The lumped mass is located at a distance x1 from the fixed end. The magnitude of the lumped mass is denoted by M1 and the rotatory inertia by J1 . Damping is neglected. The extended Hamilton’s principle for the system is written in the form

© Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_11

145

146

11

Columns under a Rocket-Based Follower Force …

Fig. 11.1 Column having a lumped mass and equipped with a rocket motor

Fig. 11.2 Mathematical model of a column having a lumped mass

Zt2

Zt2 fdðT
V þ Wc Þgdt þ

t1

dWn dt ¼ 0;

ð11:1Þ

t1

where the energy and work expressions are given by   2   2  ZL  2 @y 1 @y @ 2 y  1 @y  þ LR m dt þ M þ M1  @t 2 @t @x@t  2 @t  x¼xM1 0 x¼L  2 2   2 2  1 @ y  1 @ y  þ J ;  þ J1  2 @x@t  2 @x@t 

1 T¼ 2

x¼L

V¼

1 2



ZL EI 0

@2y @x2

2 dx;

ZL  2 1 @y P dx; Wc ¼ 2 @x 0  @y  dWn ¼
P dy @x x¼L

x¼xM1

ð11:2Þ

11.1

Finite Element Formulation and Stability Analysis

147

Here T is the kinetic energy, V is the elastic energy, Wc is the work done by the conservative component of the force P, and dWn is the virtual work done by the nonconservative component of the force P. The column is divided into N elements of equal length. Application of the standard finite element method [3] to Eq. (11.1) leads to the discretized equations of motion. The nondimensional deflection of the discretized system gðn; tÞ is expressed in the form gðn; tÞ ¼ uðnÞekt :

ð11:3Þ

The following nondimensional quantities are defined: a¼

LR ; L

m¼ n1 ¼

J J1 M M1 ; l1 ¼ ; ; m1 ¼ ; l¼ mL3 mL mL3 mL x1 ; L

p¼

PL2 ; EI

k2 ¼

mL2 s2 : EI

ð11:4Þ

The characteristic equation takes the form 

 k2 M þ K u ¼ 0:

ð11:5Þ

Damping is neglected in the present stability analysis. The numerical results in this chapter have been obtained by employing 20 finite elements, and so the discretized system has 40 degrees of freedom (20 lateral and 20 angular displacements). The present FEM analysis gives p ¼ 20:045 as the flutter limit for Beck’s column.

11.2

Rocket Motors

There were some problems connected with the rocket motors used in the flutter experiment in Chap. 10. First, the rocket motors were massive and heavy, as the motor case was one that was used for conventional combustion tests, not specially designed for the experiments. Second, there was an initial peak in the thrust curve. Third, the thrust curve was not flat, but rather increasing with burning time. Fourth, the average thrust did not remain constant at each test run. For more reliable experiments with a rocket-based follower force, it was requested that refined rocket motors be prepared, ones that might possibly be more compatible with the aim of the experiment. Figure 11.3 shows the thrust curve of the refined rocket motors used in the present experiment. Burning tests of the three refined motors showed that the drift in the average thrust was small. The average mass of the motor was M ¼ 4:05 kg, the rotatory inertia was J1 ¼ 0:0284 kg m2 , and the size of the motor was LR ¼ 0:153 m. The nominal (average) thrust was P ¼ 62:1 kgf ð503 NÞ. The burnout time was tf ¼ 3:2 s.

148

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Columns under a Rocket-Based Follower Force …

Fig. 11.3 Thrust curve of the refined motor [4–6]

11.3

Estimate of the Effect of a Lumped Mass on the Flutter Limit

Knowing the magnitude of the nominal thrust, let us consider the effect of a lumped mass on the flutter limit of a standard column of length L ¼ 1100 mm and with a cross-section of breadth b ¼ 30:0 mm and thickness h ¼ 7:0 mm. The mass per unit length m ¼ 0:567 and the bending stiffness EI = 6.00 kgf m2. Figure 11.4 shows a stability chart for the effect of a (assumed) lumped mass on the flutter limit of the standard column, in which the rotatory inertia of the lumped mass is neglected ðm1 ¼ 0Þ. It is found that a lumped mass is destabilizing when it is located along most of the span of the column, but stabilizing when it is located just near the tip end. The highest destabilizing effect takes place when the mass is mounted around the mid-span. Figure 11.4 suggests the scheme of the present experiment to demonstrate the destabilizing effect of a lumped mass (at the mid-span) on the flutter load of a column subjected to a rocket-based follower force. Two brass blocks, block A and block B, with hollow rectangular cross-sections (as shown in Fig. 11.1) were prepared as lumped masses for the present flutter test. Block A has mass M1 ¼ 1:0 kg (l1 ¼ 1:6 for L ¼ 1:1 m) and rotatory inertia J1 ¼ 5:74  10
4 kg m2 (m1 ¼ 7:6  10
4 ), while block B has mass M1 ¼ 2:3 kg (l1 ¼ 3:7 for L ¼ 1:1 m) and rotatory inertia J1 ¼ 2:03  10
2 kg m2 (m1 ¼ 3:0 10
2 ). It can be seen that the order of magnitude of the rotatory inertia of the designed blocks is much smaller than that of the lumped mass. The photograph in Fig. 11.1 shows the test column mounted with block B. Figure 11.5 shows the effect of the (designed) lumped mass on the critical thrust of the test column of length L ¼ 1100 mm. The ordinate is the ratio between the critical force with a lumped mass, p , and the critical force with no mass, p . It is seen that the highest destabilizing effect of the attached lumped mass takes place when the mass is located at n1  0:6.

11.4

Flutter Test

149

Fig. 11.4 Stability chart on the effect of a lumped mass on the flutter limit [4–6]

Fig. 11.5 Effect of the designed lumped mass (L ¼ 1100 mm) [4–6]

11.4

Flutter Test

Knowing from Figs. 11.4 and 11.5 that the most effective destabilizing action of a lumped mass takes place when the lumped mass is located at n1 ¼ 0:6 with the mass l1 ¼ 3:7, the scheme of the present test to verify the destabilizing effect of a lumped mass is given by the following two steps: Step 1: Find the length of the test column, with no lumped mass, at which the column is critical for flutter, as “critical” as possible. Step 2: Show that the column of the same length and with a lumped mass is unstable by flutter. Run No. 1: The test column of length L ¼ 1200 mm, with no lumped mass, lost its stability by violent flutter. The dynamic response of the column is shown in Fig. 11.6a.

150

11

Columns under a Rocket-Based Follower Force …

Run No. 2: The test column of length L ¼ 1100 mm, with no lumped mass, remained stable. The dynamic deflection is shown in Fig. 11.6b. Run No. 3: The column of length L ¼ 1100 mm, with the lumped mass l1 ¼ 3:7 located at n1 ¼ 0:6, lost its stability by flutter. The motion of the column is shown in Fig. 11.6c. Run No. 4: The column of length L ¼ 1100 mm, with the lumped mass l1 ¼ 1:6 located at l1 ¼ 1:6, demonstrated its critical motion, as shown in Fig. 11.6d.

(a) Run No.1

(b) Run No. 2

(c) Run No. 3

(d) Run No.4 Fig. 11.6 Dynamic responses in test runs [4, 6]

11.4

Flutter Test

151

Test runs Nos. 3 and 4 demonstrate the destabilizing effect of the lumped mass on the flutter limit of the columns. The observed flutter motion in Run No. 3 is shown in Fig. 11.7. The sequential photographs of the motion were taken every 1=6 of a second. Fig. 11.7 Observed flutter motion in Run No. 3 [6]

(a)

(b)

(c)

(d)

152

11

Columns under a Rocket-Based Follower Force …

Fig. 11.8 Stability chart and the test run points [4–6]

Figure 11.8 shows the stability chart for the test columns in their four test runs, including the four run points. The flutter limits in the chart were obtained by taking account of the measured data of the columns, including M; J1 and LR . The FEM results predict the flutter limit considerably well. The flutter test gives experimental evidence that a lumped mass may destabilize the cantilevered column under a follower force.

11.5

Discussion

In Chaps. 10 and 11, and the next Chap. 12 as well, cantilevered columns with a solid body mounted at their tip are dealt with theoretically and experimentally. In these cases, it is noted that the effect of rotatory inertia of the solid body must be considered properly. These three chapters describe the rocket motor-applied tests. Additional tests with additional rocket motors might give the more precise experimental evidence. However, it is recalled that rocket motors are pyrotechnic devices. The motors for the tests had to be prepared specially for each test. The tests using rocket motors could only be conducted in a licensed (for pyrotechnic devices) facility with help of licensed (for pyrotechnic devices) technical staff. Thus, the tests were costly, and were allowed only within a strict time limit. A movie of the dynamic responses of the test columns, as descibed in this chapter, is available in the Supplementary Materials (online) in Appendix B.

References

153

References 1. Kounadis, A. N., & Katsikadelis, J. T. (1980). On the discontinuity of the flutter load for various types of cantilevers. International Journal of Solid and Structures, 16, 375–383. 2. Park, Y. P., & Mote, C. D., Jr. (1985). The maximum controlled follower force on a free-free beam carrying a concentrated mass. Journal of Sound and Vibration, 98, 247–256. 3. Barsoum, R. S. (1971). Finite element method applied to the problem of stability of a nonconservative systems. International Journal for Numerical Methods in Engineering, 3(1), 63–87. 4. Sugiyama, Y., Matsuike, J., Ryu, B.-J., Katayama, K., Kinoi, S., & Enomoto, H. (1994). Flutter of cantilevered columns subjected to a rocket end thrust and having an intermediate concentrated mass. In AIAA-94-1622, AIAA/ASME/ASCE/AHS/ASC 35th Structures, Structural Dynamics, and Materials Conference (Hilton Head) (pp. 2419–2427). 5. Sugiyama, Y., Matsuike, J., Ryu, B.-J., Katayama, K., Kikoi, S., & Enomoto, N. (1995). Effect of concentrated mass on stability of cantilevers under rocket thrust. AIAA Journal, 33(3), 499–503. 6. Katayama, K. (1997). Experimental verification of the effect of rocket thrust on the dynamic stability of cantilevered columns. Doctoral Thesis, Department of Aerospace Engineering, Osaka Prefecture University.

Chapter 12

Columns under a Rocket-Based Subtangential Follower Force

The stability of columns under conservative forces has been the basis of structural stability theory, while the stability of columns under nonconservative forces has been only of recent interest in regard to structural stability. The stability of columns under the combined action of conservative and nonconservative forces has been an interesting topic in the field of structural stability problems, as it bridges the gap between the stability with conservative forces and the stability with nonconservative forces [1–8]. The combined action of (conservative) dead loads and (nonconservative) follower forces results in the concept of subtangential follower forces. The objective of the present chapter is to give an experimental proof of the concept of a subtangential follower force applied to a standing cantilevered column. The subtangential follower force in this chapter is realized through a combination of the weight of a rocket motor and the thrust induced by the same motor [9–11].

12.1

Mathematical Model and Finite Element Formulation

Figure 12.1a shows a photograph of the setup for the present experiment. Figure 12.1b shows a sketch of the experimental setup. A column was cantilevered vertically and equipped with a solid rocket motor at its tip end. The motor was loosely harnessed by two thin wires to prevent the test column from swaying in an extreme manner while still allowing it to oscillate freely with small and moderate amplitude. Sensors were installed on the test column for axial strain and lateral displacement. The dynamic responses of the test column were recorded by a video camera and a motor-driven camera. Figure 12.2 shows the corresponding mathematical model of a column subjected to the combined action of a vertical force Px and a tangential follower force Pt . On the assumption of small deflection, the resultant force P ¼ Px þ Pt constitutes a © Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_12

155

156

12

(a) Experimental setup

Columns under a Rocket-Based Subtangential Follower Force

(b) Schematic of the setup

Fig. 12.1 Vertically cantilevered column under a subtangential force [9, 10]

Fig. 12.2 Mathematical model of the standing column [9, 10]

subtangential force, the direction of which is specified by au, as illustrated in Fig. 12.2. The angle u is the angle of the tangent of the column at the tip end ðu  1Þ to the vertical axis. The parameter a specifies the inclination of the resultant force P with the vertical axis and is referred to as the tangency coefficient. When the coefficient a ¼ 0, the direction of the force is vertical, and so the force is conservative. When a 6¼ 1:0, the force is nonconservative. The force is tangential

12.1

Mathematical Model and Finite Element Formulation

157

for a ¼ 1:0. Physically, the coefficient a expresses the degree of nonconservativeness of the applied force, and given by a¼

Pt ; P

P ¼ Px þ Pt :

ð12:1Þ

To derive the equation of motion, let us start with the extended Hamilton’s principle: Zt2

Zt2 ðT  V þ Wc þ Wsw Þdt þ

d t1

dWnc dt ¼ 0:

ð12:2Þ

t1

The energy expressions are given as 1 T¼ 2

 2  2 2 ZL  2 @y 1 @y @2y 1 @ y þ LR m dx þ M þ J ; @t 2 @t @t@x x¼L 2 @t@x x¼L 0

V¼

1 2



ZL EI 0

@2y @x2

2 dx;

ZL  2 1 @y P dx; Wc ¼ 2 @x

ð12:3Þ

0

 2 @y mgðL  xÞ dx; Wsw @x 0   @y dy dWnc ¼ Pt : @x x¼L 1 ¼ 2

ZL

Here T is the kinetic energy, V is the elastic energy, Wc is the work done by the conservative component of the subtangential force, Wsw is the work done by the self-weight of the column, and dWnc is the virtual work done by the nonconservative component of the applied force. For discretization of Eq. (12.2), the column is divided into 20 finite elements of equal length with cubic shape functions. A standard FEM formulation yields the equation of motion in matrix form, M€v þ Kv ¼ 0;

ð12:4Þ

where M and K denote the global mass matrix and the global stiffness matrix, respectively. The generalized nodal displacements are denoted by the vector v. The nodal displacements and rotations are assumed to vary in time as

158

12

Columns under a Rocket-Based Subtangential Follower Force

v ¼ v0 ekt ;

ðk ¼ r þ ixÞ

ð12:5Þ

With Eq. (12.5), Eq. (12.4) results in an eigenvalue problem in the form  2  k M þ K v0 ¼ 0:

ð12:6Þ

The eigenvalues are determined numerically. The stability of the system is determined by the concept of asymptotic stability.

12.2

Rocket Motors

Figure 12.3 shows the thrust curve of the rocket motor used in the experiment. The average thrust is assumed to be constant and took the value 40.0 kgf (392 N) for about 4 s. The rotatory inertia of the motor was J ¼ 0:12 kg m2 and the distance from the tip end of the column to the center of gravity of the motor was LR ¼ 0:20 m, just as they were in the first experiment described in Chap. 10. As the aim of the experiment was to realize the subtangential follower force, a considerable conservative force component is essential. Therefore, the massive, conventional motor case, as used in Chap. 10, was used for the present experiment. The motors were weighed to find the nominal initial mass of 14.65 kg, including the mass of propellant of 0.90 kg. The average mass of the motor was thus assumed to be 14.2 kg.

Fig. 12.3 Thrust curve

12.3

12.3

Test Columns

159

Test Columns

Only four rocket motors were provided for the present experiment, so only four test runs were planned. Thus, detailed and careful considerations were made regarding the design of the test columns. The columns were made of aluminium with the measured density q ¼ 2; 670 kg=m3 . The bending stiffness EI was obtained by a static bending test for two types of columns having a cross-section with b ¼ 30:0 mm  h ¼ 8:0 mm and b ¼ 30:0 mm  h ¼ 9:0 m. The measured bending stiffness EI gave the experimental Young’s modulus E ¼ 6:90  103 kgf=mm2 ð67:6 GPaÞ. The divergence and flutter forces of the candidate test columns of different dimensions were estimated for the purpose of selecting four test columns: column for run No. 1 with dimensions 1040  30  9 mm (L  b  h), column for run No. 2 with dimensions 1130  30  9 mm, column for run No. 3 with dimensions 1330  30  9 mm, and column for run No. 4 with dimensions 1125  30  8 mm.

12.4

Stability Estimates

First, let us discuss the dynamics of the test columns. The first two eigenfrequencies are obtained through the characteristic equation (12.6). Figure 12.4 shows the first and second eigenfrequency of column No. 2, with and without the rocket thrust. Points C1 and C2 denote the first and second eigenfrequency of the column under no axial loading. The curve C1 D1 shows the first eigenfrequency under a conservative loading (a ¼ 0), while the curve C1 F shows the same frequency under a subtangential force with a ¼ 0:74: As it is assumed that the rocket thrust Pt is constant and equal to 40:0 kgf ð392NÞ, the tangency coefficient during the application of the thrust is given by a ¼ 40:0=ð14:2 þ 40:0Þ ¼ 0:74. It is seen in Fig. 12.4 that the first eigenfrequency in the case of conservative loading, that is, a ¼ 0, becomes lower with increasing loading and ultimately vanishes. On the other Fig. 12.4 First and second eigenfrequencies of test column No. 2

160

12

Columns under a Rocket-Based Subtangential Follower Force

Fig. 12.5 Stability map of test column No. 2 [9, 10]

hand, as the subtangential load with a ¼ 0:74 increases, the first eigenfrequency does not decrease; rather, it slightly increases. It is seen here that the term “stabilizing effect due to a subtangential force” (exactly a tangential follower force) implies a higher eigenfrequency. It is known that that type of instability, and thus the stability limit, of a cantilevered column under a subtangential force (a generalized Beck’s column) depends on the tangency coefficient; as the coefficient a varies from zero to unity, instability by divergence in the first mode takes place for a  0:5, while instability by flutter occurs for 0:5\a  1:0 [6]. Figure 12.5 shows the stability map of test column No. 2 (the maps for other test columns are similar to this one). The map is obtained by the assumption that the dead load is fixed as Px ¼ 14:2 kgf, while the thrust Pt is increasing. In Fig. 12.5, the design point A designates the column under the dead load, while point B designates the column under the subtangential load with a ¼ 0:74. It can be seen that the application of the rocket thrust, in addition to the dead load, changes the type of instability from static to dynamic. As design point B is much lower than the flutter limit, the column under the subtangential force will not lose its stability by flutter, but rather will remain stable dynamically. This is the physical mechanism of the stabilizing effect of the rocket thrust.

12.5

Experiment with Columns under a Rocket-Based Subtangential Follower Force

On the basis of a thorough consideration of the test columns with different dimensions and their stability maps, the goal of the present experiment was to focus on the following two events: Event I: The test column under a dead load sways with a low frequency initially, as the load is close to, but lower than, the buckling load. Then, the rocket thrust of 40 kgf will be applied to the column in addition to the dead load. The column may

12.5

Experiment with Columns under a Rocket-Based …

161

possibly oscillate with a higher frequency during the burning of the motor (propellant). After the burnout of the motor (propellant), the column will sway again with a lower frequency. Event II: The column under the buckling load is at rest in its buckled form. Then, the thrust is applied to the column. The application of the thrust may possibly make the column dynamically stable when the thrust is active. After burnout, the column will return to the buckled form. It is noted that, in Chap. 10, we only considered the mass of the rocket motor, as the test column with a motor was cantilevered horizontally and thus nullifying the effect of gravity. In the present experiment, however, the test column is cantilevered vertically, and thus it is necessary to consider the weight of the rocket motor as a dead load. It is noted that the nominal weight of the motor before ignition is 14.65 kgf, while the weight of the propellant is 0.9 kgf. The weight of the motor after burnout is thus 13.75 kgf; this is the dead load after burnout of the motor. The average weight of the motor during burning is 14.2 kgf (139 N). It is assumed that the column is subjected to the average dead load Px , during burning of the motor. The rocket thrust Pt is assumed to be constant and equal to 40 kgf ð392 NÞ during the 4-second burning period. Thus, the total load in form of a subtangential follower force is assumed to be 54.2 kgf (=14.2 kgf + 40 kgf). Four test runs were carried out in the present experiment: Run No. 1: A test column (No. 1) with dimensions 1040  30  9.0 mm (L  b  h) was cantilevered vertically. The column without rocket thrust vibrated with a low frequency and with a moderate amplitude. Under the action of the rocket thrust, it oscillated with a higher frequency and a smaller amplitude. After the burnout of the rocket motor, it vibrated as it did before ignition. Run No. 2: A test column (No. 2) of dimensions 1.130  30  9 mm was installed. As the test column was longer than the test column in Run No. 1, it oscillated with a lower frequency. The recorded displacement and axial strain are shown in Fig. 12.6a. It is seen from this figure that the test column under a dead load oscillated with the low frequency of 0.29 Hz [0.227 Hz predicted by Eq. (12.6)], while, under the combined load with the thrust, it oscillated with the

(a) Run No. 2

(b) Run No. 4

Fig. 12.6 Axial strain and displacement recorded in run Nos. 2 and 4 [9, 10]

162

12

Columns under a Rocket-Based Subtangential Follower Force

Fig. 12.7 First eigenfrequencies when a ¼ 0 and a ¼ 0:74 in run No. 2

higher frequency of 0.66 Hz [0.544 Hz by predicted by Eq. (12.6)]. Details of the first eigenfrequencies (in the low frequency region of Fig. 12.4) for a ¼ 0 and a ¼ 0:74 are plotted in Fig. 12.8. Points A and B in Fig. 12.7 show the experimental results, which are compared with the numerical results. Run No. 3: A column (No. 3) of dimensions 1330  30  9 mm was installed. The length of the column was determined such that it was just slightly longer than the “buckling length”. The column with no thrust was in the state of a bent (buckled) configuration. Application of the thrust yielded a total (average) compressive load of 54.2 kgf (531 N). At the moment of application of the thrust, the bent column straightened up and began to oscillate with a moderate frequency. After the burnout of the rocket motor, the column oscillated with a very low frequency. The reason why the test column did not retain its bent form was that the dead load after the burnout (13.75 kgf) was smaller than that before the ignition (14.65 kgf), as the propellant (0.9 kgf) was consumed. Run No. 4: A column (No. 4) of dimensions 1.125  30  8.0 mm was installed. The dimensions were determined so as to realize a clear buckled form of the column at its initial state. Under the combined action of the motor weight and its thrust, the column oscillated around its undeformed axis. After the rocket motor was burned out, the column returned to its bent form. The recorded displacement and axial strain are shown in Fig. 12.6b. A sequence of frames of the column’s response observed in run No. 4 is shown in Fig. 12.8. Figures 12.6, 12.7 and 12.8 testify to the stabilizing effect of a rocket thrust on the dynamics of the column under a dead load. Next we will discuss the observed response in Fig. 12.8, together with the response recorded in Fig. 12.6b. Before ignition, and thus with no rocket thrust, the test column was buckled and at rest in its (static) bent form. During burning of the rocket motor, from ignition to burnout, the column oscillated about its straight configuration. After burnout of the rocket motor, the column again lost its stability by divergence and finally took a buckled configuration. As the amplitude of the test column was constrained to a not very large amplitude by the harness wires on both

12.5

Experiment with Columns under a Rocket-Based …

163

Fig. 12.8 Sequential frames of observed motion of the column in Run No. 4 [9, 10]. a Before the ignition; buckled form with no thrust, b at ignition, and j a while later after the burnout; buckled form with zero thrust

sides, as seen in Fig. 12.1, the divergent motion on one side would soon be pulled back to the opposite side due to the elasticity of the stretched harness wire on the other side, and so on. In this way, the essentially divergent motion responded just like a vibro-impact motion. The motion ultimately settled into a buckled form, due to air resistance.

12.6

Discussion

This chapter has discussed a column subjected to the combined action of a conservative loading and a rocket-based follower force. It has been demonstrated that the action of a follower force stabilizes the column, which initially buckled under a

164

12

Columns under a Rocket-Based Subtangential Follower Force

conservative loading, as the application of a follower force changes the nature of the stability of the column from static to dynamic instability. It is noted the subtangential follower force realized in the present experiment is valid only when the column is discussed under the assumption of small deflection. A movie of the dynamic responses of the test columns, as described in this chapter, is available in the Supplementary Material (online) in Appendix B. The present experiment have recently been discussed in Ref. [11]. Another interesting topic in the field of nonconservative stability problems is the optimum design of structural systems under nonconservative (follower) forces [12–19]. Optimum design of cantilevered columns under the combined action of conservative and nonconservative forces has been discussed by Langthjem and Sugiyama [20, 21]. The experimental verification of shape-optimized cantilevered columns under a rocket-based follower force was presented by Sugiyama et al. [22].

References 1. Leipholz, H. (1962). Die Knicklast des eiseitig eigespannten Stabes mit gleichmäßig verteiler, tangentialer Längsbelastung. Zeitschrift für Angewandte Mathematik und Physik, 6, 581–589. 2. Sugiyama, Y., Katayama, T., & Sekiya, T. (1969). Studies on nonconservative problems of instability of columns by difference method. In Proceedings of the 19th Japan National Congress for Applied Mechanics, 1969 (pp. 23–31). Tokyo: The University of Tokyo Press. 3. McGill, D. J. (1971). Column instability under weight and follower loads. Journal of the Engineering Mechanics Division, American Society of Civil Engineers, 97, 629–635. 4. Sugiyama, Y., Kawagoe, H., & Tanimoto, Y. (1974). Stability of elastic columns subjected to distributed sub-tangential follower forces. Reports of the Faculty of Engineering (vol. 5-1, pp. 1–10). Tottori University. 5. Sugiyama, Y., & Kawagoe, H. (1975). Vibration and stability of elastic columns under the combined action of uniformly distributed vertical and tangential forces. Journal of Sound and Vibration, 38(3), 341–355. 6. Celep, Z. (1977). On the vibration and stability of Beck’s column subjected to vertical and follower forces. Zeitschrift für Angewandte Mathematik und Mechanik, 57, 555–557. 7. Leipholz, H. (1980). Stability of Elastic Systems. Alphen aan den Rijn: Sijthoff & Noordhoff. 8. Sugiyama, Y., & Mladenov, K. A. (1983). Vibration and stability of elastic columns subjected to triangularly distributed sub-tangential forces. Journal of Sound and Vibration, 88(4), 447–457. 9. Katayama, K. (1997). Experimental verification of the effect of rocket thrust on the dynamic stability of cantilevered columns. Doctoral Thesis, Department of Aerospace Engineering, Osaka Prefecture University. 10. Sugiyama, Y., Katayama, K., Kiriyama, K., & Ryu, B.-J. (2000). Experimental verification of dynamic stability of vertical cantilevered columns subjected to a sub-tangential force. Journal of Sound and Vibration, 236(2), 193–207. 11. Mutyalarao, M., Bharathi, D., Narayana, K. L., & Nageswara Rao, B. (2017). How valid are Sugiyama’s experiments on follower forces? Letter to the Editor, International Journal of Non-Linear Mechanics, 93, 122–125. 12. Plaut, R. H. (1975). Optimal design for stability under dissipative, gyroscopic, or circulatory loads. In A. Sawczuk, Z. Mroz (Eds.), Optimization in structural design. Berlin: Springer. 13. Odeh, F., & Tadjbakhsh, I. (1975). The shape of the strongest column with a follower load. Journal of Optimization Theory and Applications, 15, 103–118.

References

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14. Claudon, J. L. (1975). Characteristic curves and optimum design of two structures subjected to circulatory loads. Journal de Mécanique, 14, 531–543. 15. Hanaoka, M., & Washizu, K. (1980). Optimum design of Beck’s column. Computers & Structures, 11, 473–480. 16. Tada, Y., Seguchi, Y., & Kema, K. (1985). Shape determination of nonconservative structural systems by the inverse variational principle. Memoirs of the Faculty of Engineering (vol. 32, pp. 45–61). Kobe University. 17. Gutkowski, W., Mahrenholz, O., Pyrz, O. (1983). Minimum weight design of structures under nonconservative forces. In G. I. N. Rozvany (Ed.), Optimization of large structural systems (pp. 1087–1100). Dortrecht: Kluwer. 18. Ringertz, U. T. (1994). On the design of Beck’s column. Structural Optimization, 8, 120–124. 19. Langthjem, M. A., & Sugiyama, Y. (1999). Optimum design of Beck’s column with a constraint on the static buckling load. Structural Optimization, 18, 228–235. 20. Langthjem, M. A., & Sugiyama, Y. (2000). Optimum design of cantilevered columns under the combined action of conservative and nonconservative loads. Part I: The undamped case, Computers & Structures, 74, 385–398. 21. Langthjem, M. A., & Sugiyama, Y. (2000). Optimum design of cantilevered columns under the combined action of conservative and nonconservative loads. Part II: The damped case, Computers & Structures, 74, 399–408. 22. Sugiyama, Y., Langthjem, M. A., Iwama, T., Kobayashi, M., Katayama, K., & Yutani, H. (2012). Shape optimization of cantilevered columns subjected to a rocket-based follower force and its experimental verification. Structural and Multidisciplinary Optimization, 46, 829–838.

Chapter 13

Pinned-Pinned Columns under a Pulsating Axial Force

There are many examples of physical systems, including structural systems, that are subject to time-varying excitations. One example is a string subjected to a pulsating axial tension. The string loses its stability by so-called parametric resonance, which occurs primarily when the excitation frequency h is twice the string’s eigenfrequency xo , that is, h ¼ 2xo . This phenomenon is known as Melde’s effect [1, 2]. This chapter gives an introduction to parametric resonances of a column under a pulsating axial force. The loss of stability of structural systems under pulsating forces has been an area of considerable interest in the field of structural mechanics and engineering. As a simple case, a pinned-pinned column under a harmonically varying force is discussed in this chapter. The force is instationary, hence the system is nonconservative. The dynamic responses of the column near the resonance boundaries are discussed in detail to show how the column loses its stability dynamically. A systematic discussion on the dynamic stability of elastic systems has been given by Bolotin [3].

13.1

The Mathieu Equation

Let us consider a general differential equation for a physical system subjected to a harmonic excitation in the form d2y dy þ 2l þ x2o ð1 þ c cos HtÞy ¼ 0: dt2 dt

ð13:1Þ

This type of equation is known as a Mathieu equation. We shall rely here on approximate solutions to this equation. In most of the specialist literature on this equation, the solution is obtained by applying Floquet’s theory, assuming periodic solutions on the boundaries of resonances [3]. © Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_13

167

168

13

13.2

Pinned-Pinned Columns under a Pulsating Axial Force

Stability of the Solution to the Mathieu Equation

First, let us consider the undamped case, l ¼ 0. We shall discuss the stability of the solution for small forcing ð0\c  1Þ when the forcing frequency H is close to the double of the natural frequency xo . We set H ¼ 2xo þ e, where e is a small “detuning” parameter. For convenience, we let H ¼ 2h, that is, h ¼ xo þ ð1=2Þe. Consider the reduced equation d2y þ x2o ð1 þ c cos 2htÞy ¼ 0: dt2

ð13:2Þ

We assume, as a first approximation, a solution in the form yðtÞ ¼ aðtÞ cos ht þ bðtÞ sin ht:

ð13:3Þ

It is assumed that da=dt is of same order of magnitude as ea, that is, da=dt ¼ a_ ¼ OðeaÞ, and the same for b. Then we obtain

 d2 y ¼ a x2o þ exo cos htb x2o þ exo sin ht 2 dt
  2xo a_ sin ht  2xo b_ cos ht þ O e2 a; e2 b :

ð13:4Þ

The terms of order e2 a; e2 b will be dropped from now on. As we are interested in a simple solution valid for just small values of c as stated above, rather than a numerically accurate one, we will also discard terms proportional to sin 3ht; cos 3ht in the expansion of yðtÞ cos 2ht. Substituting Eqs. (13.3) and (13.4) into Eq. (13.2), we obtain     1 1 _ 2a_ þ hb þ cxo b xo sin ht þ 2b  ea þ cxo a xo cos ht ¼ 0: 2 2

ð13:5Þ

For a solution to exist, it is necessary for both terms in round brackets ( ) to be equal to zero. A solution to the resulting system of two coupled differential equations can be written in the form faðtÞ; bðtÞg ¼ fao ; bo gexpðstÞ, where ao ; bo , and s are real constants. The system of differential equations is written in the reduced matrix form 

2s 1 cx oe 2

1 2 cxo

2s

þe

 

a0 b0



 0 ¼ : 0

ð13:6Þ

For a non-trivial solution, the determinant of the coefficient matrix needs to be zero. We obtain

13.2

Stability of the Solution to the Mathieu Equation

1 s ¼ 4

"

2

1 xo c 2

169

#

2

e :

ð13:7Þ

2

Recalling the amplification factor expðstÞ, a solution is stable if s\0 and unstable if s [ 0. The condition for an unstable solution, that is, for parametric resonance, is thus obtained as 1 1  xo c\e\ xo c: 2 2

ð13:8Þ

Within the frequency range, the condition for resonance is given by c H c \1 þ : 1 \ 4 2xo 4

ð13:9Þ

This condition gives the principal (primary) region of parametric resonance in the first order approximation, where the instability is strongest. Parametric resonance will occur when H ¼ 2xo =n; n ¼ 1; 2; 3; . . .; however, the width of the regions of instability decrease rapidly with increasing value of n. It is noted that many texts write the Mathieu equation in the following standard form: d2 y þ ða þ b cos sÞy ¼ 0: ds2

ð13:10Þ

This corresponds to Eq. (13.2) when a ¼ ðxo =HÞ2 , b ¼ cðxo =HÞ2 and s ¼ Ht. In this expression, the instability regions will have their boundaries at a ¼ ðn=2Þ2 ; n ¼ 1; 2; 3; . . .. Now let us consider how the inclusion of damping ðl [ 0Þ in Eq. (13.1) affects the principal region of parametric resonance (instability). When the system is free from the excitation, h ¼ 0, it is known that the solution will be proportional to expðltÞ, that is, the amplitude will decay by this factor. We can assume, as a first approximation, that aðtÞ and bðtÞ in Eq. (13.1) will take the form faðtÞ; bðtÞg ¼ fao ; bo gexpðltÞexpðstÞ ¼ fao ; bo gexp½ðs  lÞt. This means that the “s” in Eq. (13.7) will be replaced by s  l. We then have the determinant condition for the damped case 1 ðs  lÞ ¼ 4 2

"

1 xo c 2

#

2

e : 2

ð13:11Þ

The limit of instability is now characterized by s  l ¼ 0. With s given by Eq. (13.7), the condition of dynamic instability is given by

170

13

Pinned-Pinned Columns under a Pulsating Axial Force

s sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2  2 1 1 xo c 4l2 \e\ xo c 4l2 :  2 2

ð13:12Þ

Accordingly, the formula for the damped case within the frequency range is written as c 1 4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     4l 2 H c 4l 2 1 1 \ : \1 þ xo c 2xo 4 xo c

ð13:13Þ

It is seen from this formula that the width of the instability region is reduced due to damping. Since the expression in the square roots must be positive, instability sets in only when c [

4l : xo

ð13:14Þ

Below the critical exciting amplitude c , the damped system is stable. It is interesting to consider the critical exciting level, as given by Eq. (13.14), through an energy analysis. To this end, we will simplify the computations by setting e ¼ 0 in the expressions of H and h for Eqs. (13.1) and (13.3). Now let us assume that a and b in Eq. (13.3) are constants. We then consider the equation d2y dy þ 2l þ x2o ð1 þ c cos 2xo tÞy ¼ 0; dt2 dt

ð13:15Þ

and assume a solution in the form yðtÞ ¼ a cos xo t þ b sin xo t:

ð13:16Þ

An energy balance equation is obtained by multiplying Eq. (13.15) by the velocity dy=dt, followed by integration over one period T, 0  t  T, where T ¼ 2p=xo . The energy gained by the system under harmonic forcing is given by ZT

dy ydt ¼ cx2o abp: dt

ð13:17Þ

ZT  2
 dy DD ¼ 2l dt ¼ 2lxo p a2 þ b2 : dt

ð13:18Þ

DW ¼

cx2o

cos 2xo t 0

The energy dissipated by the damping is

0

13.2

Stability of the Solution to the Mathieu Equation

171

The motion of the system is unstable when DW [ DD. The instability takes place when the amplitude of excitation satisfies the condition c[

  2l a b þ : xo b a

ð13:19Þ

The condition in Eq. (13.19) is identical to Eq. (13.14) when a ¼ b. It is emphasized that the solutions discussed here are approximations. The correct solutions to Eq. (13.2), for example, will include higher order harmonics as well, that is, cosine and sine functions with arguments 2ht; 3ht; . . ., (a full Fourier series in general) [1]. This applies also in the consideration of energy; and there is an additional approximation in assuming that a and b are constants, as it has been demonstrated earlier that a and b are functions of time. Nonetheless, the simple computations give an estimate of the stability properties of the solution and a qualitative description of how the inclusion of damping affects the regions of parametric resonances with a single mode.

13.3

Pinned-Pinned Columns

The dynamic stability of a column subjected to a periodic axial force was first studied by Beliaev in 1924 [4]. He discussed a column pinned (simply supported) at both ends, as shown in Fig. 13.1. It is assumed that the time-varying component of the force is harmonic.

Fig. 13.1 Pinned-pinned column under a harmonically pulsating force (Beliaev’s column)

172

13

Pinned-Pinned Columns under a Pulsating Axial Force

The equation of motion of the column is given by m

@2y @2y @4y þ ð P þ P cos ht Þ þ EI ¼ 0; o t @t2 @x2 @x4

ð13:20Þ

where Po is the constant component of the axial force, Pt is the amplitude of the periodic component, and h is the circular frequency of the periodic force. Let us write the solution to Eq. (13.20) in the form yðx; tÞ ¼ fk ðtÞ sin

kpx ; L

ð13:21Þ

where k is an integer ðk ¼ 1; 2; . . .Þ that designates the kth vibration mode of a column pinned at both ends. Substitution of Eq. (13.21) into Eq. (13.20) leads to d 2 fk þ x2k ð1  ck cos htÞfk ¼ 0; dt2

ð13:22Þ

where xk ¼ xk

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Po 1 ; Pk

ck ¼

Pt k2 p2 ; xk ¼ 2 Pk  P0 L

rffiffiffiffiffi EI k 2 p2 EI : ; Pk ¼ m L2 ð13:23Þ

Here xk is the kth eigenfrequency of the column loaded by the force Po, ck is the nondimensional amplitude of the periodic force, xk is the kth natural frequency of the unloaded column, and Pk is the kth Euler buckling load. We restrict our interest to the principal region of simple resonance, which occurs with a single mode. The resonance there takes place when the exciting frequency h is equal to the double of the frequency xk of the free vibration of the column. An expression for the boundaries of the principal region was given by Beliaev [3, 4] in the form h c ¼1 k: 2xk 4

ð13:24Þ

Nishino [5] has given the expression # ¼ 2xk

rffiffiffiffiffiffiffiffiffiffiffiffiffi c 1  k: 2

ð13:25Þ

Bolotin [3] has given the following expression, which includes the second approximation:

13.3

Pinned-Pinned Columns

# ¼ 2xk

173

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ðck =2Þ2 : 1 k þ 2 8  9ck =2

ð13:26Þ

These expressions have been obtained under the assumption that periodic motions exist on the boundaries of dynamic instability and that these motions are close to being harmonic.

13.4

Vibrations in the Vicinity of Upper and Lower Boundaries I

Let us discuss the vibrations of the pinned-pinned column (Beliaev’s column) in the vicinity of the stability boundaries, observed in an analog computer-based experiment. Figure 13.2 shows the analog circuit for the Mathieu equation (13.22), in which the upper circuit represents the dynamical system, while the lower represents the excitation. The initial conditions are taken to be fk ð0Þ ¼ 1:0 and dfk ð0Þ=dt ¼ 0. Within the context of linear vibrations, the term ‘dynamic instability’ usually means vibrations with unbounded amplitude after infinite time. In the present analog experiment, however, the following relaxed qualitative criterion is adopted: the system is defined as being dynamically unstable if, under a specified loading condition, an exponentially increasing amplitude of the response is observed within forty periods of excitation. The region obtained by the present experiment is shown

(a) Circuit Fig. 13.2 Analog circuit and symbols

(b) Symbols and functions

174

13

Pinned-Pinned Columns under a Pulsating Axial Force

Fig. 13.3 Principal region of resonance for Beliaev’s column [6]

in Fig. 13.3, and compared there with formula (13.25), which is designated as Eq. (5) in Fig. 13.3. The motions observed at the points A, B, C, and D, going from a stable to an unstable state in the vicinity of the lower bound, are shown in Fig. 13.4. The upper records show the instationary motions of the column, while the lower ones show the harmonic excitations. The beating motions with an amplitude smaller than the initial deflection, as shown in Figs. 13.4a, b, lead to a harmonic motion, as in Fig. 13.4c, and finally to an exponentially increasing motion, as shown in Fig. 13.4d. It is seen that a harmonic motion takes place on the lower boundary for resonance. The motions observed at the points E, F, G and H, going from an unstable to a stable state in the vicinity of the upper boundary, are shown in Fig. 13.5. It is seen that the motions just above the upper boundary are beating vibrations with an amplitude larger than the initial deflection. It is interesting to note that no harmonic motion is observed to define the upper boundary. In other words, a harmonic motion does not necessarily occur on the upper boundary, as it does on the lower boundary. It is interesting to recall some experiments conducted earlier by Bolotin [3], Somerset and Evan-Iwanowski [7], and Evan-Iwanowski [8]. They also observed that beats with maximum amplitude larger than an initial displacement may occur when the excitation parameters are located just above the upper boundary.

13.4

Vibrations in the Vicinity of Upper and Lower Boundaries I

(a) Point A

(b) Point B

(c) Point C

(d) Point D

175

Fig. 13.4 Vibrations in the vicinity of the lower boundary [6]. Points in ðh=2xk ; ck Þ plane: Point A (0.818, 0.6), Point B (0.840, 0.6), Point C (0.850, 0.6), Point D (0.852, 0.6)

It is noted that the phase angle between the column response and the excitation is different on the lower and upper boundaries. If one takes careful notice of the corresponding relation between the responses and the excitations, it can be seen that, on the lower boundary, the maximum amplitudes of the responses correspond to compressive components, while on the upper boundary, they correspond to tensile components. This phenomenon will be discussed analytically later in Sect. 13.6.

176

13

Pinned-Pinned Columns under a Pulsating Axial Force

(a) Point E

(b) Point F

(c) Point G

(d) Point H

Fig. 13.5 Vibrations in the vicinity of the upper boundary [6] Points in ðh=2xk ; ck Þ plane: Point E (1.140, 0.6), Point F (1.142, 0.6), Point G (1.150, 0.6), Point H (1.170, 0.6)

13.5

Vibrations in the Vicinity of Upper and Lower Boundaries II

Let us discuss the vibrations near the upper and lower boundaries, observed in another analog experiment, in more detail. Here, three quantitative criteria of dynamic instability are applied: a system under a given loading is defined to be unstable if the response of the system exceeds two times, ten times and one hundred times the initial deflection for forty cycles of the excitation frequency. The initial conditions are the same as those given in the previous section. Now let us consider a phase difference in the excitation. A modified version of Eq. (13.22) having a phase difference p is given in the form

13.5

Vibrations in the Vicinity of Upper and Lower Boundaries II

177

Fig. 13.6 Boundaries by three different criteria [6]

d 2 fk þ x2k f1  ck cosðht þ pÞgfk ¼ 0: dt2

ð13:27Þ

The results obtained by the three criteria are shown in Fig. 13.6. Stability maps obtained for Eqs. (13.22) and (13.27) are compared to present the effect of the phase difference (of excitation) on the responses. It is noted that in Fig. 13.6, Eq. (13.22) is designated as Eq. (6), while Eq. (13.27) is designated as Eq. (7). The regions of simple resonance are indicated by hatched oblique lines, which are determined by the hundred-times criterion. Beating bands determined by the two-times and hundred-times criteria are indicated by hatched horizontal lines. The beating band exists above the upper boundary in the case of Eq. (13.22) and below the lower boundary in the case of Eq. (13.27) [9, 10]. Figure 13.7 shows the two types of beating vibration observed in the vicinity of the lower and upper boundaries of the region for Eq. (13.22). There are very narrow bands determined by the two-times and the hundred-times criteria on the lower boundary for Eq. (13.22) and on the upper boundary for Eq. (13.27). These bands are not hatched in Fig. 13.6, in order to avoid confusion, because they are not attributable to a beating vibration, but rather to a small increasing rate of amplitude of a quasi-unstable vibration. The region of resonance is not affected by the phase difference of excitation. It is reasonable that the stronger criterion gives a narrower domain for instability. It is seen that there exist beating vibrations having amplitude smaller than the initial deflection, resulting in harmonic motion on the boundaries, on the lower boundary

178

13

Pinned-Pinned Columns under a Pulsating Axial Force

(a) Point A’

(b) Point B’

Fig. 13.7 Beating vibrations in the vicinity of the lower and upper boundaries of the resonance region, as shown in Fig. 13.6 [in the case of Eq. (13.27)] [6] (Point A′; in the vicinity of the lower boundary, Point B′; in the vicinity of the upper boundary)

in the case of Eq. (13.22) and on the upper one in the case of Eq. (13.27). It is seen that the vibrational process from stability to instability does not always take the passage through harmonic motion, although it is correct to assume harmonic motion on the boundaries of resonance. In other words, a harmonic vibration can exist only on the stability boundaries. For more precise observation of the dynamical behavior of solutions to the Mathieu equations, and on parametric resonances, it is suggested that a numerical simulation approach to Eqs. (13.8)–(13.9) be applied.

13.6

Effect of a Phase Angle in Excitation

Let us consider a slightly generalized version of Eq. (13.27), with a phase angle a in excitation, in the form d 2 fk þ x2k f1  ck cosðht þ aÞgfk ¼ 0: dt2

ð13:28Þ

It was shown in the previous section that there exists a nearly harmonic motion on the boundaries of simple resonance. Knowing that the excitation frequency is double the frequency of the free vibration of the column, we may express an approximate solution to Eq. (13.28) in the form 

 ht þb : fk ðtÞ ¼ ae cos 2 mt

ð13:29Þ

Substitution of Eq. (13.29) into Eq. (13.28) yields 



   2 c x2 c x2  h4 þ x2k  k2 k C1 cos ht2 þ b þ mh  k2 k :C2 sin ht2 þ b

3ht  c x2 ¼ 0;  k2 k C1 cos 3ht 2 þ 3b þ C2 sin 2 þ 3b

ð13:30Þ

13.6

Effect of a Phase Angle in Excitation

179

where C1 ¼ cos a cos 2b þ sin a cos 2b; C2 ¼ cos a sin 2b  sin a cos 2b:

ð13:31Þ

The first approximation of the solution for the principal region requires the following conditions: C1 ¼

  2 h2 1 2 ; ck 4xk

ð13:32Þ

ck x2k  C2 : 2h

ð13:33Þ

m¼

For any combination of the phase angle a in the excitation and the phase angle b in the response, using the relation C12 þ C22 ¼ 1, we obtain c2 x 2 m2 ¼ k 2k 4h

(

 2 ) 4 h2 1 2 1 2 : ck 4xk

ð13:34Þ

If the condition  2 4 h2 1 2 1 2 [0 4xk ck

ð13:35Þ

holds, m has a positive value and the system is dynamically unstable. Thus, we obtain the expression for the principal region of simple resonance h ¼ 2x

rffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 k: 2

ð13:36Þ

Equation (13.36) is the expression given by Nishino [5] (Eq. 13.25). Substituting Eq. (13.36) into Eq. (13.32), we obtain C1 ¼ 1

ð13:37Þ

C1 ¼ 1

ð13:38Þ

on the lower boundary and

on the upper boundary. For b ¼ 0, we obtain a ¼ 0 on the lower boundary and a ¼ p on the upper boundary. For b ¼ 0; the case in which the periodic solution is close to a sin ht=2 corresponds to the initial conditions that f ð0Þ ¼ 0 and df ð0Þ=dt ¼ 1, and thus a ¼ p on the lower boundary and a ¼ 0 on

180

13

Pinned-Pinned Columns under a Pulsating Axial Force

the upper boundary. The periodic solution in the form of Eq. (13.29) with m ¼ 0 exists when the initial phases a and b satisfy the conditions in Eqs. (13.37) and (13.38).

13.7

Discussion

The assumption that the periodic solutions on the boundary are close to harmonic vibration is very powerful in theoretically determining the boundaries for resonances in the first approximation. However, it is noticed that the vibrations near the boundaries, particularly in simulation and experiment, are not always close to harmonic vibration, as the character of instationary vibrations near the boundaries depends on the initial conditions, say, on the phase difference in excitation. This fact suggests some difficulties in determining a clear/unique boundary of resonances in experimental and simulation-based approaches to parametric resonances. As an example of a recent approach to the dynamic instability of columns under an arbitrary periodic load, the works by Song et al. [11] and Huang et al. [12] can be mentioned.

References 1. Melde, F. (1860). Über die Erregung stehender Wellen eines fadenförmigen Körpers. Annalen der Physik und Chemie, 109, 193–215. 2. Rayleigh, J. W. S. (1945). The Theory of Sound (Vol. I). New York: Dover Publications, Inc. 3. Bolotin, V. V. (1964). The dynamic stability of elastic systems. San Francisco: Holden-Day Inc. 4. Beliaev, N. M. (1924). Stability of prismatic rods subjected to variable longitudinal forces. In Collection of Papers; Engineering Constructions and Structural Mechanics, Leningrad (p. 149, in Russian). 5. Nishino, K. (1939). Vibrational stability of a bar under periodic longitudinal forces. Journal of the Aeronautical Research Institute, Tokyo Imperial University, 176, p. 93 (in Japanese). 6. Sugiyama, Y., Iwatsubo, T., & Ishihara, K. (1972). On nonsteady-state motions in the vicinity of the boundaries of the principal region of dynamic instability. Report of the Faculty of Engineering, Tottori University, 2(2), 28–37. 7. Somerset, J. H., & Evan-Iwanowski, R. M. (1964). Experiments on parametric instability of columns, development of theoretical and applied mechanics. In South-Eastern Conference on Theoretical and Applied Mechanics. 8. Evan-Iwanowski, R. M. (1965). On the parametric response of structures. Applied Mechanics Review, 18(9), p. 699. 9. Sugiyama, Y., Kawagoe, H., & Iwatsubo, T. (1972). On instability criteria in digital simulation of parametric instability. Reports of the Faculty of Engineering, Tottori University, 3(1), 1–9. 10. Iwatsubo, T., Sugiyama, Y., & Ishihara, K. (1972). Stability and non-stationary vibration of columns under periodic loads. Journal of Sound and Vibration, 23(2), 245–257.

References

181

11. Song, Z., Chen, Z., Li, W., & Chai, Y. (2016). Parametric instability analysis of a rotating shaft subjected to a periodic axial force by using discrete singular convolution method. Meccanica, 51(6), 1–15. 12. Huang, Y., Liu, A., Pi, Y., Lu, H., & Gao, W. (2017). Assessment of lateral dynamic instability of columns under an arbitrary axial load owing to parametric resonance. Journal of Sound and Vibration, 395, 272–293.

Chapter 14

Parametric Resonances of Columns

This chapter aims to give a general overview of the parametric resonances of columns under a harmonically pulsating force. In addition to simple resonance, combination resonances of sum and difference type are introduced, with columns having various kinds of boundary condition, other than pinned-pinned ends. For simplicity, it is assumed that damping is absent. Hsu’s conditions for resonances are introduced to give the first estimate of the principal regions of instability. Experiments with columns having clamped-clamped and clamped-pinned ends are presented, to demonstrate combination resonances of sum type. Combination resonances of difference type are introduced through an analog computer-based experiment of a cantilevered column subjected to a pulsating follower force.

14.1

Mathieu-Hill Equations

The stability of columns, and structures in general, subjected to time-varying excitations is governed by partial differential equations with time-varying coefficients. In the case of columns, a typical approach is to apply a discretization method (Galerkin, finite difference, finite element method, etc.) such that the equations are reduced to a set of coupled ordinary differential equations, which, in matrix form, can be written as M€y þ Cy_ þ fKo þ Kt ðtÞgy ¼ 0:

ð14:1Þ

In the case of a uniform column, the N  N system Eq. (14.1) can be transformed into N coupled equations in the form ( €ym þ cm y_ m þ kom þ

N X

) ktn ðtÞ yn ¼ 0;

m ¼ 1; 2; . . .; N:

ð14:2Þ

n¼1

© Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_14

183

184

14

Parametric Resonances of Columns

When the time-dependent coefficients ktn ðtÞ are periodic, they can be expanded into a Fourier series in the form ktn ðtÞ ¼

1 X

ðanr sin rht þ bnr cos rhtÞ:

ð14:3Þ

r¼1

In this case, the differential equations in the form of Eq. (14.2) are referred to as N coupled Hill equations, or Mathieu-Hill equations. A special but typical case is when the force is pulsating with a single frequency h such that, say, anr ¼ 0. The equations in the form of Eq. (14.2) take the expression ( €ym þ cm y_ m þ k0m þ

N X

) ktn cos ht yn ¼ 0;

m ¼ 1; 2; . . .; N:

ð14:4Þ

n¼1

The differential equations in the form of (14.4) are called N coupled Mathieu equations. Even the solutions and stability properties of the solutions to a single Mathieu-Hill equation, ( ) 1 X d2 y dy þ c þ ao þ ðar sin rht þ br cos rhtÞ y ¼ 0; dt2 dt r¼1

ð14:5Þ

or, especially, a single Mathieu equation, d2 y dy þ c þ ðao þ at cos htÞy ¼ 0; dt2 dt

ð14:6Þ

are very complicated mathematically [1–5].

14.2

Hsu’s Approach

Hsu [6] has proposed a perturbation expansion approach to an equation in the form of Eq. (14.1), under the assumption of small periodic perturbations, in other words, the governing equation is assumed to be in the form M€y þ Cy_ þ fKo þ eKt ðtÞgy ¼ 0;

ð14:7Þ

where e is a small parameter ð0\e  1Þ. An asymptotic expansion of the form ym ðtÞ ¼ cm ðtÞ sin xm t þ dm ðtÞ cos xm t þ

1 X q¼1

eq yðmqÞ ðtÞ

ð14:8Þ

14.2

Hsu’s Approach

185

is assumed for the elements ym ðtÞ of the solution vector yðtÞ. The unknown functions yðmqÞ ðtÞ in Eq. (14.8) are determined via application of the Krylov-Bogolyubov-van der Pol method [4]. This method is now more commonly known as the method of averaging. However, Hsu has taken his approximation only up to order e1 . As stated by Hsu in his paper [6], this approach was first applied to a single Hill equation, the Mathieu equation, by Struble and Fletcher [7]. The extension of the method to a matrix Hill equation is not trivial, however, due to the possibility of internal resonance between the individual modes, and due to the lengthy algebra involved. Cases of internal resonance are investigated in greater detail in another paper by Hsu [8], however, still only up to the first order in the small parameter e.

14.3

Coupled Mathieu Equation of Columns

When a column is pinned (simply supported) at both ends and subjected to a harmonically periodic force, the reduced system of equations is expressed by a system of uncoupled Mathieu equations. In this case, only simple resonances with a single mode occur. However, if the columns have other types of end condition, the situation is quite different. Then the reduced system of equations of motion for the columns is given by a system of coupled Mathieu equations. In this case, combination resonances with multiple modes may take place, in addition to simple resonance.

Fig. 14.1 Columns under a periodic axial force

(a) Clamped-clamped column (b) Clamped-pinned column

186

14

Parametric Resonances of Columns

Let us consider two columns subjected to a harmonically time-varying axial force and supported by (i) clamped-clamped and (ii) clamped-pinned ends, as shown in Fig. 14.1. For simplicity, it is assumed that damping is absent. The equation of motion for the columns is written as m

@2y @2y @4y þ P cos ht þ EI ¼ 0: t @t2 @x2 @x4

ð14:9Þ

The boundary conditions are: For clamped-clamped ends: yð0; tÞ ¼

@yð0; tÞ @yðL; tÞ ¼ yðL; tÞ ¼ ¼ 0; @x @x

ð14:10Þ

@yð0; tÞ @ 2 yðL; tÞ ¼ yðL; tÞ ¼ ¼ 0: @x @x2

ð14:11Þ

For clamped-pinned ends: yð0; tÞ ¼

Applying, for example, the finite difference method, the Galerkin method, or the finite element method to the partial differential equation of motion (together with the corresponding boundary conditions), we obtain a system of ordinary differential equations in matrix form   d2 y Pt 2 þ x K þ F cos ht y ¼ 0: o dt2 P

ð14:12Þ

The coupled Mathieu equations (14.12) can be transformed from the generalized coordinates y to the normal coordinates x by diagonalization of the matrix K. Then, we obtain   d2 x Pt 0 2 0ðoÞ D þ x B þ cos ht x ¼ 0; o dt2 P

ð14:13Þ

where B0ðoÞ is a diagonal matrix. Equation (14.13) is ultimately expressed in the form  d2 x  ðoÞ þ B þ eD cos ht x ¼ 0; dt2

ð14:14Þ

where e¼

Pt ; P

BðoÞ ¼ x2o B0ðoÞ ;

and

D ¼ x2o D0

ð14:15Þ

14.3

Coupled Mathieu Equation of Columns

187

In Eq. (14.14), x is a column matrix of n dependent variables. BðoÞ is a diagonal matrix whose elements are all positive and equal to the second power of the natural frequencies of the unloaded system, x21 ; x22 ; . . .; x2n , assuming that x21 \x22 \. . .\x2n . D is a n  n matrix with elements dij .

14.4

Hsu’s Resonance Conditions

Hsu [6] has presented some simple, and very useful, resonance conditions for an undamped dynamical system having multiple degrees of freedom and when the magnitude of the harmonic excitation is small (of order of magnitude e; e  1). Hsu’s approach to the Mathieu-Hill equations will be discussed in detail in the next chapter (see Sect. 15.2). Equation (14.4) is rewritten in the form ! n X d2 x i 2 þ xi þ e dij cos ht xj ¼ 0; dt2 j¼1

ði; j ¼ 1. . .; nÞ:

ð14:16Þ

Hsu’s conditions for the principal regions of resonances for the system expressed in Eq. (14.16) are given in the forms [6] (i) h ¼ 2xk þ ek, 2xk  (ii) h ¼ xk þ xl þ ek; ðxk þ xl Þ 

e jdkk j e jdkk j \h\2xk þ ; 2 xk 2 xk

k 6¼ l;

    e dkl dlk 1=2 e dkl dlk 1=2 \h\ðxk þ xl Þ þ ; 2 xk xl 2 xk xl

(iii) h ¼ xl  xk þ ek;

ð14:17Þ

ð14:18Þ

l [ k;

    e dkl dlk 1=2 e dkl dlk 1=2   ðxl  xk Þ  \h\ðxl  xk Þ þ ; ð14:19Þ 2 2 xk xl xk xl where xk is the kth natural frequency of the unloaded system and k is an arbitrary real number. The above expressions indicate that when the elements dkl and dlk ðk 6¼ lÞ are of the same signs, combination resonances of sum type occur, while when they take opposite signs, combination resonances of difference type occur. The simple resonances given by Eq. (14.17) are understood as special cases of combination resonances of sum type by Eq. (14.18) when k ¼ l. Theoretically, there are predicted the first, second, third, etc. regions of parametric resonance related to the same eigenfrequencies [2]. The first region is

188

14

Parametric Resonances of Columns

referred to as principal region which is predicted by the first-order approximation, while the second is referred to as secondary region which is predictable through the second-order approximation, and so on. It is noted that Hsu’s conditions are the first order approximations and so predict only principal regions of dynamic instability, while laboratory experiment and numerical simulation (and an analog computer-based experiment in Sect. 14.7) may present secondary regions.

14.5

Estimate of the Principal Regions of Resonances

Figure 14.2 shows the principal regions of instability of the column having clamped-clamped ends. Combination resonances of the sum type are possible only between the kth and ðk  2mÞth modes. Figure 14.3 shows the instability regions for the column having clamped-pinned ends. Combination resonances of sum type between any two different modes can occur. This is attributable to the unsymmetrical character of the modes (coordinate functions) with respect to the midpoint of the column.

Fig. 14.2 Regions of parametric resonances of the clamped-clamped column [9]

Fig. 14.3 Regions of parametric resonances of the clamped-pinned column [9]

14.6

14.6

Experiment with Columns Having Clamped-Clamped and Clamped-Pinned Ends

189

Experiment with Columns Having Clamped-Clamped and Clamped-Pinned Ends

A schematic of the experimental setup is shown in Fig. 14.4. Figure 14.5 shows the actual ends devised for clamped and pinned boundary conditions. The periodic axial force was applied to the specimen by a 50 kgf (490 N) electromagnetic shaker. The axial force was measured by both a load cell (② in Fig. 14.4) and strain gauges (③ in Fig. 14.4). Two capacity-type micrometers were mounted on the test column at two different positions for measurement of displacements. All outputs from the load cell, the strain gauges and micrometers were sent to a synchro-scope for observation and recording. Two steel specimens, of length L = 22.7 mm, for the clamped-clamped column and L = 21.8 mm for the clamped-pinned column, were used. The two specimens had the same breadth b = 1.0 mm, and thickness h = 0.070 mm. The Young’s modulus of the specimens was E = 2.1  106 kg/cm2 (200 GPa) and the mass per unit length m = 5.61  10−7 kgf/cm (5.50  10−6 kg/cm). Theoretically estimated values for the first three natural frequencies of the clamped-clamped column were cc cc xcc 1 ¼ 70:9 Hz; x2 ¼ 198 Hz; x3 ¼ 388 Hz, while those of clamped-pinned cs cs column were x1 ¼ 53:2 Hz; xcs 2 ¼ 173 Hz; x3 ¼ 360 Hz. The theoretical cc value of the Euler buckling force was P = 4.62 kgf (45.3 N) for the

Fig. 14.4 Schematic of the experimental setup [10]. ①: Specimen, ②: Load cell, ③: Strain gauges, ④: Upper displacement sensor, ⑤: Lower displacement sensor, ⑥: Lower end support

190

14

(a) Clamped end

Parametric Resonances of Columns

(b) Pinned end

Fig. 14.5 Clamped and pinned ends [10]

clamped-clamped column and Pcs  ¼ 2:54 kgf (24.9 N) for the clamped-pinned column. The regions of dynamic instability obtained by the experiment are shown in Fig. 14.6. The experimental instability regions are hatched and are compared with the theoretical estimates obtained by Hsu’s conditions (solid lines) and the estimates obtained by a numerical simulation procedure developed by Iwatsubo et al. [11] (dotted lines). Figure 14.6a shows the regions of dynamic instability of the clamped-clamped column. In Fig. 14.6a, two principal regions of simple resonances, in the first and second mode, can be observed. It is interesting to note that the principal region in the second mode is wider than that in the first mode. This is the reason why the vertical ordinate is normalized by the second eigenfrequency xcc 2 . In addition to the secondary regions of simple resonance in the first and second modes, a region of combination resonance of sum type ðh ¼ xk þ xl þ ekÞ of the first and third modes was observed as well. It is interesting to note that the region of combination resonance is just as important as the secondary regions of simple resonance when h ¼ xk þ ek. Typical photographic records of unstable responses at points A, B and C in each principal region in Fig. 14.6a are shown in Fig. 14.7. The upper and middle waves are the outputs (the responses of the column) from the upper and lower pickups (strain gauges), respectively. The lowest waves are the outputs (excitations) from the shaker (it is noted that the uppermost wave was recorded with its phase being shifted by p radians due to a trivial technical mistake in the experiment). Figure 14.6b shows the regions of dynamic instability of the clamped-pinned column. The photographic records at the points D, E, F and G in Fig. 14.6b are shown in Fig. 14.8. In Fig. 14.6b, in addition to the simple resonances,

14.6

Experiment with Columns Having Clamped-Clamped and Clamped-Pinned Ends

(a) Clamped-clamped column

191

(b) Clamped –pinned column

Fig. 14.6 Regions of parametric resonances of columns under pulsating forcing [10]

(a) Point A

(b) Point B

(c) Point C

Fig. 14.7 Unstable responses observed at points A, B and C in Fig. 14.6a [10]

combination resonance of sum type in the first and second modes is observed as well. However, the present experiment failed to observe the combination resonance of sum type in the first and third modes. This is probably because the third mode was prone to be damped out more rapidly than the second mode. It is interesting to note that a beating band (refer to Sect. 13.5) is observed just above the principal

192

14

Parametric Resonances of Columns

Fig. 14.8 Unstable vibrations of the clamped-pinned column [10]

(a) Point D

(b) Point E

(c) Point F

(d) Point G

region of simple resonance in the first mode. The boundary of the beating band is represented by open circles. Figure 14.8c shows a beating vibration observed in the beating band. If we compare Fig. 14.6a, the instability map for the clamped-clamped column, and Fig. 14.6b, the instability map for the clamped-pinned column, more disagreements (of the first and second eigenfrequencies given by theory and experiment) are found in Fig. 14.6b than in Fig. 14.6a. The theoretical estimates of the first and second eigenfrequencies were obtained by applying the bending stiffness EI, which was calculated by using nominal data for Young’s modulus, etc. Another question on the regions of unstable vibrations involves the fact that a wide region of disturbed vibrations above the principal region of the first mode was observed in the case of the clamped-pinned column, as shown in Fig. 14.6b. The vibrations looked like forced vibrations. The region with such vibrations is designated by special symbols (x). No clear reasoning why the forced-like vibrations took place has been revealed. It was supposed that these disagreements in the case of the clamped-pinned column might be attributable to the pinned end realized in the present experiment. The pinned end was comprised of a pin supported by ball bearings, as shown in Fig. 14.5b.

14.7

Columns under a Pulsating Follower Force

Let us consider an elastic cantilevered column as shown in Fig. 14.9. It is assumed that the cantilever is subjected to a pulsating follower force. This column was first discussed by Bolotin in his monograph [3]. He obtained the principal regions of simple resonance in the first and second modes. The column may be referred to as Bolotin’s column.

14.7

Columns under a Pulsating Follower Force

193

Fig. 14.9 Column under a pulsating follower force (Bolotin’s column)

The equation of motion is m

@2y @2y @4y þ P cos ht þ EI ¼ 0: t @t2 @x2 @x4

ð14:20Þ

Applying the same finite difference approximation to Eq. (14.20), with 6 segments, as in Chap. 2, we obtain the discrete system of equations of motion

y2 y3 4 d2 y4 + 6 4 EI dt 2 L m y5 y6 −2 1 1 −2 PL2 + 2 0 1 6 EI 0 0 0 0

6 −4 1 0 0 0 1 −2 1 0

−4 6 −4 1 0 0 0 1 −2 0

1 0 −4 1 6 −4 −4 5 1 −2 0 0 0 cosθ t 1 0

0 0 1 −2 1 y2 y3 y4 = 0, y5 y6

ð14:21Þ

where yi ði ¼ 2; 3; . . .; 6Þ is the displacement at the center of the ith segment of the divided column. If we let h ¼ 0, the system expresses the reduced expression for Beck’s column. It is noted that the stiffness matrix in Eq. (14.20), related to the applied load, is non-symmetric. For application of Hsu’s resonance conditions to Eq. (14.20), we obtain the normalized form

194

14 y2 y

Parametric Resonances of Columns

0.9824 × 10 − 2

0 0.38 20 × 10 0

d2 3 6 4 EI y4 + dt 2 y L4 m

0.2618 × 10 1 0.7605 × 10 1

5

y6

0 +

PL2 6 4 6 2 EI L4

0.21 92 × 10 1 0.4912 × 10 − 1 0.3547 × 10 − 1 − 0.18 77 × 10 − 1 0.9288 × 10 − 2 0.52 23 × 10 0 − 0.3569 × 10 0 0.6365 × 10 − 1 − 0.2610 × 10 1 − 0.98 06 × 10 − 2

0.1339 × 10 2 − 0.3063 × 10 0 0.5791 × 10 0 − 0.4165 × 10 0 − 0.15 34 × 10 0 0.58 60 × 10 − 1 − 0.13 81 × 10 1 − 0.8000 × 10 − 1 − 0.3733 × 10 − 1 0.26 85 × 10 − 1 0.6760 × 10 − 1 y2 0.3429 × 10 0 y3 − 0.1590 × 10 0.1529 × 10 − 1 cosθ t y4 = 0. y5 0.1306 × 10 − 1 y6 − 0.3614 × 10 1

ð14:22Þ It is found that the first natural frequency is given approximately by x21 ¼ 12:7EI=mL4 and the second by x22 ¼ 495EI=mL4 . The difference between the 6-segment approximation and the exact value is 1.5% for x1 and 0.97% for x2 . This fact implies that the reduced system of Eq. (14.22) can represent the corresponding column expressed by Eq. (14.20) with good accuracy [12]. It is interesting to note that b12 b21 \0, b23 b32 \0, b34 b43 \0 etc. This suggests, as Eq. (14.19) has suggested, that the combination resonances of difference type ðh ¼ xl  xk þ ekÞ between x2 and x1 , x3 and x2 , x4 and x3 , etc., take place. The instability regions obtained by applying Hsu’s conditions, Eqs. (14.17), (14.18) and (14.19), to Eq. (14.22) are shown in Fig. 14.10 by solid lines. The dotted lines show the boundaries given by Bolotin [3]. The solid lines with open circles indicate the results obtained by an analog computer-based experiment, as described in Chaps. 2 and 3. The amplitude of the pulsating force is normalized by the flutter limit P for Beck’s column. Figure 14.11 shows the unstable vibrations observed in the analog experiment conducted by Sugiyama et al. [13]. Now, the nature of the unstable vibration in each region can be discussed. Region A: The unstable vibrations are due to the simple resonance of the second mode. The exciting frequency coincides with the double of the second eigenfrequency, h ¼ 2x2 þ ek. Region B: The unstable vibrations are due to the combination resonance of difference type between the third and second modes, h ¼ x3  x2 þ ek. The growth rate of the amplitude is smaller than that for simple resonance. Region C: The unstable vibrations are caused by combination resonance of sum type between the first and third modes. The excitation frequency matches the half of sum of the first and the third eigenfrequency, h ¼ ðx1 þ x3 Þ=2 þ ek. It is

14.7

Columns under a Pulsating Follower Force

195

Fig. 14.10 Regions of parametric resonances of Bolotin’ s column [12]

interesting to find that the first mode is more dominant than the third, and the growth rate is rather small. Region D: The unstable vibrations are attributable to the simple resonance of the second mode. The exciting frequency matches the second eigenfrequency, h ¼ x2 þ ek. Region E: The unstable vibrations are caused by the combination resonance of difference type between the second and first modes, h ¼ x2  x1 þ ek. The growth rate of the amplitude is large. Region F: The unstable vibrations are due to simple resonance of the first mode, h ¼ 2x1 þ ek. It is noteworthy that instability regions B and E (of combination resonances of difference type) are found in the case of a cantilevered column under a pulsating follower force, Bolotin’s column. It is also interesting to confirm that the principal region of combination resonance of difference type of the second and first eigenfrequency (Region E) is more important than the principal region of simple resonance of the first eigenfrequency (Region F).

196

14

Parametric Resonances of Columns

(a) Region A

(b) Region B

(c) Region C

(d) Region D

(e) Region E

(f) Region F

Fig. 14.11 Unstable vibrations of Bolotin’s column observed in the analog experiment [13]

14.8

Discussion

In the theoretical approach to parametric resonance of columns, based on their mathematical models, it has been assumed that columns with different end conditions are subjected to an axial pulsating force, without mentioning how to realize such a type of end condition and how to apply such a force. However, if one tries to take an experimental approach to these columns, one will face some difficulties in realizing an experimental counterpart to the mathematical model. This chapter (Sect. 14.6 in particular) has discussed that the realization of a (perfect) pinned end is not so easy in the case of the dynamic stability of columns as it may be in the case of the static stability. Section 14.7 showed theoretically, with an analog computer-based experiment, that a cantilevered column under a pulsating follower force (Bolotin’s column) loses its stability by combination resonances of difference type, in addition to simple resonances and combination resonances of sum type. As to the realization of combination resonances of difference type in a laboratory, an experimental approach to such a type of resonance will be presented in Chap. 16. For a detailed mathematical discussion, see Refs. [1–8].

References

197

References 1. Arscott, F. M. (1964). Periodic differential equations. Oxford: Pergamon Press. 2. Magnus, W., & Winkler, S. (1966). Hill’s equation. New York: Dover Publications, Inc. 3. Bolotin, V. V. (1964). The dynamic stability of elastic systems. San Francisco: Holden-Day, Inc. 4. Nayfeh, A. H., & Mook, D. T. (1995). Nonlinear oscillations. New York: Wiley. 5. Stoker, J. J. (1950). Nonlinear vibrations in mechanical and electrical systems. New York: Interscience Publishers. 6. Hsu, C. S. (1963). On the parametric excitation of a dynamical system having multiple degrees of freedom. Journal of Applied Mechanics, 30, 367–372. 7. Struble, R. A., & Fletcher, J. E. (1962). General perturbation solution of the Mathieu equation. Journal of the Society for Industrial and Applied Mathematics, 10, 314–328. 8. Hsu, C. S. (1965). Further results on parametric excitation of a dynamical systems. Journal of Applied Mechanics, 32, 373–377. 9. Iwatsubo, T., Sugiyama, Y., & Ogino, S. (1974). Simple and combination resonances of columns under periodic axial loads. Journal of Sound and Vibration, 33(2), 211–221. 10. Iwatsubo, T., Saigo, O., & Sugiyama, Y. (1973). Parametric instability of clamped-clamped and clamped-simply supported columns under periodic axial load. Journal of Sound and Vibration, 30(1), 65–77. 11. Iwatsubo, T., Sugiyama, Y., & Ishihara, K. (1972). Stability and non-stationary vibration of columns under periodic loads. Journal of Sound and Vibration, 23, 245–257. 12. Sugiyama, Y., Iwatsubo, Y., & Ishihara, K. (1982). Parametric resonances of a cantilevered column under a periodic tangential force. Journal of Sound and Vibration, 84, 301–303. 13. Sugiyama, Y., Fujiwara, N., & Sekiya, T. (1970). Studies on non-conservative problems of instability of columns by means of analogue computer. In Proceedings of the Eighteenth Japan National Congress for Applied Mechanics (pp. 113–126). Tokyo: The University of Tokyo Press.

Chapter 15

Parametric Resonances of Columns with Damping

This chapter discusses the mathematical aspects of parametric resonances of columns with damping, dealing generally with dynamical systems with damping. In addition to the first-order approximation for the resonance boundaries of Mathieu-Hill equations, an approach to the second-order approximation is suggested.

15.1

Approaches to Mathieu-Hill Equations

Several classic books that deal extensively with single equations of the Mathieu-Hill type have been published, such as Arscott [1] and Magnus and Winkler [2]. In comparison, much less has been written about coupled equations. Coupled Mathieu-Hill equations have mostly been investigated by engineers and applied mathematicians [2]. Examples of application-oriented monographs are those by Bolotin [3], Nayfeh and Mook [4], and Seyranian and Mailybaev [5]. Bolotin developed a generalization of the method of Hill’s determinants. In his method, the solution is expressed in terms of a Fourier expansion. Equating the coefficients of the corresponding sine and cosine terms, he obtains two infinite determinants, one for sine and one for cosine. Alternatively, the solution can be expressed in a complex form, as a doubly infinite exponential expansion. In the direct form of this approach, each element in the two determinants is itself a matrix, and this may result in laborious computations. Accordingly, Bolotin suggests diagonalization, to what is now known as a Jordan normal form, and applies approximation methods in which most off-diagonal terms are discarded. Nayfeh and Mook [4] have employed the method of multiple scales. Their results, obtained by a first order expansion, agree with Hsu’s conditons [6]. They have proceeded to consider the second-order approximation as well.

© Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_15

199

200

15.2

15

Parametric Resonances of Columns with Damping

Hsu’s Approach to Coupled Hill Equations

Following Hsu’s approach [6], let us consider the coupled Hill equations d2y dy þ ½S þ PðtÞy ¼ 0: þ RðtÞ 2 dt dt

ð15:1Þ

The excitation matrix PðtÞ can be written in the form PðtÞ ¼ P0 þ P1 ðtÞ:

ð15:2Þ

If Eq. (15.1) is normalized, in the sense that the stiffness matrix S is diagonalized, we have d2 x dx þ fK þ Pt ðtÞgx ¼ 0: þ Q ðt Þ dt2 dt

ð15:3Þ

Now, it is assumed that the damping and the periodic coefficients are small, and thus Eq. (15.3) can be written in the form d2x dx þ ½K þ eDðsÞx ¼ 0; þ eFðsÞ ds2 dt

ð15:4Þ

where eð [ 0Þ is a small parameter. x and K are given in the forms
 x ¼ ðx1 ; x2 ; . . .; xn ÞT ; FðsÞ ¼ fij ðsÞ ;  
 K ¼ diag x21 ; x22 ; . . .; x2n ; DðsÞ ¼ dij ðsÞ :

ð15:5Þ

Hsu [6] discusses Eq. (15.4) according to the method first used by Struble and Fletcher [7]. Then, the conditions for instability are discussed. However, the mathematical procedures for the solutions and instability conditions are very complicated. Thus, hereafter, only the main results of the procedures are described. For simplicity, let us now assume that the damping matrix is constant and positive definite. It is assumed that the excitation force is harmonic. Thus, we have a set of equations, specifically n coupled Mathieu equations, ! n n X X d 2 xi dxi þ x2i þ e þe fij dij  cos hs xi ¼ 0: ds2 ds j¼1 j¼1

ð15:6Þ

Starting with the general expression in Eq. (15.1), Hsu discusses the equation in general. However, he has given conditions for the regions of resonances in a special

15.2

Hsu’s Approach to Coupled Hill Equations

201

case in which damping is absent ðfij ¼ 0Þ and all of the parametric excitations are in phase. He has given, as described in Sect. 14.4, the first approximation of the principal region of parametric resonances, of the first order of e, as follows:    h  xi  xj \Dhij e;

ð15:7Þ

where 1 Dhij ¼ 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi dij dji  : xi xj

ð15:8Þ

The regions of simple resonances are special cases of Eq. (15.7) when i ¼ j: The regions of combination resonances of sum type are given by Eqs. (15.7) and (15.8) with the positive sign chosen from “” and that of difference type with the negative sign. If we deal with Eq. (15.6) by an extension of Hsu’s approach when damping is present, we obtain the regions of parametric resonances in the form      h  xi þ xj \ 1 fij þ fji e 4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi dij dji fij 4 : xi xj fij fji e

ð15:9Þ

Now let us consider the case with internal and external damping and put fii ¼ ci ¼ j þ cx2i ;

fij ¼ 0ði 6¼ jÞ;

ð15:10Þ

where j is the nondimensional coefficient of external damping and c is the nondimensional coefficient of internal damping, as described earlier in Chap. 1. The equation of motion of the column with damping is written as ! n X d 2 Xi dXi 2 þ xi þ e þ eci dij  cos hs Xi ¼ 0: ds2 ds j¼1

ð15:11Þ

Equation (15.9) then reduces to the form    h  xi þ xj \ 1 ðci þ cjÞe 4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c 2 dij dji i : 4 xi xj ci cj e

ð15:12Þ

Equation (15.12) can be written in the form    h  xi  xj \dDh0 e;

ð15:13Þ

202

15

Parametric Resonances of Columns with Damping

where 1 þ gij cj j þ cx2j 1 ; Dh0 ¼ d ¼ pffiffiffiffiffi ; gij ¼ ¼ 2 2 gij ci j þ cx2i

15.3

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c 2 dij dji i  :  4gij xi xj e

ð15:14Þ

Effect of Damping

Now that we have obtained the formulas for a damped column under a harmonic excitation, let us discuss the effect of damping on the regions of parametric resonances. In the case of simple resonance ði ¼ jÞ, we have gii ¼ 1. In the case of combination resonances ði 6¼ jÞ, when the internal damping is absent ðc ¼ 0Þ, or when the external damping is much larger than the internal damping, we have gij ffi 1. When the external damping is absent ðj ¼ 0Þ, or when the internal damping is much larger than the external damping, we may have gij ffi

x2j x2i

:

ð15:15Þ

It is interesting to note that gij ði 6¼ jÞ may take a value larger than unity. For example, in the case of a cantilevered column, g12 ¼ 39:3, g13 ¼ 308, and g23 ¼ 7:84, etc. When there is small internal damping, but also a comparative amount of external damping, we have, in general, gij [ 1 and d [ 1. The regions of resonances can exist only when  

2  dij dji     4gij ci [ 0: x x  e i

ð15:16Þ

j

Thus there is the critical excitation parameter, ecr , a minimum amplitude of the parametric excitation, smaller than that at which the column is stable, ecr ¼

pffiffiffiffiffi gij ci : Dhij

ð15:17Þ

The effect of damping on the region of parametric resonance is illustrated in Fig. 15.1. Figure 15.1a shows a simple effect of damping in which the region degenerates as a whole as a result. This type of effect takes place in the case of simple resonance of columns with internal and external damping, and in the case of combination resonance of columns with dominating external damping. However, in the case of combination resonances of the columns with damping, the effect of damping is stabilizing in the sense that the starting point of the region degenerates,

15.3

Effect of Damping

203

Fig. 15.1 Schematic explanations of the effect of damping on the regions of parametric resonances

and destabilizing in the sense that the area of the region grows wider than that for the undamped case, as shown in Fig. 15.1b.

15.4

Second-Order Approximation

Let us consider n coupled Mathieu equations in the form n n X d 2 xi X dxi þ x2i xi þ ps coshs þ fij dij xi ¼ 0; 2 ds ds j¼1 j¼1

ð15:18Þ

ðj ¼ 1; 2; . . .; nÞ: It is assumed that the damping and the excitation are small. Equation (15.18) can then be written in the form n n X X d2x dxi 2 þ x þ 2e l x þ 2ecoshs dij xi ¼ 0; ij i i ds2 ds j¼1 j¼1

ð15:19Þ

where fij ¼ 2elij ;

ps ¼ 2e:

ð15:20Þ

For the approximation conditions for the region of resonances up to the order of e2 , let us introduce three time-scales, as used in the concept of multiple scales [4]: T0 ¼ s; T1 ¼ es; T2 ¼ e2 s:

ð15:21Þ

204

15

Parametric Resonances of Columns with Damping

Here T0 is a fast time scale, T1 is a slow time scale, and T2 is a ‘very slow’ time scale. The derivatives with respect to s are now expanded in the form  2 d @ @ d2 @2 @2 @ @2 2 @ 2 ¼ þe þe ; ¼ þ 2e þe þ2 : ds @T0 @T1 @T2 ds2 @T02 @T0 @T1 @T0 @T2 @T12 ð15:22Þ Following the procedure given by Nayfeh and Mook [4], applying the conditions of elimination of the secular terms, and considering the conditions for non-trivial solutions, after a lengthy and complicated mathematical processing, we arrive at second-order approximations for the principal regions of resonances in the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    1   p2s dij dji h  xi þ xj \ fii þ fjj 1 2 4xi xj fii fjj !     p2 fii  fjj dji Y~i  dij Y~j 1 Zi Zj p2 Y i Yj  þ þ ; þ s  s 8 xi uj 8 xi xj 16xi xj fii fjj

ð15:23Þ

where Z i ¼ fii2 þ 4x2i

X k6¼i

fik fki ; x2i  x2k

X X dki dik dij dji dik dki Yi ¼   ;  2 2 2 4xi xj 2xi þ xj x2k k6¼j xj  xk k X fik dkj X dik fkj  xj : Y~i ¼ xi 2 2 xi  xk x2j  x2k k6¼i k6¼j

ð15:24Þ

Z j , Y j , and Y~j are given by Eq. (15.24) with j in place of i. The conditions for combination resonances of difference type can be obtained by Eqs. (15.23) and (15.24), with xi in place of xi . Those for simple resonances are given by these conditions when i ¼ j. Proceeding with the second approximation, we can obtain the conditions for the secondary regions of resonances in the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      1   p4s Zij Zji 1 h  xj  xi \ fii þ fjj 1   4 2 xi xj fii fjj   1 Zi Zj p2 Z~i Z~j  þ þ þ s ; 16 xi xj 4 xi xj

ð15:25Þ

15.4

Second-Order Approximation

205

where X

dim dmj ; 2 xi  xj 4x2m m X fik fki Z i ¼ fii2 þ 4x2i ; x2i  x2k k6¼i X X dim dmi dim dmi ~i ¼ Z þ :  2  2 2 xi  xj 4xm 3xi þ xj 4x2m m m

Zij ¼



ð15:26Þ

As to the secondary regions of other types of resonance, the conditions for combination resonances of difference type can be obtained by Eqs. (15.25) and (15.26), with “ xi ” in place of xi . Those for simple resonances are given by these conditions when i ¼ j.

15.5

Discussion

This chapter has given an overview of the theory of parametric resonances applied to dynamical systems. It has been shown that the damping has a destabilizing effect on the regions of combination resonances of columns, in the sense that the regions are widened. The destabilizing effect of damping in the case of combination resonances was first discussed by Weidenhammser [8], and by Schmidt and Weidenhammer [9], and later, for example, by Yamamoto and Saito [10] for general mechanical systems with damping, without classifying damping into internal and external damping.

References 1. Arscott, F. M. (1964). Periodic differential equations. Oxford: Pergamon Press. 2. Magnus, W., & Winkler, S. (1966). Hill’s equation. New York: Wiley. 3. Bolotin, V. V. (1964). The Dynamic Stability of Elastic Systems. San Francisco: Holden-Day, Inc. 4. Nayfeh, A. H., & Mook, D. T. (1995). Nonlinear oscillations. New York: Wiley. 5. Seyranian A. P. & Mailybaev, A. A. (2003). Multiparameter Stability Theory with Mechanical Applications. Singapore: World Scientific Publishing. 6. Hsu, C. S. (1963). On the parametric excitation of a dynamical systems having multiple degrees of freedom. Journal of Applied Mechanics, 30, 367–372. 7. Struble, R. A., & Fletcher, J. E. (1962). General perturbation solution of the Mathieu equation. Journal of the Society for Industrial and Applied Mathematics, 10, 314–328. 8. Weidenhammer, F. (1951). Der eingespannte, axial-pulsierend belaste Stab als stabilitätsproblem. Ingenieur-Archiv, 19, 162–191.

206

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Parametric Resonances of Columns with Damping

9. Schmidt, G., & Weidenhammer, F. (1961). Instabilitäten gedämpten rheolinearer Schwingungen. Mathematischen Nachrichten, 23, 301–318. 10. Yamamoto, T. & Saito, A. (1967). On the oscillations of summed and differential types under parametric excitation (vibratory systems with damping), Transactions of the Japan Society of Mechanical Engineers, 33, 905–914 (in Japanese).

Chapter 16

Columns under a Pulsating Reut Force

Combination resonances of difference type may possibly be one of the most interesting topics in the field of structural dynamics. This chapter describes an experimental approach to this topic.

16.1

Columns under a Pulsating Generalized Reut Force

It is assumed that a cantilevered column is subjected to a pulsating generalized Reut force, PðtÞ ¼ Po þ Pt cos Ht, as shown in Fig. 16.1. Internal damping ðE  Þ is considered, in addition to external damping (C). It is assumed that the attachment carries a finite-sized mass M having rotatory inertia J, as these properties are to be considered in the corresponding experiment. The equation of motion of the column is given by m

@2y @y @5y @2y @4y þ E I 4 þ PðtÞ 2 þ EI 4 ¼ 0: þC 2 @x @t @x @t @x @x

ð16:1Þ

The boundary conditions are @yð0; tÞ ¼ 0; @x 2 3 @ yðL; tÞ @ 3 yðL; tÞ  @ yðL; t Þ EI þ J þ E I þ PðtÞyðL; tÞ ¼ 0; @x2 @x2 @t @x@t2 4 @ 3 yðL; tÞ @ 2 yðL; tÞ @yðL; tÞ  @ yðL; t Þ ¼ 0:  M þ E I þ PðtÞð1  aÞ EI @x3 @x3 @t @t2 @x yð0; tÞ ¼ 0;

© Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_16

ð16:2Þ

207

208

16

Columns under a Pulsating Reut Force

Fig. 16.1 Column under a generalized pulsating Reut force

The following nondimensional variables and parameters are defined: rffiffiffiffiffi y x t EI ; g ¼ ;n ¼ ;s ¼ 2 L L L m rffiffiffiffiffiffiffi rffiffiffiffiffi CL2 E I m 2 ; h ¼ HL j ¼ pffiffiffiffiffiffiffiffiffi ; c ¼ 2 ; mE EI L mEI M J ; m¼ : l¼ mL mL3 Po L2 Pt L2 ; ps ¼ ; p ¼ po þ ps cos hs: po ¼ EI EI

ð16:3Þ

Equations (16.1) and (16.2) can then be written in the nondimensional form @2g @g @5g @2g @4g þc þj þ pðsÞ 2 þ 4 ¼ 0; 4 2 @s @s @s@n @n @n

ð16:4Þ

16.1

Columns under a Pulsating Generalized Reut Force

209

@gð0; sÞ ¼ 0; @n @ 2 gð1; sÞ @ 3 gð1; sÞ @ 3 gð1; sÞ þm þc þ pðsÞgð1; sÞ ¼ 0; 2 2 @n@s2 @n @n @s @ 3 gð1; sÞ @ 4 gð1; sÞ @ 2 gð1; sÞ @gð1; sÞ ¼ 0:  l þ c þ pð s Þ ð 1  aÞ 3 3 2 @s @n @n @n @s

gð0; sÞ ¼ 0;

16.2

ð16:5Þ

Finite Difference Formulation and Stability Analysis

Noticing that the boundary conditions contain a term with a time-varying coefficient, we will not rely on the Galerkin method, as it is difficult to obtain base-functions that satisfy the boundary conditions. Instead we apply the finite difference method (FDM) in order to discretize Eqs. (16.4) and (16.5), with an n segment approximation. We then obtain the discretized equations of motion in the matrix form     M€ g þ n4 cB þ jI g_ þ n4 K þ n2 po F g þ n2 ps cos hsFg ¼ 0:

ð16:6Þ

Transformation of Eq. (16.6) into normalized coordinates yields the following form: € þ Cu_ þ Ku þ ps cos hsDu ¼ 0; u

ð16:7Þ

where K is a diagonal matrix having its elements of squared eigenfrequencies of the column when ps ¼ 0. In order to obtain a theoretical overview of parametric resonance of the column, for simplicity, it is assumed that the attachment is a massless rigid plate ðl ¼ m ¼ 0Þ. Then, the system of equations of motion is written in the reduced form n X d 2 ui dui 2 þ x þ c u þ p cos hs dij uj ¼ 0; i i s i ds2 ds j¼1

ði; j ¼ 1; 2; :::; nÞ;

ð16:8Þ

where ci ¼ j þ cx2i . In the following, 20 segments are employed in the FDM approximation, i.e., n ¼ 20. Now let us apply the resonance conditions obtained by the second-order approximation, as given in Eq. (15.23) (Chap. 15). Figure 16.2 shows the regions of resonances of the column when the nonconservativeness parameter a ¼ 1:0, and when the mass of the attachment is neglected. It is noted that if the force is steady, the column is conservative and loses its stability by divergence. The nondimensional divergence limit is given by po ¼ 7:831 [see Eq. (9.12)]. The pulsating

210

16

(a) Undamped case (

Columns under a Pulsating Reut Force

(b) Damped case (

Fig. 16.2 Regions of resonances when a ¼ 1:0 [1, 2]

(a) Undamped case (

(b) Damped case (

Fig. 16.3 Region of resonances when a ¼ 0 [1, 2]

amplitude is normalized by this divergence limit. It is seen in Fig. 16.2 that combination resonances of sum type occur, in addition to simple resonances, when the direction parameter a ¼ 1:0. Figure 16.3 shows the regions of resonances of the column when the nonconservativeness parameter a ¼ 0. It is seen in Fig. 16.3 that when a ¼ 0, combination resonances of difference type occur, in addition to the combination resonances of sum type and simple resonances. Figure 16.3 suggests that if one would like to

16.2

Finite Difference Formulation and Stability Analysis

211

carry out an experimental verification of combination resonances of difference type, a clever way is to conduct an experiment with a column under a pulsating Reut force. As to the effect of damping on the regions of parametric resonances, it is seen that damping has a stabilizing effect on the regions of simple resonances, reducing these regions as a whole. However, it has a destabilizing effect on the regions of combination resonances in the sense that the instability regions are widened, while it is stabilizing in the sense that there exists a minimum amplitude of pulsating force. There is a gap between the regions of resonances for the undamped case and those for the damped case. It is recommended that this gap is to be investigated in detail in future, both theoretically and experimentally.

16.3

Experiment with Columns under a Pulsating Reut Force

16.3.1 Experimental System The present experiment is based on the test equipment for the generalized Reut’s column, as described in Chap. 9. The experimental setup is modified in the sense that the applied force is not steady, but rather pulsating. Figure 16.4 shows a schematic diagram of the experimental system. It is composed of a system for generation of a pulsating force, a test column system, and a measurement system. A photograph of the setup is shown in Fig. 16.5.

16.3.2 System for a Pulsating Reut Force A pulsating Reut force is realized by a pulsating air jet impinging onto the attachment mounted on a test column at its tip end. The pulsating air flow is generated by means of an electromagnetic valve, which is on-off-controlled by the signal from Fig. 16.4 Schematic diagram of the experimental setup for a column under a pulsating Reut force [1, 2]

212

16

Columns under a Pulsating Reut Force

Fig. 16.5 Overview of the experimental setup [1, 2]

Fig. 16.6 Time history of a pulsating force [1, 2]

a function generator. The flow rate of the air jet, and hence the force applied to the column, is controlled by the main valve. The nominal flow rate is realized when the electromagnetic valve is open. It is noted that the nominal flow rate may not be exactly equal to the actual rate when the on-off valve is active. The excitation frequency h can be controlled by a function generator. Figure 16.6 shows the time-varying force realized in the present test at the nominal flow rate of 30 L/min. The force is not exactly harmonic, but rather periodic. It was assumed, through Fig. 16.6, that the constant force component Po and the amplitude of pulsating component Pt take the same value, P0 ¼ Pt . It was assumed that they were constant in the test run.

16.3.3 Test Pieces The test pieces were cut from an acrylic plate of thickness 2.0 mm. The dimensions were: length L = 1000 mm, breadth b = 10 mm and thickness h = 2.0 mm. The mass per unit length of the column was m ¼ 2:6x 102 kg=m. To control the nonconservativeness parameter a of the Reut force, the attachment was covered with a porous (nonwoven) material, so as to realize a rough surface of the attachment, and with a glass plate to realize the smoothest surface. Figure 16.7

16.3

Experiment with Columns under a Pulsating Reut Force

(a) Bird’s-eye view of the setup

213

(b) Nozzle and the attachment

Fig. 16.7 Test column on the test bed [1, 2]

shows the test column set on the test bed. The test column was fixed horizontally at the end and suspended by long and thin threads from the ceiling (Fig. 16.7a). Only small-amplitude motion of the column was allowed in the horizontal plane.

16.3.4 Attachment Figure 16.7b shows the attachment with a cover of unwoven material (refer to Sect. 9.2). With this material, it was possible to realize the small nonconservative parameter a ¼ 0:01. The attachment with the cover had the dimensions 87.0  27.0  10.0 mm, and mass MA ¼ 2:8  103 kg. The nonconservativeness parameter a was determined according to the procedure described in Chap. 9. It is noted that, hereafter, the mass of the attachment is considered in the theoretical prediction of instability regions.

16.3.5 Eigenfrequencies The first two natural frequencies of the test column without the attachment were obtained as xo1 ¼ 0:67 Hz and xo2 ¼ 3:81 Hz. The first two natural frequencies of the column in the presence of attachment A were measured to be xA1 ¼ 0:58 Hz and xA2 ¼ 3:44 Hz, while those with attachment B were xB1 ¼ 0:53 Hz and xB2 ¼ 3:34 Hz. The variation of the first two eigenfrequencies with an increasing constant force Po (gf) is shown in Fig. 16.8.

214

16

Columns under a Pulsating Reut Force

Fig. 16.8 Variation of eigenfrequencies ða ¼ 0:01Þ [1, 2]

It is seen that the first eigenfrequency increases as the force increases, while the second eigenfrequency decreases. It is found that they can be separated by a threshold of 2.0 Hz, which was used as the separation frequency between the low-pass filter and the high-pass filter.

16.3.6 Regions of Parametric Resonances A pulsating Reut force was applied to the test column. The pulsating frequency and the flow rate at which the resonance occurred were recorded. The experimental limits of resonance are plotted with open circles in the ps  h plane in Fig. 16.9 ðthe parameter a ¼ 0:01Þ. Fig. 16.9 Regions of resonances of the column with the parameter a ¼ 0:01 [1, 2]

16.3

Experiment with Columns under a Pulsating Reut Force

215

Fig. 16.10 Unstable vibration configurations observed in Region C [1, 2]

The regions enclosed by solid lines are those predicted by the resonance conditions given in Chap. 15. Four regions of parametric resonances are found. Region A is the principal region of simple resonance in the second mode. Region B is the secondary region of simple resonance in the second mode. Region C is the principal region of combination resonance of difference type in the second and first modes. Region D is the principal region of simple resonance in the first mode. Theoretical predictions of the second approximation were made by assuming that c ¼ m ¼ 103 . Now let us turn our attention to Region C. Figure 16.10 shows the photograph of the unstable vibration of the test column in Region C. Unstable vibrations of the test column were recorded after filtering them through high-pass/low-pass filters, in which the separation frequency was 2.0 Hz (refer to Fig. 16.8). Figure 16.11 shows the records of the outputs (vibration) of Region C, by the flow rate 25 L/min. and H ¼ 2:4 Hz. Figure 16.11a shows the signal given to the electromagnetic valve. Figure 16.11b shows the vibration of the column without filtering. Figure 16.11c shows the response obtained through the low-pass filter ðH\2:0 HzÞ, that is, the first mode component, while Fig. 16.11(d) displays the response through the high-pass filter ðH [ 2:0 HzÞ, that is, the second mode component. By filtering the combination response of difference type between the second and first, the first mode component and the second mode component are clearly separated. It is seen that both the first mode component and the second mode component are increasing.

16.4

Discussion

The present experiment was a rather qualitative attempt to realize combination resonances of difference type (of columns) in a laboratory. Many points have been left to be refined in future. The destabilizing effect of damping on combination resonances of difference type has been found in theory in the sense that the regions are widened. It is suggested that the role of damping on the resonances may be discussed through energy consideration, as discussed in Chap. 5, to clear the mechanism of the role of damping and to bridge the gap between the undamped

216

16

Columns under a Pulsating Reut Force

Fig. 16.11 Unstable responses in Region C [1, 2]

case and the damped case. The combination resonances in an engineering application were discussed by Beal [3], in the case of a flexible missile under a pulsating thrust. Another experimental approach to combination resonances of difference type has been dealt with in the case of the dynamic stability of cantilevered pipes conveying pulsating fluid. The topic has been worked on by Pauïdoussis and his colleagues, and their achievements have been compiled in book form [4]. Parametric resonances in an annular disc, including the effects of friction and damping, have been discussed by Mottershead et al. [5].

References

217

References 1. Naitou, H. (1990). Studies on summed and differential combination resonances of columns under a pulsating Reut force (Master Thesis). Department of Aerospace Engineering, Osaka Prefecture University. (in Japanese). 2. Sugiyama, Y. (1992). Summed and differential combination resonances of cantilevered columns under a pulsating Reut force. In P. S. Theocaris (Ed.) Proceedings of the 3rd National Congress on Mechanics. Hellenic Society for Theoretical and Applied Mechanics, Vol. 1, pp. 70–77. 3. Beal, T. R. (1965). Dynamic stability of a flexible missile under constant and pulsating thrusts. AIAA Journal, 3(3), 486–494. 4. Païdoussis, M. P. (1998). Fluid-structure interactions, slender structures and axial flow (Vol. 1). New York: Academic Press. 5. Mottershead, J. E., Ouyang, H., Cartmell, M. P., & Friswell, M. (1997). Parametric resonances in an annular disc, with a rotating system of distributed mass and elasticity; and the effects of friction and damping. Proceedings of the Royal Society of London, A, 453, 1–19.

Chapter 17

Remarks about Approaches to the Dynamic Stability of Structures

Different approaches to the dynamic stability of structures, and of columns in particular, are considered in the present book. The typical approaches are analytical (mathematical analysis-based), computational (numerical), and experimental approaches. The analytical approach has traditionally been considered as the most powerful and fundamental, and the other two approaches have played important roles in corroborating the results. Each approach has its advantages and disadvantages and plays a complementary role to the others. The mutual complemental roles of these three typical approaches are schematized in Fig. 17.1. This trilateral approach may ensure a sound development of the theory of the dynamic stability of structures. A large number of papers on the dynamic stability of columns under follower forces appeared after Bolotin’s book (the English version), published in 1963 [1]. However, the majority of these papers have relied on analytical approaches. D. D. Joseph wrote in his chapter, in 1981, [2]: “It seems likely to me that the theory of [hydrodynamic] stability, bifurcation and transition will, in the future, come to rely increasingly on abstract methods for qualitative analysis and on numerical analysis for explicit results. I would expect an important role, but a decreasingly important one, to be played by the traditional method of applied analysis.” In the traditional method of applied analysis, the analytical approach, one typically identifies the terms in the governing equations that are of importance and leaves out the terms that are non-essential. This step alone, which comes before actually solving the problem, often provides much new insight and understanding. Moving on, then, to solve the problem, an analytical solution approach has the advantage of giving the complete solution to the problem. However, with many problems, especially those of a practical nature, an exact analytical solution often becomes overwhelmingly complicated. Hence, the analytical approach has lost much ground to numerical approaches in recent years, as Joseph predicted it would, more than 35 years ago [2]. This is unfortunate for the progress of science. The importance of the ability to express the behavior of a complicated mechanical system in a few mathematical equations cannot be overestimated. Howe wrote [3]: © Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6_17

219

220

17

Remarks about Approaches to the Dynamic Stability of Structures

Analytical approach

Dynamic stability of structures

Computational approach

Experimental approach

Fig. 17.1 Trilateral approach to the dynamic stability of structures

“A simple formula is often worth a thousand numerical simulations, and can reveal connections between different control parameters that might otherwise take hours or weeks to deduce from a computational analysis.” In this book, some analog computer-based experiments (simulations) have been presented to highlight interesting aspects of the dynamic stability of columns. As to analog computer experiments, use of an analog computer, and thus analog experiments, was a new (at the time) and very exciting possibility in engineering laboratories from the 1960s to the earlier part of the 1970s, before modern computers became common in laboratories. These now old-fashioned analog experiments are presented in this book to encourage the readers to conduct computational (numerical) experiments of the dynamic stability of structural systems. As to the experimental approach to nonconservative stability problems, it is interesting to recall Bolotin’s suggestion, given at the end of his monograph. Bolotin wrote [2]: “It is felt that the principal lines along which future research in this field should be directed must aim, not at increasing the number of purely academic problems solved, but at providing an answer to the question of the degree to which “follower” forces can satisfactorily represent actual forces encountered in practice. Here experimental investigations are of prime importance.” As to the role of experiment, D. C. Drucker’s statement, given in his general lecture [4], is suggestive. He concluded that, “Experiment is essential, it is vital, and it is creative. Over the years, experiment alone provides the basis for the refinement and extension of existing theory and the development of new theory.”

References 1. Bolotin, V. V. (1963). Nonconservative problems of the theory of elastic stability. New York: Pergamon Press.

References

221

2. Joseph, D. D. (1981). Hydrodynamic stability and bifurcation. Topics in Applied Physics, 45. In H. L. Swinney & J. Gollub (Eds.) Hydrodynamic Instabilities and the Transition to Turbulence (Berlin: Springer). 3. Howe, M. (2016). Mathematical methods for mechanical sciences. London: Imperial College Press. 4. Drucker, D. C. (1962). On the role of experiment in the development of theory. In Proceedings of the Fourth U.S. National Congress of Applied Mechanics, Vol. 1 (Held at the University of California, Berkeley), ASME, New York, pp. 15–33.

Appendix A

Suggested Exercises

Below are listed some exercises that are recommended to students, engineers and young scientists who are interested in and studying the dynamic stability of structures. The exercises are fundamental and have been selected such that they convey the enjoyment of studying dynamic phenomena related to the present topic. The answers are not given here, as these depend on the personal degree of eagerness and interest in the topic, and thus they shall be given by the readers themselves, enjoying discussions with their colleagues, and possibly with their advisors. Category A: Experiences A-1: Prepare an elastic bar, say, a long ruler, for an instant buckling test. Subject it to axial compression. Observe buckling of the bar. A-2: Prepare a long flexible bar for an instant divergence test. Set the bar standing upward (vertically) by fixing it with a clamp. Observe oscillations by making the bar longer and longer, step by step, until it loses stability by divergence. Measure the period of motion with a stopwatch to find the frequency of the oscillations. Plot the relation between the length and the frequency. A-3: Prepare a long rubber hose for a flutter demonstration, with flutter resulting from an internal fluid flow. Fix the hose to a faucet at one end and hold it downward vertically. Open the tap fully to initiate flutter. Prepare a very long hose and place it in a pool. Open the tap fully to observe the hose dancing in the water, realizing flutter motion with large amplitude. A-4: Sit in a swing and excite it at both top points for amplified amplitude. In this way, you will experience simple resonance. In terms of parametric resonance, think about the optimal timing to squat and to rise (stand up). Category B: Experiments B-1: Prepare an elastic bar for a static bending test. Set up a simple test rig through the use of a clamp. Fix one end of the bar with the clamp and hold the bar horizontally. Place a load at the tip end. Consider at least three or more different © Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6

223

224

Appendix A: Suggested Exercises

loads and measure the corresponding deflections with a ruler. Find the bending stiffness and compare it with your theoretical prediction. B-2: Prepare an elastic bar for a free vibration test. Set up a simple test rig through the use of a clamp. Fix one end of the bar with the clamp and hold it horizontally. Give the bar a small disturbance and measure the natural frequency with a stopwatch. Find the bending stiffness and compare it with that obtained in the static test. B-3: Prepare a beam specimen for a decaying motion test. Secure the upper end of the beam with a pin and let the beam hang down (under gravity) such that it can move like a pendulum. Make a record of the decaying pendulum motion to find the coefficient of external damping due to air resistance. B-4: Prepare a flexible beam for a decaying free vibration test. Make a record of the decaying free vibration of the beam to find the coefficient of internal/structural damping. The coefficient of internal damping is obtained by subtracting the contribution of external damping from the total logarithmic decrement. B-5: Prepare a flexible column equipped with a small rigid body at its tip. Keep the column standing upward (vertically) by fixing it with a clamp. Measure the frequency of the column by making the column longer and longer, step by step, until it loses stability by divergence. Measure the period of lateral vibration with a stopwatch to find the frequency of the column. Plot the relation between the length and the frequency. Estimate the experimental buckling length due to self-weight and compare it with the theoretical value. B-6: Conduct thought experiments on the dynamics of Euler’s column and Beck’s column in their original modeling. Discuss realistic modified versions of these columns that are realizable in the laboratory. Category C: Computational Approaches C-1: C-2: C-3: C-4:

Conduct Conduct Conduct Conduct

a a a a

computational computational computational computational

experiment experiment experiment experiment

on on on on

Euler’s column. Beck’s column. Beliaev’s column. Bolotin’s column.

Category D: Mathematics D-1: Derive the equation of motion of a uniform beam with internal damping. Assume that the material obeys the Kelvin-Voigt viscoelastic constitutive law. Apply Galerkin’s method, with a two-term approximation, to the equation of motion to obtain a reduced system of ordinary differential equations. D-2: Derive the equation of motion of the same beam subjected to external damping. Apply Galerkin’s method, with a two-term approximation, to the equation of motion to obtain a reduced system of ordinary differential equations.

Appendix A: Suggested Exercises

225

D-3: Check the numerical accuracy of the central finite difference method applied to the free vibrations of beams with different boundary conditions. Take the mesh points through the concept of dynamics and then through the concept of statics. In the former case, the mesh points must be taken at the center of each cell, while in the latter case, the mesh points are at the sides of the cells. D-4: Derive Hsu’s conditions for the principal regions of resonances, accurate to order e1 , with damping excluded. Here, e is the nondimensional amplitude of the time-varying component. D-5: Extend Hsu’s procedure so as to obtain the conditions for the principal regions of resonances, when damping is considered. D-6: Derive the conditions for the regions of resonances up to order e2 .

Appendix B

Movie of Test Runs in Chapters 10, 11 and 12

Experiment 1: Test runs in Chap. 10: Flutter of cantilevered column Test #1 for Run No. 4, Test #2 for Run No. 5, and Test #3 for Run No. 7: Refer to Fig. 10.4 and Sect. 10.5.2 in Chap. 10. Experiment 2: Test runs in Chap. 11: Effect of a lumped mass Test #1 for Run No. 1, Test #2 for Run No. 2, Test #3 for Run No. 3, and Test #4 for Run No. 4: Refer to Sect. 11.4 and Fig. 11.8. Experiment 3: Test runs in Chap. 12: Stabilizing effect of rocket thrust Test #1 for Run No. 2, and Test #2 for Run No. 4: Refer to Fig. 12.5 and Sect. 12.5 in Chap. 12. Note: A movie showing three experiments is given in the Supplementary Material (online).

© Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6

227

Appendix C

Flutter Motion of a Damped Column under a Rocket-Based Follower Force

The three experiments in Chaps. 10–12 used test columns made of aluminium with nondimensional damping of c = 1.0  10−3. An extra run with an acrylic column with nondimensional damping of c = 2.0  10−2 was conducted to watch flutter motion under a follower force. The acrylic column was designed to keep the same bending stiffness and length with the corresponding aluminium columns. This extra run was conducted in the course of a series of test runs as described in the paper by Sugiyama, Y., Langthjem, M. A., Iwama, T., Kobayashi, K., Katayama, K., and Yutani, H., (2012): Shape optimization of cantilevered column subjected to a rocket-based follower force and its experimental verification, Structures and Multidisciplinary Optimization. 46. 826–838 (refer to Sect. 12.6).

© Springer Nature Switzerland AG 2019 Y. Sugiyama et al., Dynamic Stability of Columns under Nonconservative Forces, Solid Mechanics and Its Applications 255, https://doi.org/10.1007/978-3-030-00572-6

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Appendix C: Flutter Motion of a Damped Column under a Rocket-based …

Setup of an acrylic test column

A movie of flutter of the acrylic column under a rocket-based follower force of nominal thrust of 580 N (available in the Supplementary Material (online)): Among many flutter motions observed in a series of flutter tests in Chaps. 10–12 and some other tests, here is seen a beautiful flutter motion of a damped column under a follower thrust.

Appendix C: Flutter Motion of a Damped Column under a Rocket-based …

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Dynamic response recorded in the test shows a harmonic motion with increasing amplitude, which confirms that the column lost stability by flutter

Index

A Adjoint boundary value problem. See self-adjointness and non-self-adjointness Adjointness. See self-adjointness and non-self-adjointness Analog computer-based experiment, 14, 17, 28, 173, 183, 188, 194, 196 Approach analytical, 219 experimental, 31, 145, 196, 207, 216, 219, 220 Hsu’s, 184, 187, 200, 201 numerical, 219 Approximation first-order, 188, 199 second-order, 188, 199, 203, 204, 209 Argand diagram, 28, 42, 43, 46, 75, 142, 143 Asymptotic stability. See stability, asymptotic Attachment, 103, 115, 118, 120–124, 207, 209, 211–213 B Beam, definition, 1, 2, 11 Beat, 174 Beating band, 177, 191, 192 Beating vibration, 43, 177, 192 Beck’s column, definition, 25, 30 Beliaev’s column, 171, 173, 174, 224 Bending stiffness, 1, 3, 5–7, 10, 21, 71, 122, 133, 134, 159, 192, 224 Bolotin’s column, 192, 193, 195, 196, 224 Boundary conditions, beam, column clamped-clamed, 10, 13, 15, 186 clamped-free (cantilevered), 10 clamped-pinned, 186

pinned-pinned, 183 Boundary value problem, 89, 104, 111–113 Branch flutter, 78 nominal, 78 Buckled configuration, 162 Buckling, 2, 5, 10, 11, 13, 18, 19, 23, 90, 134, 136, 138, 160, 161, 223, 224 See also divergence Buckling test, 134–136, 223 C Calculus of variation, 49 Cantilevered column, 3, 6, 10, 13, 20, 25, 32, 37, 103, 127, 129, 130, 134, 135, 140, 142, 145, 152, 155, 156, 160, 183, 192, 195, 196, 202, 207 Cantilevered pipe, 71, 72, 78, 85, 87, 88, 90, 93, 94, 96, 101 Cantilever, standing, 20, 155 Characteristic equation, 7, 27, 38, 39, 77, 90, 107, 110, 118, 131, 147 Characteristic root, 39, 77 Clamped-clamped column, 188–190, 192 Clamped-pinned column, 189, 190, 192 Column, definition, 1, 11 Combination resonance of difference type, 183, 194, 195, 215 of sum type, 183, 187, 188, 190, 191, 194, 196, 201, 210 Conservative force, 10, 120, 158 Coriolis force, 56, 73, 81, 87, 97 Coupled Hill’s equations. See Hill’s equations, coupled

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234 Coupled Mathieu equations, 184–186, 200, 203 Critical flow speed, 75, 77, 78, 85, 91 D d’Alembert’s principle, 72, 104 Damped Beck’s column, 39, 66, 67 Damper, 87, 96–101, 124–126 Damping, external, internal, 40, 41, 60, 87, 97, 126, 201, 202, 207, 224 Damping ratio, 8, 9, 100 Dashpot. See damper Deflection test, 5, 6, 133 Destabilizing effect of damping, by follower force by parametric excitation, 201, 202 Device calibration, 120, 121 pyrotechnic, 134, 152 Diagonalization, 186, 199 Dirac’s delta function, 87 Displacement sensor, 189 Dissipative work, 54 Divergence, definition, 4, 19 Dual role of damping, 67 E Eigenfrequency, 17–19, 78, 82, 100, 106, 132, 159, 160, 167, 172, 190, 194, 195, 214 Eigenfunction, 9, 89 Eigenvalue, 4, 7, 27, 39, 42, 44, 51, 53, 65, 75, 78–80, 82, 108–110, 158 Eigenvalue branch, 43, 83, 142 Eigenvalue curve, 27, 108 Eigenvector, 51, 57, 77 Energy kinetic, 19, 28, 52, 75, 147, 157 mechanical, 11, 17, 19, 52 potential, 19, 75 Energy balance, 49, 52, 54, 57, 60, 61, 170 Energy consideration, 49, 50, 215 Energy growth rate, 62, 63 Euler-Bernoulli beam theory, 1, 73 Euler buckling, 1, 17, 112, 172, 189 Euler’s column, 13–15, 17, 28, 224 External damping, 3, 6–9, 11, 37–41, 43, 53, 57, 60, 61, 63, 87, 97, 101, 104, 115, 124, 126, 129, 134, 201, 202, 205, 207, 224

Index F Feodosiev’s beam, 31 Finite difference method, 3, 14, 186, 209, 225 Finite element method, 3, 38, 51, 76, 145, 147, 183, 186 Fixed node, 67 Floquet’s theory, 167 Fluid-conveying pipe, 50, 80, 85, 87 Fluid-structure interaction problem, 11, 49, 71 Flutter branch, 78 Flutter configurations, fluid-conveying pipe, 80 follower force-loaded cantilever with damping, 50 Flutter, definition, 4 Flutter test, 134, 136, 137, 139, 148, 149, 152 Flying beam, 31, 32 Follower force, definition pulsating, 183, 192 subtangential, 155 Force balance, 72 Fourier expansion, 199 G Galerkin method, 49, 89, 108, 109, 186, 209 Galerkin representation, 50, 66 Generalized coordinate, 3, 4, 186 Growth rate, 5, 43, 44, 194, 195 H Hamilton’s principle, 49, 75, 145, 157 Harmonically pulsating force, 171, 183 Harmonic motion, 28, 174, 177, 178 Hill’s equations, coupled, 184, 199, 200 Hsu’s approach, 184, 187, 200, 201 Hsu’s condition, 183, 187, 188, 190, 194, 225 I Infinite time, 5, 19, 29, 141, 173 Initial condition, 4, 173, 176, 179, 180 Internal damping, 3, 9, 11, 37, 39–43, 47, 49–51, 53, 54, 56, 57, 59–67, 75, 78, 82–84, 87, 91, 95, 97, 118, 126, 141, 142, 201, 202, 207, 224 Internal resonance, 29, 185 K Kelvin-Voigt constitutive law, 3, 129 Kelvin-Voigt damping, 57 Krylov-Bogolyubov-van der Pol method, 185

Index L Lagrange’s equation of motion, 49 Logarithmic decrement, 8, 9, 101, 224 Lumped mass. See mass, lumped Lyapunov’s stability criterion, 3 M Mass, lumped, 87, 92, 94–97, 99–101, 145, 146, 148–152 Mass ratio, 75, 78, 85, 93, 97, 99 Mathematical model, 124, 129, 130, 145, 146, 155, 156, 196 Mathieu equation, 167–169, 184, 185 Mathieu-Hill equation, 184 Matrix damping, 38, 77, 200 load, 38, 51, 77 mass, 38, 51, 77, 157 stiffness, 38, 51, 77, 157, 193, 200 Melde’s effect, 167 Method of averaging, 185 Modal function, cantilevered column, 66 Modulus of elasticity, 5 of viscosity, 3, 9, 134 Motion dragging, 49 harmonic, 28, 174, 177, 178 non-harmonic, 17 Moving node, 67, 81, 82 Multiple scales, method of, 199 N Natural frequency, 9, 16, 17, 57, 108, 168, 172, 187, 194, 224 Nominal branch, 78 Nominal order, 78 Nonconservative force, definition, 11, 12, 120, 128, 155, 164 Nonconservative nature of a follower force, 32, 34 Nonconservativeness, 25, 104, 115, 157 Nonconservativeness parameter, 121, 123, 210, 212 Nondimensionalization, 20, 26, 27, 37, 42, 50, 74, 76–78, 88, 89, 94–97, 106, 108, 117, 118, 131–134, 142, 147, 172, 201, 208, 209, 225 Non-self-adjointness, 111 Non-trivial solution, 90, 107, 110, 118, 131, 168 Nonwoven fabric, 121, 123

235 O Optimum design, 164 Order first, 34, 169, 185, 188, 199, 201 oscillatory mode, 80 second, 34 P Parametric resonance, 167, 169, 187, 196, 202, 209, 223 Pflüger’s column, definition, 37 Phase angle gradient, 54, 56, 60, 61, 64–66 Pinned-pinned column, 18, 167, 171, 173 Pipe conveying fluid power equation, 71, 74, 90, 101 Preliminary test, 5, 6, 10, 123, 133 Principle of virtual work, 104 Pulsating force realization, 118, 196 Pulsating Reut force, 128, 207, 208, 211, 214 Pyrotechnic device, 136, 152 Q QR-algorithm, 39, 77 Quasi-mode, 81, 83 R Rate of amplitude growth, 5, 43, 44, 132, 142 Region of resonance principal, 174, 177 secondary, 190 Resonance combination, 57, 183, 185, 187, 188, 190, 191, 194, 196, 201, 202, 204, 205, 210, 211, 215, 216 simple, 172, 177–179, 185, 187, 190, 192, 194, 196, 201, 202, 204, 205, 210, 211, 215, 223 Reut force calibration system See device, 121 generalized, 119, 121, 127, 207 pulsating, 128, 207, 208, 211, 214 realization, 118 Reut’s column, definition, 103 Rocket-based follower force, 129, 139, 145, 147, 148, 163, 164 Rocket motor, standard refined, 147 Role of damping, 34, 39, 49, 50, 215 Rotatory inertia, 1, 115, 122, 124, 130, 133, 145, 147, 148, 152, 158, 207

236 S Self-adjointness, 111 Spring support, 91, 96 Stability, asymptotic definition, 4, 29 in a finite time interval, 29, 30, 43, 44, 47, 141, 142 Stability criteria, 2, 3, 11, 43, 44 Stability map, for fluid-conveying pipe for partial follower force, 50 Stabilizing effect damping, 40, 95 no damping, 67 Standing cantilevered column. See cantilever, standing Subtangential follower force. See follower force, subtangential T Tangency coefficient, 156, 159, 160 Test bending, 122, 159, 223 buckling, 134, 135, 223 decaying beam motion, 9, 11 decaying pendulum motion, 7, 11

Index deflection, 5, 6 flutter, 134, 136, 137, 139, 148, 152 preliminary, 5, 6, 10, 123, 133 vibration, 5, 7, 133, 224 Thought experiment, 32, 34 Thrust curve, 132, 147, 148, 158 Transfer. See branch Two-mode-coupling, 27, 30, 42, 43 V Variational principle, 49 Velocity-dependent force, 73, 81 Vibration configuration, 34, 66 Virtual work, 75, 106, 147, 157 W Wave, progressive travelling, 56 Work, of a conservative force of a follower force, 164 Y Young’s modulus, 3, 5, 75, 159, 189, 192