Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6: Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 [1st ed.] 9783030477202, 9783030477219

Table of contents :
Front Matter ....Pages i-viii
Paper Machine Winder Vibration Testing Using TVDFT (Jake Zwart, Derek Lindgren)....Pages 1-16
Measurement Methods for Evaluating the Frequency Response Function of a Torque Converter Clutch (Luke Jurmu, Darrell Robinette, Jason Blough, Eric Mordorski, Craig Reynolds)....Pages 17-25
Frequency-Domain Triangulation of Spatial Harmonic Motion for Single-Camera Operating Deflection Shape Measurement (Domen Gorjup, Janko Slavič, Miha Boltežar)....Pages 27-30
Investigating the Feasibility of Laser-Doppler Vibrometry for Vibrational Analysis of Living Mammalian Cells (Sascha Schwarz, Stefanie Kiderlen, Robert Moerl, Stefanie Sudhop, Hauke Clausen-Schaumann, Daniel J. Rixen)....Pages 31-36
Visio-Acoustic Data Fusion for Structural Health Monitoring Applications (Chad R. Samuelson, Caitrin A. Duffy-Deno, Christopher B. Whitworth, David D. L. Mascareñas, Jeffery D. Tippmann, Alessandro Cattaneo)....Pages 37-51
Automatic Interpolation for the Animation of Unmeasured Nodes with Differential Geometric Methods (Daniel Herfert, Kai Henning, Jan Heimann)....Pages 53-59
Measurement and Analysis of the Nonlinear Stiffness of CFRP Components during Vibration Fatigue Testing (Michele Peluzzo, Dario Di Maio, Andrea Cammarano, Paolo Castellini)....Pages 61-71
Model Reduction of Electric Rotors Subjected to PWM Excitation for Structural Dynamics Design (Margaux Topenot, Gaël Chevallier, Morvan Ouisse, Damien Vaillant)....Pages 73-78
Fast Computation of Laser Vibrometer Alignment Using Photogrammetric Techniques (Daniel P. Rohe, Bryan L. Witt)....Pages 79-85
Photogrammetry-Based Structural Damage Detection by Tracking a Laser Line (Y. F. Xu)....Pages 87-99
Measuring Aero-Engine Pipe Vibration with a 3D Scanning Laser Doppler Vibrometer (Christoph W. Schwingshackl)....Pages 101-104
A “Mechanical” Vision of Image-Based Identification methods in Structural Dynamics (Paolo Castellini, Emanuele Zappa)....Pages 105-108
Modelling of Guided Waves in a Composite Plate Through a Combination of Physical Knowledge and Regression Analysis (Marcus Haywood-Alexander, Tim J. Rogers, Keith Worden, Robert J. Barthorpe, Elizabeth J. Cross, Nikolaos Dervilis)....Pages 109-114
Improved FRF Estimation from Noisy High-Speed Camera Data Using SEMM (Tomaž Bregar, Klemen Zaletelj, Gregor Čepon, Janko Slavič, Miha Boltežar)....Pages 115-118
Full-Field Modal Analysis by Using Digital Image Correlation Technique (Emilio Di Lorenzo, Davide Mastrodicasa, Lukas Wittevrongel, Pascal Lava, Bart Peeters)....Pages 119-130
Method for Selecting Rotor Suspension Design Criteria (Brian Damiano)....Pages 131-138
Implementation of Total Variation Applied to Motion Magnification for Structural Dynamic Identification (Nicholas A. Valente, Zhu Mao, Matthew Southwick, Christopher Niezrecki)....Pages 139-144
A Complex Convolution Based Optical Displacement Sensor (Matthew Southwick, Zhu Mao, Nicholas A. Valente, Christopher Niezrecki)....Pages 145-153
Current Methods for Operational Modal Analysis of Rotating Machinery and Prospects of Machine Learning (German Sternharz, Tatiana Kalganova)....Pages 155-163

Citation preview

Conference Proceedings of the Society for Experimental Mechanics Series

Dario Di Maio · Javad Baqersad  Editors

Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research.

More information about this series at http://www.springer.com/series/8922

Dario Di Maio • Javad Baqersad Editors

Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020

Editors Dario Di Maio University of Twente Enschede, Twente, The Netherlands

Javad Baqersad Kettering University Flint, MI, USA

ISSN 2191-5644 ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-030-47720-2 ISBN 978-3-030-47721-9 (eBook) https://doi.org/10.1007/978-3-030-47721-9 © The Society for Experimental Mechanics, Inc. 2020 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Rotating Machinery, Optical Methods & Scanning LDV Methods represent one of eight volumes of technical papers presented at the 38th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Houston, Texas, February 10–13, 2020. The full proceedings also include volumes on Nonlinear Structures and Systems; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamic Substructures; Special Topics in Structural Dynamics & Experimental Techniques; Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing; and Topics in Modal Analysis &Testing. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Enschede, Twente, The Netherlands Flint, MI, USA

Dario Di Maio Javad Baqersad

v

Contents

1

Paper Machine Winder Vibration Testing Using TVDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jake Zwart and Derek Lindgren

1

2

Measurement Methods for Evaluating the Frequency Response Function of a Torque Converter Clutch. . . . Luke Jurmu, Darrell Robinette, Jason Blough, Eric Mordorski, and Craig Reynolds

17

3

Frequency-Domain Triangulation of Spatial Harmonic Motion for Single-Camera Operating Deflection Shape Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domen Gorjup, Janko Slaviˇc, and Miha Boltežar

4

Investigating the Feasibility of Laser-Doppler Vibrometry for Vibrational Analysis of Living Mammalian Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sascha Schwarz, Stefanie Kiderlen, Robert Moerl, Stefanie Sudhop, Hauke Clausen-Schaumann, and Daniel J. Rixen

27

31

5

Visio-Acoustic Data Fusion for Structural Health Monitoring Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chad R. Samuelson, Caitrin A. Duffy-Deno, Christopher B. Whitworth, David D. L. Mascareñas, Jeffery D. Tippmann, and Alessandro Cattaneo

37

6

Automatic Interpolation for the Animation of Unmeasured Nodes with Differential Geometric Methods . . . Daniel Herfert, Kai Henning, and Jan Heimann

53

7

Measurement and Analysis of the Nonlinear Stiffness of CFRP Components during Vibration Fatigue Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michele Peluzzo, Dario Di Maio, Andrea Cammarano, and Paolo Castellini

61

8

Model Reduction of Electric Rotors Subjected to PWM Excitation for Structural Dynamics Design . . . . . . . . . Margaux Topenot, Gaël Chevallier, Morvan Ouisse, and Damien Vaillant

73

9

Fast Computation of Laser Vibrometer Alignment Using Photogrammetric Techniques. . . . . . . . . . . . . . . . . . . . . . . Daniel P. Rohe and Bryan L. Witt

79

10

Photogrammetry-Based Structural Damage Detection by Tracking a Laser Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. F. Xu

87

11

Measuring Aero-Engine Pipe Vibration with a 3D Scanning Laser Doppler Vibrometer . . . . . . . . . . . . . . . . . . . . . . 101 Christoph W. Schwingshackl

12

A “Mechanical” Vision of Image-Based Identification methods in Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . 105 Paolo Castellini and Emanuele Zappa

13

Modelling of Guided Waves in a Composite Plate Through a Combination of Physical Knowledge and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Marcus Haywood-Alexander, Tim J. Rogers, Keith Worden, Robert J. Barthorpe, Elizabeth J. Cross, and Nikolaos Dervilis

vii

viii

Contents

14

Improved FRF Estimation from Noisy High-Speed Camera Data Using SEMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 ˇ Tomaž Bregar, Klemen Zaletelj, Gregor Cepon, Janko Slaviˇc, and Miha Boltežar

15

Full-Field Modal Analysis by Using Digital Image Correlation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Emilio Di Lorenzo, Davide Mastrodicasa, Lukas Wittevrongel, Pascal Lava, and Bart Peeters

16

Method for Selecting Rotor Suspension Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Brian Damiano

17

Implementation of Total Variation Applied to Motion Magnification for Structural Dynamic Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Nicholas A. Valente, Zhu Mao, Matthew Southwick, and Christopher Niezrecki

18

A Complex Convolution Based Optical Displacement Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Matthew Southwick, Zhu Mao, Nicholas A. Valente, and Christopher Niezrecki

19

Current Methods for Operational Modal Analysis of Rotating Machinery and Prospects of Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 German Sternharz and Tatiana Kalganova

Chapter 1

Paper Machine Winder Vibration Testing Using TVDFT Jake Zwart and Derek Lindgren

Abstract Paper machine winders take jumbo reels of paper that are up to 9 m wide and 3 m in diameter, slit them to pressroom widths and rewind the paper to pressroom roll diameters. With the paper rolls continually changing in diameter, their rotational speeds also change continually. The rotational speed of the rolls in contact with the paper web, that either support the paper set or guide it through the process, is controlled by their fixed diameters and the paper speed. Thus, the speed of the paper set starts at a high rotational speed, but continually slows as the set increases in diameter, with its rotational speed crossing the speed of its supporting rolls. A high vibration in the winder system can have a number of possible sources. A vibration monitoring technique using multiple tachometers, multiple triaxial accelerometers, spectral mapping and order tracking using the TVDFT is used to determine whether the signature of the high vibration matches a harmonic of any of the rotational speeds of the rotating components of the winder. Keywords Paper machine · Winder vibration · TVDFT · Roll throw-outs · Transient vibration · Multi-tachometer order tracking · Order track · Crossing orders · TVDFT order

1.1 Introduction 1.1.1 Winder Basics Modern paper machines produce jumbo reels of paper up to 9 m wide and 3 m in diameter as shown in Fig. 1.1. This is far too large for any printing press to use, so it is slit into the required pressroom width and wound into the required pressroom diameter with the winder as shown in Fig. 1.2. After the paper is slit and spread, it is built into a set of rolls as shown in Fig. 1.3. This set of paper rolls ready for wrapping and shipping is shown in Fig. 1.4. The paper thickness for newsprint is typically about 75 μm, about the thickness of a human hair. The newest winders have an operating speed of 2500 mpm (150 kph or 95 mph). Hence, we have a thin tender web that is travelling phenomenally fast. To ensure it does not break, or roll defects do not occur, the speeds and torques on each of the rolls must be carefully controlled. Since winding is a batch process, where the set is removed while the winder is at a standstill, there is a period of acceleration, then constant speed operation followed by deceleration, all referenced to the web speed. If there are extenuating circumstances, there can be more speed changes before the set is complete. Without paper in the winder, the winder will have its own set of structural resonances. Each roll will have its critical speed and the structure of the winder will have resonances as well. To complicate matters, each nip (location where a roll touches the paper roll) can be modelled as a spring and damper. The paper roll is supported by two drums beneath it, while the rider roll above it provides the needed stability to remain on the two drums. As the roll builds, the nip contact width increases, increasing the spring constant of that nip. The mass of the paper starts off as just the mass of the cores, and then continually increases as the paper set builds. Thus, the parameters that affect the resonant frequencies continually change.

J. Zwart () Spectrum Technologies, Puslinch, ON, Canada e-mail: [email protected] D. Lindgren Resolute Forest Products, Augusta Newsprint Division, Augusta, GA, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_1

1

2

Fig. 1.1 Paper reels produced on a modern paper machine

Fig. 1.2 Winder used to slit the reel into pressroom sized widths and diameters

J. Zwart and D. Lindgren

1 Paper Machine Winder Vibration Testing Using TVDFT

Fig. 1.3 A set of paper rolls building in the winder

Fig. 1.4 A set of paper rolls ready to be wrapped and shipped

3

4

J. Zwart and D. Lindgren

Since all the winder rolls have fixed diameters, their rotational speeds are determined by the web speed, assuming that there is no slippage between the paper web and the rolls. However, the parent reel continuously decreases in diameter, while the set of paper rolls continuously increases in diameter. Thus, even at constant web speed the rotational speeds of the parent and the set rolls continuously change. The diameter of the paper set starts off very small and increases until it is larger than the diameter of the drums supporting the paper set, resulting in their rotational speeds crossing over as well. Since the paper web is delicate, and defects in the paper roll structure lead to pressroom breaks, all winder parameters affecting roll structure, including the winder speed, must be carefully controlled. The speed of each roll is referenced to the master speed controller. Anytime the acceleration rate of the winder is changed, the speed goes through an S-curve. This limits the jerk, which places smaller dynamic loads on the paper roll and contributes to smoother operation. Normally the S-curve is characterized by the number of seconds it takes to go from one level of acceleration to another. The winder typically makes paper rolls for many different pressrooms, not all of the rolls in the same set need be the same width. For manufacturing convenience, there is no shaft in the cores holding them together. Each roll of paper in the set will have four dominant rigid body modes of vibration. The frequency of these modes will depend upon the dimensions of the roll, its mass and the spring constant and damping at the nip, along with the dynamics of the winder. Thus, the diameter and width of the paper rolls along with the nip load between the paper and each roll will affect the dynamics. There is extensive literature on the theory and practice of modern winding machines. In many cases, the practice comes first, problems become evident and the theory is studied to understand the observed phenomena. An excellent summary of the different types of winding machines, as well as the theory of winding along with pertinent paper properties, has been summarized in the book, “Winding: Machines, Mechanics and Measurements” [1]. Paper is a viscoelastic material. In the machine direction and cross machine direction a first order model of linear elasticity works quite well. However, when a stack of paper is compressed, the paper modulus is non-linear, with a few different models used to describe this non-linearity. These material properties, have been used to develop models for the structure of wound rolls. The current model of choice was developed by Hakiel [2], which describes the radial structure of the roll quite well, using basic material properties of the roll. The structure of a wound roll of paper not only depends upon the properties of the paper, but also the winding conditions and set diameter. The set diameter building from 100 mm to perhaps 1500 mm is the largest change in conditions. Jorkama [3] has created an analytical model of winding dynamics, assuming the spring constant between the roll and set is modeled by Hertzian contact mechanics. These results show changing eigenfrequencies with increasing set diameter. The most important winding conditions are the tension on the paper as it is being wound onto the roll, the nip load between the paper roll and the rear drum, and the torque differential between the front drum and the rear drum. The roll speed, the load at the other nips and the harmonic mean of the roll radii are secondary conditions. There are times when the winding process can go dramatically awry. In the worst-case scenario, the rolls start bouncing as they are being wound and are thrown out of the pocket of the winder as reported by Zwart. [4] Without adequate guarding, the roll can do much damage to machinery or people in its trajectory. With a complex system like this, where the speed of multiple components changes, a transient vibration analysis technique is the ideal vibration trouble-shooting tool.

1.2 Transient Vibration Analysis When a system is undergoing speed changes, the ideal measurement procedure consists of measuring the vibration at the locations of interest along with the rotational speeds of the likely drivers of the vibration. The analysis technique starts with a spectral map of the STFT to determine how the frequencies of vibration change with time and the rotational speed of the various rolls causing the vibration. From the spectral map, the basic problem areas are determined. This could be resonance or a forced vibration at a specific rotational speed, or some combination of the two. The rotational speeds of the rolls that are suspected of being the source of the vibration are measured, normally with a 1 pulse per revolution (PPR) tachometer. Once the spectral map has determined the primary likely driving forces causing the vibration, order slices give more detailed information. In the case of winders, where the front and back drum often are nominally the same diameter, the frequency resolution must be extremely high to determine which drum is the predominate source of the vibration. During the constant speed portion of the wind, synchronous time averaging utilizing a tachometer from the front drum and the rear drum, or extremely high resolution FFT analysis, can be used to determine the problem. However, a method that will work while the speed is varying is preferred.

1 Paper Machine Winder Vibration Testing Using TVDFT

5

1.2.1 Rotational Speed Determination For order tracking methods to work, the angle of shaft rotation must be related to the vibration. This requires an accurate method to determine this angle. The normal procedure used is to obtain a square wave tachometer signal, often at 1 PPR. The timing between pulses is determined, Eq. 1.1, and the inverse of this time is the rotational speed of the roll. f=

1 PPR · t

(1.1)

where f is the rotational frequency, and t is the time increment between pulses. Usually the time increment is an integer count of a high-speed clock pulse stream. The higher the clock rate, the more precise the speed will be. Often the timing of the tachometer signal will be measured by the same data acquisition front end that measures the vibration signals. In this case the digitization rate must be rapid enough to obtain a reasonable estimate of the time between tach pulses. This time estimate can be greatly improved by taking a linear interpolation of the actual tach threshold value and the sample values on each side. In principle a sinc filter would give a more precise [5] estimate, however it requires many more points. My testing has shown worse results in this interpolation method. A least square cubic spline (LSCS) is normally fit to the speed to smooth the jitter, leaving a highly accurate speed map. Often the number of knots in the LSCS is varied to increase the accuracy of the interpolation. Uniformly spaced knots work well for speed maps like those shown in this paper. When there are sudden changes in speed, such as a car that is shifting, or when a piece of equipment is put through discrete speed steps, knots should be placed at these speed changes. An algorithm that puts more knots at locations where there is a greater acceleration or jerk, while limiting the number of knots so as not to follow the jitter of the tach, is useful.

1.2.2 Time Varying Discrete Fourier Transform Basics With the resampling-based order tracking method the data is collected as equal increments in time and is then resampled into equal angle increments based on the rotational speed. When multiple tachometers are used with a resampling procedure, the data will need to be resampled for each tachometer. Not only is this computationally intensive, but there is no means to separate out the effect from different tachometers for crossing orders. To overcome this problem, Blough [6] developed the Time Varying Discrete Fourier Transform (TVDFT) in his PhD thesis. There have been numerous papers published since that time based on this order tracking method. The TVDFT algorithm can use multiple orders per tachometer and multiple tachometers. It can handle crossing orders. The raw data utilized by the TVDFT is evenly spaced in time, the norm for instrumented data collection. When the TVDFT algorithm is combined with the Orthogonality Compensation Matrix (OCM), all orders from all tachometers are calculated simultaneously, resulting in excellent discrimination between the orders even when they cross. Fourier analysis starts from the fundamental equations, Eq. 1.2, where x(t) is signal amplitude, f is frequency and t is time. With sampled data, the continuous integral is rewritten into a discretized form using the midpoint rule to perform numerical integration. Instead of integrating from negative infinity to positive infinity, the integration is done over one block of data from time 0 to time T.  ∞ S (f) = x (t) · exp (−j · 2 · π · f · t) dt  x (t) =

−∞ ∞ −∞

(1.2) S (f) · exp (j · 2 · π · f · t) dt

The phase in the exponential term in the discretized form of the above equations can be written in a variety of formats, Eq. 1.3, where rps is the rotational speed in Hz. The rightmost expression in this equation again uses the midpoint rule to perform the numerical integration of the phase. Higher order integration schemes such as the trapezoidal rule or Simpson’s rule can also be used with little increase in computational cost. For the purposes of this derivation, the integration is shown as a summation. When the summation in this equation is evaluated inside the global summation of the discrete Fourier

6

J. Zwart and D. Lindgren

Transform, then each subsequent value of n is just one more term in the inner summation. Thus, a cumulative summation such as the Matlab cumsum function works well. Since it is an integration a cumtrapz or cumsimps (not a built-in function) function will also work.  n·t n  ϕ (n · t) = 2π · f · n · t = 2π · oi rps dt = 2π · oi · t · rps (k · t) (1.3) 0

k=1

As with conventional signal processing, when using the TVDFT, the forward transform is divided by the time length of data transformed, and the inverse transform is multiplied by the time length. The transform pair is shown in Eq. 1.4.    N n  1  Se (oi ) = · rps (k · t) x (n · t) · exp −j · 2π · oi · t · N n=1 k=1 (1.4)    N n   x (t) = rps (k · t) S (oi ) · exp j · 2π · oi · t · n=1

k=1

where: Se – the estimated order spectrum for the given order and block of data. N – number of samples in the block. x(n* t) – sample oi – order of interest j – imaginary operator rps – instantaneous rotational speed [Hz]. This forward transform can be combined with windowing in a conventional fashion as shown in Eq. 1.5 with the window correction factor given in Eq. 1.6. N K  Se (oi ) = · N





x (n · t) · exp −j · 2π · oi · t ·

n=1

n 

 rps (n · t) · Window (n)

 (1.5)

k=1

and K=

1 N 

(1.6)

Window (n)

n=1

where: Window(n) – is the nth element in the window. Any window used in conventional Fourier Analysis can be used with the same amplitude and frequency trade offs. The exponential part of this equation is referred to as the kernel for a specific order. The inner summation for this kernel only needs to be calculated once for each tachometer. The kernel then needs to be recalculated for each order associated with that tachometer but remains constant for each channel of data, thus only needs to be calculated once for all the channels. As seen in Eq. 1.5, the window can be multiplied by the exponential term along with the window correction factor. The kernel is specific for each order and needs to be recalculated for each block of data. There are a number of methods for selecting the length of a block of data with more details given in Blough’s thesis. For one rotating element, the number of time samples per block can be selected for a constant number of revolutions in the time data. If this can be arranged to be an integer number of cycles for the order being evaluated, there will be less leakage. Alternatively, the number of data points per block can be kept constant.

1 Paper Machine Winder Vibration Testing Using TVDFT

7

When there are closely spaced orders, or crossing orders, energy from one order can leak into another order. To minimize this effect Blough introduced the orthogonality compensation matrix, where leakage between orders is corrected. The form of this compensation matrix for the simple case of three orders is shown in Eq. 1.7. This can be expanded to include all the orders of concern. For this scheme to work the orders with significant energy must be included. In practice, it is the orders with significant energy that are closely spaced or are crossing that require compensation. ⎛

⎞ ⎛ ⎞ ⎛ ⎞ e1,1 e1,2 e1,3 S1 Se1 ⎝ e2,1 e2,2 e2,3 ⎠ · ⎝ S2 ⎠ = ⎝ Se2 ⎠ e3,1 e3,2 e3,3 S3 Se3

(1.7)

where Se – the estimated order obtained from Eq. 1.5. S – the compensated order ei,j – the individual terms of the correction matrix as given by Eq. 1.8. The cross-orthogonality terms are calculated using Eq. 1.8, where the line over a term indicates the complex conjugate of that term. When this matrix is computed, the diagonal terms will be one. The off-diagonal terms provide the correction. When the orders are well separated there is very little correction required. When the orders are close or crossing the correction is much more significant. ⎛   ⎞  N n n    K ⎝exp −j · 2π · oi · t · eij = · rps (k · t) · Window (n) · exp −j · 2π · oj · t · rps (k · t) ⎠ N n=1

k=1

(1.8)

k=1

1.3 Using TVDFT to Analyze Winder Vibration A newsprint winder was having excessive vibration. In this installation a pair of accelerometers were permanently installed near the center of the rider roll beam, one measuring in the vertical direction and the other in the machine direction. The signals from these accelerometers were integrated to velocity and then rms averaged. When these signals reach a predetermined level, the PLC sends a signal to the winder drive to reduce the winder speed by 500 fpm. If the operator determines that the vibration has been reduced by a sufficient amount, he will increase the winder speed again. For this paper making operation the winder was the bottleneck in paper production, partly caused by its speed being limited by the vibration levels. For the diagnostic work the winder was instrumented with eight triaxial accelerometers. One accelerometer was mounted on each of the front and back drum bearings. One was mounted on each core chuck, and finally an accelerometer was mounted on each end of the rider roll beam. Optical tachometers were used to measure the main rotational speeds. One was used for the paper set rotational speed, one for each of the front and rear drums and one for the unwind stand. For this work, the maximum frequency of interest was determined to be 100 Hz. For a margin of safety, the actual maximum frequency selected was 200 Hz, resulting in a sampling rate of 512 Hz. The tachometers used were 1 PPR tachometers which give good results. When using a DSA to collect square wave tachometer pulse data, the sampling rate should be at least 10–20 times the rate at which the tachometer pulses arrive. This limits the number of PPR when using the DSA to collect the tachometer signals. The vibration data was collected for periods ranging from hours to overnight and then post-processed. The initial processing of the data calculated the winder speed and the rms vibration level of one channel of data as shown in Fig. 1.5. From this information the data from a single set was selected to analyze in detail. This detailed analysis can then be repeated for other sets as required.

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The data from the tachometers is processed to rotational speed of the set and drum and is shown in the top two plots of Fig. 1.6. These speeds in conjunction with the drum diameter are then used to determine the set diameter as shown in the bottom plot of Fig. 1.6. In this particular example, the automatic winder speed reduction occurred four times due to high vibration levels. Because the acceleration and jerk (rate of change of acceleration) is controlled to stay within limits, using a slice of data at the roll rotational speeds from a STFT is reasonably valid. There is greater difficulty in dealing with the region where the set diameter is similar to the drum diameter. TVDFT order analysis is an improvement on slicing the STFT to obtain an order track. For many winders the drums are nominally the same diameter, which in practice means a very small difference in diameter. With varying speed and such closely spaced orders, separating out the contribution of each would be quite challenging with conventional FFT analysis. In the early stages of analysis, it is always wise to obtain a good overview of the data. A spectral map of the STFT is an excellent place to start. Figures 1.7 and 1.8 show examples of this for the rider roll tending side vibration in the vertical and machine direction. Since the rider roll rides on the paper set, it is flexibly mounted and expected to have a higher level of vibration than more rigidly mounted rolls. The main vibration seen in this plot is occurring at the drum rotational speeds and also at the rider roll rotational speed. Note that the vibration at the drum roll rotational speed increases and decreases in a beating type of effect, indicating a possible interaction between the front drum and the rear drum. Figures 1.9 and 1.10 show the order slice extracted from the STFT at the back drum rotational speed and measured at the rider roll tending side in the vertical and machine directions. These show a significant to high level of vibration with the highest level exceeding 9 mm/s rms. The vibration level would have been higher had the winder speed not been automatically slowed down.

1 Paper Machine Winder Vibration Testing Using TVDFT Set Rot Speed - Maximum winder speed is 7988 fpm

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When this vibration level is compared with the results from the TVDFT analysis, Figs. 1.11 and 1.12, the contribution of the vibration at these same locations is shown as a function of the orders of a number of rotating components. Immediately it can be seen that in the vertical direction the vibration due to the back drum is lower than the vibration in the order slice taken from the STFT. The picket fence effect has also been removed. The vibration contribution from the front drum is much less than from the rear drum. Hence it is clear that the picket fence effect in the STFT order slice is due to the beating effect between the vibration contributions from the front drum and the back drum. In the machine direction, at about 120 s, the vibration due to the back drum is slightly over 10 mm/s, about 10% higher than shown in the STFT based order slice. This is likely due to the removal of the effect of the front drum vibration in the TVDFT analysis. The core chucks hold the core of the set at each end of the winder. The vertical vibration from the tending side and drive side are shown in Figs. 1.13 and 1.14, Here it is plain that the major contributor to the vibration is the set, as the majority of the vibration occurs at the set rotational speed. The highest level of vibration is about 50 mm/s rms and occurs at about the time where the winder deceleration starts and the set rotational speed is approximately 10 Hz. It is also interesting to note that the phase changes continuously on the tending side but remains relatively constant on the drive side. The testing was performed with the tachometer mounted on the drive side. Presumably the diameter of the set was slightly different on the tending side as compared to the drive side, resulting in the continuous phase change on the tending side.

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Fig. 1.7 Spectral map of the tending side rider roll vertical vibration

Fig. 1.8 Spectral map of the tending side rider roll machine direction vibration

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Figures 1.15, 1.16, 1.17, and 1.18 show the order tracking results for the vertical vibration on the bearings of the back drum tending side and drive side and the front drum tending side and drive side respectively. Winder drums showed vibration at less than 1 mm/s rms. On the back drum the vibration easily exceeds the maximum desired vibration, sometimes by a factor of 2 or more. The order plots make it obvious that the high vibration corresponds to the back drum rotational speed. The peak vibration levels on the tending side, when compared with the speed map, are shown to be very speed dependent. When the speed is reduced, the vibration levels go down, and when the speed levels are increased again, the vibration levels increase. The vibration phase for this order is constant. The vibration on the front drum is lower, and thus the vibration contributors are more easily seen in the plot. This includes order 1 and 3 of the front drum and order 1 on the back drum. In the initial portion of the set, where the set mass is lower, there is also a significant contribution at the rider roll rotational speed, noted on the graph as ‘Tach BD Order 2.9408’, since the rider roll rotates at 2.9408 times the back drum rotational speed. These results clearly indicate, that the back drum has the highest contribution to the vibration. These results were validated by the mill, when, based on this work, they replaced the back drum and the resultant vibration levels were much lower.

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Fig. 1.18 TVDFT order slices of FDDS vertical velocity

1.4 Conclusion The TVDFT has proven to be a robust and powerful tool to analyze vibrations in machines with many orders, some extremely closely spaced with speeds that are constantly changing. In this test case, the TVDFT showed a major contributor to the vibration was the winder back drum. Based on this the back drum was replaced, allowing the winder to operate at higher speeds.

References 1. Good, J.K., Roisum, D.R.: Winding: machines, mechanics, and measurements. Tappi Press and DEStech Publications Inc. (2008) 2. Hakiel, Z.: Nonlinear model for wound roll stresses. TAPPI J. 70(5), 113–117 (1987) 3. Jorkama, M.: The role of analytical winding dynamics in winder design. TAPPI J. (1998) 4. Zwart, J., Tarnowsky, W.: Winder vibration related to set throw-outs. Pulp & Paper Canada. (2008) 5. Lipshitz, S.P., Vanderkooy, J.: Pulse-code modulation—an overview. J. Audio Eng. Soc. 52(3), 200–215 (2004) 6. Blough, J.R.: Improving the Analysis of Operating Data on Rotating Automotive Components. Dissertation, University of Cincinnati, Ph.D (1998) Jake Zwart MASc, P.Eng, works for Spectrum Technologies and specializes in bringing advanced vibration test techniques into heavy industry, combining advanced vibration knowledge with intimate process knowledge. He has developed unique test instrumentation and implemented advanced analysis algorithms to aid in finding solutions to difficult problem. Derrick Lindgren is the mill manager at Augusta Newsprint. He brings a strong technical background and management skills to ensure the paper mill remains productive.

Chapter 2

Measurement Methods for Evaluating the Frequency Response Function of a Torque Converter Clutch Luke Jurmu, Darrell Robinette, Jason Blough, Eric Mordorski, and Craig Reynolds

Abstract In developing new torque converter clutch (TCC) technology, it is desirable to have accurate models to have confidence in performance predictions. To develop and calibrate TCC models, a test cell has been developed to measure the torsional vibration isolation performance of TCCs. The isolation performance is defined as the ratio of output torque to input torque, or torque transmissibility. This test cell uses an electric motor as a torsional exciter and a secondary motor (absorbing dyno) to control the output speed of the TCC. This loading condition, input torque-output speed, replicates the loading seen in a vehicle drivetrain where the engine provides a torque to the input of the torque converter and the vehicle’s wheels provide the speed boundary condition to the output. The torque transmissibility plot is acquired using a stepped sine approach with three frequencies per measurement. Two other excitation methods will be investigated to further reduce testing time. – Step inputs (both up and down) – Pseudo-random excitation (constant amplitude, random phase) The torque transmissibility results of the new excitation methods will be compared to the previous measurement method for validation. Keywords Torque converter clutch · Torsional vibration · Torque transmissibility · Pseudo random · Torsional excitation

2.1 Introduction Development of accurate models to simulate torsional vibration isolation of the TCC requires accurate measurements of the vibration isolation. In this paper, the torsional vibration isolation performance will be measured in terms of torque transmissibility. A test cell has been developed that has the capability to evaluate the torque transmissibility of a TCC while replicating the loading seen in a vehicle drivetrain. While there have been other test cells developed for testing TCCs [1–5], the published results either don’t fully characterize the TCC resonance, the test setup doesn’t use the same input torque, output speed boundary conditions, or the test methods don’t yield as clean of a result as a stepped sine approach. The main problem with evaluating the torque transmissibility using a stepped sine approach is the amount of testing time it takes to acquire the data. The goal of this paper is to evaluate different torsional excitation techniques in order to reduce the current testing time, while still maintaining the quality of results.

2.2 Methods To test the TCC, the torque converter is installed into the fixture which is outfitted with a hydraulic manifold that allows for the state of the TCC to be controlled [6]. The torque transmissibility of the TCC is measured in the drivetrain dynamometer

L. Jurmu () · D. Robinette · J. Blough · E. Mordorski Department of Mechanical Engineering, Michigan Technological University, Houghton, MI, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] C. Reynolds General Motors Company, Detroit, MI, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_2

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Fig. 2.1 Drivetrain dynamometer used to measure the torque transmissibility of a TCC

Fig. 2.2 Signal processing to calculate the torque transmissibility of a TCC. Time domain torque signals are acquired, then the auto powers are computed, and the ratio of these auto powers compute the torque transmissibility

with the clutch locked (Fig. 2.1). An electric drive motor and an absorbing dyno are used to mimic the loading of a TCC in a vehicle. The drive motor provides a mean and dynamic torque to the input of the TCC fixture, and the absorbing dyno controls the speed of the output shafting. The torque transmissibility of the TCC is evaluated by measuring the output and input torque across a frequency range of dynamic torque excitation.

2.2.1 Tri-Tone Method An example of the signal processing used to compute the torque transmissibility, using the tri-tone method is detailed in Fig. 2.2. Time domain torque signals were acquired with the excitation active. In this example, the drive motor is providing a mean torque of 150 Nm with three sine tones of 0.2, 8.6, and 17 Hz superimposed (all of 15 Nm amplitude). From the

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Table 2.1 Test conditions for acquiring torque transmissibility Test method Tri tone Torque step Speed step Pseudo random

6 tone 10 tone 22 tone 44 tone 88 tone

Mean input torque (Nm) 150 – – 150 150 150 150 150

Dynamic input torque (Nm) 15 – – 15 15 5 5 4

Output speed (rpm) 1500 1500 1500 1500 1500 1500 1500 1500

Fig. 2.3 Top: Input speed step response of the drive motor. Bottom: Input torque signal as a result of step speed command

acquired torque signals, linear scaled auto power spectra were computed, the amplitudes at the frequencies of excitation extracted, and the torque transmissibility (output/input torque) computed. For the purposes of this work, all torque transmissibility measurements were taken at a mean operating speed of 1500 rpm. Depending on the test method, a variety of mean and dynamic torques were used to measure the torque transmissibility (Table 2.1). As previously mentioned, it is of interest to acquire the same torque transmissibility plot with reduced testing time. Three different test methods are compared: input torque step, input speed step, and pseudo random.

2.2.2 Speed Step Method The idea behind the speed step method was that a torque impulse is required to achieve a step change in speed. The drive motor is typically operated in a torque control mode, but it also has the capability to operate in a speed control mode. For this test, the absorbing dyno was spinning freely (no control on the output shafting), and the drive motor commanded a step speed change. Since the motor wasn’t able to track a step perfectly, a perfect torque impulse wasn’t achieved (Fig. 2.3). However, with enough bandwidth in the frequency domain, it may be possible to measure the torque transmissibility. To compute the torque transmissibility, the H1 frequency response estimation was used.

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2.2.3 Torque Step Method A signal that satisfies all Dirichlet conditions for completing the Fourier transform (FFT) is a totally observed transient. For this reason, a step command in torque is another excitation signal proposed to evaluate the torque transmissibility of the TCC (Fig. 2.4). In this test, the drive motor was commanded a step change while the absorbing dyno was holding to a constant output speed of 1500 rpm. Similar to the speed step method, the H1 frequency response estimation was used for the torque transmissibility.

2.2.4 Pseudo Random Excitation Pseudo random excitation is a form of broadband excitation in which every discrete frequency in a frequency range is excited at the same amplitude, but the phase of each frequency is random [7]. This is different from a pure random excitation, which is defined as random amplitude and phase. This method of excitation is similar to the current tri-tone method. The only differences are the number of tones and randomized phase. Since these types of excitation haven’t been requested of the drive motor before, the bandwidth of frequency content in the excitation signal will be increased incrementally (6, 10, 22, 44, and 88 tones). The pseudo random test was done two different ways. The first was to excite with sine tones of sequential frequencies i.e. the frequencies were next to each other on the frequency axis. For example, for 10 tones, frequencies between 0.2 and 2 Hz were excited in 0.2 Hz increments. The second way spaced the frequency of each tone far apart from each other. With the spaced version, only 6 and 10 tones were input per measurement because above 10 tones, the frequencies would be too close together. When tones were spaced apart, the test could be designed such that only one frequency was exciting about the resonance per measurement. When several tones excite the resonance at once, the response amplitudes become large.

2.3 Results The baseline for comparing the different excitation methods was the tri-tone method (Fig. 2.5). This method has been used previously, and has yielded clean results that are useful for developing TCC models.

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Fig. 2.5 Torque transmissibility and coherence acquired with the tri tone method

2.3.1 Speed Step Method The first alternative test method used was the speed step method. First a step speed change of 100 rpm (1500–1600 rpm) was executed, and then a step speed change of 200 rpm (1400–1600 rpm) was tried. The initial tests used 10 averages to compute torque transmissibility, but when that yielded a noisy result, 20 and 50 averages were tried. The torque transmissibility results of the speed step method are noisy compared to the tri-tone method (Fig. 2.6). The TCC resonance appears in the data, but the coherence of all speed step tests is only accurate to about 2 Hz. The step speed method gives a poor characterization of the TCC resonance.

2.3.2 Torque Step Method Next, the torque step method was tried. Two variations of the torque step were investigated. The first was a 50 Nm step increase (from 10 to 60 Nm), and second a 100 Nm decrease (from 150 to 50 Nm). The step down had a larger amplitude torque change as it required less effort from the drive motor. In the torque step method (Fig. 2.7), the torque transmissibility was well defined right up to the resonance. The coherence was also at 1 up until 12 Hz for the step down, while the coherence for the step increase wasn’t as high in the 0–12 Hz range. There also was good agreement between the step torque method and the tri-tone method in the 0–10 Hz range, however, a difference in natural frequency for the 50 Nm step, 100 Nm step, and tri-tone methods were attributed to non-linear behavior of the TCC. The torque step method gave a better characterization of the TCC than the speed step method but also became noisy beyond the TCC resonance.

2.3.3 Pseudo Random Method The first pseudo random tests that were executed were the sequential tone tests (Fig. 2.8). These tests achieved good agreement with the tri-tone method in most of the frequency range tested. The main difference existed about the natural

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frequency of the TCC. The amplitude of the torque transmissibility increased at the resonance as the number of tones was increased, and the resonance itself shifted up in frequency. This would alter the predicted TCC stiffness and damping parameters, but gives insight into the non-linear behavior of this particular TCC. This TCC contains a Belleville spring which introduces a large amount of hysteresis, and the non-linear effect dominate the torque transmissibility at the resonance. The pseudo random excitation method provided better coherence than the previous step input methods, but as the number of tones increased to 22 and beyond, the coherence suffered. For all numbers of tones, the coherence was close to one for frequencies below 30 Hz. For 6 and 10 tones, the coherence was near one for the whole frequency range (0–100 Hz). The second version of the pseudo random method yielded the best comparison to the tri-tone test method (Fig. 2.9). The coherence was good for the whole frequency range, and the only real obvious differences were again about the TCC resonance. When comparing the sequential pseudo random tests versus the spaced pseudo random tests, it was apparent that the spaced tests matched the tri-tone test better in the 20–100 Hz range. As already discussed, the TCC that was tested has a high amount of hysteresis, and its torque transmissibility varies due to input amplitudes and frequency content as well. While the 10 tone spaced test had a big difference about the resonance, the 6 tone spaced test results were closer to the baseline tri-tone test. Looking closer at the shape of the peak reveals the 6 tone test peak was more ragged than the tri-tone, and therefore didn’t lend itself as useful as the tri-tone test for parameter estimation. However, these new tests at different loading cases will be useful for estimating the hysteresis behavior in the TCC.

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Fig. 2.7 Torque step method vs. tri-tone method. Torque transmissibility and coherence

Fig. 2.8 Pseudo random vs tri-tone methods. Sequential frequencies of excitation. Left two plots show the whole frequency range. Right two plots zoom in on 0–30 Hz

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Fig. 2.9 Pseudo random vs. tri-tone methods. Excitation frequencies spaced apart from each other. Left two plots show the whole frequency range. Right two plots zoom in on 0–30 Hz Table 2.2 Total measurement time of each test method Test method Tri tone 5avg Speed step Torque step 10 averages

Pseudo random

Number of averages 5 10 20 50 10 5 5 5 5 5

Tones per measurement 3 NA NA NA NA 6 10 22 44 88

Total measurement time (seconds/minutes) 1350/22.5 270/4.5 498.5/8.31 1350/22.5 230/3.83 600/10 450/7.5 180/3 90/2 45/0.75

2.4 Discussion Coming back to the motivation behind this work, did any of these test methods reduce testing time while still adequately representing the TCC resonance? Table 2.2 demonstrates the total measurement time for the tri-tone method was 22.5 min and 45 s for the pseudo random (88 tones) method. Thus, from a time savings perspective alone the pseudo random method was best. Validity of the pseudo random method only extended to 10 tones to achieve good coherence. The test time for pseudo random with 10 tones input per measurement was 1/3 (450 s/1350 s) of the tri-tone test time. From this testing it was clear that the pseudo random method better matches the tri-tone torque transmissibility than the step input methods. The one drawback with comparing the tri-tone method and pseudo random method was that the test

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article demonstrated non-linear behavior with respect to torque. When characterizing a non-linear system, it is important to acquire data at different loading cases to fully understand the non-linearity. Even though the tri-tone and pseudo random tests produce differences in torque transmissibility results, both methods gave a useful answer from a TCC modelling standpoint.

2.5 Conclusion Historically, a tri-tone excitation scheme was used to excite a TCC. Three alternative torsional excitation methods (speed step, torque step, pseudo random) were tested to determine if testing time could be reduced while maintaining valid results. However, the TCC was a non-linear system, so the new test methods were not able to replicate the tri-tone test results. Of the three, the pseudo random method produced nearest to valid results. The pseudo random method using 10 spaced tones per measurement saved the most time while maintaining good coherence and best matches the tri-tone test results. To further this work, it would be beneficial to try the pseudo random method with more tonal resolution (4–10 tones). Another type of excitation not yet experimented with and possible with the drivetrain dynamometer would be a torque pulse. A torque pulse might excite a larger frequency range than the step input methods, which could make it useful for characterizing the TCC.

References 1. Aurora-Smith, A., The Simulation and Experimental Characterisation of the Torque Converter Damper System (2017) 2. Kim, G.-W., Shin, S.-C.: Research on the torque transmissibility of the passive torsional vibration isolator in an automotive clutch damper. Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 229(13), 1840–1847 (2015) 3. Ngoc, M.V., Shin, S.C., Kim, G.W.: Comparative study on non-traditional torsional vibration isolators for automotive clutch dampers. Noise Control Eng. J. 66(6), 541–550 (2018) 4. Krak, M., Dreyer, J., Singh, R.: Development of a non-linear clutch damper experiment exhibiting transient dynamics. SAE Int. 8(2), 754–761 (2015) 5. Gi-Woo, K., Jung-June, K.: Non-Traditional Torsional Isolators for Transmission Clutch Damper. FISITA (UK) Limited: Stansted, Busan (2016) 6. Mordorski, E.R.: Development of a dynamic torsional actuator for torque converter clutch characterization. Michigan Technological University (2018) 7. Phillips, A.W., Zucker, A.T., Allemang, R.J.: A comparison of MIMO-FRF excitation/averaging techniques on heavily and lightly damped structures. In: Proceedings of the 17th International Modal Analysis Conference, p. 1395 (1999) Luke Jurmu is a PhD candidate in the Mechanical Engineering department at Michigan Tech.

Chapter 3

Frequency-Domain Triangulation of Spatial Harmonic Motion for Single-Camera Operating Deflection Shape Measurement Domen Gorjup, Janko Slaviˇc, and Miha Boltežar

Abstract Optical vibration measurement methods have, due to their non-contacting nature and high spatial resolution, steadily been gaining in adoption with the developments in high-speed camera technology in recent years. In high-frequency applications, however, where observed displacements of the vibrating structure are particularly small in amplitude, the relatively high noise-floor of high-speed camera measurements still presents a major limitation. Taking into account the properties of the vibration response of linear mechanical systems, the negative effects of noise on the displacement measurement uncertainty can be mitigated in the frequency domain. In this work, a novel method for frequency-domain triangulation of displacement data, acquired by a multiview high-speed camera system, is presented. For linear, time-invariant mechanical systems under stationary excitation, frequency-domain triangulation enables us to perform spatial vibration response measurements using only a single, moving high-speed camera. The measurement field-of-view can thus be extended for objects with complex 3D geometry, utilizing a relatively simple and cost-effective imaging system. The resulting spatial displacement spectra are used in full-field deflection shape identification. Keywords Frequency domain · Triangulation · High-speed camera · Vibration measurement · Multiview geometry

3.1 Introduction High-speed camera systems are a well-established alternative to traditional vibration measurement techniques. With the introduction of 3D digital image correlation technique some of the traditional limitations of 2D imaging systems are eliminated, but the limited field of view of stereo camera pairs remain problematic in some applications [1]. In recent years, various methods have been proposed that extend the use of digital cameras for displacement measurements to objects of arbitrary shapes and dimensions. These methods employ the principles of multiview geometry and triangulation [2] to extract spatial information from simultaneously acquired image sequences of the observed mechanical process. Data acquired by a moving stereo-pair of high-speed cameras can also be used to extend the field of view of a 3D DIC measurement in a process called surface stitching [1]. Distortions in the optical systems and synchronization errors of multiple cameras can negatively affect multi-camera measurements [3]. Single-camera multiview measurement methods have gained in popularity in recent years, due to their lower complexity and cost-effectiveness. Such systems usually require additional light-splitting elements such as mirror adapters to project multiple views of the observed object on a single image sensor [3, 4].

3.2 Frequency Domain Triangulation for Single-Camera 3D ODS Measurement A method of multiview deflection identification using a single monochrome camera and spatial triangulation in the frequency domain is proposed [5]. Using a single, moving high-speed camera, multiple image sequences of a vibrating linear, timeinvariant mechanical structure under stationary broadband excitation are acquired from various viewpoints. Spatial operating deflection shapes X(ω) of a linear, time-invariant structure can be reconstructed by frequency domain triangulation of image-

D. Gorjup () · J. Slaviˇc · M. Boltežar Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_3

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based displacement data, in contrast to the well-established 3D DIC method, where triangulation is performed on timesynchronized multiview measurements.

3.3 Multiview Imaging A single view in a multiview measurement setup is defined by a transformation matrix P, projecting the coordinates of a point in space U at a particular point in time t into an image of the point u [2]: point in space U at a particular point in time t into an image of the point u [2]: u(t) =

1 P U (t) w

(3.1)

P = K [R| t] is a 3 × 4 projective transform matrix, describing the optical properties, position and orientation in space of a particular viewpoint, and w is a perspective scaling factor, w = p3 U, where p3 denotes the third row of the camera matrix P. The camera projection is therefore a non-linear operation in the Euclidean coordinate system [2]. By identifying the position of a single point in space U in multiple images, we arrive to a system of equations: u1 = P1 U u2 = P2 U

(3.2)

where u1 and u2 are the two views of the point U in space. Additional views add more equations to the already overdetermined system (3.2), which can be solved in a least-squares sense for the unknown coordinates U in a process called triangulation [6].

3.4 Frequency-Domain Triangulation of Small Harmonic Motion To reconstruct spatial harmonic motion spectra by triangulating image information, frequency-domain images of the motion must be constructed for each camera view. It can been shown that the perspective scaling factor (3.1) can be considered constant for small harmonic motion [5], and frequency-domain images of the observed spatial motion can be constructed: u (ω) = Δu (ω) + uREF =

1  P ΔU (ω) + U REF w

(3.3)

where Δu(ω) denotes the identified image-based displacements, transformed into the frequency domain, uREF the initial position of the observed point in an image, ΔU(ω) the 3D motion amplitude spectra, and UREF the initial spatial position of the observed point. From the results of multiview triangulation (3.2) of u(ω) for each observed point on the surface of the structure, the spatial ODS, X(ω), can therefore be obtained, as illustrated in Fig. 3.1.

3.5 Experiment A concave steel object was placed on a LDS V555 electrodynamic shaker, approximately 30 angular degrees off the shaker’s vertical axis, and excited with a broadband stationary profile. A high contrast speckle pattern was applied to the object’s three visible faces, and the scene was illuminated using two 50 kLumen LED reflectors. Rotating the specimen by approximately 60 degrees around the vertical axis between each consecutive video acquisition period, six video sequences were acquired using a stationary Photron Fastcam SA-Z high-speed camera. The multiview system was calibrated using the Perspective N-point algorithm [2] in post-processing, and approximately 2700 points on the surface of the specimen were located and analyzed in each image sequence using the Simplified gradientbased optical flow method [7]. The obtained image-plane displacements Δu(t) were transformed into frequency domain using DFT. Multiple views of the displaced points were used in the triangulation step to obtain the ODS X(ω) using the proposed frequency-domain triangulation procedure.

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Fig. 3.1 Frequency domain ODS triangulation procedure

Fig. 3.2 Examples of spatial ODS, identified using frequency-domain triangulation

3.6 Results and Conclusion Four examples of the obtained spatial ODS are shown in Fig. 3.2. A single high-speed camera method for 3D ODS identification is introduced. Spatial operating deflection shapes of a vibrating 3D object were identified in the presented experiment. The proposed method utilizes frequency-domain triangulation of image-based displacement in a stationary mechanical process to facilitate full-field single-camera multiview measurements without the need for precise time-synchronization of multiple video sequences or any additional specialized equipment.

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References 1. Patil, K., Srivastava, V., Baqersad, J.: A multi-view optical technique to obtain mode shapes of structures. Measurement. 122, 358–367 (2018) 2. Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, New York (2003) 3. Yu, L., Pan, B.: Single-camera high-speed stereo-digital image correlation for full-field vibration measurement. Mech. Syst. Signal Process. 94, 374–383 (2017) 4. Durand-Texte, T., Simonetto, E., Durand, S., Melon, M., Moulet, M.-H.: Vibration measurement using a pseudo-stereo system, target tracking and vision methods. Mech. Syst. Signal Process. 118, 30–40 (2019) 5. Gorjup, D., Slaviˇc, J., Boltežar, M.: Frequency domain triangulation for full-field 3D operating-deflection-shape identification. Mech. Syst. Signal Process. 133, 106287 (2019) 6. Hartley, R.I., Sturm, P.: Triangulation. Comput. Vis. Image Underst. 68(2), 146–157 (1997) 7. Javh, J., Slaviˇc, J., Boltežar, M.: The subpixel resolution of optical-flow-based modal analysis. Mech. Syst. Signal Process. 88, 89–99 (2017) Domen Gorjup Born in 1992 in Celje, Slovenia. A final-year PhD student and assistant at Laboratory for Dynamics of Machines and Structures (LADISK), Faculty of Mechanical Engineering, University of Ljubljana. Research work mainly focused on image-based vibration measurement and experimental modal analysis. Janko Slaviˇc is a Full Professor at the Faculty of Mechanical Engineering at the Univ of Ljubljana. He is currently supervising 8 PhD students. His research focuses are high-speed image processing structural dynamics, high cycle vibration fatigue, and most recently self-aware smart structures. Miha Boltežar is a Professor of Mechanics and head of Laboratory for dynamics of machines and structures at University of Ljubljana, Faculty of Mechanical Engineering.

Chapter 4

Investigating the Feasibility of Laser-Doppler Vibrometry for Vibrational Analysis of Living Mammalian Cells Sascha Schwarz, Stefanie Kiderlen, Robert Moerl, Stefanie Sudhop, Hauke Clausen-Schaumann, and Daniel J. Rixen

Abstract The mechanical properties of cells are key indicators of the cell’s developmental state, their viability or can be a biomarker for pathogenesis such as cancer. State-of-the-art methods to analyze the biomechanical properties of cells are often contact-based like the atomic force microscopy. Laser-Doppler vibrometry (LDV), which optically determines the characteristic vibrational spectrum of investigated objects could be an alternative, non-invasive and non-destructive method to determine mechanical properties of cells and other biological samples. LDV is well established in the field of mechanical engineering, but only rarely used in other fields. Here we investigated whether LDV can be used to characterize and discriminate the mechanical properties of living mammalian cells. Using our current setup, which uses an LDV with a wavelength of 532 nm, we were not able to determine the cell’s mechanical properties, because of the low reflectivity of the low reflectivity of the cell in aqueous media at this wavelength. Nevertheless, by using confocal LDV setups or by staining the plasma membrane and using suitable bandpass filters in the optical path of the LDV, measuring the mechanical properties of living cells by LDV might be possible in future attempts. Keywords Laser-Doppler Vibrometry · Tissue engineering · Vibrational analysis · Biomechanics · Fibroblasts

4.1 Introduction The mechanical properties of cells vary with their physiological function in the human body [1, 2]. Injury or pathological progression is often accompanied by a change in biomechanical properties of cells and tissues [3]. In the research of cancer development and of metastases formation, cell stiffness was recently identified as a suitable biomarker to discriminate healthy from abnormal cells [4–7]. Current methods for assessing mechanical properties of cells are for example flow-based microfluidics [8, 9] and contact-based methods like atomic force microscopy (AFM) [10–14]. Even though AFM measurements allow very sensitive probing of the sample’s topography and mechanical properties, the cell must be in physical contact with the AFM tip, which may lead to mechanical damage and also poses the risk of contamination. In addition, this method is time-consuming and can only be used for samples exhibiting height variations below approximately 5 μm in z-direction and the scan area is typically limited to one or a few hundred micrometers in x and y-direction. Microfluidic-based approaches, on the other hand, allow high throughput measurements, but only on suspended cells. Adherent cells or multicellular tissue components can not be examined with this method [15].

S. Schwarz () Center for Applied Tissue Engineering and Regenerative Medicine – CANTER, Munich University of Applied Sciences, Munich, Germany Department of Mechanical Engineering, Technical University of Munich, Garching, Germany e-mail: [email protected] S. Kiderlen · S. Sudhop · H. Clausen-Schaumann Center for Applied Tissue Engineering and Regenerative Medicine – CANTER, Munich University of Applied Sciences, Munich, Germany e-mail: [email protected]; [email protected]; [email protected] R. Moerl Polytec GmbH, Waldbronn, Germany e-mail: [email protected] D. J. Rixen Department of Mechanical Engineering, Technical University of Munich, Garching, Germany e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_4

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As a laser-based, non-destructive and high-resolution measurement technique, Laser-Doppler vibrometry (LDV) is an established method for a broad range of technical applications [16, 17]. Contactless non-destructive measurements are particularly useful for investigating soft, or fragile samples, such as biomaterials or tissues. A further advantage of is the scalability of the experiments regarding high-throughput experiments and the type of sample that can be measured: singlesuspended or adherent cells up to large tissue samples. Regarding the functionality of biological systems in the area of biomedical engineering, LDV may provide new insights and help to develop new methods [18–20]. To our knowledge, this is the first study investigating the feasibility of using LDV for vibrational analysis of living cells. In this study, we examined, whether it is possible to use a LDV, here with a wavelength of 532 nm, for measuring the mechanical vibrations of living adherent mammalian cells, while they are completely submerged in culture medium.

4.2 Materials and Methods 4.2.1 3D-Printing of Sample Holder For attaching the Petri dish directly on the platform of the LDV, a sample holder was custom-designed and 3D-printed using an Ultimaker 2+ (Ultimaker, Utrecht, Netherlands) fused deposition modelling 3D-printer and polylactide filament. The design was adapted to a standard Petri dish with an outer diameter of 35 mm.

4.2.2 Cell Culture Murine NIH3T3 fibroblast wildtype (WT) cells were cultured in Dulbecco’s Modified Eagle Medium (F0445, Biochrom, Berlin, Germany) supplemented with 10% fetal calf serum (S0115, Biochrom GmbH, Berlin, Germany), 1% GlutaMax (5050–038, Gibco/Thermo Fisher Scientific, Waltham, USA) and 1% Penicillin/Streptomycin (A2212, Biochrom, Berlin, Germany) at 37 ◦ C and 10% CO2 in a humidified cell culture incubator. For LDV experiments, the cells were seeded in 35 mm cell culture dishes (Petri dishes) and cultivated for 2 days. Experiments were carried out with 70–80% confluence, meaning that around 20% of the Petri dish’s surface was not overgrown by cells and available for measurements.

4.2.3 Experimental Procedure and Setup Experiments were carried out using a micro system analyzer (MSA-600, Polytec GmbH, Waldbronn, Germany) and a standard loudspeaker for the excitation. As an excitation signal, a frequency sweep from 0 to 800 Hz (periodic chirp) was applied using the built-in signal generator. To avoid reflections from the air-water interface, we used a 25x water immersion objective (420852-9871, Carl Zeiss Microimaging GmbH, Jena, Germany) with a numerical aperture of 0.8. Focusing was done manually using the fine drive of the microscope. A region was identified, in which a single cell was clearly recognizable and where a free Petri dish surface area was located nearby, to be used as a reference. Using the scanning unit both regions, cell surface and Petri dish could be measured in one test-run using the measurement laser. As long as the samples were not used for measurements, they were stored in an incubator unit at 37 ◦ C and 10% CO2 concentration. Because the LDVmicroscope setup had no incubation unit, the measurement time was limited to a maximum of up to 5 min (Fig. 4.1).

4.3 Results After immersing the microscope objective into the cell culture medium and focusing on the cells, individual cells could clearly be identified in the live-video signal of the camera and a grid of LDV measurement points could be positioned with measurement points on the cell and on the substrate (Petri dish) surface (see Fig. 4.2). Figure 4.3 shows the velocity amplitude determined by LDV on a measurement point on the substrate (green line) and on the cell (dashed blue line). As can be seen from the signal on the substrate’s surface (green line), the excitation with an external loudspeaker renders sufficient input signal, to excite the Petri dish surface, with several resonance peaks appearing

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Fig. 4.1 Experimental setup showing the cell culture dish filled with culture medium and cells positioned directly on the LDV sample stage with a custom made sample holder (see Materials & Methods). A commercial loudspeaker was attached directly to the LDV sample stage and used for vibration excitation. During the LDV measurement, the objective of the microscope was immersed into the cell culture medium

Fig. 4.2 Bright field microscopy image of a murine fibroblast. The measuring grid of the LDV (yellow dots) was placed at the edge of the cell (blue line) in order to measure areas on the bare Petri dish (input) and the cell (output)

between 35 and 520 Hz. However, the vibrational amplitude recorded on the cell covered area (dashed blue line) is virtually indistinguishable from the signal recorded on the pure substrate. If the transmissibility is calculated by dividing the signal recorded at the cell covered area (“Output” in Fig. 4.2) through the signal of the pure Petri dish (“Input” in Fig. 4.2), its value remains close to one between 35 and approximately 520 Hz, and only at low frequencies (520 Hz) range of the spectrum, the transmissibility becomes larger than 1. However, because below 35 and above 520 H a very small velocity amplitude of only a few hundred nanometers per second was observed, both on the cell covered and the uncovered areas, this part of the spectrum most likely contains merely background noise and cannot be used to draw any conclusions regarding the biomechanical properties of the cell. The fact that measurements recorded with other cells with distinctly different mechanical properties showed similar results (data not shown), corroborates the assumption, that the recorded signals do not reflect the biomechanical properties of the cell.

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Fig. 4.3 Amplitude velocity signals of the cell (dashed blue curve, left axis) and the bare Petri dish (green curve, left axis), as well as the resulting transmissibility (orange, right axis)

In addition to LDV measurements on adherent cells, experiments with Latrunculin A (Lat. A) were also carried out to specifically influence the mechanical properties of the cells. Lat. A is an actin cytoskeleton disrupting drug that binds monomeric actin and thus inhibits further polymerization. Therefore, Lat. A modifies the mechanical properties of the cell in a dose-dependent manner within minutes [21, 22]. Again, no significant differences could be observed between treated and untreated cells and the Petri dish (data not shown).

4.4 Conclusion Using our current setup, which uses an LDV laser wavelength of 532 nm, we were not able to determine the cells mechanical properties, most likely, because of the small refractive index difference between the cells (n = 1.34–1.38) [23] and surrounding aqueous medium (n ≈ 1.34) and the resulting low reflectivity of the cell at this wavelength. Due to the much higher refractive index difference between medium and plastic Petri dish, at normal incidence, the intensity of light which is reflected from the substrate surface is much higher, than the intensity of light which is reflected at the plasma membrane of the cell. This may explain the similarity between the signals recorded on cell covered areas and on bare substrate. Nevertheless, by using confocal LDV setups [24] or by staining the plasma membrane and using suitable bandpass filters in the optical path of the LDV, measuring the mechanical properties of living cells by LDV might be possible in future attempts. Acknowledgements The authors acknowledge financial support through the research focus “Herstellung und biophysikalische Charakterisierung drei-dimensionaler Gewebe – CANTER” of the Bavarian State Ministry for Science and Education and the financial support through the “BayWISS – Ressourceneffizienz und Werkstoffe” program.

References 1. Swift, J., Ivanovska, I.L., Buxboim, A., Harada, T., Dingal, P.C.D.P., Pinter, J., Pajerowski, J.D., Spinler, K.R., Shin, J.-W., Tewari, M., Rehfeldt, F., Speicher, D.W., Discher, D.E.: Nuclear lamin-A scales with tissue stiffness and enhances matrix-directed differentiation. Science. 341, 1240104 (2013). https://doi.org/10.1126/science.1240104 2. Jakab, K., Norotte, C., Marga, F., Murphy, K., Vunjak-Novakovic, G., Forgacs, G.: Tissue engineering by self-assembly and bio-printing of living cells. Biofabrication. 2(2), 022001 (2010). https://doi.org/10.1088/1758-5082/2/2/022001 3. Holle, A.W., Young, J.L., Van Vliet, K.J., Kamm, R.D., Discher, D., Janmey, P., Spatz, J.P., Saif, T.: Cell–extracellular matrix Mechanobiology: forceful tools and emerging needs for basic and translational research. Nano Lett. 18, 1–8 (2017). https://doi.org/10.1021/acs.nanolett.7b04982

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4. Weder, G., Hendriks-Balk, M.C., Smajda, R., Rimoldi, D., Liley, M., Heinzelmann, H., Meister, A., Mariotti, A.: Increased plasticity of the stiffness of melanoma cells correlates with their acquisition of metastatic properties. Nanomed. Nanotechnol. Biol. Med. 10, 141–148 (2014). https://doi.org/10.1016/j.nano.2013.07.007 5. Kalli, M., Stylianopoulos, T.: Defining the role of solid stress and matrix stiffness in cancer cell proliferation and metastasis. Front. Oncol. 8, 55 (2018). https://doi.org/10.3389/fonc.2018.00055 6. Rianna, C., Radmacher, M.: Comparison of viscoelastic properties of cancer and normal thyroid cells on different stiffness substrates. Eur. Biophys. J. 46, 309–324 (2017). https://doi.org/10.1007/s00249-016-1168-4 7. Xu, W., Mezencev, R., Kim, B., Wang, L., McDonald, J., Sulchek, T.: Cell stiffness is a biomarker of the metastatic potential of ovarian cancer cells. PLoS One. 7, e46609 (2012). https://doi.org/10.1371/journal.pone.0046609 8. Islam, M., Mezencev, R., McFarland, B., Brink, H., Campbell, B., Tasadduq, B., Waller, E.K., Lam, W., Alexeev, A., Sulchek, T.: Microfluidic cell sorting by stiffness to examine heterogenic responses of cancer cells to chemotherapy. Cell Death Dis. 9, (2018). https://doi.org/10.1038/ s41419-018-0266-x 9. Guck, J., Ananthakrishnan, R., Mahmood, H., Moon, T.J., Cunningham, C.C., Käs, J.: The optical stretcher: a novel laser tool to micromanipulate cells. Biophys. J. 81, 767–784 (2001). https://doi.org/10.1016/S0006-3495(01)75740-2 10. Reuten, R., Patel, T.R., McDougall, M., Rama, N., Nikodemus, D., Gibert, B., Delcros, J.-G., Prein, C., Meier, M., Metzger, S., Zhou, Z., Kaltenberg, J., McKee, K.K., Bald, T., Tüting, T., Zigrino, P., Djonov, V., Bloch, W., Clausen-Schaumann, H., Poschl, E., Yurchenco, P.D., Ehrbar, M., Mehlen, P., Stetefeld, J., Koch, M.: Structural decoding of netrin-4 reveals a regulatory function towards mature basement membranes. Nat. Commun. 7, 13515 (2016). https://doi.org/10.1038/ncomms13515 11. Kim, T.-H., Rowat, A.C., Sloan, E.K.: Neural regulation of cancer: from mechanobiology to inflammation. Clin. Transl. Immunol. 5, e78 (2016). https://doi.org/10.1038/cti.2016.18 12. Docheva, D., Padula, D., Schieker, M., Clausen-Schaumann, H.: Effect of collagen I and fibronectin on the adhesion, elasticity and cytoskeletal organization of prostate cancer cells. Biochem. Biophys. Res. Commun. 402, 361–366 (2010). https://doi.org/10.1016/j.bbrc.2010.10.034 13. Docheva, D., Padula, D., Popov, C., Mutschler, W., Clausen-Schaumann, H., Schieker, M.: Researching into the cellular shape, volume and elasticity of mesenchymal stem cells, osteoblasts and osteosarcoma cells by atomic force microscopy: stem cells. J. Cell. Mol. Med. 12, 537–552 (2008). https://doi.org/10.1111/j.1582-4934.2007.00138.x 14. Viji Babu, P.K., Rianna, C., Mirastschijski, U., Radmacher, M.: Nano-mechanical mapping of interdependent cell and ECM mechanics by AFM force spectroscopy. Sci. Rep. 9, 12317 (2019). https://doi.org/10.1038/s41598-019-48566-7 15. Fregin, B., Czerwinski, F., Biedenweg, D., Girardo, S., Gross, S., Aurich, K., Otto, O.: High-throughput single-cell rheology in complex samples by dynamic real-time deformability cytometry. Nat. Commun. 10, 415 (2019). https://doi.org/10.1038/s41467-019-08370-3 16. Rothberg, S.J., Allen, M.S., Castellini, P., Di Maio, D., Dirckx, J.J.J., Ewins, D.J., Halkon, B.J., Muyshondt, P., Paone, N., Ryan, T., Steger, H., Tomasini, E.P., Vanlanduit, S., Vignola, J.F.: An international review of Laser Doppler Vibrometry: making light work of vibration measurement. Opt. Lasers Eng. 99, 11–22 (2016). https://doi.org/10.1016/j.optlaseng.2016.10.023 17. Castellini, P., Martarelli, M., Tomasini, E.P.: Laser Doppler Vibrometry: development of advanced solutions answering to technology’s needs. Mech. Syst. Signal Process. 20, 1265–1285 (2006). https://doi.org/10.1016/j.ymssp.2005.11.015 18. Rosowski, J.J., Mehta, R.P., Merchant, S.P.: Diagnostic utility of Laser-Doppler Vibrometry in conductive hearing loss with normal tympanic membrane. Otol. Neurotol. 24, 165–175 (2003). https://doi.org/10.1097/00129492-200303000-00008 19. Schuurman, T., Rixen, D.J., Swenne, C.A., Hinnen, J.W.: Feasibility of Laser Doppler Vibrometry as potential diagnostic tool for patients with abdominal aortic aneurysms. J. Biomech. 46, 1113–1120 (2013). https://doi.org/10.1016/j.jbiomech.2013.01.013 20. Conza, N.E., Rixen, D.J., Plomp, S.: Vibration testing of a fresh-frozen human pelvis: the role of the pelvic ligaments. J. Biomech. 40, 1599– 1605 (2007). https://doi.org/10.1016/j.jbiomech.2006.07.001 21. Antonacci, G., Braakman, S.: Biomechanics of subcellular structures by non-invasive Brillouin microscopy. Sci. Rep. 7, 46789 (2017). https:/ /doi.org/10.1038/srep46789 22. Kohler, J., Popov, C., Klotz, B., Alberton, P., Prall, W.C., Haasters, F., Müller-Deubert, S., Ebert, R., Klein-Hitpass, L., Jakob, F., Schieker, M., Docheva, D.: Uncovering the cellular and molecular changes in tendon stem/progenitor cells attributed to tendon aging and degeneration. Aging Cell. 12, 988–999 (2013). https://doi.org/10.1111/acel.12124 23. Kim, G., Lee, M., Youn, S.Y., Lee, E.T., Kwon, D., Shin, J., Lee, S.Y., Lee, Y.S., Park, Y.K.: Measurements of three-dimensional refractive index tomography and membrane deformability of live erythrocytes from Pelophylax nigromaculatus. Sci. Rep. 8, 1–8 (2018). https://doi.org/ 10.1038/s41598-018-25886-8 24. Rembe, C., Dräbenstedt, A.: Laser-scanning confocal vibrometer microscope: theory and experiments. Rev. Sci. Instrum. 77, 1–11 (2006). https://doi.org/10.1063/1.2336103 Sascha Schwarz is a Ph.D. candidate at the Center for Applied Tissue Engineering and Regenerative Medicine (CANTER) at the University of Applied Sciences Munich and the Institute for Applied Mechanics at the Technical University of Munich. Research focus lies on the mechanical characterization of biological materials using laser-Doppler vibrometry and the fabrication of functional tissue equivalents with the help of biofabrication technologies. Stefanie Kiderlen is a Postdoc at the LMU – Soft condensed matter group and former PhD student at the Center for Applied Tissue Engineering and Regenerative Medicine (CANTER) and the Laboratory for Nanoanalytics and Biophysics at the University of Applied Sciences Munich. Research focus lies on the characterization of the mechanical properties of single cells and the extracellular matrix (ECM) using atomic force microscopy (AFM). Robert Moerl is a graduate engineer for mechatronics, works in sales for Polytec GmbH. Stefanie Sudhop is scientist at the Center for Applied Tissue Engineering and Regenerative Medicine (CANTER) at the Munich University of Applied Sciences. She is interested in Tissue Engineering, mechanobiology and cell biology.

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Prof. Hauke Clausen-Schaumann is biophysicist and member of the Center for Applied Tissue Engineering and Regenerative Medicine at Munich University of Applied Sciences. His research focuses on the mechanical characterization of biological systems on the single molecule, single cell and cells and tissue level. Prof. Daniel J. Rixen is chair of the Applied Mechanics group at the Technical University of Munich since 2012. His research interest focus on experimental and numerical methods for structural dynamics and in robotics.

Chapter 5

Visio-Acoustic Data Fusion for Structural Health Monitoring Applications Chad R. Samuelson, Caitrin A. Duffy-Deno, Christopher B. Whitworth, David D. L. Mascareñas, Jeffery D. Tippmann, and Alessandro Cattaneo

Abstract Structural health monitoring has been an expanding discipline due to its potential to decrease maintenance and downtime costs, detect failure early, extend life spans, and fulfill the increased need of safety and security. Acoustic source identification techniques’ have been used for remote structural health monitoring, but the applicability of each technique has been limited by factors ranging from achievable spatial resolution to hardware costs. This paper aims to mitigate current acoustic techniques’ limitations by exploring the possibility of fusing acoustic and video data. This paper focuses on combining microphone acoustic measurements with vibrational information recovered from video-based measurements. Among acoustic methods, acoustic arrays have been used for remotely detecting, localizing and characterizing acoustic sources. Acoustic-array based techniques are limited in their ability to discriminate multiple closely-spaced acoustic sources from far-field acoustic pressure signals. On the contrary, video-based techniques have shown the ability to recover fullfield, high resolution mode shapes, and the associated frequencies and damping ratios with virtually no dependence with the distance from the target. The challenge with video methods, applied to acoustic source identification, is that acoustic sources may occur in the kilohertz range requiring a higher frame per second sampling rate than most low cost cameras. Acoustic measurements provide additional information content that is used to recover the correct frequency content of an acoustically radiating structure from temporally-aliased (sub-Nyquist) video measurements. Experiments are conducted to show how combining acoustic and video data relaxes the hardware requirements for acoustic source detection and localization applications. Keywords Structural health monitoring · Vibro-acoustics · Video processing · Signal aliasing · Sub-Nyquist sampling

5.1 Introduction 5.1.1 Background In this work, we focus on the development of visio-acoustic data fusion techniques to bring together the complementary advantages of acoustics and video-based structural dynamics to enable new structural characterization techniques. The presence of damage on a structure may result in the change of its vibrational characteristics [1]. Many different vibroacoustics methods have been developed to capture a change in the vibrational response of a structure. Microphone array-based techniques have the ability to detect, localize and characterize acoustically radiating structures [2, 3]. The acoustic techniques suffer both economical and technical limitations. Off-the-shelf microphone-arrays and the commercial license of a given

C. R. Samuelson Department of Mechanical Engineering, Brigham Young University, Provo, UT, USA C. A. Duffy-Deno Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA e-mail: [email protected] C. B. Whitworth Department of Mechanical Engineering, The University of Texas at Dallas, Richardson, TX, USA e-mail: [email protected] D. D. L. Mascareñas · J. D. Tippmann · A. Cattaneo () Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_5

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acoustic technique cost several thousand dollars. In addition, the spatial resolution achievable with acoustic techniques depends on the ability to acquire near field information (i.e. evanescent waves) [4]. To obtain high-resolution spatial acoustic characterization, the microphone-array needs to be placed well within 2 wavelengths from the acoustic source [3], with important limitations on the measurement setup. Laser Doppler vibrometers have been shown to capture accurate full-field structural vibration data [5], but they are only available at the cost of ten of thousands of dollars. Complex measurements setups implementing off-axis digital holography and advanced signal post-processing techniques [6] have been developed to image vibration fields. Another method, in order to obtain full-field information, involves attaching many accelerometers on a structure. Direct acceleration measurements cost an unnecessary amount of time and hardware, furthermore they might create unwanted load effects on lightweight and lightly damped structures. In recent years, at the Los Alamos National Laboratory (LANL), research has been undergone to use both high-frame rate [7] and sub-Nyquist [8] video measurements to identify and characterize vibrational sources. The results achieved at LANL added original contributions to the other works available in the literature that proved the possibility to recover high-spatial resolution vibrations [9, 10] and even sound [11] from video measurements. The cost and limitations of these vibro-acoustic methods are the driving motivators for our fusion idea.

5.1.2 Idea We explored the possibility to combine video-based imaging techniques and microphone-data to identify which parts in a vibrating structure are contributing to sound radiation. Although video measurements enable to analyze/extract vibrations from a scene, not all vibrations in a scene are radiating sound waves. For this reason, it is not immediate to determine which vibrations are actually responsible to generate the sound field. In this work, we used the acoustic data collected with one microphone to identify which vibrations recorded in a high-frame-rate video of the scene correspond with the actual acoustic sound being emitted from the acoustic source. When combined, high-frame-rate videos and acoustic data enabled to create a high spatial resolution acoustic map of the scene. Further, we applied our visio-acoustic fusion technique to videos and sounds recorded with a commercial smartphone. We leveraged the information collected by the smartphone’s microphone to produce acoustic maps from aliased videos. Capabilities and limitations of the proposed counter-aliasing technique are analyzed in light of the results obtained using high-frame-rate camera and smartphone’s videos.

5.2 Method 5.2.1 Process Microphone Data Find Acoustic Source Frequencies The proposed method starts by transforming the microphone acoustic data from the temporal domain into the Fourier domain. The frequencies with the largest amplitudes are the most audible frequencies in an acoustic scene. Depending on the number of acoustic sources in the scene, a different number of peak frequencies in the microphone data can be found. In the experimental setups used in this work, no more than 5 different acoustic sources were present. We will move forward with the assumption that we are only detecting the 5 maximum frequencies in our data sets, but again this number can be adjusted depending on the acoustic scene.

Compute Aliased Frequencies Most audible acoustic sources radiate in the lower end of the kHz frequency range. Since most video data is being sampled below the kHz frequency range (less than 1000 frames per second), aliasing will occur when detecting the motion of these vibro-acoustic objects in the scene. To overcome aliasing an expensive high-frame rate video camera can be used, as done in the Blind-identification paper [7], or some research has been done to predict at what frequency an aliased frequency actually is radiating at [8]. A typical low-cost microphone has a minimum sampling rate of 44.1 kHz. We plan to overcome the aliasing affecting sub-Nyquist sampled video data by utilizing the high-sampling rate microphone data.

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This is done by finding the main peaks in the spectrum of the microphone data that correlate to the vibrating objects in the video’s camera scene. Based on the frame rate of the video, the peaks aliased frequency locations can be calculated as follows: A=

micpeaks (f )

(5.1)

fs 2

where fs is the sampling rate of the camera. Micpeaks (f) is the peak frequencies found in Sect. 5.2.1. The integer portion of A, Aint , is the number of times that the peak frequency is folded over to end up in the Nyquist frequency range. The decimal portion of A, Adec , represents the fraction of fs (i.e. where that frequency will show up in the aliased video data). Depending on if the integer portion of A is even or odd, multiply the decimal portion by fs or 1 minus the decimal as shown below: If Aint is even: micalias (f ) = Adec ∗ (

fs ) 2

(5.2)

If Aint is odd: micalias (f ) = (1 − Adec ) ∗ (

fs ) 2

(5.3)

where micalias (f ) is the new calculated aliased frequency.1

5.2.2 Process Video Data Obtain Motion from Video Vibrational motion is obtained based on a similar technique discussed in the Blind Identification paper [7]. The complex steerable pyramid [12] is applied to each frame of the video to filter it. Building a steerable pyramid and then collapsing it yields I (x + δ(x, t)) =

∞  ω=−∞

Rω (x, t) =

∞ 

ρω (x, t)ej 2π ω0 (x+δ(x,t))

(5.4)

ω=−∞

where x is the pixel location, t is the frame number (time index), δ contains the displacement of the structure, ω0 is the spatial frequency, and Rω (x, t) is the subband representation on the spatial scale ω defined as Rω (x, t) = ρω (x, t)ej 2π ω0 (x+δ(x,t))

(5.5)

with the local amplitude ρω (x, t) (corresponding to edge strength) and the local phase ψ(x, t) = 2π ω0 (x + δ(x, t)) = 2π ω0 x + 2π ω0 δ(x, t) that encodes the temporal vibration motion of the structure. The specific subband, ω, is then chosen in the direction of motion most relevant for the specific recorded scene. The temporal mean 2π ω0 x is subtracted from ψ(x, t) to obtain only the displacement of the object from the reference phase, 2π ω0 δ(x, t). Lastly, combining δ(x, t) for every frame builds a 3-D Phase Matrix containing the vibrational motion of the structure.

Filter Video Data Based on Aliased Frequencies from Microphone Data In order to properly segment the vibro-acoustic scene, the 3-D Phase Matrix must be filtered at each aliased frequency found using the procedure described in Sect. 5.2.1. The method proposed in our work applies a temporal bandpass filter [10] to the

1 Note:

The process is repeated for all the main frequency peaks identified in the spectrum.

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3-D Phase Matrix at each found aliased frequency. Essentially a 4-D matrix is made where the first and second dimensions describe the pixel location, x, the third dimension contains all frames, and the fourth dimension index refers to the frequency filtered. In the case of finding 5 frequency peaks in the spectrum of the microphone data, the size of the fourth dimension is 5.

5.2.3 Segment Video Data Use Level Set Method To segment counter-aliased video data, the level set method [13] can be employed to locate which pixels in the video have a certain power value. Power values can be visualized via a 3-D surface plot containing the locations of the pixels of the recorded scene on the 2-D plane and the vertical axis being the values of power for a given frequency band. To implement the level set method, the temporally filtered Phase Matrix is used to find the energy spectral density [14] for each pixel via S(ω) = |Y (ω)|2

(5.6)

where ω represents frequency in radians per interval of sampling and Y is the amplitude of one pixel from the temporally filtered Phase Matrix in the Fourier domain. Parseval’s Theorem [14], as shown below ∞ 

1 |y(t)| = 2π t=−∞ 2



π −π

S(ω)dω

(5.7)

where y(t) is the signal in the time domain, can be applied to each pixel’s displacement to find their average power across a frequency band. These power values would then be plotted in 3-D to show the magnitude of power with peaks indicating where the greatest vibration is occurring in the recorded scene for a specific frequency range.

Apply Segmentation Based on Power Values The generated visio acoustic power plots can then be made into 2-D images/heat maps with the colors corresponding to the varying values of power in comparison to the lowest and maximum power values that are represented by blue and yellow colors, respectively. Segmentation of the peaks in the power data can then be applied based on these colors. This process includes using the k-means clustering method to find pixels that are similar in color to form a “mask”. Edge detection is used on the mask to create an outline of the non-blue color regions. These outlines are overlapped with the power heat map to isolate the peak power areas which is then overlapped with an image of the recorded scene from the video to completely visually represent the results.

5.3 Testing Performed There were two different experimental set-ups in testing the visio-acoustic technique proposed in this paper. Our first experiment involved multiple tuning forks (i.e. 128, 512, and 2048 Hz) vibrating at the same time recorded by the hi-speed camera and the smartphone, but in different tests. The second experiment had three tests which were recordings made by both the hi-speed camera (Edgertronic SC2+) and the smartphone (Samsung S9) at the same time of a vibrating 512 Hz tuning fork.

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5.3.1 Experimental Set-Ups for the Recordings of Multiple Tuning Forks Experimental Set-up for High-Frame Rate Camera The first set-up was for using the hi-frame rate camera and a prepolarized, condenser microphone PCB 130A23 (see Fig. 5.1). The camera was spaced about 1.8 m from the tuning forks. The microphone was spaced within the near field, 2λ (λ represents acoustic wavelength), of the 2048 Hz fork, because this fork was the highest frequency fork and thus had the smallest wavelength (smaller near field). It follows that the microphone was within the near field of all tuning forks. To avoid aliasing with the initial high-frame rate video, the video was sampled at 4940 frames per second (FPS). This avoided aliasing because double 2048 Hz (frequency of the highest frequency tuning fork) was still less than 4940 Hz. This video data was processed by the above method just skipping Sect. 5.2.1 and changing Sect. 5.2.2 to filtering based on original found frequencies from the microphone data. Results are shown in Fig. 5.6. The 4940 FPS video data was then down sampled by a down sampling factor of 6 reducing the frames per second from 4940 to 824 FPS. The downsampling caused the 512 and 2048 Hz acoustic information to be aliased mimicking a low-frame rate video. The above method was then implemented on the down sampled video data to produce the results shown in Figs. 5.7 and 5.8.

Experimental Set-Up for Smartphone (Low-Frame Rate Camera and High Sampling Rate Microphone) The second set-up was designed for using a low-frame rate smartphone camera and the smartphone’s accompanying hisampling rate microphone (see Figs. 5.2 and 5.3). Only two (128 and 512 Hz) of the three tuning forks were recorded in this experiment because we found from the first experiment that the 2048 Hz tuning fork wasn’t vibrating enough for the camera to recognize its motion (more explained in the Results section). Again, when recording the tuning fork data with the microphone, the smartphone was placed in the near field of the 512 Hz tuning fork (the highest frequency tuning fork) (see Fig. 5.3). The visio-acoustic technique proposed in this work was then implemented on the smartphone data. The smartphone experiment was repeated at 60, 120, 240 FPS. Results are shown in Figs. 5.9, 5.10, and 5.11.

Fig. 5.1 Experimental set-up for hi-frame rate camera. 4940 FPS used and down sampled to 823 FPS. Edgertronic SC2+ camera was used along with center microphone in microphone array. Record data based on vibration of 3 tuning forks with resonant frequencies (128 Hz – left, 512 Hz – center, 2048 Hz – right)

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Fig. 5.2 Video experimental set-up for smartphone (Low-frame rate camera) with two tuning forks (512 Hz – left, 128 Hz – right)

Fig. 5.3 Smartphone microphone experimental set-up with two tuning forks (512 Hz – left, 128 Hz – right)

5.3.2 Experimental Set-Up for Direct Comparison of Simultaneous Recordings A set of tests were performed with both the smartphone and the hi-speed camera recording a vibrating 512 Hz tuning fork with the settings of each camera being varied (see Fig. 5.4). The distances that the cameras were placed from the tuning fork were kept constant for all tests and was found by matching the areas covered in the physical scene for each pixel for each camera. The number of pixels in the frame for the hi-speed camera were maximized, having a horizontal count of 1280 and a vertical count of 864 pixels. The smartphone was set to have a resolution of 4k where the number of horizontal pixels is 2160 and the number of vertical pixels is 3840. From these calculations, it was found that with a fixed distance for the smartphone of 525 mm, the distance that the hispeed camera should be and was placed was 624 mm. These distances were maintained the same for all the tests that were performed. External lighting was used and was focused on the tuning fork. The tuning fork was hit with the rubber end of a hammer so as to decrease the magnitude of other modal frequencies besides 512 Hz that radiate. For recording the sound, both the

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Fig. 5.4 Experimental set-up for the recording of a 512 Hz tuning fork with both cameras recording simultaneously

Fig. 5.5 Set-up of simultaneously recording microphones

smartphone microphone and the PCB microphone corresponding to the hi-speed camera were placed close to the tuning fork to assure that they were in the near field (see Fig. 5.5).

Experimental Set-Up for First Test Performed For the first test, both cameras had frame rates of 60 fps and shutter speeds of 1/24,000 s. The smartphone had an ISO value of 500 and the hi-speed camera was set to an ISO value of 2500.

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Experimental Set-Up for Second Test Performed The second test had all the same parameters as the first test except that the hi-speed camera had a frame rate of 2400 fps. The purpose of setting the frame rate on the hi-speed camera to 2400 fps was to compare segmentation results with the smartphone which was set at 60 fps. To process the video data of the hi-speed camera, the data was downsampled by a factor of 40 to equal the sampling rate of the smartphone.

Experimental Set-Up for Third Test Performed The third test that was performed had both cameras’ frame rates set to 60 fps and their shutter speeds set to 1/24,000 s while the ISO value on the smartphone was set to its maximum value of 800 and the hi-speed camera’s ISO value was set to 10000. This was to see whether varying the amount of light that is captured while filming affects the motion detection by each pixel.

5.4 Results 5.4.1 Experimental Results for the Recordings of Multiple Vibrating Tuning Forks Experimental Results for High-Frame Rate Camera We were able to obtain segmentation results by leveraging the data obtained from the single microphone to identify the location and intensities of vibrations contained in the scene (see Fig. 5.6). For the 4940 frames-per-second video, we located peaks in vibrations at 128 and 512 Hz in each power spectrum that overlapped the tuning forks when segmented. There is a noticeable lack of peaks in the power spectrum for 2048 Hz and thus segmentation was not possible for that frequency (Fig. 5.6). Similarly, for the downsampled and counter-aliased video, we were able to obtain similar results in that segmentation occurred as predicted for 128 and 512 Hz but not for 2048 Hz (see Fig. 5.8).

Experimental Results for Smartphone, Low-Frame Rate Camera and High Sampling Rate Microphone When using the smartphone at 60 frames per second, we were able to segment at 128 Hz. However, segmenting did not work for the 512 Hz (see Fig. 5.9). We believe this may be due to the minimal amount of movement captured by the video which could be caused by the smartphone camera possibly being set to a low shutter speed and/or resolution. The video taken at 120 frames per second (Fig. 5.10) produced similar results as the 60 frames per second, however there was more noise and less noticeable peaks in the power spectrum. We believe the unsatisfactory outcome obtained at 120 frames per second could have been a result of not hitting the 512 Hz tuning fork with enough force. Noise appears at 60 and 120 frames per

Fig. 5.6 4940 FPS hi-frame rate camera segmentation results (128 Hz – left, 512 Hz – center, 2048 Hz – right)

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Fig. 5.7 7 peaks found in microphone data and then aliased frequencies computed based on 824 FPS down sample camera data. Peaks 1, 7, and 4 correspond to 128, 512, and 2048 Hz forks

Fig. 5.8 824 FPS down sampled hi-frame rate camera segmentation results (128 Hz – left, 512 Hz – center, 2048 Hz – right)

second for the 128 Hz segmentation image due to the corresponding aliased frequency being between 7 and 8 Hz. This due to noise generally being more prevalent at lower frequencies. The 240 Hz filter was able to give a more detailed segmentation of the 128 Hz tuning fork and displayed less noise. The better results at 240 frames per second (Fig. 5.11) are due to its corresponding aliased frequency being at 112.2 Hz where typically less noise occurs at higher frequencies.

5.4.2 Experimental Results for Direct Comparison Experiments Test 1 The results show that segmentation of the peaks in power at 512 Hz occurs for the videos captured by both the smartphone and the hi-speed camera. As seen in Figs. 5.12 and 5.13, the areas that are segmented are not consecutive and are minute in size. Most likely, the scattered segmentation results are due to the low frame rate that the videos were recorded at. The outcome of Test 1 does indicate though that at nearly identical camera settings, the low performing smartphone can produce results that are similar to the hi-speed camera when it records at a low frame rate, in this case 60 fps. The difference in locations of the segmented areas on the tuning fork is most likely due to the fact that the cameras were directed towards the tuning fork at varying angles.

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Fig. 5.9 60 FPS smartphone segmentation results (128 Hz – left, 512 Hz – right)

Fig. 5.10 120 FPS smartphone segmentation results (128 Hz – left, 512 Hz – right)

Test 2 For the results of the second test, segmentation of detected peaks in vibrations at 512 Hz was possible for both cameras, but there is a noticeable difference. The hi-speed camera video has segmentation of distinct, continuous regions of peaks in power values while the smartphone has sparse peaks. These results can be seen in Figs. 5.14 and 5.15, respectively.

Test 3 The results show that for the smartphone, the segmentation at 512 Hz is worse when the video is recorded with the maximum ISO value possible than the segmentation of the video recorded in test two that had an ISO value of 500. The results are shown in Figs. 5.16 and 5.17 for both cameras.

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Fig. 5.11 240 FPS smartphone segmentation results (128 Hz – left, 512 Hz – right) Fig. 5.12 Test 1 smartphone camera segmentation results at 512 Hz (60 FPS, shutter speeds of 1/24,000 s, ISO 500)

5.5 Conclusion This paper explored a new vibro-acoustic structural health monitoring technique that is unsupervised, less costly and uses more accessible equipment than conventional vibro-acoutic techniques. Additionally, a data fusion algorithm was developed to enable high-spatial resolution, localization, characterization, and visualization of acoustic sources. The algorithm combines acoustic measurements with sub-Nyquist video measurements to counter the effects of aliasing in videos. From the results, we can conclude that our counter-aliasing technique was successful in segmenting the predicted areas of the scene at each frequency. This conclusion is based on comparing the segmented images from the original, high speed video using the hi-speed camera to the segmented images from the downsampled video as the 128 and 512 Hz tuning forks were

48 Fig. 5.13 Test 1 hi-speed camera segmentation results at 512 Hz (60 FPS, shutter speeds of 1/24,000 s, ISO 2500)

Fig. 5.14 Test 2 smartphone camera segmentation results at 512 Hz (60 FPS, shutter speeds of 1/24,000 s, ISO 500)

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Fig. 5.15 Test 2 hi-speed camera segmentation results at 512 Hz (2400 FPS downsampled by a factor of 40, shutter speeds of 1/24,000 s, ISO 2500)

Fig. 5.16 Test 3 smartphone camera segmentation results at 512 Hz (60 FPS, shutter speeds of 1/24,000 s, ISO 800)

segmented for both video types. Being able to segment with the smartphone videos shows that a smartphone can indeed be used as a vibro-acoustic diagnostic tool and that a video with frame rates starting as low as 60 frames per second can be used. Multiple conclusions were made for both the high-speed camera and the smartphone which include them both being limited by the degree of motion that is captured and that using a higher frame rate improves spatial resolution. It was also concluded

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Fig. 5.17 Test 3 hi-speed camera segmentation results at 512 Hz (60 FPS, shutter speeds of 1/24,000 s, ISO 10000)

that the created algorithm which generated the segmented images is limited in its segmentation ability for the smartphone as the results for each camera of the second experiment’s test two greatly vary.

5.6 Future Work Narrow band acoustic source signals are signals with only one frequency, while broadband signals are signals where multiple frequencies are embedded into one acoustic source. The counter-aliasing technique proposed in our work was developed for narrow band signals (just finding natural frequencies of tuning forks). Future work will be done in expanding the method to broad band signals. A microphone array can also be used, instead of one microphone, to apply beamforming when recording the microphone data to reduce noise. Suppressing aliasing could also be improved using coded exposure. By randomizing the shutter speed on the camera, coded exposure effectively changes the windowing function applied to the recorded video. Coded exposure has found to reduce blurring in video data [15], and could help produce more detailed results with less noise in the vibro-acoustic segmented scene. Acknowledgments The authors are grateful for the support of Los Alamos National Laboratory and the Los Alamos Dynamics Summer School program. We thank Los Alamos National Laboratory for the fellowship funding provided for doing this research. Wer would also like to acknowledge Yongchao Yang and Wesley Scott for their previous work done in the area of motion identification and image processing here at Los Alamos National Laboratory along with David Mascareñas.

References 1. Farrar, C.R., Doebling, S.W., Nix, D.A.: Vibrationbased structural damage identification. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 359(1778), 131–149 (2001) 2. Cigada, A., Ripamonti, F., Vanali, M.: The delay & sum algorithm applied to microphone array measurements: numerical analysis and experimental validation. Mech. Syst. Signal Process. 21(6), 2645–2664 (2007) 3. Veronesi, W.A., Maynard, J.D.: Nearfield acoustic holography (nah) II. Holographic reconstruction algorithms and computer implementation. J. Acoust. Soc. Am. 81(5), 1307–1322 (1987) 4. Williams, E.G.: Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography. Elsevier Science, San Diego (1999) 5. Nassif, H.H., Gindy, M., Davis, J.: Comparison of laser doppler vibrometer with contact sensors for monitoring bridge deflection and vibration. NDT E Int. 38(3), 213–218 (2005)

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6. Rajput, S.K., Matoba, O., Awatsuji, Y.: Characteristics of vibration frequency measurement based on sound field imaging by digital holography. OSA Contin. 1(1), 200 (2018) 7. Yang, Y., Dorn, C., Mancini, T., Talken, Z., Kenyon, G., Farrar, C., Mascareñas, D.: Blind identification of full-field vibration modes from video measurements with phase-based video motion magnification. Mech. Syst. Signal Process. 85(C), 567–590 (2017) 8. Yang, Y., Dorn, C., Mancini, T., Talken, Z., Nagarajaiah, S., Kenyon, G., Farrar, C., Mascareñas, D.: Blind identification of full-field vibration modes of output-only structures from uniformly-sampled, possibly temporally-aliased (sub-nyquist), video measurements. J. Sound Vib. 390(C), 232–256 (2017) 9. Wu, H.-Y., Rubinstein, M., Shih, E., Guttag, J., Durand, F., Freeman, W.: Eulerian video magnification for revealing subtle changes in the world. ACM Trans. Graph. (TOG) 31(4), 1–8 (2012) 10. Wadhwa, N., Rubinstein, M., Durand, F., Freeman, W.: Phase-based video motion processing. ACM Trans. Graph. (TOG) 32(4), 1–10 (2013) 11. Davis, A., Rubinstein, M., Wadhwa, N., Mysore, G., Durand, F., Freeman, W.: The visual microphone: passive recovery of sound from video. ACM Trans. Graph. (TOG) 33(4), 1–10 (2014) 12. Simoncelli, E.P., Freeman, W.T.: Steerable pyramid: a flexible architecture for multi-scale derivative computation. In: IEEE International Conference on Image Processing, Vol. 3, pp. 444–447. IEEE (1995) 13. Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988) 14. Stoica, P.: Spectral Analysis of Signals. Pearson Prentice Hall, Upper Saddle River (2005) 15. Raskar, R., Agrawal, A., Tumblin, J.: Coded exposure photography: motion deblurring using fluttered shutter. ACM Trans. Graph. (TOG) 25(3), 795–804 (2006) Chad R. Samuelson, Caitrin A. Duffy-Deno, Christopher B. Whitworth developed this work during the 20th Los Alamos Dynamics Summer School at the Los Alamos National Laboratory (LANL). Chad, Katie and Chris’ extra-curricular activities include designing the science module sector of a MARS Rover, programming Arduinos and designing wind turbines. Alessandro Cattaneo, Jeffery D. Tippmann and David D. L. Mascareñas mentored Chad, Katie and Chris throughout the execution of this project.

Chapter 6

Automatic Interpolation for the Animation of Unmeasured Nodes with Differential Geometric Methods Daniel Herfert, Kai Henning, and Jan Heimann

Abstract In the field of structural dynamics, more and more demands are made on a realistic representation of mode and operating deflection shapes. This work deals specifically with the problem of interpolating sensor data for unmeasured points. This is applied to 3D geometries with large vertex sets and few measurement points. The interpolation of very complex 3D geometry objects (e.g. 3d Scans or CAD programs) is very important for the realistic visualization of vibrations in structural dynamics. The presented interpolation method respects the connectivity of the mesh and additionally adds boundary conditions and curvature of the geometry. Thereby we expect more realistic shapes for unmeasured points. The developed method is based on the mathematical framework of discrete differential geometry and provides a simple interface that requires only the 3D geometry and sensor data, resulting in a sparse linear system. The solution of this linear system then can be efficiently split into two parts; factorization and back substitution. Thus, users can interact in realtime with the animated sensor data visualization. This gives the user the ability to explore measured data in a more realistic way. For validation, we tested the interpolation method with measured data of representative 3D geometries. Keywords Interpolation unmeasured nodes · Mode shapes · Differential geometry · Modal analysis · Computer graphics

6.1 Introduction In the field of structural dynamics, more and more demands are made on a realistic representation of mode and operating deflection shapes. This realistic representation is required in order to enable the best possible comparability to simulated results with very high resolution and to give the possibility for a precise analysis of the structure. The improving visualization techniques of 3D animations, especially computer games and films, lead to a higher and higher expectation on the engineering software. A realistic representation can be measured in two different ways. The time-consuming option would be the measurement of many measuring points for a higher resolution. In order to achieve a fast measurement and still having a high measuring resolution, the alternative would be an interpolation of few measuring points. Therefore, the problem of interpolating sensor data on unmeasured points is specifically discussed in this paper. The two main challenges are complex geometries on the one hand and geometries with few measuring points on the other. The first challenge appears increasingly often, for 3D scans and geometries from CAD programs. Particularly 3D scans are more and more used to create geometries. To provide realistic visualization the presented interpolation method respects the connectivity of the mesh, additional boundary conditions and curvature of the geometry. We achieve more realistic shapes for unmeasured points with this approach. The developed method is based on the mathematical framework of discrete differential geometry and provides a simple interface that requires only the 3D geometry and sensor data, resulting in a sparse linear system. The solution of this linear system then can be efficiently split into two parts; factorization and back substitution. Thus, users can interact in realtime with the animated sensor data visualization. This gives the user the ability to explore measured data in a more realistic way. For validation, we tested the interpolation method with measured data of representative 3D geometries. The measurements were performed with a Laser Doppler Vibrometer to obtain a very high resolution. This is advantageous for later validation to check the interpolation with many sampling points and high resolution. The interpolation is validated in this publication

D. Herfert () · K. Henning · J. Heimann Society for the Advancement of Applied Computer Science, Department of Structural Dynamics/Pattern Recognition, Berlin, Germany e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_6

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exclusively by examples of experimental modal analysis. However, it can also be used in the software for applications of operational modal analysis and operational deflection shape analysis. This approach for the interpolation of unmeasured nodes is integrated in the vibration analysis software WaveImage Modal [1]. This software provides all the essential components for performing a dynamic structural analysis. Starting with simulation, a full recorder module for performing vibration measurements, various options for analyzing the measurement data and the model updating component to fit FE simulation to measured results. The operational and experimental modal analysis, the operating deflection shapes and the order analysis are available for classical modal analysis. In addition, a large number of algorithms for signal processing is available. WaveImage is specialized in processing large amounts of data of several Gigabytes (e.g. Laser Doppler Measurements).

6.2 Background In this paper we consider the problem of function interpolation as the formulation of a differential equation, more precisely

 as the so-called poisson problem. In the classical sense, the poisson problem is to find a function u C 2 (Ω) ∩ C 0 Ω in a domain ⊂R2 with prescribed values on the boundary of the domain . This is mathematically written as Δu = f, in Ω

(6.1)

u = g on ∂Ω

(6.2)

Where f is the real-valued source function on the interior of and g is of the boundary function on ∂ . For the boundary condition we use the absolute value of u on ∂Ω, this is called dirichlet condition [2]. For the discretization of Eq. 6.1 on arbitrary meshes, we directly use results from differential geometry [3] and we will only review the needed discretization for our application of function interpolation. We will not go into details for solving differential equations on arbitrary surfaces or general knowledge for weak solutions of PDE’s. For the details there is [4], a very good introduction to this topic. It should be noted that we only search for functions u that actually arise from measurements, so we also need not to argue the existence of u. Our domain will be triangular meshes M = (V , E, F ) with n vertices, where V is the set of vertices, E is the set of edges and F is the set of Faces. Each vertex vi V is a point on the surface M ⊂ R3 represented as Cartesian coordinate vi = (xi , yi , zi ) [3]. To apply the theory provided from differential geometry an important condition on M is that it is a sub-manifold of R3 , such that a mapping from M to R2 locally exits. For more on this refer [4]. The Laplace operator required for the interpolation problem results from the derivation of the so-called differential or δcoordinates that are defined as the difference between the absolute coordinates of vi and the center of mass of its immediate neighbors in the mesh [3]. 

1  y δi = δix , δi , δiz = vi − vj di

(6.3)

j N (i)

Where N(i) = {j | (i, j) E} and di = |N(i)| is the number of immediate neighbors of vertex vi (degree or valence of vi ). The transformation from Cartesian coordinates to δ-coordinates can be represented in matrix form. For this representation we construct the adjacency (connectivity) matrix of the mesh  Aij =

1 (i, j ) E 0 otherwise

and the diagonal matrix D such that Dii = di . Then the matrix transforming the absolute coordinates to relative coordinates is L = I − D −1 A.

(6.4)

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The matrix L is called the topological or graph laplacian [5]. Often the symmetric version Ls = DL = D − A is used and it’s entries are ⎧ ⎨ di i = j (6.5) (Ls )ij − 1 (i, j ) E ⎩ 0 otherwise Now the δ coordinates can be computed by Ls x = Dδ x for the x coordinate and analogously for the y and z components. From a differential geometry perspective, the δ coordinates can be viewed as a discretization of the continuous laplacebeltrami operator [5], if assumed that our mesh is a piecewise-linear approximation of a smooth surface. The laplace-beltrami operator is the laplace operator on curved spaces. For our implementation we use the discretization of the graph laplacian since it does not require any special condition on the triangle meshM, except the manifold condition, and is very robust. Respecting the local geometry of the triangle mesh would give even more accurate discretization’s of the laplacian operator [3] but require also more quality of the triangulation. Now, for interpolating unmeasured nodes on triangle meshes we discretize Eq. 6.1 by setting the rows of the laplacian matrix L corresponding to known function values with all zeroes and a 1 along the diagonal, and their δ coordinates are set to be equal to the known function values at those points. The same procedure applies to the boundary conditions that are only special points of u. Now with a discretization of the laplace operator on a triangular surface, we can build the system Lu = f

(6.6)

Where L contains informations about our mesh and f provides information about our measured data. The solution can be done efficiently since the resulting system is very sparse [3]. For the solution in our implementation we use the Eigen C++ library that provides standard numerical solvers for sparse linear systems.

6.3 Measurement Setup Two structures were measured and simulated as part of the publication. These are a flange and a UAV rotor blade. These measurements were performed by Jan Heimann as part of a master thesis in the Department of Structural Dynamics/Pattern Recognition [6]. The vibration response of a stainless steel flange (120 mm × 14 mm) to a force excitation was investigated. The flange was attached to a frame with a rubber band. Thus free boundary conditions can be assumed on the entire surface. The flange was excited by a shaker (PCB SmartShaker with integrated power amplifier, model K2007E01) via a thin stinger (diameter approx. 2 mm). In order to get a force transmission only normal to the surface the stinger with force sensor (PCB type 208C02) was mounted on the back side at the outer edge using glue, see Fig. 6.1. The system response of the flange to the force excitation was measured with a 3D-Laser Doppler Vibrometer (3D-LSV) measurement sytem (PSV-500, Polytec). As test signal a periodic chirp was used. Based on simulation results, the frequency range of the excitation was limited to 4–12 kHz. The surface velocities in all three directions (x, y,z) were scanned using an unevenly distributed mesh grid with more than 200 measuring points over the whole surface of the flange. The correlation between excitation and response leads to the individual frequency response function for each measuring point. This is used to identify the modal parameters. The system response of a carbon fiber reinforced polymer (CFRP) rotor blade was also investigated. The rotor blade (274 mm length) was fixed at its root on one side in order to realize the actual installation situation. For this purpose the rotor blade was mounted to a frame. The light structure was excited broadly by an automatic modal hammer (WaveHit, gfai tech GmbH) to avoid additional mass coupling by a sensor. In order to achieve a sufficient force transmission the location of the impact have been chosen to be on the free end of the blade tip, see Fig. 6.1. The modal hammer, as a full automatic device, was synchronized with 3D-LSV data acquisition system, which measured the individual system response in all three directions (x, y, z) at 171 unevenly distributed measuring points on the whole surface of the structure.

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Fig. 6.1 Left: Setup for the Experimental Modal Analysis on a flange. The structure with free boundary conditions is excited by a shaker between 4–12 kHz. Acquisition of the system response via 3D-LSV. Right: Setup for the Experimental Modal Analysis on a rotor blade. The structure fixed on one side was excited broadly by an automatic modal hammer (WaveHit, gfai tech GmbH). Acquisition of the system response via 3D-LSV Table 6.1 MAC- Value for the comparison of measured mode shape with 226 measurement points and different gradations of interpolation over all modes for the flange Interpolation flange with 142 measurement points Interpolation flange with 110 measurement points

4608 Hz 0.996 0.99

7590 Hz 0.94 0.93

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9977 Hz 0.98 0.95

6.4 Analysis The experimental modal analysis component from the Software WaveImage Modal (Fig. 6.2) was used to calculate the modal parameters for both measurements. For this purpose the Laser Doppler Vibrometer SIMO-Measurements were used. With the CMIF-Algorithm 4 Modes were found for the flange and 10 Modes for the UAV blade. From both measurements we use the first four and six modes for the validation of the interpolation. The interpolation was carried out in two gradations, which differ in the number of interpolated points (Fig. 6.3). For the flange we have 226 measurement points. In the interpolation we reduced the measurement points in two steps. In both steps, all measuring points at the edge and a few additional measuring points in the middle of the structure were retained. In addition to a high visual similarity in the images, the amplitude of the original mode shapes could also be fully preserved. The results of the interpolation are very satisfying, because the holes in the structure place an additional demand on the interpolation. For the UAV Blade we have 171 measurement points. In the interpolation we reduced the measurement points in two gradations. In contrast to the flange, here a random distribution of the measuring points was used to have a more difficult starting position for the interpolation. Since the blade oscillates very identically to a beam, the use of the measuring points at the edge would have been trivial (Fig. 6.4). In this application it was also possible to interpolate visually very similar mode shapes with very few randomly distributed measuring points. Due to the random selection of the measuring points, some modes showed stronger deviations from the original natural mode, because the measuring points did not occupy all significant points of the mode. If the measurement omits essential points (e.g. mountains and valleys), these can of course not be compensated by interpolation. In addition to the visual comparison the modal assurance criterion (MAC) between the original measurement and the interpolated solution was calculated to determine the quality of the interpolation. These indicator gives the degree of consistency between two mode shapes. The MAC values are between 0 and 1. Where 1 means there a identical and 0 means a strong dissimilarity of mode shapes. In Table 6.1 the visual similarity between the fully measured mode shapes and the interpolated mode shapes described above is again confirmed by a very high Mac value across all modes. Due to the already mentioned random distribution of the measuring points for interpolation, the MAC values in Table 6.2 are lower compared to the flange. Nevertheless, they are very similar in most modes, even with very few measuring points. With an even more optimized selection of the measurement points, higher MAC value and fewer measuring points would be

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Fig. 6.2 Screenshot of the WaveImage Experimental Modal Analysis Component, Modes are found with CMIF-Algorithm Left: Results of flange, Right: Results of UAV blade

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Fig. 6.3 Gradations of interpolation. Exemplary at the first mode of flange. Original Mode Shape with 226 measurement points, first Interpolation with 142 measurement points, second interpolation with 110 measurement points. The Measurement Points for the Interpolation are presented with red points. The other points are interpolated

Fig. 6.4 Gradations of interpolation. Exemplary at the first mode of UAV blade. Original Mode Shape with 171 measurement points, first interpolation with 91 measurement points, second interpolation with 11 measurement points. The Measurement Points for the Interpolation are presented with red points. The other points are interpolated Table 6.2 MAC- Value for the comparison of measured mode shape with 171 measurement points and different gradations of interpolation over the first six modes for the UAV blade Interpolation UAV blade with 91 measurement points Interpolation 2 UAV blade with 11 measurement points

106 Hz 0.96 0.87

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possible. Corresponding investigations will be carried out in the future. In principle, the results obtained are very satisfying and will be further optimized to the application in modal analysis in the future.

6.5 Conclusion As a main property, the presented interpolation scheme allows interpolation along a surface and does not interpolate unconnected points across the surrounding space. Furthermore, the implementation of our method is relatively simple and requires only the solution of a sparse linear system of equations, which can be done efficiently [3]. Additionally the user is not used to must provide any configuration step

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As with any interpolation scheme, it is necessary to have a good idea of the function to be searched for in order to obtain good results. In this paper, we have tried to show the limitations of the interpolation method presented by providing as less information as possible about the function in question. In further investigations, it may be possible to reduce the memory for storing modes by prescribing f with precise information’s. For this purpose, further techniques from modal analysis shall be used. It may also be possible to find optimal sensor positions on a given geometry by inverting the problem as optimization. Due to the very efficient solving on modern computer systems the presented method may applicable for Augmented Reality and Virtual Reality environments or other interactive application with sensors.

References 1. Homepage of the Software WaveImage. Modal https://wave-image.com/modalanalysis/?lang=en2019 (2019) 2. Bronstein I., Semendjajew K.: Taschenbuch der Mathematik (2000) 3. Sorkine O.: Laplacian Mesh Processing (2005) 4. Crane K.: Discrete Differential Geometry: An Applied Introduction (2019) 5. Fiedler, M.: Algebraic connectivity of graphs. Czech. Math. J. 23, 298–305 (1973) 6. Heimann J.: Experimentelle Validierung eines Eigenwert-Optimierers sowie Erweiterung des Parameterraums zur Berücksichtigung von Temperatureinflüssen. Master Thesis Technical University Berlin (2019) Daniel Herfert Studied Computer Science at the Humboldt University of Berlin with focus on Artificial Intelligence, Assistant at the Chair of Artificial Intelligence and Robotics at the Humboldt University of Berlin for 4 years, GFaI employees since 2010, Head of Department Structure Dynamics/Pattern Recognition, Founder of WaveImage software for modal analysis. Kai Henning B. Sc. in Computer Science at University of Applied Science Lübeck with focus on Software Architecture and Embedded Systems. GFaI employees since 2018, Software Engineering for Computer Graphics and Virtual Environments in context of applications for Structural Dynamics. Jan Heimann M. Sc. in Engineering Science at the Technical University of Berlin with focus on Technical Acoustics and Structural Dynamics. Employed as an acoustic and vibration engineer at the company gfai tech in the department for Consulting and Services since 2019.

Chapter 7

Measurement and Analysis of the Nonlinear Stiffness of CFRP Components during Vibration Fatigue Testing Michele Peluzzo, Dario Di Maio, Andrea Cammarano, and Paolo Castellini

Abstract One of the typical fatigue failures of Carbon Fibre Reinforced Plastic (CFRP) components undergoing vibration loading is by delamination. Vibration fatigue tests are carried out at amplitude levels at which a sample undergoes to large deflections, and so high strains, to initiate the fatigue process. However, the higher the vibration amplitudes the higher the nonlinear vibration spectral contents will be. Furthermore, the occurrence of delamination generates additional nonlinear harmonics which are excitation cycles dependent, which means that the nonlinear stiffness changes with the component time to failure. This research work will attempt to create a multi-dimension stiffness map which is both function of the vibration amplitude and the number of cycles. The restoring surface map will be used to calculate the stiffness amplitude dependent as the fatigue process develops until the termination of the test. The ability to continuously monitor the nonlinear stiffness will allow to characterize the crack as it develops from a non-critical to a critical delamination size. Keywords SLDV · CFRP · Vibration fatigue · Nonlinear

7.1 Introduction This research work represents an improvement of a recent developments carried on failure criteria of composites components under vibration fatigue. Magi’s work [1] delivered a robust method for testing composites under high cycle fatigue (HCF) that is able to identify a critical moment in the fatigue life of a component. It was observed how the structural behaviour changes during a fatigue test by measuring the response phase under fixed vibration amplitude and excitation frequency. By identifying a critical event, in the fatigue life, it was possible to define the transition from damage tolerance and damage propagation. The critical event describes a critical delamination size which was a helpful criterion for producing the S-N curves. The resonance frequency is directly related to the stiffness of the component, which changes with the evolution of microcracking and delamination. Therefore, resonant testing at constant excitation frequency was adopted as a first approach, measuring the phase difference between excitation and response, because keeping a constant phase was crucial in order to have a direct measurement of frequency shift and to allow large frequency changes, as reported in some publications [2]. Obviously, once the critical event occurred, the growth of delamination reduced the stiffness of the structure, resulting in a significant shift in resonance frequency. Magi, in his work [1], explains the reason why one can measure the phase decay in order to observe a stiffness degradation: considering the phase lag in a base excitation between the base and mass displacement, the first order Taylor expansion of the equation shows that the phase is directly proportional to the stiffness for small variations at constant excitation frequency. Nevertheless, measuring the phase decay could not be a very reliable method due to the strongly dependence from modal parameters m, k and c, respectively modal mass, stiffness and damping.

M. Peluzzo · D. Di Maio () Department of Applied Mechanics, Mechanical Engineering, University of Twente, Enschede, The Netherlands e-mail: [email protected] A. Cammarano Department of Mechanical Engineering, University of Glasgow, Glasgow, UK e-mail: [email protected] P. Castellini Department of Industrial Engineering and Mathematical Science, Polytechnic University of Marche, Ancona, Italy e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_7

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Furthermore, recent experiments highlighted that the crack opening is associated with a typical nonlinear response due to the appearance of nonlinear harmonics as results of the vibration amplitude. A new approach is presented in this paper, which adopts the restoring force surface method to study the elastic behaviour of a CFRP component during a vibration fatigue test. The technique described in this research work allows to directly find both stiffness and damping in a nonlinear system, by processing acceleration and velocity time signal data from the test set up (see Sect. 7.4). The objective of this work is to understand the character of the nonlinearity showed by a component under vibration fatigue and to directly observe the stiffness change as an output of the test. This work should enable the achievement of a further goal such as the development of a relationship between the stiffness degradation and an opening delamination under vibration loading. The goal is to aid the characterization of mechanical property of CFRP specimens currently carried out by time consuming ASTM standard tests, such as ASTM E647 for measurement of fatigue crack growth rate.

7.2 Test Structure, Setup and Experimental Method Restoring force method offers an efficient and reliable identification approach of nonlinear single-degree of freedom systems developed by Masri and Caughey [3]. However, this method may be extended to a multi-degree of freedom systems using a modal space model [4]. The restoring force surface method is based on Newton’s second law: mx(t) ¨ + f (x(t), x(t)) ˙ = p(t)

(7.1)

where p(t) is the applied force and f (x, x) ˙ is the restoring force, i.e. a nonlinear function of displacement and velocity. Since the function f is assumed to be dependent only on x and x, ˙ it can be represented by a surface over the phase plane (x, x). ˙ One can adjusts the equation as following: f (x(t), x(t)) ˙ = p(t) − mx(t) ¨

(7.2)

If the mass m, the excitation p(t) and the acceleration x(t) ¨ are known, all quantities of this equation are known and hence so is function f. In practice, one would measure the acceleration and numerically integrate it to find velocity and displacement. From Eq. (7.1), it is possible to find the restoring force defined as fi = pi − mx¨i where subscript i refers to the sampled value. Thus, for each sampling instant a triplet (x, x, ˙ f ) is found, i.e. the value of the restoring force is known for each point of the phase plane (x, x). ˙

7.3 Base Excitation Figure 7.1 shows a schematic model of the system under study consisting in a CFRP specimen attached to a moving base. The specimen is excited around its first resonance and one can describe it with the following equation:

Fig. 7.1 Schematic of setup under study

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mx¨ = −fr (u, u) ˙

(7.3)

where m is the effective mass, x¨ is the absolute acceleration of the sample, u is the relative displacement which is indicated as μ = x − xb

(7.4)

The basic idea behind the RFS method is that the measured acceleration is proportional to the force applied to a SDOF system [5]. Thus, if the displacement and velocity can also be obtained by time-domain integration from acceleration, then the functional relationship can be evaluated. Considering the dynamic equation (7.3) the linear terms can be extracted from the total restoring force so the unknown nonlinear function fnl contains only nonlinear terms:

x¨ = −2ζ ωn u˙ − ω2 n u −

fnl (u, u) ˙ m

(7.5)

Where ζ and ωn are the damping ratio and the natural frequency respectively. All time signal data measured during the test will be processed by the restoring force surface method. Schematic model in the figure above refers to the real test setup which is described in Sect. 7.4, where one can see the type of measurement carried out for this research work.

7.4 Experimental Test 7.4.1 CFRP Specimens The experimental campaign was designed to initiate the crack at a specific location of the sample and allow the crack to open in delamination. This was achieved by creating thermoset panels (named SxNP5, where x is for the specimen classification) made of T700/M21 from Hexcel with a stacking sequence [0, 90-cut, 0-cut, 90, 0, 0, 90, 0, 90, 0] and thermoplastic panels made of 16-layered polyphenylene sulphide (PPS) laminate reinforced by unidirectional carbon fibres (Fig. 7.2). This peculiar stacking sequence has got a 0 degrees ply on the top surface, which bears tensile and compressive loads during bending, and two cut-plies underneath. The crack path is therefore forced to follow a predefined direction, which is transversal to the ply orientation and then parallel to them. Figures 7.3 and 7.4 show the samples with the position of the cut-ply (dashed line) with respect to the clamping position (solid line), 10 mm, and the measurement position marked on each sample by marker and then covered by reflective tape

10 mm

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0 90 0 90 0 0 90 0 90 0

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0 0 0 0 0 0 0 0

Fig. 7.2 CFRP thermoset (on the left) and thermoplastic (right) specimen configuration: stacking sequence, position of the cut-ply and direction of the crack path. Clamping position is 10 mm from the cut ply zone

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Fig. 7.3 Thermoset sample. It shows the measurement point (patch), clamping zone (solid line) and cut-ply position (dashed line)

Fig. 7.4 Thermoplastic sample

̇

LDV

ODS

̈

Fig. 7.5 Test setup and type of measurement

after the sample was secured in the fixture. Attention was paid to the actual clamping condition which should be uniform across the sample width to avoid undesired dynamic effects.

7.4.2 Test Setup and Program Figures 7.5 and 7.6 illustrate the experimental setup which was made of an electromagnetic shaker which mounted a knifeedge fixture made by two silver steel rods to hold the specimen at its mid length. A frame was built around the shaker to support a single point Laser Doppler Vibrometer to measure the vibration response (velocity), whereas the base excitation was measured by an accelerometer. Moreover, a Scanner LDV was used to capture the operating deflection shape (ODS) in order to be able to look for something that shows a delamination zone during the fatigue test. Data from LDV and accelerometer was used as an input for data processing in MONTEVERDI, a LabView code, which returned the following output: • • • • • • •

Vibration amplitude [g]; Phase response tracing [rad]; Excitation frequency [Hz]; Nonlinear harmonics from vibration response [m/s]; Velocity and base acceleration time signals; FRF and modal analysis parameters; Response from a modal test using a chirp signal.

The velocity response of the specimen and the base acceleration response data can be used in a post-processing MATLAB code in order to apply the restoring force surface method, following the aforementioned equations in Sect. 7.2.

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Fig. 7.6 Termoset specimen with LDV laser spot, on test setup

7.4.3 Test Sequence To get all the output mentioned in Sect. 7.4.2, test has been run following this sequence for both thermoset and thermoplastic specimens: 1. Initial FRF and modal analysis to calculate resonance frequency and loss factor; 2. Modal test with chirp signal at different output voltage amplitudes (0.1-6 V or higher) in order to visualize stiffness relationship from small to large displacement which should be nonlinear at high amplitude; 3. Fatigue test (HCF) run with excitation frequency close to the resonance and vibration amplitude which can get fatigue damage, for 10 Million of cycles; 4. Modal test with chirp signal at different amplitude in order to find out differences in stiffness relationship after a delamination when it occurred; 5. Final FRF and modal analysis to calculate resonance frequency after fatigue test and the resonance decay. During these tests is important to follow simple rules about specimen mounting precision and a checked laser focus on the reflective tape to avoid noisy data.

7.4.4 Data Processing Validation As a way of validating the reliability of post-processing code, numerical and experimental tests have been done. Displacement and velocity data from a simulated 2 DOF system with cubic stiffness has been used: the aim of this step was to understand if time signal integration and derivation could introduce errors or wrong data. Input data is from a chirp signal. Figure 7.7 shows that data elaboration is correct, and one can observe the cubic stiffness very clear. Polynomial coefficients have been calculated by curve fitting, referring to equation ax3 + bx2 + cx + d = f(x): a = 4.9 × 1016 ;

b = −1.08 × 105 ;

c = −2.857 × 104 ;

d∼0

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Fig. 7.7 Force-Displacement relationship (on the left) and Restoring Force Surface (on the right) of simulated 2DOF system with cubic stiffness

Moreover, it was important to understand if the stiffness decay value (dK) could be reliable: vibration fatigue test is able to monitor linear stiffness as a first attempt and the rate dK/dN represents an important value in order to correlate stiffness degradation to the crack growth rate. Thus, experimental stiffness has been calculated with a 3-point bending test, before and after the HCF test. Then, the stiffness decay from vibrations and from experimental values have been compared. Calibration tests were run with other CFRP specimens with different stacking sequence which did not hindered the calibration attempt. Results were very promising: • Stiffness decay from vibration data: 0.37 N/mm (−4.16%); • Stiffness decay from 3 PB tests: 0.28 N/mm (−3.5%). The difference is lower than 1% which was considered acceptable for this type of test. Furthermore, this difference could be related to the least square fitting method. Nevertheless, vibration testing could be used to estimate the stiffness while an endurance tests is carried out on the specimen without a standard three-point bending test.

7.5 Results and Analysis As the test sequence has been completed, one has to analyse the phase decay curve and the harmonics magnitude along number of cycles in order to find out any delamination presence. One can do this observing two relevant events: • An abrupt decay of phase response which indicates a delamination phenomenon; • A nonlinear slope of harmonics, especially the fifth and the seventh which, from recent works, are more representative about microcracking or delamination occurrence.

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Fig. 7.8 Phase response along number of cycles. Decay indicates damage growth in the specimen, but it there is not the occurence of critical moment

After detecting this, it is needed to compare modal test data from chirp signal in order to understand the consequences of the composite damage i.e. stiffness relationship change. Hence, one can quantify the stiffness decay along the number of cycles in terms of dK (N/mm) and percentage of decreasing, correlating this to the resonance change value from FRF data acquired before and after the HCF test. Due to a great amount of acquired data, RFS plotted from HCF data corresponds to 100 sample along the whole testing cycles.

7.5.1 Thermoplastic Specimen Test has been run with a 0.7 mm constant displacement amplitude at the measurement point, which is typically few tens of mm away from the clamp, at constant excitation frequency (close to the 1st bending resonance) and for 106 cycles. Figure 7.8 shows the phase response decay and one can see that something occurred at last cycles, due to microcracking propagation in the specimen accordingly with harmonics magnitude trend (Fig. 7.9). Nevertheless, there is not clearly evidence of nonlinear behaviour which can describe a strong delamination phenomenon, but one can observe the stiffness degradation of 2.7% caused by fatigue damage, as reported in Fig. 7.11. Restoring force surfaces obtained also illustrates how during the fatigue testing the damping relationship has been changed due to a different shape of surface from perfect elliptical to distorted one.

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7.5.2 Thermoset Specimen Thermoset specimen has been tested with 1.7 mm of displacement amplitude for 10 million cycles and one cannot observe a substantial change in stiffness or damping relationship because damage occurred is very small, surfaces remain elliptical (Fig. 7.12) and, moreover, stiffness decay is about less of 1% (Fig. 7.13). Although, analysing chirp signal data one can understand how the greater the excitation amplitude the stronger the nonlinearity of force-displacement relationship, as one can see in Fig. 7.14.

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RF Surface

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7.6 Conclusions This research work introduces an innovative technique to investigate on fatigue life in composite applications, using output from vibration testing. The new approach allows to monitor stiffness and damping behaviour adopting the Restoring Force Surface method processing vibration data i.e. base acceleration and velocity response of the specimen. However, this paper takes a different approach to the one commonly found in literature. The RSF is produced during the HCF trial by using the excitation frequency, which is kept constant and not time varying as in a chirp signal. This testing method enables to directly trace the stiffness as function of number of cycles. A shortfall was identified during the post-processing of the HCF date. Since the RFS is made by two single sinewaves, the plane is described a single circle which does not provide enough data points to observe distortion caused by the nonlinear stiffness. Nevertheless, this paper presents an advancement to the vibration testing techniques so far ever attempted.

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Stiffness decay (averaged) 39.55 39.5

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39.45 39.4 39.35 39.3 39.25 39.2 0

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Fig. 7.13 Stiffness degradation during vibration fatigue testing

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Fig. 7.14 RF Surfaces 2D view, from chirp signal data. At higher displacement, the relationship becomes nonlinear (on the right)

Acknowledgements Di Maio wishes to acknowledge HEGEL project (Grant Agreement no: 738130) for the composite thermoset material and for TPRC for the thermoplastic one.

References 1. Magi, F.: Vibration fatigue testing for identification of damage initiation in composites. PhD Dissertation, University of Bristol (2016) 2. Lazan, B. et al.: Dynamic testing of materials and structures with a new resonance-vibration exciter and controller. Technical report, Wright Air Development Center, Ohio (1952) 3. Caughey, T.K., Masri, S.F.: A nonparametric identification technique for nonlinear dynamic system. J. Appl. Mech. 46, 433–447 (1979) 4. Dimitriadis, G., Cooper, J., Platten, M., Wright, J.: Identification of multi-degree of freedom nonlinear system using an extended modal space model. Mech. Syst. Signal Process. 23(1), 8–29 (2009) 5. Epp, D., Allen, M, Sumali, H.: Restoring force surface analysis of nonlinear vibration data from micro-cantilever beam. ASME International Mechanical Engineering Congress and Exposition (2006) Dario Di Maio Graduated in Mechanical Engineering at the University Politecnica delle Marche. PhD at Imperial College London. PostDoc and Assistant Prof. in Dynamics at Bristol University. Currently, Assistant Prof. at Twente University.

Chapter 8

Model Reduction of Electric Rotors Subjected to PWM Excitation for Structural Dynamics Design Margaux Topenot, Gaël Chevallier, Morvan Ouisse, and Damien Vaillant

Abstract Rotors of asynchronous machines can be subjected to risk of failure due to vibratory fatigue. This is caused by the way electric motors are powered. Pulse Width Modulation (PWM) is the control strategy of the traction chain. This signal is composed by a fundamental and numerous harmonics of voltage and current that induce harmonics on the torque signal resulting in huge torque oscillations. It can lead to repeated torsional resonance when coincidences occur. This can induce severe damages and even lead to rupture if electric excitations are not taken into account at the design stage. In this work, a magnetic finite element model is built by using Fourier decomposition in order to take into account harmonics due to PWM. Pressures exported from this model are used as inputs for mechanical FEM. A mechanical reduced order model is also proposed in order to compute stress in rotating part. This second model allows to reduce time computation and then to consider several operating points to build a complete speed up. A correlation is performed between these two models and rotating tests in order to discuss the relevance of these approaches to design rotor parts. Keywords Electric motor · Vibratory fatigue · Pulse width modulation · Structural dynamic design · Model reduction

8.1 Introduction and State of the Art Trains are set into motion through electrical energy. From a theoretical point of view, electric machines can be driven by sine wave signals but practically, due to signal synthesis issues and power limitations, motors are driven using Pulse Width Modulation (PWM). This allows to manage easily the variation of the rotating speed. The PWM is based on a fundamental frequency equal to the sine signal frequency. However, PWM generates small-amplitude oscillations at frequencies corresponding to the harmonics, inducing vibrations and spurious mechanical stress. Figure 8.1 illustrates the difference between PWM and sine waveform signals. This results in torque oscillations over a wide frequency range and possibly large amplitudes, sometimes leading to the rupture of several rotating parts due to resonance [1]. Wachel plots interference diagram in 1993 in order to check for coincidence of the torsional mode with critical operating speed [2]. Song-Manguelle proves in [3] and Feese in [4] that the variable frequency drive leads to shaft failures due to pulsating torque. Other rotating parts can be subjected to failures like fans [5, 6]. The authors of [7] recommend to pay attention of the drive’s voltage spectrum to design mechanical parts. Electromagnetic and mechanical phenomenon should be taken into account, this is why several authors propose to couple the two physics: Hallal and Pellerey propose a weakly coupled methodology in [8] and [9]. As Hallal, Dupont proposes in [10] a methodology to project magnetic load on mechanical mesh. Journeaux exposes in [11] the advantages and the drawbacks of weak and strong coupling. Delforge alerts on the huge computational time to perform such a study in [12]. Van der Giet and al. make the comparison between 2D and

M. Topenot () Department of Applied Mechanics, FEMTO-ST Institute, University Bourgogne Franche-Comté, Besancon, France Alstom Transport, Ornans, France e-mail: [email protected] G. Chevallier · M. Ouisse Department of Applied Mechanics, University Bourgogne Franche-Comté, FEMTO-ST Institute, Besancon, France e-mail: [email protected]; [email protected] D. Vaillant Alstom Transport, Ornans, France e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_8

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Fig. 8.1 Electric current intensity waveform with PWM or sine

3D coupled electromagnetic and structural dynamic simulation in [13]. From the mechanical point of view, the rotor is a complex assembly. Its modeling is subjected to various assumptions. In particular, the stack lamination is relatively hard to model. Millithaler proposes in [14] a methodology to model laminated structures. An identification methodology is proposed by Mogenier in [15] to get material parameters in rotor subjected to bending modes. The aim of this study is to propose and compare rotors dynamic models able to capture the resonance phenomenon due to PWM excitation. Two models are proposed: the first one consists in weakly coupling magnetic and mechanical finite elements models by using Fourier expansion in order to take into account harmonics due to PWM whereas the second one uses reduced order modelling in order to obtain the structural response faster. The scale of refinement which is necessary to describe the problem with a good accuracy will be discussed after the presentation of the results. Rotating tests have been performed in order to measure the strain in rotating parts for a fixe speed that corresponds to the resonance speed. Figure 8.2 shows a block diagram that explains how models is compared to reference data.

8.2 PWM Synthesis and Stress Analysis Electro-technical engineers are responsible for the PWM synthesis in the inverter in order to power trains. Several strategies are availables for signal generation [16] according to industrial requirements. This mission requires to wisely choose the way that impulses are created by avoiding losses and increasing performance. It can directly impact the traction chain but also the dynamic behavior of electric rotors. Among all the strategies, asynchronous, synchronous and space vector modulation are the most common. For the two first one, carrier signal is compared to a reference sine waveform signal. This comparison leads to the opening or the closing of switches (IGBT in the inverter). The difference between both lies in the ratio of carrier frequency and the sine frequency. Space vector modulation is a really interesting way to get PWM signals. Indeed, this strategy allows to choose the switching angles (corresponding to switching times). This means that it is possible to change harmonics frequencies in the spectrum of the signal. Due to the performance requirements, it is difficult to completely avoid harmonics but a good knowledge of the studied structure can help to avoid structural dynamic problems. In this paper, the electromagnetic part of the motor is modeled by an equivalent electric circuit with voltage drive. The voltage source corresponds to the sum of the harmonics of the PWM signals by using Fourier expansion. Authors alert on the limitation of the number of variables that can be set and therefore on the limitation of the number of harmonics taken into account in commercial calculation codes. In general, the carrier frequency is really high compared to frequencies of interest for structural dynamic design. It means that filters can be applied to lower the number of harmonics, but the user

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Fig. 8.2 Block diagram illustrating the methodology

Fig. 8.3 Rotor divided in 9 sections

must be aware that the reconstructed signal is changed. In addition, if no filter is applied, the maximal frequency is the carrier frequency and then it could lead to very small time steps and then to huge computational time and results files. A 3D mechanical finite element model of an electric rotor is considered as reference. The torque provided by the electromagnetic simulation is applied at the rotor center and is also used as load for a reduced order model (see Fig. 8.2). An electromechanical lumped model has been published by Bruzzese in 2016 [17]. He managed to define an electrotechnical model by solving equivalent electrical circuit and applying magnetic torque on a simplified mechanical model composed by masses, springs and dampers. Inertia and stiffness matrices were evaluated from a lumped representation. Here, the mechanical reduced order model is built from the modal basis and nodal coordinates of sections coming from the 3D FEM in order to have a more accurate mechanical representation of the real rotor compared to the work proposed in [17]. The key of such a model is the transformation matrix that allows the reduction. Starting from a full 3D model composed of thousands of degrees of freedom, the model is reduced to only nine rotational degrees of freedom. This work being focused on torsion behavior of the electric machine, only rotation degrees of freedom around motor axis are considered. The rotor is decomposed in nine sections as shown in Fig. 8.3. The reduction matrix is built by modal projection using the link between displacement and rotational degrees of freedom of rotor sections. Writted on the modal basis, the dynamic equation in generalized coordinates {q} is {q} ¨ + []{q} ˙ + []{q} = {f } ⎡ with [] = ⎣

..

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⎤

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. 2ξi ωi ..

⎤ ⎥ ⎦, [] being the modal basis

. ωi2 ..

.

and [K] the stiffness matrix, {f } = modal stresses:

..

[]T {F },

.

{F } being the load vector. The mechanical stresses are obtained by using the {σ } = [σ ]{q}.

(8.2)

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Fig. 8.4 Torque oscillations due to PWM (left side) and FFT of this signal (right side)

8.3 Results Load excitations and modal behavior should be mastered to predict the dynamic response of the rotor. Figure 8.2 shows what quantities should be compared. The torque signal and the spectrum exported from electromagnetic simulation is illustrated in Fig. 8.4. This is the input load for mechanical computation. Numerically, high torque harmonics are close to the natural frequency of the rotor. But experimentaly, the torque has been measured with a device that allows only to measure a signal until 250 Hz. This means that torque spectrums can not be compared. However, mean values of experimental and numerical torque can be compared in order to validate the simulation methodology. The 5% gap between the two indicates that the model is representative of the test conditions. A coincidence between the load harmonics and the natural frequency of the torsion mode is observed in the test data. In order to reproduce this effect with models, the maximal torque value in the spectrum has been spread around the torsional eigen frequency in numerical models. This spreading is necessary because a small error in models can cause the absence of the coincidence. It results in 4% of relative error on the maximal stress at the torsional resonance between the full model and the reduced order model. This indicates that reduced order model is efficient to describe the behavior of the structure and allows to consider a full speed up. The computation of the modal basis takes 8 min on a calculator with 16 CPU, 3.2 GHz and 225692 MB RAM and then the calculation of rotation and stresses thanks to reduced model takes 2 min. In comparison, the magnetic computation with magnetic solver takes 5 h on a laptop with 8 CPU, 2.7 GHz and 16 Go RAM and then the mechanical computation with mechanical FE solver takes 2 h on the calculator. The Fig. 8.5 illustrates qualitatively the comparison of the measured waterfall of the stress (left side) and the computed one (right side). Resonance clearly appears when load harmonics cross the eigen frequency. It remains to calibrate the model to fit the rotating test data. Several aspects should be taken into account: • the damping ratio of the rotor has been measured in free-free condition and may be very different in the full motor in rotating condition; • even if the constant value of the computed torque fits well the average value of the experimental mechanical torque, the computed torque is the electromagnetic torque and due to losses, it might not be identical to the mechanical torque. The inertia of the rotor might filter the torque; • uncertainties about the location and the orientation of the gauges can lead to a bad correlation between measured and computed strain; • during tests, the speed is oscillating and it can lead to a rotor that is not excited exactly at the resonance, or the slip changes during tests due to temperature that not allows to stay at the resonance for a long time. All these aspects can explain differences between tests and models. Modelling assumptions must be carefully chosen.

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Fig. 8.5 Comparison of stress waterfall obtained by measurement (left side) and by computation with the reduced order model (right side)

8.4 Conclusion This paper compares two computational methodologies to investigate the vibratory response of electric rotors subjected to PWM excitation. The first one combines two Finite Elements Models in order to solve the multiphysic problem. The torque computed from the magnetic solver with test PWM signal is the input of the computation of the structural response to determine the mechanical stresses. Secondly, a model reduction is proposed to fasten the computation time during design phase. The 4% difference on maximal stress at the torsional resonance between the full model and the reduced order model allows to conclude that the reduction is accurate enough and efficient to compute the stress along a speed up. However, it remains to calibrate models to tests data. Future work will focus on the understanding of phenomenon and assumptions to be choose to fit tests data. Model updating will allow a better prediction of rotor vibratory response.

References 1. Bruzzese, C., Tessarolo, A., Santini, E.: Failure root-cause analysis of end-ring torsional resonances and bar breakages in fabricated-cage induction motors. In: 2016 XXII International Conference on Electrical Machines (ICEM), pp. 2251–2258. IEEE (Sept, 2016) 2. Wachel, J.C., Szenasi, F.R.: Analysis of torsional vibrations in rotating machinery. In: Proceedings of the 22th Turbomachinery Symposium. Texas A&M University. Turbomachinery Laboratories (1993) 3. Song-Manguelle, J., Schroder, S., Geyer, T., Ekemb, G., Nyobe-Yome, J.-M.: Prediction of mechanical shaft failures due to pulsating torques of variable-frequency drives. IEEE Trans. Indus. Appl. 46, 1979–1988 (2010) 4. Feese, T., Maxfield, R.: Torsional vibration problem with motor/ID fan system due to PWM variable frequency drive. In: Proceedings of the 37th Turbomachinery Symposium (2008) 5. Kreitzer, S., Obermeyer, J., Mistry, R.: The effects of structural and localized resonances on induction motor performance. IEEE Trans. Indus. Appl. 44, 1367–1375 (2008) 6. Holdrege, J.H., Subler, W., Frasier, W.E.: AC induction motor torsional vibration consideration – a case study. IEEE Trans. Indus. Appl. IA-19, 68–73 (1983) 7. Kerkman, R.J., Theisen, J., Shah, K.: PWM inverters producing torsional components in AC motors. In: 2008 55th IEEE Petroleum and Chemical Industry Technical Conference, (Cincinnati, OH), pp. 1–9. IEEE (Sept, 2008) 8. Hallal, J.: Études des vibrations d’origine électromagnétique d’une machine électrique: conception optimisée et variabilité du comportement vibratoire. Ph.D. thesis. Compiègne (2014) 9. Pellerey, P., Lanfranchi, V., Friedrich, G.: Coupled numerical simulation between electromagnetic and structural models. Influence of the supply harmonics for synchronous machine vibrations. IEEE Trans. Mag. 48, 983–986 (2012) 10. Dupont, J.-B., Bouvet, P., Humbert, L.: Vibroacoustic simulation of an electric motor methodology and focus on the structural FEM representativity. In: International Conference on Electrical Machines, Marseille, pp. 1–7 (2012) 11. Journeaux, A.: Modélisation multi-physique en génie électrique. Application au couplage magnéto-thermo-mécanique. Ph.D. thesis. Université Paris-Sud (2013) 12. Delforge, C., Lemaire-Semail, B.: Induction machine modeling using finite element and permeance network methods. IEEE Trans. Mag. 31, 2092–2095 (1995)

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13. Van der Giet, M., Schlensok, C., Schmulling, B., Hameyer, K.: Comparison of 2-D and 3-D coupled electromagnetic and structure-dynamic simulation of electrical machines. IEEE Trans. Mag. 44, 1594–1597 (2008) 14. Millithaler, P., Sadoulet-Reboul, E., Ouisse, M., Dupont, J.B., Bouhaddi, N.: Structural dynamics of electric machine stators: modelling guidelines and identification of three-dimensional equivalent material properties for multi-layered orthotropic laminates. J. Sound Vib. 348, 185–205 (2015) 15. Mogenier, G.: Identification et prévision du comportement dynamique des rotors feuilletés en flexion. Ph.D. thesis. Institut National des Sciences Appliquées de Lyon (2011) 16. Bowes, S., Clements, R.: Computer-aided design of PWM inverter systems. IEE Proc. B Elect. Power Appl. 129(1), 1 (1982) 17. Bruzzese, C., Santini, E.: Electromechanical modeling of a railway induction drive prone to cage vibration failures. In: IECON 2016–42nd Annual Conference of the IEEE Industrial Electronics Society. IEEE (2016) Margaux Topenot Currently at the beginning of her 3rd years of thesis at the Department of Applied Mechanics in FEMTO-ST institute in France, Margaux TOPENOT is collaborating with Alstom transport in order to improve design methodologies of electric rotors.

Chapter 9

Fast Computation of Laser Vibrometer Alignment Using Photogrammetric Techniques Daniel P. Rohe and Bryan L. Witt

Abstract Laser vibrometry has become a mature technology for structural dynamics testing, enabling many measurements to be obtained in a short amount of time without mass-loading the part. Recently multi-point laser vibrometers consisting of 48 or more measurement channels have been introduced to overcome some of the limitations of scanning systems, namely the inability to measure multiple data points simultaneously. However, measuring or estimating the alignment (Euler angles) of many laser beams for a given test setup remains tedious and can require a significant amount of time to complete and adds an unquantified source of uncertainty to the measurement. This paper introduces an alignment technique for the multipoint vibrometer system that utilizes photogrammetry to triangulate laser spots from which the Euler angles of each laser head relative to the test coordinate system can be determined. The generated laser beam vectors can be used to automatically create a test geometry and channel table. While the approach described was performed manually for proof of concept, it could be automated using the scripting tools within the vibrometer system. Keywords Multi-point · Laser vibrometry · Photogrammetry · Alignment · Automation

9.1 Introduction Laser Doppler Vibrometry (LDV) is a mature technology that has been used to overcome numerous testing challenges [1]. It has the ability to measure dynamic motion without mass-loading the test article, which can be important for small components and high spatial density measurements. It also has a very high frequency range compared to traditional instrumentation. Typically, laser vibrometers have been fielded in single-point or scanning variants, with the scanning LDVs being very successful with their ability to measure a number of points quickly and accurately. Unfortunately, this multi-point data is not obtained simultaneously, which can limit the testing that can be performed. Recently, the Multi-Point Vibrometer (MPV) has been introduced, which can measure multiple channels of laser vibrometer data simultaneously, either in triaxial (three beams per measurement point) or uniaxial (one beam per measurement point) configurations. From a user’s perspective, this system is effectively a large number of single-point vibrometers, which need to be positioned and oriented manually. While the scanning systems had feedback with the mirror servos so the laser beam geometry could be automatically generated, the orientation of each laser beam must be given manually to the MPV systems. This paper introduces an approach to quickly generate laser beam orientations for each of the MPV measurement beams using 3D photogrammetric techniques. While the proof-of-concept presented in this paper was performed manually, it could be easily automated using the macro capabilities of the MPV system.

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. D. P. Rohe () · B. L. Witt Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_9

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Fig. 9.1 Test setup showing the test article in front of the MPV system (a) as well as the photogrammetry setup showing calibration target and cameras mounted to the tall vertical posts (b). (a) Beam test article with retro-reflective tape at laser positions. (b) Photogrammetry setup

9.2 Demonstration Experiment The process described in this paper was validated using the test setup shown in Fig. 9.1 using a Polytec MPV-800 system with 24 laser heads. The test article is a rectangular beam with retro-reflective tape at 18 measurement points. The 24 laser heads were arranged to provide 15 uniaxial and 3 triaxial measurements. Two Canon Powershot G1X point-and-shoot cameras were mounted to tall vertical posts above the test from which they could view the part and laser beams. A stereo camera calibration was performed using a Correlated Solutions 10 × 14 dot 40 mm calibration target.

9.3 Laser Beams in 3D Space A laser beam from an LDV system can be modeled geometrically as a vector in 3D space originating from the laser head and passing through a given measurement location on the object under test. The LDV measures the velocity of the object at the point where the laser beam intersects the part, in the direction along the laser beam. Therefore, if only uniaxial measurements at each measurement point are obtained, the test geometry can be constructed solely from the point where the laser intersects the part and its orientation in 3D space. The situation is slightly more complicated for a triaxial measurements where 3 beams intersect the test article at the same point. Each of the 3 beams measures a different component of the 3D velocity at that point. The transformation between the 3D velocity, [vx , vy , vz ], in the object’s coordinate system and the velocities measured by each laser beam, [v1 , v2 , v3 ], is given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ v1 l1x l1y l1z vx ⎣v2 ⎦ = ⎣l2x l2y l2z ⎦ ⎣vy ⎦ v3 l3x l3y l3z vz

(9.1)

where lij is the cosine of the angle between the ith laser head and the coordinate direction j . The matrix of direction cosines is simply inverted to compute the 3D velocity in the object’s coordinate system from the independent laser beam measurements. Note that each row of the matrix is the unit vector of the laser beam in the xyz coordinate system. Therefore, for both uniaxial and triaxial measurements, the vector formed by the laser beam is the key to creating a test geometry.

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9.4 Determining Laser Geometry Using Photogrammetry Two points are sufficient to define a vector in 3D space, so for this work, the positions of two points on each laser beam will be found, from which the laser vector can be computed. One of the points will be the position of the laser spot on the part, which will define the measurement point position in space as well as aid in defining the laser orientation. The second point on each laser beam was created by placing a large plane (in this case the calibration target) in front of the part; the position of each laser spot on this intermediate plane provides a second point on the vector. Note that if the part was complex, or there wasn’t a camera view that could capture all laser spots simultaneously, the intermediate plane approach could be used twice. However, if test geometry was required, a third measurement would need to be performed to identify the locations of the laser spots on the part, as two planes will not give the location of the measurement point on the part. There are many methods of measuring geometry. The simplest methods may involve a ruler and protractor. Coordinate measuring machines may also be used for more complex geometry; however they are not easily automated. In the case of this work, geometry of the laser beam will be measured using 3D photogrammetric techniques. 3D photogrammetry involves capturing images of a scene using a pair of stereo-calibrated cameras. The stereo camera pair needs to initially be calibrated so intrinsic and extrinsic parameters of the stereo camera rig, which define the cameras’ properties and relative positioning to each other, can be computed. Individual features in each stereo image pair can be triangulated to determine a point in 3D space [2]. 3D photogrammetry was chosen for its ease of automation in software. Once the camera images are obtained, the images can be sequentially loaded into memory, the laser spots can be identified, and triangulation can be performed. The taking of the images can also be automated depending on what kind of cameras are available. A pair of webcams, for example, is easily automatable with the Open Computer Vision (OpenCV) library [3]. For this proof of concept, however, a pair of digital cameras were used and the images were captured manually by pressing the shutter buttons.

9.4.1 Camera Calibration Camera calibration was performed using a Correlated Solutions 10 × 14 dot-grid calibration target with 40 mm grid spacing. Figure 9.2 shows the pair of calibration images used for this work. The OpenCV stereoCalibrate function was used to calibrate the stereo pair of cameras. Alternatively, a photogrammetry or digital image correlation software package could be used, though it may be more difficult to automate this step if a commercial package is used.

Fig. 9.2 Calibration images captured by the digital cameras. (a) Left calibration image. (b) Right calibration image

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9.4.2 Laser Spot Identification on Image The first step to automating the triangulation process is to identify a single laser spot in an image. This was performed by closing the laser shutter for all lasers, capturing an “all off” image, then opening the shutter of just one laser at a time and capturing an image of the updated scene. If the lighting in the room is constant and the scene is static, the only difference between the two images should be the appearance of a laser spot. A simple subtraction of the two images clearly identifies the laser spot in the difference image. Figure 9.3 shows this process. The difference image is thresholded to produce a binary black and white image. Contours are found in the image using the OpenCV findContours function, and the center of the contour is treated as the laser spot center. This process is repeated for each laser head individually. Images are captured for each laser head intersecting the object, and again for each laser head intersection the intermediate plane (calibration target plate).

9.4.3 Triangulation With camera matrices from the calibration and laser points identified on each image, the points could be triangulated in 3D space. Triangulation was performed using the OpenCV triangulatePoints function, which uses the Direct Linear Transform algorithm from [2]. After triangulation, a vector representing each beam can be computed in 3D space using the point triangulated on the intermediate plane image and the point triangulated on the part image. To verify the triangulation, the extracted laser beams can be projected back into image space using the OpenCV projectPoints function, which projects 3D points to the image plane. Figure 9.4 shows this verification step. Note that the vectors produced by triangulation will likely be defined in an arbitrary coordinate system defined by the camera calibration (Fig. 9.5), so it will be useful to transform them to the part coordinate system. The measurement points have just been triangulated in a camera coordinate system, so if the (x, y, z) coordinates of the measurement points are known in the part coordinate system, a transformation between those sets of points can be directly solved for via e.g. [4] or a similar method. If the measurement point coordinates are not known in the part coordinate system, some other features can be used. In the present case, the coordinates of the beam corners are known, so they are used in the transformation. The corner coordinates in the camera coordinate system can be triangulated using the same approach that was done for the laser spots, and then the rigid transformation can be solved for. After transformation, the laser vectors should align with the part. Figure 9.6 shows the laser beams aligned with the part model after transformation into the part coordinate system. The vector definition should be sufficient to create a test geometry consisting of measurement point locations and directions; however, some users may wish to import the laser geometry into other software, which may involve specifying Euler Angles to define the orientation. Euler angles can be readily computed from rotation matrices (see e.g. [5]). However, the unit vector formed by the laser beam consists of only one row or column of the rotation matrix, so it may not be clear

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Fig. 9.4 Laser vectors projected back onto the camera image showing approximate path of laser beam from laser head to part

Fig. 9.5 Raw laser vectors computed from the triangulation. They are defined in the left camera coordinate system, which is arbitrarily defined with respect to the part

how to proceed. One solution to this is to form a reference vector vref that is meaningful to the part or measurement system coordinates. For example if Euler angles are defined using a yaw and pitch rotation, an “up” direction may be meaningful, as it will be the vector about which yaw is defined. If the laser vector vl is defined as the first axis of the coordinate system, the third axis of the coordinate system v⊥ can be defined using the direction of the vector resulting from the cross produce of the laser vector and the reference vector, as the cross produce will produce a vector that is perpendicular to both reference and laser vectors. The laser vector may not be perpendicular to the reference vector, so to produce the true second coordinate system axis, the reference vector should be redefined as the cross product of the third vector and the laser vector vref⊥ . This will result in three orthogonal directions based on the direction of the laser vector, shown schematically in Fig. 9.7, which can be assembled into a rotation matrix. Euler angles can then be computed from this matrix. v⊥ = vl × vref

(9.2)

vref⊥ = v⊥ × vl

(9.3)

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Fig. 9.6 Laser beams aligned with the part

z vref

v⊥

vref ⊥

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y vl

Fig. 9.7 Coordinate systems reconstructed from a laser vector

9.5 Conclusions This paper presented a technique to more accurately estimate laser angles from a MPV test setup. It involves highlyautomatable photogrammetry process that could be seamlessly integrated into the MPV workflow via software macros. As the authors only had a limited time with an MPV system, a quick proof-of-concept was instead performed. The technique was demonstrated on a beam test setup, and laser beams were accurately reproduced and plotted with a model of the part.

References 1. Castellini, P., Martarelli, M., Tomasini, E.P.: Laser doppler vibrometry: development of advanced solutions answering to technology’s needs. Mech. Syst. Signal Process. 20, 1265–1285 (2006) 2. Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2003) 3. Bradski, G.: The OpenCV library. Dr. Dobb’s J. Softw. Tools. 25, 120–125 (2000) 4. Arun, K.S., Huang, T.S., Blostein, S.D.: Least-squares fitting of two 3-D point sets. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 698–700 (1987) 5. Eberly, D.: Euler angle formulas. Technical report, Geometry Tools, Redmond, 98052 (Mar 2014)

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Daniel P. Rohe Dan got his Master’s degree at the University of Wisconsin. For the last 7 years, he has been at Sandia National Laboratories, with the last 6 being in a testing role in Experimental Structural Dynamics. Dan’s research interests lie at the intersection of test and analysis, currently working to use models to improve optical testing.

Chapter 10

Photogrammetry-Based Structural Damage Detection by Tracking a Laser Line Y. F. Xu

Abstract Researches on photogrammetry have received tremendous attention in the past few decades. In this paper, a photogrammetry-based structural damage detection method is developed, where a visible laser line is projected to a surface of a structure, serving as an exterior feature to be tracked. A laser-line-tracking technique is proposed to track the projected laser line on captured digital images. Modal parameters of a target line corresponding to the projected laser line can be estimated by conducting experimental modal analysis. By identifying anomalies in curvature mode shapes of the target line and mapping the anomalies to the projected laser line, structural damage can be detected with identified positions and sizes. An experimental investigation of the damage detection method was conducted on a damaged beam. Modal parameters of a target line corresponding to a projected laser line were estimated, which compared well with those from a finite element model of the damaged beam. Experimental damage detection results were validated by numerical ones from the finite element model. Keywords Structural damage detection · Photogrammetry · Vision-based method · Laser-line-tracking technique · Experimental modal analysis

10.1 Introduction Vibration-based damage detection is one application of structural dynamics. An assumption of the application is that changes in structural properties of a structure, such as mass and stiffness, due to occurrence of damage can cause changes in its modal parameters, including natural frequencies, modal damping ratios and mode shapes. Inversely, one can detect the damage by identifying and processing the changes in modal parameters. In practice, workability of a vibration-based damage detection method largely depends on accuracy of modal parameter estimation. Recent advances of camera and computer technologies have benefited usage of photogrammetry for structural dynamics and its application to vibration-based damage detection. By tracking movements of exterior features of a structure that appear on digital images captured by cameras, its full-field displacement and deformation can be measured in a fully non-contact manner, and its modal parameters can be estimated for damage detection. In the field of experimental structural dynamics, digital image correlation (DIC) techniques have been used to measure vibration and estimate modal parameters [1–3, 10, 16, 19]; they have also been used for structural damage detection. Ghorbani et al. [6] generated detailed cracked maps of a full-scale masonry wall based on its surface deformation fields measured by use of a three-dimensional DIC technique. Seguel and Meruane [13] identified debonding in an aluminum honeycomb sandwich panel based on its full-field vibration shapes measured by use of a three-dimensional DIC system. Before applying a DIC technique, a surface of a structure is usually prepared by painting a high contrast speckle pattern so that its digital images captured at different instants can be compared to track deformation of the surface. When certain locations of a structure instead of a surface are of interest, one can attach visible targets to the structure and displacement of the locations can be obtained by tracking the attached targets [4, 5, 20]. Lecompte et al. [8] developed a point-tracking technique to measure static deformation of a cracked concrete beam and detect its cracks, where infrared light-emitting diodes (LEDs) were attached to the beam and served as targets. Compared with results from a DIC system, the density of displacement measured by use of the point-tracking technique was low as it depended on the number of attached LEDs. Wahbeh et al. [17] mounted LEDs

Y. F. Xu () Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_10

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to a structure and measured its two-dimensional vibration. The LEDs that served as targets were tracked by locating their centers on each captured digital image. Rucka and Wilde [12] attached circular points to a lateral surface of a cracked beam that underwent static deformation, which was measured by tracking the attached points on captured digital images. Wavelet transforms of the measured deformation were used to detect the crack of the beam. Feng and Feng [4] tracked vibration of multiple black dots attached to a lateral surface of a beam and the tracked vibration compared well with those measured by use of laser displacement sensors and accelerometers. In this work, a photogrammetry-based structural damage detection method is developed. A digital camera is used to capture a series of digital images of a visible laser line projected to a surface of a structure and a target line corresponding to the projected laser line is tracked. Modal parameters of the target line are estimated by use of the experimental modal analysis technique, and positions and sizes of structural damage can be identified based on estimated mode shapes of the target line. The developed structural damage detection method was experimentally investigated on a damaged cantilever beam. Modal parameters of a target line estimated by the experimental modal analysis technique were compared with those from a finite element model of the beam. Experimental damage detection results were validated by those from the finite element model.

10.2 Methodology 10.2.1 Laser-Line-Tracking Technique A three-dimensional schematic of the laser-line-tracking technique is shown in Fig. 10.1a. In this technique, a digital camera is facing a surface of a beam and a field-of-view (FOV) plane is formed on a portion of the beam surface. A line laser is projecting a straight visible laser line to the beam surface along the length of the beam. The portion of the projected laser line that lies on the FOV plane is termed as the FOV laser line. The line laser is placed at a position with an orientation so that the plane, which is defined by the line laser and FOV laser line, is perpendicular to the beam surface. Based on a pinhole perspective imaging model of a camera [15], an image plane can be defined in front of the lens of the camera. A target line corresponding to the FOV laser line is formed on the image plane. It is the target line along with its background mapped to the image plane that appears on digital images captured by the camera. As shown in Fig. 10.1b, when the beam undergoes transverse deformation, the target line corresponding to the FOV laser line will undergo in-plane deformation on the image plane, i.e., on a captured digital image. An intensity profile of a target line corresponding to a FOV laser line resembles a one-dimensional Gaussian distribution [14]: − I˜ (y) = Ae

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(10.1)

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where y denotes the y-coordinate of a two-dimensional Cartesian coordinate system on an image plane, A is the amplitude of the distribution, y˜0 is the y-coordinate of the center of the intensity profile, and σ determines the width of the intensity profile; I attains its maximum value when y = y0 . Note that the x- and y-axes of the coordinate system are in the horizontal and vertical directions of the image plane, respectively. When the centers are tracked along x-axis of the image plane, a gray-gravity rule [9] can be used, where y0 associated with an intensity profile is calculated by y0 (x) =

 [I (x, y) × y] y

(10.2)

where I (x, y) denotes intensity values at (x, y) and  denotes summation over all pixels of the intensity profile at x along y-axis. A laser-line-tracking technique is proposed that a FOV laser line be tracked by tracking centers of intensity profiles of its corresponding target line on a captured image.

10.2.2 Experimental Modal Analysis of a Target Line When the beam in Fig. 10.1b undergoes transverse vibration, the target line will undergo in-plane vibration on digital images captured by the camera, and one can track the corresponding vibrating target line by use of the gray-gravity rule. Assuming that the beam is a linear structure and its transverse vibration has a small amplitude, the time history of the tracked target line can be represented by a linear combination of mode shape vectors of the target line: y¯0 (x, t) =

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where J is the number of modes of the target line, φj is the j -th mode shape vector of the target line with φj (x) being its entry at x, and qj is the j -th modal coordinate with qj (t) being its value at time t. When the captured digital images have p × q pixels with p and q denoting numbers of pixels along x- and y-axes, respectively, φ j is a q-dimensional vector if the target line spans the length of the image plane. Assuming that the vibration of the beam is solely caused by excitation fs applied to a point on the beam, a generalized frequency response function between y¯0 at x and fs can be expressed by H (x, ω) =

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(10.4)

where Y¯0 and Fs are Fourier transforms of y¯0 and fs , respectively, and ω denotes a circular frequency. Modal parameters of the target line can be estimated by analyzing H with a modal analysis algorithm.

10.2.3 Structural Damage Detection Technique A curvature mode shape is the second-order derivative of a mode shape of a structure and damage can introduce local anomalies to curvature mode shapes [11]. On digital images captured by the camera in the setup of the laser-line-tracking technique, the local anomalies in curvature mode shape of the structure can be mapped to corresponding curvature mode shapes of the target line. Inversely, identifying the local anomalies in the curvature mode shapes of the target line can assist in detecting the damage by mapping the local anomalies to the FOV laser line. A non-model-based damage detection technique by Xu et al. [18] is used here to identify local anomalies in curvature mode shapes of a target line, by mapping which locations and sizes of damage in a structure along a corresponding FOV laser line can be identified. An overview of the damage detection technique is provided here and more details of the technique can be found in Ref. [18]. An entry of a curvature mode shape vector of a target line at x can be calculated using a central difference scheme: φj

(x) =

φj (x + h) − 2φj (x) + φj (x − h) h2

(10.5)

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where a prime denotes spatial derivative and h is the step size of φj . It is shown by Xu et al. [18] that a mode shape from a polynomial that fits a mode shape of a damaged structure can well approximate that of a corresponding undamaged structure, provided that the undamaged structure is geometrically smooth and made of materials that have no stiffness discontinuities. By comparing a curvature mode shape vector of a target line with that from a polynomial that fits the corresponding mode shape of the target line, a curvature damage index (CDI) is defined and its value at x can be expressed by 2  p

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where φj is a mode shape vector from a polynomial that fits φj ; φj (x) denotes the entry of φj at x. Locations and sizes of the damage can be identified by mapping neighborhoods with high CDI values to the FOV laser line. A polynomial fit can be expressed by φ (x) = p

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where m is the number of measurement points of a mode shape to be fit. A proper order of the polynomial fit is two plus the least k value, with which a fitting index fit (k) =

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(10.9)

is larger than 90%, where RMS (·) denotes a root-mean-square of a vector. When multiple mode shapes are available, one can identify the damage in neighborhoods with consistently high CDI values corresponding to different mode shapes. An auxiliary CDI is defined to assist identification of the neighborhoods, which can be expressed by δa (x) =  δˆj (x)

(10.10)

where  denotes summation over all available mode shapes and δˆj is a normalized CDI associated with δj in Eq. (10.6) with the maximum value of 1.

10.3 Experimental Investigation 10.3.1 Experimental Setup Experimental investigations of the laser-line-tracking, experimental modal analysis and damage detection techniques were conducted on a damaged cantilever beam. Dimensions of the beam and experimental setup of the investigations are shown in Fig. 10.2a, b, respectively. One end of the beam was clamped by a bench vice to simulate a fixed boundary. The damage was in the form of a machined area with thickness reduced by 20% on one side of the beam, as shown in Fig. 10.2c. A visible laser line was projected to the intact surface of the beam by use of a class-2 line laser, Bosch GLL 50; the projected laser line is shown in Fig. 10.2d. A series of digital images with 64 × 1024 pixels were captured with a frame rate of 500 fps by the camera. An impact hammer, PCB 086C05, was used to excite the beam on P in the form of single impacts, and the location of P is shown in Fig. 10.2a. When the camera captured images, the illumination environment was a dark indoor condition

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Fig. 10.2 (a) Dimensions of the damaged aluminum cantilever beam, (b) a setup of the beam for experimental investigations, (c) damage in the form of a machined area on a side of the beam and (d) a projected laser line along the length of the beam

without any external light sources. Hence only a target line corresponding to a FOV laser line, which was the portion of the project laser line, existed on the captured digital images. The position and length of the FOV laser line in the experimental setup are shown in Fig. 10.2a.

10.3.2 Experimental Modal Analysis Results Experimental modal analysis was conducted on the target line with single impacts to P on the target line. Single impacts fs and y¯0 of eight sampling periods were measured and the duration of each sampling period was 32 s. Sampling frequencies of fs and y¯0 had the same value as the frame rate of the camera, i.e., 500 Hz. Time histories of fs and y¯0 at x = 512 pixel in the first sampling period are shown in Fig. 10.3a, b, respectively. The single impact fs had non-zero values after its occurrence and y¯0 at x = 512 pixel did not decay to zero within the sampling period. A rectangular window was applied to fs in each sampling period so that its values before and after occurrence of an impact were zero; a exponential window was applied to y¯0 in each sampling period so that its amplitudes at all x-coordinates became almost zero at the end of a sampling period. Windowed fs and y¯0 in the first sampling period are shown in Fig. 10.3c, d. FRFs calculated from H1 formulation [7] using windowed fs and y¯0 at x = 512 pixel in the first one, two, four and eight sampling periods are shown in Fig. 10.4a. Noise floors of the FRFs were lowered by increasing the number of sampling periods. Coherence functions associated with the FRFs in Fig. 10.4a using windowed fs and y¯0 at x = 512 pixel in the first two, four and eight sampling periods are shown in Fig. 10.4b. Relatively high coherence function values could be found in neighborhoods of frequencies where four amplitude peaks existed in the FRF calculated using measured fs and y¯0 in the eight sampling periods, indicating the four frequencies could be natural frequencies of the target line and the natural frequencies could be accurately estimated when only the FRFs at x = 512 pixel were available. FRFs at x = 800 pixel in the first one, two, four and eight sampling periods are shown in Fig. 10.4c. Similar to the FRFs in Fig. 10.4a, noise floors of the frequency response functions were lowered by increasing the number of sampling periods. In coherence functions associated with the FRFs in Fig. 10.4d, relatively high coherence function values could be found in neighborhoods of frequencies

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where four amplitude peaks existed in the FRFs calculated using measured fs and y¯0 in the eight sampling periods shown in Fig. 10.4c, while relatively low coherence function values could be found in the neighborhood of frequencies where an additional amplitude peak of the FRF existed. It was indicated that the five frequencies could be natural frequencies of the target line, which could be accurately estimated; however, the mode shape corresponding to the additional mode might not be accurately estimated. In order to better identify modes of the target line, a composite FRF, i.e., the sum of magnitudes of available FRFs, was calculated and shown in Fig. 10.4e, where six peaks could be identified. A total of 1024 FRFs were analyzed using an experimental modal analysis algorithm, PolyMax in LMS Test.Lab v17, to estimate modal parameters of the target line, including natural frequencies and mode shapes. The estimated natural frequencies are listed in Table 10.1a. A finite element model of the beam was constructed with the elastic modulus and Poisson’s ratio of the beam being 68.5 GPa and 0.3, respectively. The first six natural frequencies of the beam from the finite element model are listed in Table. 10.1a to compare with those from the experimental modal analysis of the target line. The first five natural frequencies of the target line compared well with those from the finite element model. Note that the sixth measured natural frequency of the target line was 241.40 Hz and had a difference of −6.3%. The reason was that the sampling frequency of y¯0 was 500 Hz, the associated Nyquist frequency was 250 Hz and the actual sixth natural frequency of the target line was over 250 Hz. By unfolding the measured sixth natural frequency of the target line with respect to Nyquist frequency of 250 Hz, one

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Table 10.1 (a) Comparison between natural frequencies of the target line from the experimental modal analysis (experimental) and those from a finite element model of the beam (numerical) and (b) entries of the MAC matrix in percent between the first six mode shapes of the target line from the experimental modal analysis and those from the finite element model along the FOV laser line (a) Mode 1 2 3 4 5 6 Mode 1 2 3 4 5 6

Experimental (Hz) 3.10 18.71 54.86 103.96 177.50 241.40 1 99.89 73.52 10.10 19.16 11.63 2.87

Numerical (Hz) 3.08 18.86 54.40 103.96 176.01 257.52 (b)

2 76.48 99.97 2.54 37.16 0.39 0.00

3 10.57 1.60 99.80 0.97 78.21 0.58

Difference (%) 0.6 −0.8 0.8 0.0 0.8 −6.3

4 17.24 33.90 0.20 99.88 0.01 58.18

5 11.37 0.09 73.92 0.64 99.27 0.23

6 2.29 0.07 0.01 55.41 0.20 94.92

Table 10.2 Proper orders of polynomials that fit measured mode shapes of the target line (experimental) and those from the finite element model along the FOV laser line (numerical) Mode Experimental Numerical

1 5 5

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could obtain an unfolded natural frequency of 258.6 Hz, which had a difference of 0.4%, compared with the sixth natural frequency from the finite element model. Mode shapes from the finite element model of the beam along the FOV laser line are shown in Fig. 10.5, and those of the target line were mapped to the beam along the FOV laser line and shown in Fig. 10.5. A modal assurance criterion (MAC) matrix between the mode shapes from the finite element model and those of the target line is listed in Table 10.1b. Diagonal entries of the MAC matrix were all over 94%, indicating that the mode shapes of the same modes compared well with each other. Some off-diagonal entries of the MAC matrix had relatively large values since the FOV laser line was on a portion of the beam and some mode shapes of the beam along the FOV laser line were similar. Use of the sixth mode shape for damage detection of the beam would be excluded hereafter, though the diagonal entry of the MAC matrix associated with the sixth mode shape of the target line and that on the FOV laser line from the finite element model was larger than 94%.

10.3.3 Structural Damage Identification Results The first five mode shapes of the target line were fit by polynomials with proper orders determined based on fit in Eq. (10.9) and the proper orders are listed in Table 10.2. Mode shapes from the polynomial fits were mapped to the beam along the FOV laser line and shown in Fig. 10.5. The mapped mode shapes from the polynomial fits compared well with those of the target line and also well with those from the finite element model of the beam along the FOV laser line. Curvature mode shapes of the target line were calculated and mapped to the beam along the FOV laser line as shown in Fig. 10.6. Anomalies could be observed in the mapped curvature mode shapes by comparing them with those from the polynomial fits. CDIs associated with the first five mapped curvature mode shapes of the target line are shown in Fig. 10.7 and their associated auxiliary CDI is shown in Fig. 10.8. The position and size of the damage could be identified in the neighborhood with high auxiliary CDI values. The first five mode shapes of beam along the FOV laser line from the finite element model were then fit by polynomials with proper orders determined based on fit in Eq. (10.9) and the proper orders are listed in Table 10.2. Curvature mode shapes from the polynomial fits are shown in Fig. 10.6. Similar to the experimental results of the mapped curvature mode shapes of the target line, anomalies can be observed in the curvature mode shapes from the finite element model of the beam by comparing them with those from the polynomial fits. CDIs associated with the first five mode shapes from the finite element model of beam along the FOV laser line are shown in Fig. 10.7 and their associated auxiliary CDI is shown in Fig. 10.8.

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The position and size of the damage could also be identified in the neighborhood with high auxiliary CDI values. More importantly, the auxiliary CDI associated with the mapped mode shapes of the target line and mode shapes of the beam from the finite element model along FOV laser line compared well with each other, especially in the neighborhood of the damage.

10.4 Conclusions A photogrammetry-based structural damage detection method is developed and experimental investigations of the method were conducted on a damaged beam. In the developed method, a digital camera is used to measure vibration of a structure by tracking a projected laser line. Structural damage can be detected by identifying anomalies in mapped curvature mode shapes of a target line that corresponds to the projected laser line. An experimental modal analysis technique is proposed to estimate modal parameters of a target line on a series of digital images captured by a camera. Experimental damage detection results compared well with numerical ones from the finite element model.

Acknowledgment The author is grateful for the faculty startup support from the Department of Mechanical and Materials Engineering at the University of Cincinnati.

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6. Ghorbani, R., Matta, F., Sutton, M.A.: Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation. Exp. Mech. 55(1), 227–243 (2015) 7. Heylen, W., Sas, P.: Modal analysis theory and testing. Katholieke Universteit Leuven, Departement Werktuigkunde (2006) 8. Lecompte, D., Vantomme, J., Sol, H.: Crack detection in a concrete beam using two different camera techniques. Struct. Health Monitor. 5(1), 59–68 (2006) 9. Li, Y., Zhou, J., Huang, F., Liu, L.: Sub-pixel extraction of laser stripe center using an improved gray-gravity method. Sensors 17(4), 814 (2017) 10. Molina-Viedma, A., Felipe-Sesé, L., López-Alba, E., Díaz, F.: High frequency mode shapes characterisation using digital image correlation and phase-based motion magnification. Mech. Syst. Signal Process. 102, 245–261 (2018) 11. Pandey, A., Biswas, M., Samman, M.: Damage detection from changes in curvature mode shapes. J. Sound Vib. 145(2), 321–332 (1991) 12. Rucka, M., Wilde, K.: Crack identification using wavelets on experimental static deflection profiles. Eng. Struct. 28(2), 279–288 (2006) 13. Seguel, F., Meruane, V.: Damage assessment in a sandwich panel based on full-field vibration measurements. J. Sound Vib. 417, 1–18 (2018) 14. Steger, C.: Unbiased extraction of lines with parabolic and gaussian profiles. Comput. Vis. Image Understand. 117(2), 97–112 (2013) 15. Sutton, M.A., Orteu, J.J., Schreier, H.: Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications. Springer Science & Business Media, New York, NY, USA. (2009) 16. Trebuˇna, F., Hagara, M.: Experimental modal analysis performed by high-speed digital image correlation system. Measurement 50, 78–85 (2014) 17. Wahbeh, A.M., Caffrey, J.P., Masri, S.F.: A vision-based approach for the direct measurement of displacements in vibrating systems. Smart Mater. Struct. 12(5), 785 (2003) 18. Xu, Y., Zhu, W., Liu, J., Shao, Y.: Identification of embedded horizontal cracks in beams using measured mode shapes. J. Sound Vib. 333(23), 6273–6294 (2014) 19. Yu, L., Pan, B.: Single-camera high-speed stereo-digital image correlation for full-field vibration measurement. Mech. Syst. Signal Process. 94, 374–383 (2017) 20. Zeinali, Y., Li, Y., Rajan, D., Story, B.: Accurate structural dynamic response monitoring of multiple structures using one CCD camera and a novel targets configuration. In: Proceedings of the 11th International Workshop on Structural Health Monitoring, Palo Alto, pp. 12–14 (2017) Y. F. Xu is currently an assistant professor in the Department of Mechanical and Materials Engineering at the University of Cincinnati. He obtained his PhD in mechanical engineering from the University of Maryland Baltimore County.

Chapter 11

Measuring Aero-Engine Pipe Vibration with a 3D Scanning Laser Doppler Vibrometer Christoph W. Schwingshackl

Abstract The vibration of accessories in an aero engine represents a major problem in the design, since failing accessories are a source of aero engine shut downs. Understanding the vibration behaviour of the accessories is challenging. Their manufacture, assembly and maintenance introduce a large amount of uncertainty to the actual state of the system and they are often of a very complex shape. This can lead to highly three-dimensional operating deflection shapes, with potentially strong nonlinear dynamic behaviour due to a multitude of joints. The above makes the accurate capture of the vibration response of supply pipes of an aero engine quite challenging. New 3D measurement techniques, such as 3D Scanning Laser Doppler Vibrometers, can help to obtain a detailed map of the complex motion such systems experience in operation, but their tightly curved and small diameter pipes can present many challenges to an experimental setup. This paper will discuss the vibration measurement of a combustion chamber outer casing response with a 3D SLDV system, particularly focusing on some lessons learned during setup. Keywords 3D SLDV · Casing vibration · Accessories · Pipes · Operating deflection shape

11.1 Introduction Measuring the vibration response of large scale assembled structures regularly presents a challenge, since traditional vibration measurement techniques, such as accelerometers or 1D Scanning Laser Doppler Vibrometers (SLDV) are more suited for simple geometries. For more complicated geometries with often localised three dimensional vibration responses, either a very sophisticated experimental setup and a long experimental campaign is needed to capture the response, or one of the more recent full field 3D approaches (3D-Scanning LDV [1, 2], 3D-Electronic speckle interferometry [3, 4], 3D-Digital Image Correlation [5] . . . ) can be used to capture the full response. Even though the above advanced measurement systems provide advanced capabilities to measure 3D response, their application to real industrial test cases can be quite challenging. To enable measurements of structures with complex geometries, the Vibration University Technology Centre at Imperial College London has recently bought a POLYTEC PSV500 Xtra 3D SLDV system. As a first test case a Combustion Chamber Outer Casing (CCOC) was selected, which in the past has proven very challenging for capturing accurate operating deflection shapes (ODS). This paper will discuss the chosen setup for the measurement of a highly three dimensional test structure with several layers of pipes on the casing on top of each other. Some challenges and limitations to the setup will be presented, together with some very promising initial measurement results.

11.2 Test Setup The CCOC in Fig. 11.1a was chosen as the test subject due to a series of challenges for vibration measurements. It (i) provides a strongly curved surface, (ii) the main fuel supply pipe and the fuel injector pipes add three dimensionality to the structure, and (iii) from previous measurement attempts it was known that the assembly shows a highly three dimensional, sometimes

C. W. Schwingshackl () Department of Mechanical Engineering, Imperial College London, London, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_11

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Fig. 11.1 (a) Overall test setup, and (b) shaker excitation

localised, vibration response. The casing was placed on three small rubber pads on a wooden bench, which allowed the structure to vibrate in a nearly unconstrained way. A small Data Physics V4 shaker was attached to the side of the structure (see Fig. 11.1b) to provide excitation to the system, where the excitation force was measured via a PCB force transducer, placed between the stinger and the casing. The 3D-SLDV system was set up approximately 1.3 m away from the front end of the casing, and the three heads were aligned in a typical triangular configuration as shown in Fig. 11.1a, with the zeroed laser beams approximately meeting in the centre of the area of interest. The chosen stand-off distance was thereby a compromise between a large enough field of view and required measurement accuracy.

11.3 Measurement Setup After completing the experimental setup, a considerable amount of time was spent on calibrating the 3D-SLDV system and generating a measurement grid. The calibration procedure ensures that all three LDVs line up correctly at all measurement locations. Given the strong curvature of the CCOC this task proved to be quite tricky, since the visible-light range finding laser led to unreliable distance data once the angle of incident went beyond approximately ±22◦ . Unfortunately, this limited the possible measurement field to a total of 45◦ . Since the aim of the setup was to explore the capability of the system with no further surface treatment (e.g. reflective tape) the setup was continued for the 45◦ field of view. Next a coordinate system was defined. Ideally a polar coordinate system should be used for the cylindrical casing structure, with its origin on the centre line, but for this initial test a simpler cartesian coordinate system was selected, that was aligned with the front of the casing. Once the calibration process was completed, the actual measurement gird had to be defined. It was found that a combination of measurement grid options was required to cover the area of interest. The casing was thereby mainly covered with rectangular meshes, also some line elements were used close to the edges of the measurement field, while the fuel supply pipe and the pigtail pipes were modelled with line elements only. The final 2D measurement grid, containing approx. 250 points, can be seen in Fig. 11.2a. The range finding laser was then used to identify the position of each measurement point in 3D space, which turned out to be particularly challenging for the injector pipes, due to their small radii and strong curvatures. Shadowing effects complicated the measurement grid setup further, since certain points behind the pipes could not be accessed by all three laser beams. Several modifications to the original mesh eventually led to the 3D measurement grid shown in Fig. 11.2b. The curvature of the casing and the positions of the pipe work were captured correctly in this final mesh, but certain areas on the casing, especially underneath the fuel supply pipes could not be accessed due to the shadowing effect.

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Fig. 11.2 Measurement gird as (a) defined in 2D and (b) identified by the range finder in 3D

Fig. 11.3 Operating Deflection Shapes for three example modes

11.4 Vibration Measurements An initial single point frequency sweep identified a very high frequency density on the casing structure. Three frequencies where then selected to measure the ODS of the casing, fuel manifold, and the injector pipes with the measurement setup from Fig. 11.2b. The resulting plots in Fig. 11.3 for the three selected resonance frequencies demonstrates the excellent capability of the 3D SLDV system to get high quality data not only for the strongly reflective and curved casing, but also the twisted, highly three dimensional fuel supply pipes. The extracted ODS very clearly show casing and pipe dominated modes. Particularly the fuel manifold and the fuel injector pipes move with a strong 3D motion which was very difficult to capture accurately with traditional measurement techniques. Each full ODS measurement was obtained in around 15 min.

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11.5 Summary The presented results, obtained with the new 3D-SLDV system, highlight its ability to measure complicated, curved structures with attached accessories, in a reasonable amount of time. Quite a few lessons were learned in setting up the tests, particularly concerning the limits of the range finding laser on strongly curved structures, and the shadowing effect in case of multilayered structures. Acknowledgements A special thanks to Rolls-Royce plc and the EPSRC for the support under the Prosperity Partnership Grant “Cornerstone: Mechanical Engineering Science to Enable Aero Propulsion Futures”, Grant Ref: EP/R004951/1 for supporting this work and allowing to publish its outcomes.

References 1. Maguire, M., Sever, I.: Full-field strain measurements on turbomachinery components using 3D SLDV technology. AIP Conference Proceedings, v 1740, p 080001 (12 pp.), 28 June 2016 2. Alarcon, D. J.; Sampathkumar, K.R., Paeschke, K., Mallareddy, T.T., Angermann, S., Frahm, A., Ruether-Kindel, W., Blaschke, P.: Modal model validation using 3D SLDV, geometry scanning and fem of a multi-purpose drone propeller blade. In: Proceedings of the 35th IMAC, A Conference and Exposition on Structural Dynamics, v 8B, pp. 13–22 (2017) 3. Zanarini, A.: Full field ESPI vibration measurements to predict fatigue behaviour. In: Proceedings of the ASME International Mechanical Engineering Congress and Exposition, IMECE 2008, v 1, pp. 165–174 (2009) 4. Krupka, R.: Application of ESPI techniques for the study of dynamic vibrations. In: Proceedings of SPIE – The International Society for Optical Engineering, v 5503, pp. 79–84 (2004) 5. Reu, P.L., Rohe, D.P., Jacobs, L.D.: Comparison of DIC and LDV for practical vibration and modal measurements. In: Mechanical Systems and Signal Processing, v 86, pp. 2–16, 1 March 2017 Dr. Christoph W. Schwingshackl is a Senior Lecturer in the Dynamics Group at Imperial College London where he leads the Structural Dynamics Team of the Vibration University Technology Centre, sponsored by Rolls-Royce. His main research interest is structural dynamics of aircraft engines with a focus on the experimental and analytical analysis of nonlinear friction damping and rotor dynamics.

Chapter 12

A “Mechanical” Vision of Image-Based Identification methods in Structural Dynamics Paolo Castellini and Emanuele Zappa

Abstract Image-based measurements are nowadays well-established techniques, and their further development seems endless considering the fast evolution of cameras, computer hardware and software. New algorithms for structural dynamics are continuously developed and, frequently, made available in the computer community. On the other hand, experimental set-up is still an issue. This tutorial aims to focus the attention to this aspect, and will cover: • • • •

The basic of the experiment design; The selection of the hardware set-up (object preparation, lighting, cameras, acquisition parameters); Limits of the acquisition and uncertainty of the measure; New acquisition technologies

The proposed discussion gives basic and advanced considerations to improve the quality of the processing results and images obtained can be elaborated with different processing algorithms. Keywords Image processing · Cameras · Optics

12.1 Introduction In recent days image processing techniques made impressive development with algorithms very efficient and accurate [1–3]. On the other hand, it is necessary to avoid forgetting the complete pipeline of the measurement procedure, taking care the process of image formation and acquisition. As in every multistep process, the care in the early steps can significantly improve the finale results, or, mainly, errors in these steps made impossible an accurate processing of information. Finally, if quantitative data must be extracted, system calibration and measurement uncertainty are issue to much frequently underestimated. Hereafter, we will walk the complete process of image formation, following the path of the light, the real actor of the image acquisition.

12.2 Illumination Illumination is a key factor. In large structures, it is necessary to exploit natural illumination, but it can represent a serious problem due to the significant change of it among different experiment, or among different frame in the same experiment (as shown in Fig. 12.1). Feature recognition can become a serious problem.

P. Castellini () Department of Industrial Engineering and Mathematical Science, Polytechnic University of Marche, Ancona, Italy e-mail: [email protected] E. Zappa Department of Mechanics, Polytechnic of Milan, Milan, Italy © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_12

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Fig. 12.1 Natural illumination: changes between different frames

(a) #1 - SEM DIC Chall. Samp. 13

(b) #2 - SEM DIC Chall. Samp. 13

(c) #3 - Experimental

(d) #4 - Numerical pattern

Fig. 12.2 Surface preparation for DIC measurement

To overcome such problems, artificial and controlled illumination can be provided. In this case, colour, polarization, intensity, time history, coherence and direction of illumination can be selected and used to highlight specific feature of the image, or to suppress significant disturbs.

12.3 Surface Optical Behaviour and Sample Preparation Illumination light interact with the target surface. Good surfaces (usually named as “cooperative surface”) are those make easier the measurement: an example of it are mat white surfaces. When necessary and possible, surface can be prepared, accordingly with the measurement procedure selected. In Fig. 12.2 some example of surface preparation is provided.

12.4 Optica Setup and Objectives Light scattered by the surface must be collected and focused on the sensing array (the camera). The selection of the objective is very important, because it impact on the final result, but also significantly on the cost of the setup. Different kind of objective can be selected (see in Fig. 12.3 the comparison between conventional and telecentric objectives).

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Conventional Lens

Telecentric Lens

Fig. 12.3 Conventional vs telecentric objective. (Courtesy: Edmund Optics [4])

Fig. 12.4 Acquired images with different blur level

12.5 Image Quality: Focusing and Motion The best instrumentation needs to be correctly adjusted. Focusing and integration time are parameters to be set with maniacal attention. In Fig. 12.4 examples of blur images is provided. In the real world we have also to take into consideration non perfect alignment in camera set-up. Special setup, like those related to the Scheimpflug rule [5], will be discussed.

12.6 New Acquisition Technologies Camera technology is continuously improving. Pixel count is incredibly increasing with reasonably prices, but this can also represent a problem related to the corresponding increase of storage and processing needs. New technologies are now emerging, like smart cameras and event-based cameras [6]. Smart cameras are common cameras with a processing unit embedded. In this way data are processed real-time (or preprocessed and compressed) inside the camera and data flow is considerably reduced. Event-based cameras provided a pixel-per-pixel processing, defining in real-time how and if each pixel needs to be acquired, depending by the observed scene (the event). Those solutions open a new era in image processing.

12.7 Measurement Uncertainty Finally, measurement uncertainty needs to be estimated and managed. Uncertainty budget is a crucial step in experiment design, and in image processing involves the full path, from acquisition to processing. Some cases will be discussed.

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References 1. Baqersad, J., Poozesh, P., Niezrecki, C., Avitabile, P.: Photogrammetry and optical methods in structural dynamics – a review. Mech. Syst. Signal Process. 86, 17–34 (2016) 2. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn. Upper Saddle River, Prentice Hall (2008) 3. Sarrafi, A., Mao, Z., Niezrecki, C., Poozesh, P.: Vibration-based damage detection in wind turbine blades using Phase-based Motion Estimation and motion magnification. J. Sound Vib. 421, 300–318 (2018) 4. https://www.edmundoptics.com/resources/application-notes/imaging/advantages-of-telecentricity/ 5. https://it.wikipedia.org/wiki/Regola_di_Scheimpflug 6. http://rpg.ifi.uzh.ch/research_dvs.html Paolo Castellini received the PhD degree in 1996 at University of Padova, Italy. In 1997, he joined the Mechanical Department of Ancona, as an researcher and in 2011 become Associate Professor. His research interests are related to non-contact mechanical measurement techniques, mainly based on laser and microphone arrays. Emanuele Zappa received the Ph.D. degree in applied mechanics from Politecnico di Milano, Italy, in 2002. In 2001, he joined the Department of Mechanical Engineering, as a Researcher, where he is currently an Associate Professor. He has authored several publications in the field of measurements, with a focus on vision-based techniques: digital image correlation, 3-D structured light scanning, and stereoscopy.

Chapter 13

Modelling of Guided Waves in a Composite Plate Through a Combination of Physical Knowledge and Regression Analysis Marcus Haywood-Alexander, Tim J. Rogers, Keith Worden, Robert J. Barthorpe, Elizabeth J. Cross, and Nikolaos Dervilis

Abstract The use of high-frequency guided waves, such as Rayleigh and Lamb waves (actively) or acoustic emissions (passively), has become increasingly prominent in engineering applications, particularly for structural health monitoring (SHM) and, more traditionally, non-destructive evaluation (NDE). In comparison to low-frequency analysis, guided waves have the additional benefit of being able to locate damage with finer spatial resolution (controlled by the diffraction limit). This paper looks into developing a health-monitoring strategy for fibre-reinforced polymer structures using ultrasonic guided waves (UGWs); part of the remit is to determine a methodology for modelling of UGWs propagation. As fibres within such a material act as a secondary guide for these waves, time-space modelling of the waves is difficult. Presented here is a novel methodology utilising a physics-incorporated, data-driven model to determine the feature-space of UGW propagation. The method uses Gaussian processes and in this paper is made a comparison between different kernel-based methods. By careful consideration of these machine learning techniques, more robust and generalised models can be generated. Keywords Guided waves · Model generation · Machine learning · Composite modelling · Health monitoring

13.1 Introduction An increasing trend in engineering applications is to use complex materials such as composites and porous materials, a primary benefit being their increased strength to weight ratio, amongst other advantages. There have been advances in the modelling of these materials, such as in terms of fatigue damage [1]. A comparative review of state-of-the-art modelling methodologies for damage in composites has been done by Orifici et al. [2], in which they discuss many issues such as length scales and implicit modelling. In the fields of non-destructive evaluation (NDE) and structural health monitoring (SHM), the use of high-frequency stress waves are becoming increasingly adopted, due to their ability to detect damage on a small scale. One such method is through the use of ultrasonic guided waves (UGWs), these are waves which propagate through a structure that acts as a wave-guide, such as plates. Time-space modelling of guided waves can be solved analytically for isotropic materials [3] or using numerical methods for layered composites [4]; however, the complexity of these calculations becomes substantial when modelling the interaction with damage or for a fibre-composite. One primary issue with fibrecomposite materials is the secondary guide behaviour of the fibres within the material, which for quasi-isotropic materials, it may be possible to disregard. Detailed information on guided waves and their use in SHM is well described by Worden et al. [5] and Rose [6], though a short introduction to the behaviour and characteristics of these waves is given here. A key factor of guided waves in plates is the variety of wave modes that propagate within a single wave-packet, which are split into two main types; symmetric and antisymmetric, each of which has an increasing number of modes with increasing frequency-thickness. It is also important to note that the separation of wave modes can only occur along the symmetry axis in anisotropic media [7]. Mook et al. [8] showed a key effect of damage on UGWs as they propagate through a carbon-fibre reinforced polymer (CFRP) – the conversion of wave modes when interacting with local inhomogeneities. This knowledge can be useful for NDE and SHM strategies, by modelling each wave mode as it propagates, it is possible to detect damage through changes in this behaviour. A crucial characteristic of guided waves in fibre composites is the phenomena of Continuous Mode Conversion (CMC), as

M. Haywood-Alexander () · T. J. Rogers · K. Worden · R. J. Barthorpe · E. J. Cross · N. Dervilis Dynamics Research Group, Department of Mechanical Engineering, University Of Sheffield, Sheffield, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_13

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shown by Mook et al. [8] and Willberg et al. [9], where the boundaries of layers or weaves cause conversion of S0 modes to A0 in frequent enough occurrences that they can be considered continuous along the propagation path. This leads to obvious difficulties when trying to model the behaviour of UGWs in composites with fibrous layers. Along with the increase in available computing power, recent years have shown continuously more adoption of machine learning techniques applied to engineering modelling problems in order to overcome barriers in physic-based modelling, though not always successfully. Many of these methods are applied to a dataset to generate a model of certain behaviour; however, this can limit application; it is only reasonable to assume this model fits for this material/structure, and generalisation cannot be implemented so changes in design or environment would require retraining of the model. By implementing physical knowledge to constrain model generation these issues can be overcome [10]. The work presented here shows the early stages of development of a methodology for modelling the feature-space of a guided wave as it propagates through a fibre-matrix composite through the use of physics-incorporated, data-driven techniques. Once a feature-space model of selected features that are affected by inhomogeneities is generated, detection of damage can become possible.

13.2 Feature-Space Model Generation 13.2.1 Experiment Setup Ultrasonic guided waves were excited in a 1 mm thick, 300 × 300 mm carbon-fibre reinforced polymer plate using a piezoelectric transducer (PZT); this was done by inputting a square wave which has temporal width tpw . The location and setup of the PZT on the plate is shown in Fig. 13.1a, the CFRP plate is a fibre weave of layup [90/0/90]s , surrounded by an epoxy matrix. A square pulse was selected to excite a broad range of frequencies up to 1/tpw , allowing multiple wave modes to propagate throughout the material. The surface velocity of the induced bursts was then measured using a scanning laser vibrometer (SLV), with 8314 scan points covering the surface, and 50 averages were taken per scan point in order to increase the signal-to-noise-ratio. At each scan point, a 4ms burst was recorded, including 400 μs of pre-trigger, where the recording start was synchronised with the function generator attached to the PZT. The wave-packet signal measured at each location was then fed through a physics-established feature extraction process, for the purposes of this paper, one feature will be considered; the maximum amplitude of the Hilbert envelope (hm ), which is indicative of the energy of the first antisymmetric mode in the wave-packet. The values extracted from this were then used to create a feature-space map over the surface of the plate, and then fed through the model generation process outlined in Fig. 13.1b, the details of which are described below.

Fig. 13.1 (a) Diagram showing placement of PZT on 300 × 300 mm CFRP plate and locations of mounting holes where M12 bolts were used to fix the plate vertically to the wall and (b) flowchart of model development for the two-dimensional map of ultrasonic wave-packet features from experimental data and physical knowledge

13 Modelling of Guided Waves in a Composite Plate Through a Combination of Physical Knowledge and Regression Analysis Table 13.1 Range of models tested for one-dimensional feature-space model generation, along with basis parameters used for expansion before Bayesian linear regression process

Model basis A(x) = β1 x −1/2 A(x) = β1 x −1/2 + β2 A(x) = β1 x −1/2 + β2 x −1 A(x) = β1 x −1/2 + β2 x

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Basis parameters 1 = [−0.5] 2 = [−0.5, 0] 3 = [−0.5, −1] 4 = [−0.5, 1]

13.2.2 Feature-Space Mapping In order to generate a data-driven, physics-incorporated model of the feature-space from UGW propagation in the material, the experimental data was passed through a two-stage process as outlined in Fig. 13.1b. The first stage uses Bayesian linear regression (BLR) to determine the model which most accurately describes the one-dimensional attenuation of the wavepacket feature. A full description of the method of using BLR is well explained by Murphy [11], and details on how this is applied to a one-dimensional feature space can be seen in [12]. By expansion of the fundamental knowledge of the energy of a wave as it propagates through a material, four models were tested and are outlined in Table 13.1. These models are representative of secondary dissipation of energy of the wave due to additional interactions with the fibre-matrix boundaries. In order to test the generality of the models for propagation paths along or through the fibres, unidirectional datasets were gathered and grouped at {0◦ , 90◦ , 180◦ , 270◦ } and {45◦ , 135◦ , 225◦ , 315◦ } with respect to the fibre orientation. For the purposes of assessing the model the experimental dataset was split into the training set, x and cross-validation set xcv , which consisted of 75% and 25% of the total dataset respectively. There are some important values that result from the BLR approach which are useful in assessing the quality of the model generated, with respect to the data. Two such values are the minimum mean squared error (MMSE) and the cross-validation residual sum-of-squares (RSScv ). For descriptive purposes, MMSE indicates how well the most likely estimate of the mean function fits with the data used to train the model, whilst RSScv indicates how well the model fits with the cross-validation set. Another important value is the estimated noise variance, σ 2 , which gives an indication of the noise within the data stream, given the most likely model calculated. Where X & Xcv are the design matrices containing the polynomial expansion of the vector of propagation distance x for the training and cross-validation set respectively, y & ycv are the target values for the training and cross-validation set, βˆ is the estimate of the weights and n & m are the number of training and cross-validation data points respectively, the definitions of MMSE and RSScv are given by,     100 MMSE = y − Xβˆ y − Xβˆ nvar(y) RSScv

    100 Xcv βˆ − ycv Xcv βˆ − ycv = mvar(ycv )

(13.1)

(13.2)

For the generation of the two-dimensional feature-space model, a Gaussian process (GP) was used; details on this method is described by Rasmussen and Williams [13], but a short outline of the process will be described here. The primary distinction of GPs compared to other techniques is how they generate a distribution of possible functions, rather than simply the best fitting. A second distinction of GPs is that the mapping from one space to another is based on the covariance of the input data, rather than the values themselves. This is done through the use of a kernel, which is applied to the covariance matrix of the input space, and control of properties of the function which maps these spaces is done by selection and tuning of this kernel. Most commonly-used kernels are linear functions of distance metrics between input data, when using such a method the determination of the distance metric is of high importance, but can still present issues, as shown later in this paper. As the propagation of UGWs is circular from a point source, a novel kernel method, following the work of Padonou and Roustant [14], was employed to map from the polar coordinate space to the feature space and compared against model generation using the Cartesian coordinate space. The polar representation allows angular shapes to be better described and inferred in models, as well as unidirectional behaviour to be applied on the radial dimension. Inclusion of physical knowledge is done by simply subtracting the one-dimensional model from the input data along the radial dimension. Once the most accurate one-dimensional feature-space model is determined, this is then used as the physical knowledge input for generation of the two-dimensional feature-space model over the surface of the plate. This is applied by the polynomial expansion of the propagation distance (represented by radius ρ) with the basis of the chosen model .

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13.3 Results 13.3.1 One-Dimensional Model The results of the unidirectional behaviour of the chosen feature can be seen in Fig. 13.2, as well as the results of the model fit for 3 . Only this model is shown for brevity, showing the model chosen to be of best fit from quantitative and qualitative reasons. Table 13.2 shows the weights, noise and model assessment values results for each model fit. Although 2 , 3 & 4 resulted in similar values for MMSE and RSScv , and displayed similar qualities of fit on their graphs, 3 was chosen to be the model of best fit, this is because it appeared to fit the data more closely, as well as the physical explanation of the model is an additional decaying reduction in energy with propagation distance due to the fibre-matrix boundaries. As can be seen in Fig. 13.2a, the feature chosen is very noisy when extracted along the fibres, this is an artefact of the experiment setup; the CFRP weave is a black material with very low reflectivity and so the signal-to-noise ratio is very low from the laser vibrometer. Based on these results, the model A(x) = w1 x −1/2 + w2 x −1 was chosen as the physical input for the two-dimensional model generation.

Fig. 13.2 Results of BLR model generation procedure on unidirectional dataset with (a) 3 applied on propagation in direction of fibres and (b) 3 applied on propagation through fibres Table 13.2 Results of output values from BLR method applied to the four separate models for both along the direction of the fibres, and at 45◦ with respect to the fibres β1

β2

1 2 3 4

5.47e−03 6.448e−03 4.971e−03 5.889e−03

−1.741e−04 2.226e−03 −1.347e−06

1 2 3 4

2.659e−03 3.407e−03 1.886e−03 2.893e−03

−1.394e−04 3.343e−03 −8.913e−07

σ2 Along fibres 6.034e−08 5.504e−08 5.934e−08 5.341e−08 Through fibres 1.343e−08 1.006e−08 9.951e−09 1.112e−08

MMSE

RSScv

8.259e+03 7.858e+03 8.159e+03 7.740e+03

1.371e+04 1.489e+04 1.358e+04 1.545e+04

1.562e+04 1.346e+04 1.339e+04 1.415e+04

2.502e+04 2.064e+04 2.122e+04 2.175e+04

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13.3.2 Two-Dimensional Model The two-dimensional feature-space of the guided waves as they propagate throughout the plate can be seen in Fig. 13.3, as well as the results of various steps/improvements in the model generation process. As can be seen in Figs. 13.2 and 13.3a, the energy of the guided wave is greater when propagating in directions along the fibres. This is an important characteristic to be aware of when considering the modelling stages, as embedding physical properties is key to developing a robust, extendable model with machine learning methods. An interesting point to note is that Fig. 13.3b shows the results of the GP applied to the Cartesian coordinate input space with zero-mean basis; as can be seen the model generated is heavily biased from the raw data, and mimics an ‘out of focus’ copy of the raw feature map in Fig. 13.3a. Importantly, the model generated from the Cartesian input space has replicated the asymmetry in the raw data, this is an artefact of using Gaussian processes to infer models where the physics are best described in the polar coordinate space; this is discussed further in the example

Fig. 13.3 Results of the two-dimensional feature-space of log10 (hm ), for (a) raw feature data, (b) GP model from Cartesian input, (c) GP model from polar input and (d) GP model with physical model incorporated. Data is shown in log scale for purposes of illustration

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simulations of polar coordinates GPs by Padonou and Roustant [14]. In order to combat this issue, a black-box model was inferred with a GP on the polar coordinate space, the results of which can be seen in Fig. 13.3c. It can be clearly seen that this approach generates a much more generalised model, that is still true to the shape of the raw data seen in Fig. 13.3a. The results of implementation of the model 3 , chosen from the BLR process described above, are shown in Fig. 13.3d, which presents an interesting issue; the expected values show a circular ‘banding’ effect on the feature-space map. This is likely to be an issue created by the training and implementation of the one-dimensional model on the same dataset. As the trained model is implied to the GP by subtracting from the mean, the residuals of this which are fed through the GP should represent a noise structure. Thus, the GP will attempt to find structure in unstructured data, leading to the banding effect seen. This can be improved upon by implying physics within the covariance kernels of the GP, rather than subtracting from the raw data. This will allow learning of the scaling factors to of the model alongside the effect of propagation direction.

13.4 Conclusion As computational capabilities and machine learning techniques are becoming increasingly more accessible, the adoption of such methods to solve engineering problems is also becoming more prominent. This can present many underlying issues with the results, such as unreasonable assumptions, lack of extrapolation capability and computational costs. However, by implementing physical knowledge to guide learning, more robust models may be generated which can reduce many of these issues. A barrier in the progression of using guided waves in a structural health monitoring strategy is the difficulty of modelling the behaviour of these waves in complex materials. This paper shows the first steps in the progress of generating a physic-incorporated, data-driven model for the feature-space of guided waves in complex materials. Several characteristics of such a strategy, which must be carefully considered to maintain reasonability, have been discussed. The above work has shown the first steps in developing a methodology for model generation of a two-dimensional feature space for guidedwave propagation in composite materials, and though there is still further improvement required before extrapolation can be deemed feasible, the promise of this method is clear. Further work will be done to improve the model generation process through the combination of kernel design, implementation and model selection. Acknowledgments The authors would like to thank Robin Mills, from the Dynamics Research Group, for providing invaluable knowledge and for facilitating the execution of these experiments. The authors would like to acknowledge the following EPSRC grants that funded this research: EP/R004900/1, EP/R003645/1 and EP/S001565/1.

References 1. Mao, H., Mahadevan, S.: Fatigue damage modelling of composite materials. Compos. Struct. 58, 405–410 (2002) 2. Orifici, A.C., Herszberg, I., Thomson, R.S.: Review of methodologies for composite material modelling incorporating failure. Compos. Struct. 86, 194–210 (2008) 3. Viktorov, I.A.: Rayleigh and Lamb Waves: Physical Theory and Applications. Plenum Press, New York (1967) 4. Adler, E.L.: Matrix methods applied to acoustic waves in multilayers. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 485–490 (1990) 5. Worden, K.: Rayleigh and lamb waves – basic principles. Strain 37, 167–172 (2001) 6. Rose, J.L.: Ultrasonic guided waves in structural health monitoring. Key Eng. Mater. 270–273, 14–21 (2004) 7. Solie, L.P., Auld, B.A.: Elastic waves in free anisotropic plates. J. Acoust. Soc. Am. 54, 50–65 (1973) 8. Mook, G., Willberg, C., Gabbert, U., Pohl, J.: Lamb wave mode conversion in CFRP plates. In: 11th European Conference on Non-Destructive Testing (2014) 9. Willberg, C., Koch, S., Mook, G., Pohl, J., Gabbert, U.: Continuous mode conversion of lamb waves in CFRP plates. Smart Mater. Struct. 21, 075022 (2012) 10. Tulleken, H.J.A.F.: Grey-box modelling and identification using physical knowledge and bayesian techniques. Automatica 29, 285–308 (1993) 11. Murphy, K.P.: Machine Learning: A Probabilistic Perspective. The MIT Press, Cambridge (2012) 12. Haywood-Alexander, M., Worden, K., Barthorpe, R., Fuentes, R., Rogers, T., Dervilis, N.: Health monitoring of composite structures by combining ultrasonic wave data. In: Proceedings of Structural Health Monitoring (2019) 13. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. The MIT Press, Cambridge (2005) 14. Padonou, E., Roustant, O.: Polar Gaussian processes for predicting on circular domains. HAL, Hal 1119942, (2015) Marcus Haywood-Alexander PhD student currently working within the New Partnership in Offshore Wind project, with his focus being on determining the feasibility of using ultrasonic guided waves in structural health monitoring for wind turbine blades. Outside of work, has keen interests in skiing, climbing, photography & mountaineering.

Chapter 14

Improved FRF Estimation from Noisy High-Speed Camera Data Using SEMM ˇ Tomaž Bregar, Klemen Zaletelj, Gregor Cepon, Janko Slaviˇc, and Miha Boltežar

Abstract Vibration measurements with a high-speed camera are becoming a compelling alternative to accelerometers and laser vibrometers. However, estimated FRFs from the high-speed camera are usually displaying relatively high levels of noise. The noise has proven to be problematic especially in the higher frequency range, where the amplitude of the displacements are generally very small. Using the hybrid method the mode shapes can be identified even at higher frequencies, where the amplitude of the response is close to overall noise level. The identification is obtained by using eigenvalues from accelerometer in the Least Square Complex Frequency (LSCF) identification. The identified mode shapes were proven to be consistent; however, the synthesised full field FRFs after least-square frequency-domain LSFD method are erroneous, especially in the higher frequency range. In this paper the possibility of improving the estimation of full-field FRFs from noisy high-speed camera data using the System Equivalent Model Mixing (SEMM) is explored. The SEMM is normally used with a numerical model (as a parent model) with a large number of DoF which is then mixed with an experimental model which is a subset of the parent model DoF. With the proposed method the FRFs obtained from the high-speed camera data are used as a parent model and a few experimental FRFs from the accelerometers are used as an overlay model. Experimental research on the hybrid model shows an increased accuracy in the estimation of the FRFs. Keywords Frequency based substructuring · System equivalent model mixing · High-speed camera · Full-field measurements

14.1 Introduction Vibration measurement using image-based methods has become a well-established alternative to standard measurement techniques. However, the identified displacements from high-speed footage is typically heavily burdened by noise, especially in the higher frequency range, where the displacement amplitudes are generally very small [1]. It was shown that by using the Least-Squares Complex-Frequency method combined with the Least-Squares FrequencyDomain method the modeshapes identification in high-frequency range is possible by using the eigenvalues from an accelerometer [1]. Identified modeshapes are consistent even when the identified displacements are below the overall floornoise. However, the reconstructed full-field FRFs are not consistent over the whole frequency range. In this paper a methodology to increase the consistency of identified full-field FRFs is presented. The consistency is increased by combining the accelerometer FRFs with the FRFs identified from high-speed camera data. The mixing of equivalent models is achieved using System Equivalent Model Mixing (SEMM) [2].

T. Bregar Gorenje d.d., Velenje, Slovenia e-mail: [email protected] ˇ K. Zaletelj · G. Cepon () · J. Slaviˇc · M. Boltežar Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia e-mail: [email protected]; [email protected]; [email protected]; miha.boltež[email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_14

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14.2 Full-Field FRFs Estimation Using SEMM To increase the consistency of full-field FRFs a combination of different equivalent models is proposed. A schematic representation of the whole procedure is depicted in Fig. 14.1. First the modeshapes of the structures are identified through the hybrid LSCF/LSFD procedure [1] and simultaneously the accelerometer FRFs are measured. The reconstructed FRFs from high-speed camera are then mixed with the accelerometer FRFs using the SEMM [2] based on the frequency based substructuring. With that a hybrid model of full-field FRFs is obtained. In the following section a short introduction of the system equivalent model mixing is presented.

14.2.1 System Equivalent Model Mixing System Equivalent Model Mixing is a method that uses frequency based substructuring for mixing of two equivalent models, which can either be acquired from a numerical model or by direct measurements [2]. A schematic representation of SEMM is depicted in Fig. 14.2. With SEMM the model dynamics contained in the overlay model Yov are expanded on the DoFspace of the parent model Ypar . The overlay model is a subset of parent model DoF which dynamic properties have to be decoupled denoted by Yrem . An equation of motion for SEMM can be formulated as:

u = Y(f + g);

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After applying the compatibility and equilibrium conditions in accordance with the LM FBS [3] notation the following formulation is obtained: + rem par ov  Yrem + Ypar , YSEMM = Ygg − Ygg Yrem Ybb − Ybb gg bg gb where b is regarded as the boundary DoF and g as the global DoF.

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14.3 Results The proposed methodology was applied on a simple beam with dimensions 12 × 40 × 600 mm shown in Fig. 14.3. The beam was suspended with foam pads to approximate free-free conditions. Six equally spaced accelerometers were used as the overlay model Yov . Additional accelerometer was used as a reference which was excluded from the hybrid model. The parent model Ypar was acquired using the Fastcam SA-Z high-speed camera at 200,000 frames per second. To maximize pixel intensity gradient in the vertical direction, a sticker with horizontal lines was applied at the front face of the beam. Displacements from high-speed camera data were identified using the simplified optical flow method [4]. Estimated hybrid FRFs are depicted in Fig. 14.4 on Nyquist plot together with raw high-speed camera FRFs, reconstructed with the LSFD and the reference measured with accelerometer. It can be observed that the raw high-speed camera identified displacements are noisy, which is typical for the higher-frequency range. After mode shape identification the noise is filtered out; however, the reconstructed FRFs are erroneous. With hybrid FRFs obtained using SEMM the overall consistency of the FRFs is increased.

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References 1. Javh, J., Slaviˇc, J., Boltežar, M.: High frequency modal identification on noisy high-speed camera data. Mech. Syst. Signal Process. 98, 344–351 (2018) 2. Klaassen, S.W., van der Seijs, M. V., de Klerk, D.: System equivalent model mixing. Mech. Syst. Signal Process. 105, 90–112 (2018) 3. de Klerk, D., Rixen, D.J., de Jong, J.: The Frequency Based Substructuring (FBS) method reformulated according to the dual domain decomposition method. In: Proceedings of the 24th International Modal Analysis Conference, A Conference on Structural Dynamics (2006) 4. Javh, J., Slaviˇc, J., Boltežar, M.: The subpixel resolution of optical-flow-based modal analysis. Mech. Syst. Signal Process. 88, 89–99 (2017) ˇ Gregor Cepon is Associate Professor at the Faculty of Mechanical Engineering, University of Ljubljana. His research focus is orientated towards modeling the dynamics of rigid body systems and numerical and experimental substructuring.

Chapter 15

Full-Field Modal Analysis by Using Digital Image Correlation Technique Emilio Di Lorenzo, Davide Mastrodicasa, Lukas Wittevrongel, Pascal Lava, and Bart Peeters

Abstract Digital Image Correlation (DIC) is a non-contact full-field image analysis technique which allows to retrieve strains and displacements in three dimensions at the surface of any type of material and under arbitrary loading. In recent years, high-speed and high-resolution cameras have been developed for static as well as for dynamic applications. As consequence, the application fields for DIC have broadened and it has proven to be a flexible and very accurate measurement solution for deformation analysis and material characterization. In this work DIC technique is used to get the full-field displacement of the structure under test. This information could then be used to derive the modal characteristic of the structure (e.g. natural frequencies, damping ratios and full-field mode shapes). These results can be validated by using classical sensors (e.g. strain gauges, accelerometers) or other optical methods (e.g. laser doppler vibrometers). Several test cases are discussed, and two different approaches are used for combining the data obtained during the vibration tests. The most obvious approach would be the alignment of the time histories of input (shaker) and output based on reference signals for Frequency Response Functions (FRFs) calculation prior to perform any further processing. An alternative in case of broadband excitation, requires processing time data into auto and crosspowers and identify the modal parameters by using Operational Modal Analysis (OMA). Keywords Digital image correlation (DIC) · Full field modal analysis · Image processing

15.1 Introduction Nowadays, there has been an increase in applications of lightweight and composites materials in the automotive, aerospace and other advanced manufacturing industries. At the state of the art testing and validation for mechanical numerical models are mostly performed using a certain number of transducers (e.g. accelerometers, strain gauges, fiber optics, etc. . . ). However, accelerometers may mass-load the structure and can only provide measurements at discrete locations. Furthermore, placing transducers is a labour intensive and time consuming task and it could introduce electrical noise to the measured signals due to the extensive and unavoidable wiring. These are the main reasons behind the development of image processing techniques to perform modal analysis of mechanical structures without contact and without having to instrument the specimen. Optical methods such as Digital Image Correlation (DIC) has recently received special attention in the structural dynamics field because it can be used to obtain full field measurements [1, 2]. At the state of the art DIC is mostly used for static applications like structural testing and material identification. In the last years the use of DIC for measuring vibrations is gaining popularity. Particularly DIC for vibration analysis was mostly performed on aerospace and automotive components [3–7] and rotating structures [8] like turbine blades [9], helicopter rotors [1]. An interesting research topic is related to the development of techniques to use cheap, light and low speed camera to detect structure’s high frequency behaviour. Indeed, high-speed cameras are characterized by an increasing frame rate with the decreasing of the resolution. Hence, when large structures needs to be analyzed, the maximum field of view and the maximum camera resolution are suitable. This leads to the drawback of having a low frame rate, or, in other words, a low sampling frequency fs . Therefore, a small bandwidth

E. Di Lorenzo () · D. Mastrodicasa · B. Peeters Siemens Industry Software NV, Leuven, Belgium e-mail: [email protected]; [email protected]; [email protected] L. Wittevrongel · P. Lava MatchID, Gent, Belgium e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_15

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B ≤ fs /2 can be analyzed and this can bring the need to go above the Nyquist Shannon frequency in order to detect high frequency vibrations with low frame rate settings. To overcome the low sampling rate, techniques such as under-sampling and remapping the time histories to measure above the Nyquist frequency are investigated [10–12]. In this paper, DIC technique is used to get the full-field displacement time histories of the unity under test. These information are then used in combination with Siemens Simcenter Testlab to obtain the FRFs or CSD/PSD matrices for the modal parameter estimation achieved by using the well-known Polymax algorithm [13]. Two different real life examples were considered in this paper to understand the challenges still present to use DIC for structural dynamics applications: an helicopter tail blade in lab conditions and an aircraft in the hangar.

15.2 Digital Image Correlation DIC only requires a set of cameras and a specimen which has been speckled (i.e. a gradient must be present in the tested specimen, which can be created with spray paint, a marker, stickers, etc.) for point tracking. Figure 15.1 shows the DIC working principle: the first image of camera 1 is used as a reference image to which the image of the second camera (taken at the same time) is compared. The corresponding points can be triangulated into a three-dimensional point in space (denoted as [X,Y,Z]) if the orientation of the cameras is known (by performing a pre-test calibration). The same points matching can be done over time so that each measurement point can be tracked over time, leading to a total deformation tracking of the surface area of a specimen (in which the deformation is indicated as [U,V,W]). This process holds some intrinsic requirements towards the images: a gradient, as well as contrast, must be present so that features can be tracked. The size of these features lead to the minimal size of the so-called subset (also called interrogation window) indicated in yellow in Fig. 15.1, which encompasses several features and which is tracked throughout the set of images. This is needed since a single pixel is not a unique item due to the limited amount of gray scales which are stored in an image (usually 8 bit cameras are used, so a maximum of 256 grey scales are available). A cluster of pixels is however a unique set of data which can be tracked, while it will also reduce the measurement noise at the same time. This subset is restricted in the way it can deform by assigning a so-called “shape function” to it, leading to an even higher amount of robustness against noise (more information on the shape function can be found in [14, 15]). This, together with image interpolation for sub-pixel accuracy [16], leads to high-quality and precise measurements of shapes and displacements.

Fig. 15.1 Principle of DIC

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15.3 Experimental Analysis Different test beds were investigated in order to analyze the challenges related to the use of Digital Image Correlation for the dynamic characterization of aeronautical structures. More specifically, an helicopter tail blade was analyzed both using high and low speed cameras. An approach focused on multiple measurements using low speed camera to reconstruct the aliased bandwidth was examined on an aircraft wing, more specifically the F16 airplane was speckled and measured by using DIC together with accelerometers and a Scanning Laser Doppler Vibrometer (SLDV).

15.3.1 Helicopter Tail Blade The objective of this study is to perform a full-field modal analysis of an helicopter tail blade in lab conditions. The specimen on which a white base coat has been applied is shown in Fig. 15.2a. The same specimen after the application of the white coat which was needed to create enough contrast after applying the speckles (i.e. features), is shown in Fig. 15.2b. Finally, the specimen used for the DIC test is shown in Fig. 15.2c. The structure was clamped on the right hand side using an heavy mass to simulate a fixed constraint. High Speed Cameras Setup The experimental setup for the high speed cameras configuration is shown in Fig. 15.3. Two iX 720 high speed cameras were used in a stereo DIC setup to grab the pictures of the specimen during the controlled excitation. The cameras specifications and the acquisition parameters are reported in Table 15.1.

Fig. 15.2 Process of speckling. (a) Virgin structure. (b) White coated structure. (c) Speckled structure Fig. 15.3 Experimental set up

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Table 15.1 Acquisition parameters

Parameters Input type Bandwidth Resolution Pixel size Max frame rate at full resolution

Value Pull and release 4–250 Hz 2048(H) × 1536 (V) px 13.5 × 13.5 μm 6642 fps

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Parameters Subset Step Interpolation Shape function

Value 25 10 Bilinear polynomial Affine

MatchID software was used both for the image acquisition and data processing. 8253 images were acquired and 10,200 points were analyzed. Only the Z Degree-of-Freedom (DoF) (out of plane motion) was considered due to the high computational cost to process all the 3 DoFs. The geometry of the structure is obtained directly from the recorded pictures and it is shown in Fig. 15.4 together with the DIC processing parameters in Table 15.2. The blade was pulled at its free-end and the free vibrations were recorded by using the stereo setup. Operational Polymax was then used to perform Operational Modal Analysis (OMA) in order to get the modal parameters estimation with the help of Siemens Simcenter Testlab. The steps going from the data acquisition to the modal analysis results for the DIC processing are listed here: 1. Export displacement full-field of all images from MatchID software; 2. Manipulation and data reduction in Matlab to build the displacement time histories of all points measured during the entire measurements; 3. Import time histories in Simcenter Testlab for each measured Degree-of-Freedom (DoF); 4. Import the geometry from MatchID software; 5. Calculate auto/cross-powers and perform OMA; 6. Estimation of natural frequencies fn , damping ratios ζn and mode shapes; The results in Fig. 15.5, in terms of stabilization diagram and Auto-MAC, were obtained processing directly the output displacement time histories of MatchID software following the mentioned steps. From the Auto-MAC it is possible to note that all the identified mode shapes are equal to each other. The first mode (22 Hz) is dominant and by looking at the displacements instead of acceleration this is still more accentuated. Indeed only one peak is visible and, even if many modes are found, all of them resemble the first mode shape. In order to improve the results, a frequency filter between 40 and 200 Hz was used to cut out the first mode. The results are shown in Fig. 15.6. From the Auto-MAC in Fig. 15.6b it is possible to see that the first 4 mode shapes are still very similar to each others and they are mainly due to the displacements of the entire setup which was not simulating a perfect clamping. On the other

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hand, when moving to higher frequencies it is possible to distinguish the modes from each others. More in details, the 5th and the 6th mode shapes are the ones which were previously identified by means of a Scanning Laser Doppler Vibrometer (SLDV). An Optomet SLDV Short-Wavelength InfraRed (SWIR) device was also used to measure the vibrations of the helicopter blade to obtain reference results to compare with the DIC estimated modal parameters. This was done by using shaker excitation and measuring the surface velocities at several location along the upper durface of the blade. Table 15.3 and Fig. 15.7 show the comparison in terms of modal parameters and mode shapes. From Table 15.3 it is possible to see that there is a good match between DIC and LDV results in terms of natural frequencies but DIC is overestimating the damping for all the three analyzed modes. It is important to underline that the two tests were performed in two different conditions. In fact the DIC study was performed by using a so-called pull and release test, whereas the SLDV test was performed by using a shaker excitation. In any case further studies will be conducted to improve the damping estimation.

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(b)

1 Mode Shape, f n = 22.1889 Hz, ζn = 3.9472 %

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Mode 1 2 3

LDV fn (Hz) 22.36 105.40 159.90

ζn (%) 1.99 1.14 0.71

DIC fn (Hz) 22.19 102.10 158.91

ζn (%) 3.95 2.05 0.99

Low Speed Cameras Setup The experimental setup for the low speed camera configuration is shown in Fig. 15.8. Two Mako U-130B low speed camera were used in a stereo DIC setup to grab the pictures of the specimen during the excitation. The specifications of the cameras and the acquisition parameters are shown in Table 15.4. Also in this case, only the out of plane motion (Z DoF) was considered. The geometry of the measured points on the structure is shown in Fig. 15.9. 2637 images were acquired and 10,095 points were analyzed. The structure was excited providing a chirp signal between 4 and 50 Hz by using the shaker shown in Fig. 15.8. The frame rate of the camera was pushed up to the limit of 168 fps which led to an irregular time step between the grabbed pictures. Therefore, an interpolation step was needed in order to have a constant time step for the subsequent data processing steps. The shaker voltage time history was recorded and it was synchronized with the camera acquisition. This allowed to use both Experimental Modal Analysis (EMA) and Operational Modal Analysis (OMA) approaches. The following stabilization diagrams, Fig. 15.10, were obtained using the two algorithms. Unfortunately the cameras frame rate did not allow to go higher than 80 Hz, which only allows the 1st mode identification (around 20 Hz). This is explicitly shown in Fig. 15.11, where the 1st mode shape obtained by using the DIC modal analysis (both the EMA and OMA algorithms) are compared to the one obtained using the accelerometers. A conventional modal analysis performed using 10 accelerometers to measure the structural responses and a shaker as exciter was used as a reference result to compare with the DIC estimated modal parameters. In Table 15.5 and Fig. 15.11 it is shown the comparison in terms of modal parameters and mode shapes.

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Value Chirp 4–50 Hz 1280(H) × 1024(V) px 4.8 × 4.8 μm 168 fps

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From Table 15.5 it is possible to see that there is a good match between DIC and accelerometer results in terms of natural frequencies but not in terms of modal damping. Also in this case the damping obtained by using the DIC displacements is higher than the ones resulting from the classical accelerometers-based modal analysis. Further studies will be conducted to determine the cause of this difference.

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(c) Fig. 15.11 Mode shapes comparison. (a) 1st Mode shape DIC EMA (19.91 Hz, 3.55%). (b) 1st Mode shape DIC OMA (19.84 Hz, 4.39%). (c) 1st Mode shape Accelerometers (20.11 Hz, 1.78%)

15.3.2 F16 Aircraft An innovative approach is presented here aimed at measuring and identifying the full-field high frequency response using still-frame cameras. This method relies on the use of two different signals and multiple full excitation test of the structure.

15 Full-Field Modal Analysis by Using Digital Image Correlation Technique Table 15.5 Modal parameters comparison

127 Accelerometers DIC (EMA) DIC (OMA) Mode fn (Hz) ζn (%) fn (Hz) ζn (%) fn (Hz) ζn (%) 1 20.11 1.78 19.91 3.55 19.84 4.39

Fig. 15.12 Aliased bandwidth reconstruction. (a) 1st measurement. (b) 2nd measurement. (c) All measurements

Fig. 15.13 Experimental set up. (a) Virgin surface. (b) Speckled surface Table 15.6 DIC parameters

Parameters Subset Step Interpolation Shape function

Value 35 10 Bicubic spline Affine

The first signal, a sine sweep, is used for the structure excitation and the other one, a ramp, as a reference for the subsequent remapping of the camera output obtained during the different excitations of the structure. In this way it is possible to combine the information from the different tests to reconstruct the aliased spectrum and therefore to detect vibrations at a frequency higher than fs /2. The concept behind the proposed method is clearly exposed in Fig. 15.12 where the requested sampling, Fig. 15.12c, is achieved populating the signal from different full-field measurements, Fig. 15.12a, b. The opportunity to measure a F16 aircraft was used as test bed to prove the reliability and robustness of this technique. The structure was speckled using A4 paper sheets glued on the two wings and part of the fuselage as shown in Fig. 15.13. The system was also instrumented using accelerometers and using the Optomet SLDV to have a reliable modal model for the comparison with the DIC results. The DIC processing parameters together with the specifications of the cameras and the acquisition parameters are shown, respectively, in Tables 15.6 and 15.7. Also in this case, only the out of plane motion (Z DoF) was considered. 932 images were acquired and 13959 points were analyzed. Due to the used acquisition method to reconstruct the aliased bandwidth, an interpolation step was needed to reconstruct a dataset with a constant time step. A sampling rate of 10,000 Hz was used.

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Table 15.7 Acquisition parameters

Parameters Input type Bandwidth

Value Chirp 2–20 Hz

Modal Assurance Criterion (MAC) 1

60 1

0.9 0.8 0.7

40 First Mode Set

Number of poles, n

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5 0 0

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Mode 1 2

LDV+Accelerometers fn (Hz) ζn (%) 5.11 0.58 7.40 0.43

DIC fn (Hz) 5.09 7.32

ζn (%) 1.19 1.33

The following results, Fig. 15.14, in terms of stabilization diagram and Auto-MAC, were obtained processing directly the output displacement time histories. Table 15.8 and Fig. 15.15 show the comparison between LDV + accelerometers and DIC in terms of modal parameters and mode shapes. Other two modes were identified using the LDV, the 2nd and 3rd airplane bending modes at, respectively, 9.38 and 19.50 Hz. However, it was not possible to find a match of these two modes with the DIC results. This is probably due to the noise introduced by the interpolation process and the high frequency amplitude reduction problem already discussed in the other test beds.

15.4 Conclusions In this paper, the use of DIC for modal analysis is investigated. Different experimental measurements results which combine conventional modal analysis (LDV/accelerometers) with the novel DIC experimental modal analysis are presented. The quality of the measured DIC data is very good and its application in the modal analysis field in order to enrich the conventional modal analysis results with high density data is very promising. However some limitations need to be taken into account: • Data reduction techniques should be implemented to allow faster and accurate processing with current processors capabilities; • A streamline solution covering the full measurement and processing chain is needed, particularly in terms of synchronization between the two cameras and the exciter system; • Higher performance cameras need to be used for having a higher frame rate which allows to increase the maximum frequency to be analyzed;

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Fig. 15.15 Mode shapes comparison. (a) 1st Mode shape LDV (5.11 Hz, 0.58%). (b) 2nd Mode shape LDV (7.40 Hz, 0.43%). (c) 1st Mode shape DIC (5.09 Hz, 1.19%). (d) 2nd Mode shape DIC (7.32 Hz, 1.33%)

A very interesting field of application is the combination of Digital Image Correlation and conventional modal analysis to perform a very accurate validation of Finite Element (FE) models due to the very high spatial resolution. This is one of the main advantages of DIC which shows also potential for identifying defects in lightweight structures where the weight of sensors could be an issue when instrumenting the specimen for testing. Acknowledgments The authors gratefully acknowledge SIM (Strategic Initiative Materials in Flanders) and VLAIO (Flemish government agency Flanders Innovation and Entrepreneurship) for their support of the ICON project DETECT-ION, which is part of the research program MarcroModelMat (M3).

References 1. Di Lorenzo, E., Lava, P., Balcaen, R., Manzato, S., Peeters, B.: Full-field modal analysis using high-speed 3D digital image correlation. J. Phys. Conf. Ser. 1149, 012007 (2018) 2. Srivastava, V., Patil, K., Baqersad, J., Zhang, J.: A multi-view dic approach to extract operating mode shapes of structures. In: Niezrecki, C., Baqersad, J. (eds.) Structural Health Monitoring, Photogrammetry & DIC, vol. 6, pp. 43–48. Springer International Publishing, Cham (2019) 3. Molina-Viedma, A., Lopez-Alba, E., Felipe-Sese, L., Diaz, F., Rodriguez-Ahlquist, J., Iglesias-Vallejo, M.: Modal parameters evaluation in a full-scale aircraft demonstrator under different environmental conditions using HS 3D-DIC. Materials 11, 230 (2018) 4. Seguel, F., Meruane, V.: Damage assessment in a sandwich panel based on full-field vibration measurements. J. Sound Vib. 417, 1–18 (2018) 5. Bharadwaj, K., Sheidaei, A., Afshar, A., Baqersad, J.: Full-field strain prediction using mode shapes measured with digital image correlation. Measurement 139, 326–333 (2019) 6. Galeazzi, S., Chiariotti, P., Martarelli, M., Tomasini, E.: 3D digital image correlation for vibration measurement on rolling tire: procedure development and comparison with laser doppler vibrometer. J. Phys. Conf. Ser. 1149, 012010 (2018) 7. Ha, N.S., Vang, H.M., Goo, N.: Modal analysis using digital image correlation technique: an application to artificial wing mimicking beetle’s hind wing. Experimental Mechanics 55, 989–998 (2015) 8. Yashar, A., Ferguson, N., Tehrani, M.G.: Measurement of rotating beam vibration using optical (DIC) techniques. Proc. Eng. 199, 477–482 (2017). X International Conference on Structural Dynamics, EURODYN 2017

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9. Baqersad, J., Carr, J., Lundstrom, T., Niezrecki, C., Avitabile, P., Slattery, M.: Dynamic characteristics of a wind turbine blade using 3D digital image correlation. Proc. SPIE-Int. Soc. Opt. Eng. 8348, 74 (2012) 10. Barone, S., Neri, P., Paoli, A., Razionale, A.: Digital image correlation based on projected pattern for high frequency vibration measurements. Proc. Manuf. 11, 1592–1599 (2017). 27th International Conference on Flexible Automation and Intelligent Manufacturing, FAIM2017, 27–30 June 2017, Modena, Italy 11. Javh, J., Slaviˇc, J., Boltežar, M.: Experimental modal analysis on full-field DSLR camera footage using spectral optical flow imaging. J. Sound Vib. 434, 213–220 (2018) 12. Liu, Y., Gao, H., Zhuge, J., Zhao, J.: Research of under-sampling technique for digital image correlation in vibration measurement. In: Harvie, J.M., Baqersad, J. (eds.) Shock & Vibration, Aircraft/Aerospace, Energy Harvesting, Acoustics & Optics, vol. 9, pp. 49–58. Springer International Publishing, Cham (2017) 13. Peeters, B., Van der Auweraer, H., Guillaume, P., Leuridan, J.: The polymax frequency-domain method: a new standard for modal parameter estimation? Shock Vib. 11, 395–409 (2004) 14. Grédiac, M., Sur, F.: Effect of sensor noise on the resolution and spatial resolution of displacement and strain maps estimated with the grid method. Strain 50:1–27 (2014) 15. Yu, L., Pan, B.: The errors in digital image correlation due to overmatched shape functions. Measurement Science and Technology 26, 045202 (2015) 16. Schreier, H., Sutton, M.: Systematic errors in digital image correlation due to undermatched subset shape functions. Exp. Mech. 42, 303–310 (2002)

Chapter 16

Method for Selecting Rotor Suspension Design Criteria Brian Damiano

Abstract A method for selecting rotor suspension design criteria that allow rotors with internal friction to negotiate critical speeds and avoid subsynchronous internal friction instabilities is described. A rotor model of a test rotor with internal friction is created and used to calculate the test rotor’s critical speeds, mode shapes, Campbell plots, and the rotational speed corresponding to the stability boundary for a nominal set of suspension parameters. The test rotor model is then used to demonstrate the suspension design criteria selection method. Keywords Supercritical rotors · Suspensions · Design criteria · Internal friction instability · Rotor stability

16.1 Introduction It can be argued that the selection of design criteria is the most critical step in any design process because all subsequent design activities depend on these criteria. For supercritical rotors, rotor suspensions are required to allow critical speeds to be negotiated during acceleration to operating speed and must also maintain rotor stability. Stability is particularly important issue for rotors with internal friction; internal friction is a well-known cause of subsynchronous instability, having been observed and studied for nearly 100 years [1, 2]. Significant rotor internal friction often occurs in “built up” rotors, that is, rotors that are comprised of several sections. These sections are typically joined by using press fits or other mechanical means and microslip at the joints is typically the friction source [1]. Assuming an accurate model of a rotor/suspension system can be created, a relatively simple approach can be used to develop rotor suspension design criteria. The criteria come directly from an eigenvalue analysis of the model; the effects of the suspension parameters on the eigenvalues, particularly the real part, which determine modal damping and stability, are used to select suspension parameters. For supercritical rotors, the following rotor dynamic characteristics are required for successful operation: • The rotor must be able to negotiate its critical speeds as it accelerates to operating speed. For this investigation, this requirement will be translated as each negotiated critical speed must have a Q value of 20 or less. Experience at Oak Ridge National Laboratory (ORNL) has shown this criterion to be effective, however, it is recognized that other ranges of Q value may be more applicable for applications outside of those used at ORNL. • The rotor must remain stable as it accelerates to operating speed. This requirement is equivalent to saying that the real parts of the eigenvalues must remain negative over the range of operating speeds. Suspensions may be modeled as a frequency-dependent complex spring. In this formulation, the spring rate can be expressed as having both a real part and an imaginary part, or more conveniently, as having an effective weight, Wd , and

Notice of Copyright This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). B. Damiano () Oak Ridge National Laboratory, Oak Ridge, TN, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_16

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an effective damping, Cd . For a simple spring-mass-damper system connected to ground, the stiffness to ground when undergoing steady-state harmonic motion is: K (ω) = k (ω) − m (ω) ω2 + ic (ω) ω, where K(ω) is the overall frequency dependent spring rate, k(ω) and c(ω) are the frequency-dependent spring and damping rates connecting mass m to ground, i is the imaginary indicator, and ω is the frequency of the steady-state harmonic excitation. Multiplying and dividing the real part by ω2 :   k (ω) − m (ω) ω2 K (ω) = ω2 + ic (ω) ω = Wd (ω) ω2 + iCd (ω) ω ω2 Thus, Wd (ω) =



k(ω)−m(ω)ω2 ω2



ω2 and Cd (ω) = c(ω).

A rotor model description is given in the next section followed by descriptions of determining suspension criteria to allow successful negotiation of the critical speeds and to avoid internal friction-induced instability. The paper ends with a summary section.

16.2 Rotor Dynamic Analysis Code and Rotor Model Description The Cylinder rotor dynamics analysis code was used to perform the rotor dynamic analysis. Cylinder was developed at ORNL in the late 1970s and has been used for many applications over the last 40 years. Cylinder is an axisymmetric transfer matrix code that calculates natural frequencies, critical speeds, mode shapes, and steady state deflection caused by harmonic excitation such as imbalance. The results of the natural frequency or critical speed analysis (an eigenvalue analysis) are the natural frequencies or critical speeds (related to the imaginary part of the eigenvalue), the Q values (related to the real part of the eigenvalue), and a mode shape. Cylinder is currently implemented in Matlab. The test rotor model (Fig. 16.1) does not match any real rotor system and was developed specifically for this investigation. The modeled rotor is a 28-inch-long steel beam with an 0.8-inch outer diameter and a 0.75-inch inner diameter. The rotor consists of two shafts connected by a coupling located 20 inches from the left end. The coupling is represented as an infinitely stiff radial spring and by trunnion spring and damper in parallel, with a trunnion stiffness of 2000 in-lbf /rad and a trunnion damping of 0.6 in-lbf -s/rad. The trunnion damping, being in the rotating frame, is the source of the rotor instability. The rotor is connected to ground at each end by identical suspensions consisting of a bearing, Kb , with a stiffness of 10,000 lbf /in that is attached to a squeeze film damper that is represented by a spring-mass-damper system. The nominal suspension values of the effective weight, Wd , and damping values, Cd , are 0.10 lbm and 0.2 lbf -s/in. The rotor operating speed is selected to be 400 Hz. The test rotor will negotiate two flexural mode critical speeds as it accelerates to its 400 Hz operating speed. Table 16.1 lists the critical speeds and Q values for the nominal suspension values listed above. The critical speed mode shapes for the first three flexural modes are shown in Fig. 16.2.

Fig. 16.1 Lumped parameter representation of the test rotor model

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Table 16.1 Critical speeds and Q values for the test rotor model

Mode First flexural mode Second flexural mode Second flexural mode -1

0

-5

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Critical Speed (Hz) 57 298 901 -1

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1 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -30

-25

Fig. 16.2 Critical speed mode shapes for the test rotor model with nominal suspension parameters

speed line forward whirl reverse whirl

forward whirl reverse whirl

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Campbell plots showing the effects of rotational speed on the flexural mode natural frequencies of the test rotor model are shown in Figs. 16.3, 16.4, and 16.5. For each mode, a pair of plots is shown. The first plot shows the effect of rotational speed on the natural frequency and the second plot shows the effect of rotational speed on Q value. In each plot, the blue line shows the effects of rotational speed on the forward whirl mode frequency or Q value while the red line shows the effects of rotational speed on the reverse whirl mode frequency or Q value. For frequency plots, an additional black dashed line, the speed line, is included on the plot. Critical speeds occur at the intersection of the speed line and forward whirl frequency line.

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speed line forward whirl reverse whirl

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Fig. 16.4 Campbell plots for the second flexural mode of the test rotor model

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Fig. 16.5 Campbell plots for the third flexural mode of the test rotor model

The Campbell plot results for frequency are typical; as the rotational speed increases, the forward whirl mode natural frequency increases and the reverse whirl natural frequency decreases. The effect of natural frequency on Q value is similar (in this case) for the second and third flexural modes, however, the first flexural forward mode shows instability at a rotational speed of 120 Hz. In the Q value plot for the first flexural forward mode, the Q value is positive at low frequencies (indicating stability), experiences a peak and a sign change at 120 Hz (the stability boundary), and then is negative at high frequencies, indicating the mode is unstable for frequencies above the stability boundary. This result indicates that the nominal suspension values are unable to maintain stability for the first flexural forward mode beyond a 120 Hz rotational speed. The next section will describe how suspension design criteria can be defined to ensure successful negotiation of the critical speeds.

16.3 Suspension Criteria Allowing Negotiation of Critical Speeds The rotor must be able to negotiate its critical speeds as it accelerates to operating speed. For this investigation, this requirement will be translated as each negotiated critical speed must have a Q value of 20 or less. A parametric study can be performed to determine the values of effective weight and effective damping that will ensure a mode’s Q value at its critical speed is less than 20. In this study, ranges of effective weight and effective damping are selected and the critical speeds and Q value for each effective weight and damping combination are calculated. The calculated results can be displayed on an FQ plot; the plot shows the critical speeds and Q values connected by lines of constant effective weight or damping. The resulting values of effective weight or damping that result in critical speed Q values below 20 are easily picked off the plot.

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Fig. 16.6 First flexural mode FQ plot for the test rotor model

Figure 16.6 shows an FQ plot for the first flexural mode of the test rotor model. Red lines correspond to lines of constant effective damping and blue lines correspond to lines of constant effective weight. The plot can be divided into three regions. In the center is the overdamped region. In this region, increasing damping results in increasing Q values. Underdamped regions exist on each side of the overdamped region. In these regions, increasing damping results in decreasing Q values. For the first flexural mode, all Q values for the ranges of effective weight and effective damping are less than 20. Thus, the suspension criteria for negotiating the first flexural mode are −0.5 ≥ Wd ≤ 0.5 lbm and .25 ≥ Cd ≤ 1.6 lbf -s/in. Repeating the calculation for the second flexural mode results in an FQ plot for the second flexural mode (Fig. 16.7). Thus, the suspension criteria for negotiating the first flexural mode are −0.4 ≥ Wd ≤ 0.4 lbm and .5 ≥ Cd ≤ 4 lbf -s/in. The third flexural mode is not negotiated because its critical speed is above the rotor’s 400 Hz operating speed, so its FQ plot is not presented. Table 16.2 summarizes the suspension design criteria required to give a critical speed Q value less than 20 for the first two flexural modes of the test rotor model.

16.4 Suspension Criteria to Avoid Internal Friction-Induced Instability Campbell plot data shown in Fig. 16.3 indicates the first flexural mode instability is suspect and needs further investigation to determine the suspension characteristics needed to avoid instability. A parametric study comprised of the following steps was performed: • For a selected value of rotational speed and effective weight, the minimum and maximum values of effective damping were calculated that resulted in positive Q values for the first flexural mode. These effective damping values determine the stability boundary for the selected values of rotational speed and effective weight. • This calculation is repeated for a range of effective rotational speed and effective weight values, resulting in an upper and lower stability surface. The calculation results can be plotted to show the stability surface; points of rotational speed, effective weight, and effective damping inside the surface are stable, points on the surface are marginally stable, and points outside the surface are unstable. Fig. 16.8 shows the stability surface based on rotational speed, calculated for the first flexural mode. The stability surface shows that as rotational speed increases, the values of both effective weight and effective damping converge toward zero. This convergence means that as the operating speed increases, the suspension design criteria become

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Fig. 16.7 Second flexural mode FQ plot for the test rotor model Table 16.2 Suspension design criteria required to negotiate the first two flexural modes with a Q value less than 20 Mode First flexural mode Second flexural mode

Effective weight design criteria −0.5 ≥ Wd ≤ 0.5 lbm −0.4 ≥ Wd ≤ 0.4 lbm

Effective damping design criteria 0.25 ≥ Cd ≤ 1.6 lbf -s/in 0.50 ≥ Cd ≤ 4.0 lbf -s/in

Fig. 16.8 The stability surface based on rotational speed for the first flexural mode of the test rotor model

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Fig. 16.9 The stability surface based on first flexural mode natural frequency for the first flexural mode of the test rotor model Table 16.3 Suspension design criteria to maintain first flexural mode stability Frequency (Hz) 60 62 64 66 68 70 72

Effective weight (lbm) −0.150 ≥ Wd ≤ 0.100 −0.113 ≥ Wd ≤ 0.225 −0.063 ≥ Wd ≤ 0.163 −0.013 ≥ Wd ≤ 0.113 0.075 ≥ Wd ≤ 0.100 0.050 ≥ Wd ≤ 0.075 0.060 ≥ Wd ≤ 0.075

Effective damping (lbf-s/in) Cd ≤ 0.370 Cd ≤ 0.350 Cd ≤ 0.355 Cd ≤ 0.322 Cd ≤ 0.295 Cd ≤ 0.266 Cd ≤ 0.232

more stringent to maintain stability. Although not shown in Fig. 16.8, if the rotational speed were to be sufficiently increased, the stability surface would eventually contract until no stable solution would exist. The stability surface can also be shown based on modal natural frequency rather than rotational speed. The suspension properties at the natural frequencies of the rotor, not the rotational speed, are used to select the suspension design criteria. Fig. 16.9 shows the stability surface based on the first flexural mode natural frequency and a line showing the nominal suspension properties. Combinations of natural frequency, effective weight, and effective damping below the stability surface are stable. Note that the nominal suspension properties intersect the stability surface at approximately 57.7 Hz; this value of the first flexural mode natural frequency corresponds to a rotational speed of 120 Hz, the rotational speed stability boundary. Using the results shown in Fig. 16.9 to select the suspension design criteria to avoid first flexural mode instability results in the suspension criteria listed in Table 16.3.

16.5 Summary The design criteria developed for the test rotor model is summarized in Fig. 16.10. The criteria to avoid first flexural mode instability controls the design criteria; suspension criteria to negotiate the first and second flexural mode critical speeds are well outside the values needed to maintain first flexural mode stability. A method for selecting rotor suspension design criteria that allow rotors with internal friction to negotiate critical speeds and avoid subsynchronous internal friction instabilities was described. This model-based method uses parametric studies

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Fig. 16.10 Suspension design criteria values for the test rotor model

based on the rotor eigenvalues, expressed as critical speeds or natural frequencies and Q values, to determine the suspension parameters allowing the rotor to successfully negotiate critical speeds and avoid internal friction-induced instability. Acknowledgements This material was sponsored by the US Department of Energy, Office of Science. Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the US Department of Energy.

References 1. Muszy´nska, A.: Rotordynamics. CRC Press, Taylor & Francis Group, Boca Raton (2005) 2. Vance, J.M.: Rotordynamics of Turbomachinery. Wiley, New York (1988) Brian Damiano is a group leader and researcher and 38 year veteran at the Oak Ridge National Laboratory where he is currently involved in the design and development of stable isotope gas centrifuge enrichment devices. Brian has extensive experience in the area of rotor design, analysis, and balancing.

Chapter 17

Implementation of Total Variation Applied to Motion Magnification for Structural Dynamic Identification Nicholas A. Valente, Zhu Mao, Matthew Southwick, and Christopher Niezrecki

Abstract Motion magnification has gained popularity within the scientific community for its non-invasive approach to structural health monitoring. Due to the arduous task of instrumenting complex geometric structures, phase-based motion magnification (MM) permits an amplification of small motions that are not visible to the naked eye. Although MM offers a potential alternative to manual instrumentation, the extracted motion may contain artifacts due to the amplification factor. Depending upon the value of magnification, noisy displacement measurements can be present which tend to produce inconclusive results concerning frequency content. This paper presents the application of total variation (TV) to improve the amount noise present in extracted phase displacements. In the presence of large artifacts that come as a result of the magnification factor, the improvement of a spectral signal produces a more conclusive frequency response of the experimental structure. Prior work has attempted to zoom into a small range of pixels to increase spectral resolution; however, this limits the field of view and does not capture a large dynamic range of motion. Total variation has the capability of improving spectral resolution without having to spatially zoom in on a group of pixels. In this work, the modified method of motion magnification and total variation (MMTV) is applied to a simple geometric structure for structural dynamic identification. Keywords Motion magnification · Total variation · Non-invasive measurement

17.1 Introduction Non-invasive measurement has been a desired feature in structural health monitoring due to its ability to extend experimental analysis. With the growing complexity that is present in aging infrastructure, alternative methods of measurement present a manageable way to extract pertinent structural information. Vibrations that appear in larger structures are those that are normally not visible to the naked eye. Liu and Wadhwa et al. introduced the method of motion magnification in addition to improving the proposed approach for frequency domain analysis [1–4]. This method of identifying structural behavior has been widely accepted and applied for extraction of operating deflection shapes, three-dimensional mode shapes and damage detection [5–11]. Larger civil infrastructure has also been evaluated utilizing motion magnification, which ultimately provides insight into where structures lack operational integrity [12, 14–17]. Although motion magnification has aided in the simplification of testing large structures, issues can arise throughout postprocessing. The presence of ghosting artifacts appears in the magnified video, which ultimately translates to the extracted frequency content. These artifacts can provide false data; thus, not providing insight into overall structural health. The proposed approach, Motion Magnification and Total Variation (MMTV), is a way to limit the number of ghosting artifacts that appear in magnified video. The limiting of artifacts provides a more conclusive frequency spectrum, in addition to, structure behavior under dynamic loading. This paper is structured as follows: background behind the methods of (MM) and (MMTV), analysis of a simplified geometric structure, and insight into future research avenues to pursue.

N. A. Valente · Z. Mao () · M. Southwick · C. Niezrecki Department of Mechanical Engineering, University of Massachusetts Lowell, Lowell, MA, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_17

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17.2 Background To extract the relevant frequency information from a structure, one is first concerned with the amplification of subtle motion. The early stages of motion estimation saw merely a gradient based method that focused on the intensity of individual pixels. This method proved to be very sensitive to external disturbances and ultimately time consuming to gather results. To combat this, a Eulerian phase-based approach was proposed [3]. The gathering of phase information proved to be more efficient and representative of dynamic motion. The process first begins with gathering intensities, I(x,y,t), for a series of images. The intensities are brought into the complex domain C(u,v,t) using a transformation that uses a convolution between pixel intensity and Gabor wavelets. C(u,v,t) takes the form C (u, v, t) = A (u, v, t) eiφ(u,v,t)

(17.1)

The phase in the complex domain, φ(u, v, t), is band-passed at a range of frequencies and amplified by a factor of α. The filtered phase, Φ(u, v, t), now contains the magnified subtle motion that can be seen by the naked eye. Due to the choice of magnification parameters, ghosting artifacts can be seen throughout newly processed video. Total variation is a gradient based approach that was proposed by Rudin, Osher and Fatemi in the early 1990s. This method was initially used for signal denoising, but now has gained popularity in various image processing applications. Several attempts have been made to improve the distortions created by motion magnification; however, the use of median filters or linear smoothing shows minor or insignificant changes in the processed video. The method first begins by the variation of a digital signal, y, as V (y) =

 ! ! ! ! !y i+1,j − y i,j !2 + !y i,j +1 − y i,j !2

(17.2)

The sum square errors E(x,y) between the image and the digital signal is computed, which poses the optimization problem. min [E (x, y) + λV (y)]

(17.3)

The regularization parameter, λ, is the value which determines the level of de-noising that should be applied to the series of images. Following the application of total variation, the series of images show greater contrast and fewer ghosting artifacts. This makes frequency extraction and spectrum evaluation more characteristic of dynamic motion. Solving this problem is considered nontrivial; however, future work will investigate a relationship between the regularization parameter, λ, and magnification factor, α.

17.3 Analysis An experiment was conducted to test the validity of the new proposed method. The algorithm methodology follows a sequence of events while post processing video (Fig. 17.1). The total variation algorithm was following the steps outlined by Chambolle et al. [13]. The experimental setup included the use of a PCB MiniShaker, PCB accelerometer and a SONY PXW-FX9 XDCAM Full-Frame camera system. The experimental setup is depicted in Fig. 17.2. A waveform source of 0.05 (V) was applied to the shaker with a designated frequency of 25 (Hz). These settings were chosen to have the geometric cube oscillate at a frequency that could be captured by the camcorder. Slow motion video with a resolution of 1080p and sampling frequency of 120 (Hz) was used for video capture. Prior to utilizing the various computer

Motion Magnification ( )

Fig. 17.1 Computer vision algorithm

Total Variation ( )

Extracted Motion ( )

Frequency Content ( )

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Fig. 17.2 Experimental setup of test fixture

Fig. 17.3 Frequency response of PCB accelerometer

vision approaches, an accelerometer was placed on the front face of the geometric cube for frequency analysis. Figure 17.3, displays the frequency spectrum of the accelerometer while being subjected to the dynamic load. Referencing Fig. 17.3, the prescribed forcing frequency of 25 (Hz) is displayed; thus, grounding the experiment in truth prior to using alternative forms of measurement. Following the frequency estimation of the raw video, a magnified video was then evaluated using the algorithm proposed in [3]. The parameters included a frequency band, ωband , of 23–26 (Hz). Also, the corresponding pixel size, σ , and magnification factor, α, were denoted as 20 and 10 respectively. For the MMTV algorithm, a λ value of 0.1 was chosen to provide an example of drastic de-noising of an image. To compare both the MM and the proposed approach MMTV, frequency spectrums from both setups were computed to verify if there was any improvement in information provided. Figure 17.4 on the following page displays the processed images for both the MM and MMTV method.

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Fig. 17.4 Comparison of processed images for MM and MMTV approach

Fig. 17.5 Frequency spectrum of geometric cube (MM)

The figure that is representative of the MMTV method shows greater contrast than the solely magnified image. In addition, when displaying the sequence of images during oscillation there appears to be fewer ghosting artifacts present; thus, leading to a more conclusive frequency extraction. Figures 17.5 and 17.6, displays the frequency spectrum for the two varying methods. In Fig. 17.5 on the previous page, there appears to be more noise content that appears in the frequency spectrum of (u, v, t), which is a result of the magnification factor being applied. The spectrum displays a frequency of 5 (Hz), which should not show due to the prescribed forcing parameters. In Fig. 17.6, we see that this 5 (Hz) peak is contained within the noise floor, which is more representative of the dynamic motion that is applied by the shaker. The gathered data comparing the two methods provide a potential improvement to a well-known computer vision approach. This could ultimately result in a more accurate portrayal of dynamic motion of structures; therefore, providing a more conclusive depiction of structural health.

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Fig. 17.6 Frequency spectrum of geometric cube (MMTV)

17.4 Conclusion The structural health monitoring of large infrastructure has been an ongoing search due to the arduous testing setups that are needed to conduct experiments. Computer vision has permitted the simplification of experimentation; however, there are still improvements to be made with the representation of frequency estimation. The use of MM and MMTV algorithms provide insight into the manner of how these structures behave without invasive and time-consuming setup. MMTV showed improvement to the already proposed MM algorithm, which can aid in gathering more conclusive results concerning structural health. Future works will investigate a potential relationship between the regularization parameter and magnification factor to make the process of de-noising trivial. Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 1762809. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References 1. Liu, C., Torralba, A., Freeman, W.T., Durand, F., Adelson, E.: Motion magnification. ACM Trans. Graph. 24, 519–526 (2005). https://doi.org/ 10.1145/1186822.1073223 2. Wadhwa, N., Rubinstein, M., Durand, F., Freeman, W.T.: Riesz pyramids for fast phase-based video magnification. 2014 IEEE International Conference on Computational Photography (ICCP) (2014). https://doi.org/10.1109/ICCPHOT.2014.6831820 3. Wadhwa, N., Rubinstein, M., Durand, F., Freeman, T.W.: Phase-based video motion processing. ACM Trans. Graph. 32 (2013). https://doi.org/ 10.1145/2461912.2461966 4. Wadhwa, N., Wu, H.-Y., Davis, A., Rubinstein, M., Shih, E., Mysore, G.J., Chen, J.G., Buyukozturk, O., Guttag, J.V., Freeman, W.T., Durand, F.: Eulerian video magnification and analysis. Commun. ACM. 60, 87–95 (2016). https://doi.org/10.1145/3015573 5. Harmanci, Y.E., Gülan, U., Holzner, M., Chatzi, E.: A novel approach for 3D-structural identification through video recording: magnified tracking. Sensors. 19, 1229 (2019). https://doi.org/10.3390/s19051229 6. Molina-Viedma, A.J., Felipe-Sesé, L., López-Alba, E., Díaz, F.A.: 3D mode shapes characterisation using phase-based motion magnification in large structures using stereoscopic DIC. Mech. Syst. Signal Process. 108, 140–155 (2018). https://doi.org/10.1016/j.ymssp.2018.02.006

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7. Poozesh, P., Sarrafi, A., Mao, Z., Avitabile, P., Niezrecki, C.: Feasibility of extracting operating shapes using phase-based motion magnification technique and stereo-photogrammetry. J. Sound Vib. 407, 350–366 (2017). https://doi.org/10.1016/j.jsv.2017.06.003 8. Qiu, Q., Lau, D.: Defect detection in FRP-bonded structural system via phase-based motion magnification technique. Struct.l Control Health Monit. 25, e2259 (2018). https://doi.org/10.1002/stc.2259 9. Sarrafi, A., Mao, Z., Niezrecki, C., Poozesh, P.: Vibration-based damage detection in wind turbine blades using phase-based motion estimation and motion magnification. J. Sound Vib. 421, 300–318 (2018). https://doi.org/10.1016/j.jsv.2018.01.050 10. Shang, Z., Shen, Z.: Multi-point vibration measurement and mode magnification of civil structures using video-based motion processing. Autom. Constr. 93, 231–240 (2018). https://doi.org/10.1016/j.autcon.2018.05.025 11. Srivastava, V., Baqersad, J.: An optical-based technique to obtain operating deflection shapes of structures with complex geometries. Mech. Syst. Signal Process. 128, 69–81 (2019). https://doi.org/10.1016/j.ymssp.2019.03.021 12. Branch, N., Stewart, E.C.: Applications of Phase-Based Motion Processing. 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 1948 (2018). https://doi.org/10.2514/6.2018-1948 13. Chambolle, A., Caselles, V., Novaga, M., Cremers, D., Pock, T.: An introduction to total variation for image analysis. 9 (2010). https://doi.org/ 10.1515/9783110226157.263 14. Chen, J., Davis, A., Wadhwa, N., Durand, F., Freeman, W.T., Büyüköztürk, O.: Video camera–based vibration measurement for civil infrastructure applications. J. Infrastruct. Syst. 23, B4016013 (2016). https://doi.org/10.1061/(ASCE)IS.1943-555X.0000348 15. Chen, J., Wadhwa, N., Cha, Y.-J., Durand, F., Freeman, W.T., Buyukozturk, O.: Modal identification of simple structures with high-speed video using motion magnification. J. Sound Vib. 345, 58–71 (2015). https://doi.org/10.1016/j.jsv.2015.01.024 16. Wu, X., Yang, X., Jin, J., Yang, Z.: Amplitude-based filtering for video magnification in presence of large motion. Sensors. 18, 2312 (2018). https://doi.org/10.3390/s18072312 17. Fioriti, V., Roselli, I., Tatì, A., Romano, R., De Canio, G.: Motion magnification analysis for structural monitoring of ancient constructions. Measurement. 129, 375–380 (2018). https://doi.org/10.1016/j.measurement.2018.07.055 Nicholas A. Valente is a current PhD student working in the Structural Dynamics and Acoustic Systems Laboratory at the University of Massachusetts Lowell. He has earned his BS in Mechanical Engineering from Merrimack College in May, 2019. Zhu Mao received his B.S. in automotive engineering from Tsinghua University, Beijing, China, in 2002 and M.S. and Ph.D. in structural engineering from the University of California San Diego, La Jolla, CA, USA, in 2008 and 2012 respectively. He is currently on the faculty of Department of Mechanical Engineering at the University of Massachusetts Lowell, Lowell, MA, USA. He has published over 60 articles on journals and conference proceedings, and is the author of one book chapter. His research interest includes structural dynamics and health monitoring, noncontact sensing and image processing, and uncertainty quantification. Dr. Mao is the member of the American Society of Mechanical Engineers (ASME), the Society for Experimental Mechanics (SEM), and the International Society for Optics and Photonic (SPIE). He currently serves on the IMAC Advisory Board and is the Chair of the Technical Division of Model Validation and Uncertainty Quantification at SEM. He is the winner of D. J. DeMichele Scholarship Award (2011) from SEM, the recipient of the Young Investigator Award (2018) from U.S. Air Force Office of Scientific Research, and the recipient of SAGE Publishing Young Engineer Lecture Award (2019) from SEM. Matthew Southwick received the B.S. degree in mechanical engineering from University of Pittsburgh, Pittsburgh, PA, in 2019. He is currently pursuing the Ph.D. degree in mechanical engineering at University of Massachusetts Lowell, Lowell, MA, USA. His research interest includes the development of subtle motion measurement techniques for remote sensing. Christopher Niezrecki received dual B.S. degrees in mechanical and electrical engineering from the University of Connecticut, Storrs, CT, USA, in 1991. He obtained the M.S. and Ph.D. degrees in mechanical engineering from Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, in 1992 and 1999. He is a Professor and Chair of the Department of Mechanical Engineering at the University of Massachusetts Lowell, Lowell, USA. He is currently the Co-Director of the Structural Dynamics and Acoustic Systems Laboratory (http://sdasl.uml.edu/), and leads the Center for Wind Energy at UMass Lowell (www.uml.edu/windenergy). He is also the Director of the NSF-Industry/University Cooperative Research Center for Wind Energy Science Technology and Research (WindSTAR). He has been directly involved in vibrations, acoustics, smart structures, controls, and renewable energy research for over 27 years, with more than 180 publications. His areas of current research include: renewable energy systems, wind turbine dynamics, monitoring, and inspection, structural dynamic and acoustic systems, smart structures, signal processing, structural health monitoring, optical sensing, and smart materials. Over the last several years, his research has focused on using optical digital image correlation for non-contacting inspection and vibration measurement of wind turbine blades/rotating structures. Dr. Niezrecki is a member of the ASME, SPIE, and SEM, and was a 2010 recipient of a National Renewable Energy Laboratory (NREL/National Wind Technology Center) Research Participant Program Fellowship, is the 2018 Roy J. Zuckerberg Endowed Leadership Chair, and a 2019 Donahue Sustainability Fellow.

Chapter 18

A Complex Convolution Based Optical Displacement Sensor Matthew Southwick, Zhu Mao, Nicholas A. Valente, and Christopher Niezrecki

Abstract During the last decade a variety of optical displacement measurement techniques have been developed, commercialized and widely used. This paper presents a new type of optical displacement sensor that improves on the prior art with higher sub-pixel resolution, better accuracy, and better computation efficiency. The Complex Convolution Kernel Based Optical Sensor (KBOS) developed in this paper leverages the magnitude phase representation of a complex valued kernel function to achieve a superior optical displacement sensor for 2D motion measurements. The design is then validated through controlled experimentation over a range of operating conditions and prove the effectiveness of the technique. Finally, the sensor’s abilities will be compared against the state of the art as found in the literature. Keywords Displacement sensor · Convolution kernel · Optical measurement · Remote sensing

18.1 Introduction Characterization of a structure’s dynamics relies upon the accurate measurement of the structures motion through space [1]. Such measurements are commonly carried out using any number of displacement, velocity, or acceleration measurement techniques which can be broadly characterized into groups of either contact or noncontact sensors [2]. Noncontact techniques are increasingly popular as they offer the advantages or not impacting the structures mass and stiffness characteristics, requiring less instrumentation, greater portability, and allow for a denser set of measurement points [3–5]. A smaller subset of the noncontact sensor so far developed are vision based target tracking sensors which use a target affixed to the structure of interest and pattern matching algorithms to measure the targets displacements [1, 6–9]. Such sensors suffer from a number of performance shortcomings. First, such sensors have low subpixel resolution which results in a requirement for large (poorly localized) measurement areas or low precision [1, 6–8]. Also, this class of sensors requires the computation of highly up-sampled normalized cross-correlations for each displacement measurement which leads to slow computation speeds [2, 9]. To improve on these weak points, a modified Taylor Expansion interpolation approach has been used to improve the subpixel resolution and speed of the pattern matching technique [10]. In this research a novel design for a vision based target tracking sensors is presented. This sensor uses a complex convolution kernel based target design leverages the phase magnitude representation of the kernel function to achieve better sub-pixel resolution and greater computation speed than similarly employed sensors, while maintaining excellent measurement accuracy. Also, the systems accuracy and resolution will be shown to be independent of the camera orientation relative to the target. A controlled series of tests will then be carried out to evaluate the sensors performance.

M. Southwick · Z. Mao () · N. A. Valente · C. Niezrecki Department of Mechanical Engineering, University of Massachusetts Lowell, Lowell, MA, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_18

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18.2 Background 18.2.1 The Complex Convolution Kernel Based Optical Sensor (KBOS) The developed sensor uses the phase magnitude representation of a complex 2D kernel function to compute the displacements of a target present in a video frame. The sensor itself is the magnitude of the selected kernel function printed onto a sheet of sticker paper. The discrete orientation and scale of the sampled magnitude of the kernel function are then determined by an optimization of the magnitude of the wavelet transform between the sampled space and the kernel function. Once the scale and orientation are known to sufficient precision, a simple complex 2D convolution of the same pixel size as the sampled target computes a phase shift which is directly proportional to the designed targets known scale. From this, the targets 2D motion has been measured (Fig. 18.1).

Kernel Function Selection In the development of the sensor, first the kernel function (w) must be selected. The constraints placed on the kernel function are that it must be steerable, complex, periodic, and self-orthogonal at varying scales [11]. This defines a kernel which is a quadrature pair (Eq. 18.1), representable through a phase-magnitude relation [12]. wc = wr + iwi

(18.1)

These requirements ensure that by sampling the magnitude of the function as it is printed on the marker, orientation, scale and phase can all be determined with reference to the camera sampling space. The simplest function fulfilling these requirements is the complex exponential function wc = e−2π i(ux+vy)

(18.2)

In Eq. 18.2, u and v denote the horizontal and vertical scale respectively. This function will later be used to test the KBOS technique.

Physical Marker Creation Once the kernel function is selected, a target can be generated. For a 1D test, the magnitude of the kernel function is printed out with physical dimensions selected by the user. These physical dimensions give the kernel a scale (λ) in real space measurable in units of distance (mm) (Fig. 18.2). For a 2D test, the magnitude of the linear combination of the kernel function and its 90◦ rotated twin are printed out. These two functions for the basis of a 2D space defined by the plane of the target [13] (Fig. 18.3).

Sampling The marker is sampled by means of a video camera where by pixels perform a spatial averaging operation (Fig. 18.4) of the marker to discretize it in camera memory [14]. This produces a discrete function f (m, n) that is the representation of the

A

B Kernel Function Selected

C Kernel Function Magnitude Printed Out at Choice Scale

wc = e –2π i(ax+by)

Fig. 18.1 Sensor application flow

D

E

Marker Attached to

Video Recorded of

Structure

Marker

F Marker Isolated in Video Frames

H

G Optimization of Wavelet Transform Gives Orientation and Scale in Camera Space

Single 2D Convolution Gives Phase Shifts Over Time

F (a1, b1) = wc * f (x, y)

Phase Shifts are Rescaled by Known Marker Scale for Displacements

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Fig. 18.2 Physical space manifestation of kernel function as used for target

Fig. 18.3 2D target from superposition of steerable 1D kernel

physical marker in the sampling space, where m and n are the discrete indices and 0 < m < M,0 < n < N with M, N being the total number of samples in a given direction. To counter quantization errors produced by this operation, a kernel function can be selected which mostly varies linearly so that the averaging operation places points on the function. Also, it is necessary that the target function satisfies the Nyquist sampling criterion so that information is not lost due to aliasing.

Displacement Measurement With the marker now sampled over time, we turn to the calculation of its 2D motions. This is done through a three-step process. First, the scale of each of the independent discretized kernel functions is determined with reference to the sampling space. Next, the phase of each of the independent kernel functions is calculated. Finally, the phase is scaled by the known physical dimensions of the printed marker to obtain the true displacements of each function.

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Fig. 18.4 Spatial Averaging Quantization Error

Scale and Orientation Calculation The scale of the discretized kernel function is calculated with high precision by performing gradient decent to find the scale parameters that maximize the wavelet transform of the initial kernel function and the sampled marker in the first frame of measurement. 0 = ∇wc (m, n) ∗ f (m, n) ,

0 < m < M, 0 < n < N

(18.3)

The indices that make this statement true are u0 v0 and are the scale parameters that make wc a shifted version of f. This operation returns the scale of marker in the dimensions of the sampling space and is used to make the calculation of phase efficient, as once scale is known, only a single bin of the wavelet transform need be calculated to obtain phase. From scale, the orientation θ of the target kernel is determined relative the camera reference frame by Eq. 18.4. θ = tan−1

v0 u0

(18.4)

Thus, phase motions represent a projection of total 2D motions onto the orientation direction of the sampled function.

Phase Calculation With scale of the sampled marker determined, the phase shifts of the physical marker kernel are determined frame by frame through a complex 2D convolution. F = wc (m, n) ∗ f (x, y)

(18.5)

This operation does not require up-sampling in the calculation of each frames convolution and still return a subpixel representation of motion, thus offering a dramatic improvement in calculation speed over subpixel cross correlation techniques. Importantly, the calculation offers a very accurate measure of phase as long as the scale parameters of w are calculated accurately [15]. With sampling errors, the precision of this calculation is diminished, but will be shown through real world testing to still offer excellent precision. The reason for such high accuracy is principally that for a stationary signal, this

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calculation is exact, and due to the weighted averaging performed by this operation any stochastic noise has little effect on the calculation for sufficient measurement points (M*N).

Displacement Calculation The final step of measuring displacements through this method is to scale the phase ϕ to the real world units of the target. This is done using the already known target kernel scale λ and Eq. 18.6

x=

λϕ 2π

(18.6)

For the 1D case the calculated displacement represents the project of the total displacement onto the orientation direction of the sampled marker. For the 2D case, the displacements measured independently for each of the orthogonal functions allows for a calculation of the total displacement projected onto the plane of the target by Eqs. 18.7 and 18.8 x=

w1 ϕ1 w2 ϕ2 cos θ1 + sin θ2 2π 2π

(18.7)

y=

w2 ϕ2 w1 ϕ1 sin θ1 + cos θ2 2π 2π

(18.8)

18.3 Analysis 18.3.1 Implementation and Validation The effectiveness of the KBOS technique was explored through a series of controlled tests that looked to explore the techniques resolution, accuracy, and range of use.

Validation Testing The effectiveness of the KBOS system was evaluated through a series of tests using a controlled excitation of a rigid block by a small shaker. The shaker was provided with a variety of excitation waveforms and the motion was simultaneously recorded using an LVDT, Pontos point tracking system, and the KBOS sensor. The measurements were then compared to benchmark the KBOS sensors against well-established displacement measurement techniques. For the validation test, a rigid block secured to a 1 axis shaker provided controlled motion of a KBOS target as in Fig. 18.5. Then, an LVDT and a commercial point tracking was used to simultaneously record the motion of the rigid block. Appendix A includes more detailed information about the test setup used, as well as sensor information. For the tests, the KBOS sensor was printed with a 1D triangular wave with dimensions of 1in by 1in and 3 wavelengths per inch vertically as shown in Fig. 18.6. The target was printed on an inkjet printer with 200 ppi. The convolution kernel used calculation of phase for this test is a complex exponential. A variety of excitations were input to the shaker and recorded. Figures 18.7 and 18.8 shows the shaker displacements recorded using each sensor under two different broadband excitations. Tables 18.1 and 18.2 shows the TRAC values between the different sensors. This demonstrates excellent correlation between the signals and attaches a quantitative measure to the accuracy of the KBOS sensor. From these test it can be seen that the KBOS sensor was capable of accurately measuring the shaker displacements regardless of the motion bandwidth and complexity. In order to examine the noise floor of the KBOS system, a baseline test was performed were no excitation was supplied to the shaker and the motion was recorded using the KBOS sensor. Figure 18.9 shows the frequency domain response for this test.

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Sensors – 1in LVDT – 12MP PONTOS System – High speed Photon Camera & Periodic Marker Ancillary – Smart Shanker – Lighting – Dagtron Data Acquisition – High speed Cameras – –

. 500Hz for 1555 samples . 640Hz for 2048 samples Pontos system . 365Hz for 1094 samples LVDT

Fig. 18.5 Validation Test Setup 1

1in

1D Waveform

0.8 0.6 0.4 0.2 0

1in Fig. 18.6 KBOS design used in validation testing Fig. 18.7 Time response to white noise excitation

From this test, it is seen that the KBOS system appears to measure a low frequency drift that can be attributed to subtle drift between the seismically isolated shaker and camera. The frequency domain response also shows a response at 130 Hz and 140 Hz which are attributed to the cameras motion driven by its cooling fan. From the frequency domain response for the KBOS sensor it can be seen that the maximum of the noise in the frequency domain is 5e10-5 mm which represents just 1/100,000 of a pixel. In practice, a real signal sufficiently sampled ~twice as large as this value should be easily distinguishable from noise and be the minimum measureable motion. This means that from two meters away this sensor can record sub-micron level 2D displacements without need for a calibration procedure and using a single camera.

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Fig. 18.8 Time response to sin sweep excitation

Table 18.1 TRAC values for white noise response

Table 18.2 TRAC values for sin sweep response

KBOS LVDT Pontos

KBOS 1 .952 .964

LVDT .952 1 .870

Pontos .964 .870 1

KBOS LVDT Pontos

KBOS 1 .986 .993

LVDT .986 1 .997

Pontos .993 .997 1

Fig. 18.9 Baseline no excitation frequency domain measurement

18.4 Conclusion This work presents an improvement on the state of the art in optical displacement measurement sensors. The developed KBOS technique offers an improvement in the subpixel resolution of displacement measurements using a target based optical

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system. This technique also improves on the speed of displacement calculations, which could be leveraged to improve real time motion monitoring for such applications as it would be useful. In the future, this technique will be further explored as many variables in the technique remain unquantified, leaving potential room for improvements. The team plans to investigate the effectiveness of other kernel functions, the impacts of camera effects such as lens distortions and rolling shutter effects, and the impact of quantization sampling errors and pixel noise. Also, the independence of this technique on camera orientation will be demonstrated in later work alongside real world testing to demonstrate the utility of this approach. Further, the team is currently working to extend this technique to a calibration free 3D displacement measurement technique using a stereo camera system. Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 1762809. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Appendix • Validation Testing Parameters and Details – KBOS System Target Marker • Single complex exponential function v = 3 cycles/1in • Printed using generic office printer at 300 dpi • Printed on label paper Camera • • • • • •

Photon Fastcam SA2 8bit grayscale pixel depth recorded 24 mm focal length 4MP resolution ~2 m distance from marker 500 Hz sampling rate

– LVDT Shaevitz Sensors DC-EC 1000 10 V/in gain +/− 1in travel 750 Hz sampling rate – PONTOS System Bauer stereo camera pair • • • •

8bit grayscale pixel depth recorded 24 mm focal length 12MP resolution 365 Hz sampling rate

.007 pixel calibration resolution

References 1. Kohut, P., et al.: Monitoring of a civil structure’s state based on noncontact measurements. Struct. Health Monit. 12, 411–429 (2013) 2. Feng, D., Feng, M.: Computer vision for SHM of civil infrastructure: From dynamic response measurement to damage detection – a review. Eng. Strucut. 156, 105–117 (2018)

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3. Helfrick, M., et al.: 3D digital image correlation methods for full-field vibration measurement. Mech. Syst. Signal Process. 25, 917–927 (2010) 4. Yu, L., Pan, B.: Full-frame, high-speed 3D shape and deformation measurements using stereo-digital image correlation and a single color high-speed camera. Opt. Lasers Eng. 95, 17–25 (2017) 5. Poozesh, P., et al.: Large-area photogrammetry based testing of wind turbine blades. Mech. Syst. Signal Process. 86, 98–115 (2017) 6. Lee, J.J., Shinozuka, M.: A vision-based system for remote sensing of bridge displacement. NDT&E Int. 39, 425–431 (2006) 7. Sladek, J., et al.: Development of a vision based deflection measurement system and its accuracy assessment. Measurement. 46, 1237–1249 (2013) 8. Fukuda, Y., Shinozuka, M., Feng, M.: Cost-effective vision-based system for monitoring dynamic response of civil engineering structures. Struct. Control Health Monit. 17, 918–936 (2009) 9. Guizar-Sicairos, M., Thurman, S.T., Fienup, J.R.: Efficient subpixel image registration algorithms. Opt. Lett. 33, 156–158 (2008) 10. Liu, B., et al.: Vision-based displacement measurement sensor using modified Taylor approximation approach. Opt. Eng. 55, 114103 (2016) 11. Freeman, W., Abelson, E.: The design and use of steerable filters. IEEE Trans. Pattern Anal. Mach. Intell. 13, 891–906 (1991) 12. Srinivasulu, D.: 2D dual-tree complex wavelet transform. Contemp. Eng. Sci. 5, 127–136 (2012) 13. Breezer, R.: A First Course in Linear Algebra. Congruent Press, Gig Harbor (2012) 14. Lyon R.: A Brief History of ‘Pixel’. IS&T/SPIE Symposium on Electronic Imaging (2006) 15. Liao Y. Phase and Frequency Estimation: High-Accuracy and Low-Complexity Techniques. Thesis at Worcester Polytechnic Institute (2011) Matthew Southwick I am a PhD student in the mechanical engineering department at Umass Lowell researching optical measurement techniques. I graduated with my B.S. in mechanical engineering from The University of Pittsburgh in 2019.

Chapter 19

Current Methods for Operational Modal Analysis of Rotating Machinery and Prospects of Machine Learning German Sternharz and Tatiana Kalganova

Abstract Current demands for operational efficiency and safety of aircraft engines require a deep understanding of engine component behavior. Mechanical spinning tests can be utilized for this purpose by extracting modal properties of the tested hardware by Operational Modal Analysis. However, this process is challenging because the operational input force is unknown and is disturbed by influences such as harmonics in the excitation, which are characteristic for rotating hardware. To address this issue, this paper provides a qualitative review of existing methods for Operational Modal Analysis (OMA) applied to rotating machinery. A plate structure under random excitation with additional harmonic loading, which resembles the effect of rotation, is presented. The vibration response of the plate has been processed by six different OMA methods to estimate their applicability and output variance. The results of the conducted research and experimental analysis indicate that a hybrid approach based on machine learning and different vibration analysis methods can be beneficial. Specifically, data fusion and condition monitoring can be facilitated to acquire more consistent and accurate analysis results and ultimately contribute to optimized engine system designs. Keywords Operational modal analysis · Rotating machinery · Harmonic excitation · Artificial intelligence · Condition monitoring

19.1 Introduction Modern turbofan engines are exposed to high temperatures, combined loads, environmental influences and require great manufacturing effort. As engine technology evolves, requirements to data acquired by mechanical testing facilities rise as well, demanding increasing precision and comprehensive determination of the structural response at various loading scenarios. The dynamic response of any mechanical system is characterized by its modal properties, like the mode shapes, damping ratios and natural frequencies (also known as eigenfrequencies or normal frequencies). Once the modal parameters of a tested structure have been estimated, they can be used for evaluation of fatigue, condition monitoring of the component as well as verification and updating of respective numerical models. The data is also used for engine system design and performance calculations, where the dynamic behavior of multiple components has to be considered at various operating conditions. To determine modal parameters of running systems Operational Modal Analysis (OMA) must be employed, since the excitation forces are unknown. This excitation is caused by the machine operation itself and in case of rotating machinery, it can be caused by unbalance, rolling bearing elements, meshing gears, combustion and aerodynamic perturbations. This is the main difference to Experimental Modal Analysis (EMA), where the input excitation is either predefined or measurable like in the case of shaker or hammer impact testing. In order to compensate this absent information, classic OMA methods presume the excitation force to have a flat broadband frequency spectrum, i.e., white noise [1, 2]. In contrast to that, rotating systems generate predominantly harmonic, narrow-banded perturbations, which violate this assumption and can therefore result in errors. For example, harmonic peaks can be falsely interpreted as structural modes and distort or mask estimated parameters of actual structural modes [3]. These issues become more dominant the closer a harmonic and structural mode are located to each other [4, 5].

G. Sternharz () · T. Kalganova Department of Electronic and Computer Engineering, College of Engineering, Design and Physical Sciences, Brunel University London, Uxbridge, Middlesex, UK e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 D. Di Maio, J. Baqersad (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47721-9_19

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The issue of high harmonic amplitudes in the input force spectrum is especially severe in context of spinning tests. This is because individual rotating components are tested in laboratory environment without contributions from other engine subsystems or environmental excitation effects, which would increase the level of random excitation. At the same time, modal parameters from spinning tests are especially valuable because of their close representation of the rotating components in their final operation condition as opposed to measurements performed on stationary parts. The main objective of this paper is to provide a review of current methods that can be used for OMA of rotating machinery to assist an informed decision of the engineer and researcher. In the next section, such OMA methods are evaluated based on current literature and case studies. Afterwards, the results of 6 different OMA methods from a controlled test setup are presented to evaluate their applicability during minor harmonic disturbance. Finally, a discussion of the main findings and ideas for future work are given along with relevant applications of Artificial Intelligence (AI) in the last section of this article.

19.2 Existing Methods for Operational Modal Analysis of Rotating Machinery This section covers OMA methods, which consider or have been tested in conditions containing harmonics in the input excitation, since this is a main characteristic of rotating machinery vibration. The following subsections classify the methods based on the approach of how harmonics are considered within the respective methods.

19.2.1 Methods Not Specifically Adapted for Input Harmonics There is a variety of OMA methods, which do not specifically account for harmonics originating from the input load. Nevertheless, several of these methods were tested in case studies dedicated to the methods’ application in such conditions and show considerable performance in certain use cases. Stochastic Subspace Identification (SSI) algorithms form a family of time-domain OMA methods and benefit from a large application experience in research and industry. Yang et al. [6] performed a comparison of three parametric time-domain OMA methods in the presence of harmonics using a numerical model with 5 degrees of freedom (DOF). Compared to the covariance-driven Stochastic Subspace Identification (SSI-COV) and the Instrumental Variable (IV) methods, the projection-based, i.e., data-driven Stochastic Subspace Identification (SSI-DATA) provided better results. The output consisted of relatively clean stabilization diagrams, more accurate determination of harmonic frequencies/damping ratios and estimation of a structural mode even at high proximity to a harmonic peak. These observations regarding SSI-DATA methods are also largely supported by other studies [2, 3]. However, the downside of higher computational cost, required by SSI-DATA, is reported as well in [2, 7]. Unfortunately, in the publication by Yang et al. [6], a tabular (quantitative) comparison is only provided for the identified harmonics while the comparison of values for the actual structural modes is limited to illustrations of stabilization diagrams. The applicability to real operating structures with harmonics has yet to be investigated, since the provided comparison results are based on a simulated point-mass system. Another potential downside of SSI methods is the relation between the modal frequencies of interest and measurement time, requiring longer measurements for the estimation of low frequency modes [3, 8]. Moreover, it has been found that SSI is susceptible to severely under-quantified data [9]. Compared to SSI-DATA methods, the frequency-domain methods Frequency-Domain Decomposition (FDD) and Enhanced Frequency-Domain Decomposition (EFDD) are less reliable in cases with close harmonic and structural modes, potentially producing biased estimations and wrong identification [3]. The poly-reference Least-Squares Complex Frequency-Domain (pLSCF) estimation, however, has shown acceptable results in a study of a wind turbine operating at constant speed [10]. It is a widely used OMA method also known under the name of its commercial implementation “PolyMAX”. It provided approximately equal results compared to the modified Least-Squares Complex Exponential (modified LSCE) time-domain method. The modified LSCE method requires harmonic frequencies to be provided before the algorithm is applied. As such, it will be covered in a following subsection about methods with explicit consideration of input harmonics. An analysis of simulated vibration data by the pLSCF method showed that a harmonic frequency, coinciding with a structural mode, led to spurious stable poles around this frequency [11]. The authors also analyzed operational data of a large diesel engine. However, the engine produced such strong harmonic peaks, that the algorithm falsely identified them as the sole modes of the structure. Yet still, the application on in-flight vibration measurements of a helicopter showed that the

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pLSCF method was able to detect modes despite harmonic disturbances. The authors propose Time Synchronous Averaging (TSA) for reduction of harmonic disturbances from the time signal to enhance the subsequent modal estimation by the pLSCF or any other OMA method. Therefore, TSA is presented in more detail in the next subsection. Although different studies report successful utilization of SSI [6, 12] and pLSCF [4, 10, 11] methods for OMA of rotating machinery, it is believed that no direct performance comparison of these methods has been conducted yet.

19.2.2 Reduction of Harmonics by Signal Preprocessing The following methods aim to mitigate disturbing input harmonics from the measurement. While this approach does not provide modal parameters by itself, it can facilitate the estimation of modal properties with a subsequently applied OMA method. The TSA method performs resampling of the original time signal to the order domain, transforming data samples per time interval to data samples per rotation angle. Subsequent synchronous averaging is used to remove harmonic peaks from the signal. Finally, the filtered signal is resampled back to the time domain, so it can be processed further by an OMA method. A limiting factor is that TSA requires a tacho signal and an approximately constant number of revolutions per minute (rpm). In a 6 DOF simulation [11], the method was able to remove the disturbing influence of harmonics on the subsequent modal estimation with the pLSCF method despite a harmonic coinciding with a structural mode. Results from a real operational in-flight helicopter and a running diesel engine showed that TSA resulted in much cleaner stabilization diagrams and thus facilitated the estimation of structural modes. Unfortunately, the report lacks quantitative comparisons between reference parameters of the simulation model and the estimated parameters obtained using the proposed method. Moreover, the required processing time is not disclosed and could pose an issue due to the described steps of double resampling and windowed synchronous averaging. Other evaluations on operational data from wind turbines showed mixed results [10]. It is assumed, that the method is not capable of harmonic removal when the operational speed variability and spread of the harmonic peak (i.e., damping of the virtual harmonic mode) is higher than in the previously mentioned cases [13, 14]. Cepstrum Editing (CE) separates harmonic signals from the actual structural response by filtering, i.e., “liftering” of the signal in the so-called cepstrum domain. The cepstrum is usually computed from the original time signal by determining its FFT spectrum, which is subsequently log-scaled. Finally, the inverse FFT is performed to obtain the cepstrum, where different lifters are applied. In contrast to TSA, CE successfully reduced harmonics from data of a wind turbine operating at constant speed [10] and measurements from a gearbox casing at varying harmonic frequencies [14]. Similar observations in favor of CE were made when comparing it to methods based on Kurtosis and the Probability Density Function [10]. CE removed one harmonic less with otherwise similar results but performed faster than TSA when applied to measurements of helicopter fuselage at steady flight conditions [15]. However, it should be noted, that cepstrum liftering can introduce spurious numerical modes and additional damping [14]. In theory, the magnitude of this added damping can be determined and subtracted from the final results, however, a residual excess in damping values can still remain in some cases [13, 14]. Methods presented by Jacobsen et al. [16–18] utilise Kurtosis to identify potential frequencies with harmonic excitation. Afterwards, a Singular Value Decomposition (SVD) plot of the measurements is calculated and the harmonics are reduced from the plot by linear interpolation. The modified SVD functions are then used in the Enhanced Frequency-Domain Decomposition (EFDD) OMA method. In a later work [18], the authors incorporated the Curve-fitting Frequency-Domain Decomposition (CFDD) as an alternative OMA method. As a result, an improved estimation of structural modes that are close to or coincidence with a harmonic peak was achieved. However, at this point, it remains unclear how this updated method for harmonic reduction and modal estimation with CFDD compares to methods like pLSCF or SSI presented in the previous section.

19.2.3 Methods with Explicit Consideration of Input Harmonics OMA methods of this category require harmonic frequencies to be provided explicitly as input parameters and consider them in the modal analysis.

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Methods proposed by Mohanty and Rixen [5, 19, 20] extend several existing OMA algorithms to explicitly take known harmonics into account for a more truthful system identification. In shaker experiments of the authors, the modified LSCE method showed noticeable improvement compared to the polyreference LSCE (pLSCE) method in terms of identification of closely spaced structural modes and harmonics [5]. Based on simulation data, Motte et al. [10] showed that the modified LSCE mostly provides more consistent results and a slightly more accurate estimation of the natural frequency and damping ratio compared to pLSCF, i.e., PolyMAX. However, at close frequencies of a harmonic and structural mode, similar or slightly better estimation performance was observed in favor of pLSCF. It was also shown that exact knowledge of the harmonic frequency is essential for the modified LSCE method. In an experiment with an operational wind turbine, the method did not eliminate all harmonic peaks, which is explained by the imprecision of the wind turbine rpm measurements, which are required to calculate harmonic frequencies. It is also noted that the modified LSCE produces more spurious mathematical modes compared to pLSCF.

19.2.4 Methods with Implicit Consideration of Input Harmonics The OMA methods presented below are inherently not negatively affected by the presence of harmonics in the excitation spectrum. Despite this promising outlook, these methods have other limitations, which are also highlighted below. Transmissibility based OMA (TOMA) is capable of handling excitation forces that are heavily influenced by harmonic inputs or coloration of the input noise [21]. The method determines modal system parameters directly, independently from the input spectrum. However, multiple transmissibility functions are required, which must be determined at different loading conditions. In context of rotating machinery, it is questionable if a change in rotating speed provides a change in the loading conditions that is sufficient for the method. In addition, the method requires the number of uncorrelated input loads to be estimated. These factors inhibit an application in real operating scenarios, since these values are unknown. It is assumed that the performance of TOMA in real operational environment has not been demonstrated yet. Current experiments are limited to laboratory set-ups [21] and simulations [22] with a known number of spatially clearly separated and independently acting input loads. Therefore, it can be concluded that this method is not yet suitable for the intended use with rotating machinery. Order-Based Modal Analysis (OBMA) [23] is a method that follows a different approach compared to the previously presented OMA methods, in that it does not rely on broadband random excitation. Instead, OBMA utilizes harmonic input loads, originating from rotation orders of the running components. However, each of the harmonics in the input signal only covers a narrow frequency range, inhibiting the desired excitation of structural modes. Therefore, OBMA requires the vibration measurements to be performed during an acceleration or deceleration run of rotating parts in the evaluated structure. Order tracking is performed to identify the amplitude and phase of individual orders as a function of rpm. Hence, the rotating speed, i.e., tacho signal is required. Afterwards, a tracked order function is provided as input for OMA. This method has been employed in operational experiments with a wind turbine gearbox [23] and acoustic and vibrational measurements from a 4-cylinder car [24, 25]. In another comparison [26], a car mock-up was excited by a shaker in two configurations. Reference results were obtained by pLSCF from a flat broadband excitation without harmonics. For OBMA, a test run with 34 order harmonics with increasing frequencies, simulating a linear engine acceleration, was performed. A comparison of the results showed consistent natural frequencies with a maximum error of 2.6%. The estimated damping ratio and mode shapes, however, have a high variance depending on the tracked order used for the OBMA method. For example, the relative error in the damping estimate of the third mode ranges from 2.4% to 85.4%, depending on the chosen order. With the absence of sufficient random excitation in combination with an acceleration or deceleration run, it is expected that this method can outperform the previously introduced OMA methods, since OBMA prevents spurious response peaks that can lead to falsely identified modes, called “end-of-order” modes [23–25]. This is supported by a more recent study of a planetary gearbox [27], which was tested by hammer impact EMA, OMA with pLSCF and OBMA. It was found that the pLSCF method at stationary operating conditions only identified 8 out of 13 modes, which were captured by EMA and OBMA. Similar to previous observations [26], OBMA determined partly much greater damping values with errors ranging between 5% and 316% relative to EMA damping results. There is less relative error in natural frequencies at a maximum deviation of 8%.

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19.3 Experimental Case Study The described experiment was conducted by Jacobsen et al. [16] and the acquired vibration measurements were used for further analysis work below, performed by the authors of the present paper. The authors of the original experiment present techniques for harmonic identification and propose a method for reduction of identified harmonics based on the EFDD method. In contrast to that, the present paper uses the experimental data to determine and compare the general performance of several available OMA methods. An aluminum plate (Fig. 19.1a) was instrumented with 16 unidirectional accelerometers measuring vibration response normal to its surface. Random tapping on the plate was combined with a harmonic input load from a shaker to resemble an influence of rotating components. The measurement data has a sampling frequency of 4096 Hz and a duration of 60 s. The measured accelerations were analyzed by 6 different OMA methods: Enhanced Frequency-Domain Decomposition (EFDD), Curve-fit Frequency-Domain Decomposition (CFDD), SSI Principal Components (SSI-PC), SSI Unweighted Principal Components (SSI-UPC), SSI Canonical Variate Analysis (SSI-CVA), multireference Ibrahim Time-Domain (ITD) method. As introduced in the review section of this paper, EFDD and CFDD are frequency-domain methods; SSI-PC, SSIUPC and SSI-CVA can be attributed to SSI-DATA methods; finally, ITD is an algorithm closely related to SSI-COV. The ITD stabilization diagram in Fig. 19.1b clearly shows the first three structural modes. The detrimental impact of the input harmonic is also visible as it is identified as a stable mode. However, this is not necessarily problematic as long as the analyst is aware of this. Modal parameters of 8 plate modes are estimated by all 6 OMA methods and illustrated in Fig. 19.2. It is visible that each method reports different values for the natural frequency and damping ratio, although the same structure and data set were used for each OMA method. Analyzing the structural response without harmonic excitation leads to the respective natural frequency and damping ratio of 354 Hz and 0.65% for the first mode when using EFDD [16]. The estimated modal values presented below do not significantly differ from these numbers. Therefore, it can be concluded that the estimation of the first mode is not disturbed by the harmonic excitation in neither of the compared OMA methods. This is due to the setup with a single harmonic, which has a low amplitude relative to the response of the neighboring first mode as shown by Fig. 19.1b. Nevertheless, the estimated modal parameters differ overall. The maximum relative deviations of the methods per mode are limited to 0.7% in natural frequencies (at mode 8) but reach 213% for the damping ratios (at mode 7).

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19.4 Conclusions Findings form the presented review of existing OMA methods and implications of the experimental results from the previous section are discussed below. Finally, plans for future work are highlighted and the promising role of AI and Machine Learning (ML) in the field of vibration analysis is outlined.

19.4.1 Discussion of Results and Future Work There is a variety of methods, which have been used at the presence of harmonic signals in the excitation load as illustrated in in the presented review. A qualitative comparison of the discussed methods which appear to be promising for rotating machinery applications is summarized in Table 19.1.

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In many cases, there seem to be no concrete guidelines on criteria for the selection of a specific method over the other due to a lack of quantitative comparisons. The methods’ relative performances at different levels of measurement noise, sampling frequencies, different harmonic input amplitudes, proximity of response peaks, and rotating speed variability are mostly unknown. The same is true for computation durations required by individual methods. Numerical comparisons and sensitivity studies in frame of future work would address this gap. The SSI method family and pLSCF method have been identified as industry-leading due to their performance and experience with these methods from tests in a wide range of applications. In addition, preprocessing techniques, like those mentioned in the review section, can successfully supplement these methods. Despite this, no study was found, which would directly compare these OMA methods to investigate their strengths and weaknesses at specific testing conditions. Moreover, most of the presented studies, while considering harmonic inputs, only cover cases where the harmonic amplitude does not exceed the actual structural response. However, harmonics of rotating machinery can reach far higher amplitudes [11, 28], impeding modal estimation. In cases with high amplitudes of input harmonics, OBMA poses a promising approach since it utilizes the harmonics, which are otherwise often detrimental. However, this method is only applicable to acceleration/deceleration runs with sufficiently large speed ranges. Therefore, it should be combined with complementary OMA methods for the remaining operating conditions to achieve a continuous acquisition and monitoring of modal parameters. Similarly, each OMA method has its specific strengths and weaknesses as outlined by Table 19.1. Hence, for future work, a two-step approach is proposed, which would identify the current operation conditions and utilize the most suitable OMA methods, potentially using data fusion. As demonstrated, different OMA methods provide different estimations of modal parameters even in the presented experimental setup with well-spaced modes and negligible measurement noise. In case of OBMA, it was also observed, that results from a single data set vary depending on the chosen harmonic order. Thus, it is assumed that OMA would benefit from data fusion, i.e., a combination of different output sources into a single data set.

19.4.2 Overview of Machine Learning for Vibration Analysis Machine learning methods like support vector classification and Deep Boltzmann Machines (DBM) are shown to be capable of data fusion although research in this field is still limited [29]. Another important application of vibration measurements is Structural Health Monitoring (SHM) and machine condition monitoring. SHM can be based on modal properties of the monitored structure determined by OMA. Consistent and accurate parameters are crucial for timely detection of damage, emphasizing the relevance of appropriate OMA methods in combination with data fusion. While more traditional SHM methods, like Finite Element (FE) model updating require substantial computational efforts, AI-based SHM is capable of real-time analysis. Moreover, raw vibration signals can be used directly as input for SHM by the exploitation of Convolutional Neural Networks (CNN) and Auto-Encoders [30, 31]. The main advantage is that the methods autonomously learn optimal damage-sensitive features from the measurements. However, measurements of known fault conditions/locations must be available for this learning process. SHM through anomaly detection does not provide such detailed information but can be implemented with measurements solely of the nominal structural response. This is, for example, possible by data-fitting with a trained Autoregressive Exogenous Input (ARX) model [32], although the cited study focuses on civil engineering applications. Khan and Yairi [29], as well as Zhao et al. [33] review AI deep learning with a focus on structural health management. Liu et al. [34] review AI methods, including k-Nearest Neighbor, Naive Bayes classifier, Support Vector Machine, Artificial Neural Network and deep learning specifically for fault detection in rotating machinery. Because of the outlined potential of AI methods, research interest in AI-supported vibration analysis has increased over the last years. In conclusion, this field should be utilized and investigated further to facilitate future vibration-based analysis of rotating machinery. Acknowledgements We would like to express our gratitude to Cristinel Mares from Brunel University London as well as Moritz Meyeringh from Rolls-Royce Deutschland for their valuable consultations. This work was also supported by the Engineering and Physical Sciences Research Council (UK) and EXOLAUNCH GmbH (Germany).

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30. Janssens, O., Slavkovikj, V., Vervisch, B., Stockman, K., Loccufier, M., Verstockt, S., Van de Walle, R., Van Hoecke, S.: Convolutional neural network based fault detection for rotating machinery. J. Sound Vib. 377, 331–345 (2016). https://doi.org/10.1016/j.jsv.2016.05.027 31. Abdeljaber, O., Avci, O., Kiranyaz, S., Gabbouj, M., Inman, D.J.: Real-time vibration-based structural damage detection using one-dimensional convolutional neural networks. J. Sound Vib. 388, 154–170 (2017). https://doi.org/10.1016/j.jsv.2016.10.043 32. Gul, M., Catbas, F.N.: Damage assessment with ambient vibration data using a novel time series analysis methodology. J. Struct. Eng. 137, 1518–1526 (2010). https://doi.org/10.1061/(asce)st.1943-541x.0000366 33. Zhao, R., Yan, R., Chen, Z., Mao, K., Wang, P., Gao, R.X.: Deep learning and its applications to machine health monitoring. Mech. Syst. Signal Process. 115, 213–237 (2019). https://doi.org/10.1016/j.ymssp.2018.05.050 34. Liu, R., Yang, B., Zio, E., Chen, X.: Artificial intelligence for fault diagnosis of rotating machinery: a review. Mech. Syst. Signal Process. 108, 33–47 (2018). https://doi.org/10.1016/j.ymssp.2018.02.016 German Sternharz Received BSc and MSc degrees in Aerospace Engineering at the Technical University of Berlin. Worked at the Institute of Aeronautics and Astronautics, Chair of Space Technology and completed a placement at Rolls-Royce Deutschland. Afterwards, secured an EPSRC funded scholarship for the current role as Doctoral Researcher in the field of Operational Modal Analysis at Brunel University London. Dr Tatiana Kalganova (PI) BSc (Hons), PhD, is a Reader in Intelligent Systems in the Department of Electronic and Computer Engineering, PGR Director at Brunel University London. She leads the “Industrial and Applied AI” research centre within Brunel Digital Science & Technology Hub. She has over 25 years of experience in design and implementation of applied Intelligent Systems engaging particularly with Evolutionary Design and Optimisation, Evolvable hardware, Modelling and optimisation of Large Systems, Strategic and Operational research, Robotics, Swarm optimisation, CNN.