River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6 Christopher Niezrecki Javad Baqersad Dario Di Maio Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics 2019 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA
River Publishers Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6 Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics 2019 Christopher Niezrecki • Javad Baqersad • Dario Di Maio Editors
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-987-0 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2019 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction 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.
Preface Rotating Machinery, Optical Methods & Scanning LDV Methods represents one of the eight volumes of technical papers presented at the 37th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Orlando, Florida, January 28–31, 2019. The full proceedings also include volumes on Nonlinear Structures and Systems; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamics of Coupled Structures; 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. Lowell, MA, USA Christopher Niezrecki Flint, MI, USA Javad Baqersad Bristol, UK Dario Di Maio v
Contents 1 Detection of Sources of Nonlinearity in Multiple Bolted Joints by Use of Laser Vibrometer................... 1 Arnaldo delli Carri, Sante Campanelli, and Dario Di Maio 2 Single High-Speed Camera Based 3D Deflection Reconstruction in Frequency Domain........................ 15 Janko Slavicˇ, Domen Gorjup, and Miha Boltežar 3 Operational Modal Analysis of a Thin-Walled Rocket Nozzle Using Phase-Based Image Processing and Complexity Pursuit................................................................................................. 19 Marc A. Eitner, Benjamin G. Miller, Jayant Sirohi, and Charles E. Tinney 4 Full-Field Strain Shape Estimation from 3D SLDV................................................................. 31 Bryan Witt, Dan Rohe, and Tyler Schoenherr 5 Characterization of a Small Electro-Mechanical Contact Using LDV Measurement Techniques.............. 47 Kelsey M. Johnson 6 Remote Detection of Abnormal Behavior in Mechanical Systems ................................................. 59 Greta Colford, Erica Jacobson, Kaden Plewe, Eric Flynn, and Adam Wachtor 7 Modal Analysis of a High-Speed Turbomachinery for Reliable Prediction of RD Properties Throughout Operating Speed Range.................................................................................. 71 Yuhei Shindo, Kazuhiko Adachi, Satoshi Kawasaki, and Mitsuru Shimagaki 8 Mapping Motion-Magnified Videos to Operating Deflection Shape Vectors Using Particle Filters............ 75 Aral Sarrafi and Zhu Mao 9 Structural Health Monitoring of Wind Turbines Using a Digital Image Correlation System on a UAV ...... 85 Ashim Khadka, Yaomin Dong, and Javad Baqersad 10 Full-Field Mode Shape Identification of Vibrating Structures from Compressively Sampled Video .......... 93 Bridget Martinez, Yongchao Yang, Ashlee Liao, Charles Farrar, Harshini Mukundan, Pulak Nath, and David Mascareñas 11 Experimental Modal Analysis of Tumorigenesis and Cancer Metastasis ......................................... 101 Bridget Martinez, Yongchao Yang, Charles Farrar, Harshini Mukundan, Pulak Nath, and David Mascareñas 12 Full Field Strain Measurements Using 3D Laser Vibrometry...................................................... 105 Samuel Tilmann 13 Output-Only Modal Parameter Estimation Using a Continuously Scanning Laser Doppler Vibrometer System with Application to Structural Damage Detection ........................................... 113 Y. F. Xu, Da-Ming Chen, and W. D. Zhu vii
Chapter1 Detection of Sources of Nonlinearity in Multiple Bolted Joints by Use of Laser Vibrometer Arnaldo delli Carri, Sante Campanelli, and Dario Di Maio Abstract The use of non-contact measurement methods for detecting and locating sources of nonlinearities can be potentially a break-through in the nowadays experimental modal analysis. The primary goal is to define more effective test strategies, whereby contact sensors will measure the nonlinear vibration responses at the best location possible. Jointed structures are a typical example where a large number of the joint can pose the question of where what and how to measure the nonlinear response. Upon the identification of one, or more, nonlinear response mode the objective is to determine where is the source of such nonlinear vibration. Nonlinearity can be characterised when its source is well defined and can be adequately tested. This paper will attempt to detect and locate the source of nonlinearity from a multi-beam jointed assembly. The approach will be carried out by using both contact and non-contact measurement methods, the results of which will be compared and evaluated. The operator to detect the source of nonlinearity will be the coherence function applied to random response data. Keywords SLDV · Bolted joints · Nonlinear vibration testing 1.1 Introduction This paper attempts to exploit the potential of the scanning LDV system to measure vibration nonlinear response at a much greater number of locations, than it can be done by using setup based on contact sensors. The objective is to use such denser measurement grid to identify one, or more, sources of nonlinearity. It is not intended to replace the accelerometer for the characterization and quantification of a source of nonlinearity. The localization of nonlinearities is fundamental when such sources are discrete as for bolted structures. It is notorious that structures with a high number of interfaces will exhibit nonlinear responses when subjected to high amplitude of excitation forces. From a model validation viewpoint, the accurate localization, characterization and quantification of the nonlinearity can make the modelling work more effective and time efficient. Resources can be dedicated to improving the model where it is needed by inclusion of the nonlinear physics. The localization of nonlinearities was usefully carried out in [1]. It was demonstrated that by setting up a good number of accelerometers on a structure the localization can be done with a good level of accuracy. However, it looked clear that such a method of localization depends on the setup of the contact sensors, and when an engineering judgment is not based on the identification of nonlinearity it might be possible that nonlinearities stay hidden to the sensors. Following the methodology applied to locate the source of nonlinearity (explained in the following sections), it become interesting to use a different technology which would enable much denser measurement grind than offered by contact sensors. Three research works addressed the topic of localization by using the scanning laser vibrometer. One research was focussed on bolted flanges and the use of continuous scanning methods to identify the source of nonlinearity. That approach showed that by mapping the response phase of the deflection shape measured at constant frequency and several level of amplitudes one could determine the level of nonlinearity exhibited by a vibration mode [2]. The scanning laser vibrometer was used in its step-scanning mode in two attempts, where high resolution mode shapes were measured to determine which mode would exert more nonlinear A. delli Carri School of Mechanical, Aerospace and Automotive Engineering, Coventry University, Coventry, UK S. Campanelli Department of Mechanical Engineering, Universita’ Politecnica delle Marche, Ancona, Italy D. DiMaio ( ) Department of Mechanical Engineering, University of Bristol, Queens Building, Bristol, UK e-mail: aeddm@bristol.ac.uk © Society for Experimental Mechanics, Inc. 2019 C. Niezrecki et al. (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12935-4_1 1
2 A. delli Carri et al. response [3]. The outcome of that research showed that some mode shapes create high strain distribution at the flanges’ region and thus enabling interface nonlinear conditions. A more recently publication was focussed on use the stepped scanning method to measure the response phase of the deflection shapes of an aero-engine casing assembly onto which accessories were mounted to generate sources of nonlinearities [4]. The research showed that response phase of the deflection shape was a good indicator of the source of nonlinearity. The present paper aims (i) at starting from those earlier success to detect the nonlinearities of a structure, and (ii) at combining the method of localization based on accelerometers with the use of a scanning laser vibrometer. A novel structure will be used to explore these new developments. The structure is made of ten blocks each of which bolted by two bolts to form a single prismatic beam. The source of nonlinearity can be moved by opening one or more pairs of bolts. The next section will explain in more details the tests structure and setup. The paper will proceed by attempting the same localization method using a set of four reference accelerometers and the measurement made by a scanning laser vibrometer over a grid of 30 points. 1.2 Test Structure, Setup and Experimental Method The test structure was designed with the idea to repeat the same basic unit ten times. Figure 1.1 shows the basic unit which can be bolted by two pairs of M8 bolts. Figure 1.2 shows both the solid mode and the real unit made of mild steel. The full assembly of the ten units is presented in Fig. 1.3. The 18 bolts were tightened up to 20 Nm to assure full clamping conditions. The beam was suspended by strings from a frame and a shaker was installed at the bottom of the assembly, as showed in Fig. 1.4a. Thirty measurement points were marked on the beam where the laser beam would measure the vibrations. Four reference accelerometers were also included in the measurement setup, one of which at the drive point location. Figure 1.4b shows the measurement setup. A custom-made control LabView panel was used to drive the laser beam onto the measurement points and to acquire the vibration response. It was decided to avoid the use the Polytec control panel to give full accessibility to the generation of the excitation signal. Four random noise signals, with 1M samples at 10 kHz sampling frequency, were generated at four different amplitudes. These four signals were stored and used for measuring the 30 LDV measurement points and the four accelerometers. This approach was decided to allow the excitation signal from the generator to be always the same, avoiding Fig. 1.1 CAD drawing of the basic unit
1 Detection of Sources of Nonlinearity in Multiple Bolted Joints by Use of Laser Vibrometer 3 Fig. 1.2 Solid model in (a) and real unit in (b) Fig. 1.3 Solid model in (a) and real assembly in (b) variations if it were to be generated every time. The four amplitudes were identified as follows, 18 mV, 180 mV, 1080 mV and 1800 mV, respectively. The gain of the amplifier was fixed at one level and never changed. The test programme was designed as such. The first trial was focussed on assuring the complete linearity of the structure for the 20 Nm torque applied to the 18 bolts. Hence, levels 18 mV, 180 mV and 1800 mV were attempted. The second trial, labelled as configuration A, was carried out by reducing the torque from 20 Nm, to 13 Nm and 8 Nm of the pair of bolts, the fifth from the top. The third trial, labelled configuration B, was carried out using the same torque levels reducing the pair of bolts, the third from the top, and by resetting the fifth pair to 20 Nm. A total of three configuration were tried. 1.3 Theoretical Method The analysis was performed using conditioned spectral techniques from [5–8] and already applied in an earlier form in [1, 9]. The general equation of motion for a n-DOFs nonlinear system is M¨x(t) +C˙x(t) +Kx(t) + M i=1 qi· gi wix(t),wi ˙x(t) =f(t) Where M, K, Care the n ×n mass, stiffness and damping matrices, x(t) and its derivatives are the nx1 displacements, velocities and accelerations vectors, f (t) is the nx1 forcing vector. In addition to the usual linear terms, there are Mnonlinear (vectorial) terms that contribute to the system: under the summation operator one can discriminate the scaling factor qi that quantifies the strength of the nonlinearity with respect
4 A. delli Carri et al. (a) LDV point 1 LDV point 2 LDV point 30 Acc. 1 drive point unit n.10 Acc. 2 drive point unit n.7 Acc. 3 drive point unit n.4 Acc. 4 drive point unit n.1 (b) Fig. 1.4 Test setup in (a) and measurement points in (b) Fig. 1.5 Block diagram to the other linear terms, the nonlinear function gi(·, ·) that characterises the shape of the nonlinearity and the boolean-like vector wi used to describe the location of the nonlinear term. Using the Fourier TransformF[· ] to pass from the time domain to the frequency domain: K−ω 2 M+jωC X(ω) =F (ω) − M i=1 F qi· gi wix(t),wi ˙x(t) A(ω)X(ω) =F (ω) −G(ω) X(ω) =H(ω)· (F (ω) −G(ω)) This can be viewed as a set of nonlinear feedback forces acting on an underlying linear system and can be represented as a block diagram in Fig. 1.5. For the case of a 4-DOFs system with a grounded nonlinearity at DOF#1 and a non-grounded nonlinearity between DOF#3 and DOF#4 and excited at DOF#2
1 Detection of Sources of Nonlinearity in Multiple Bolted Joints by Use of Laser Vibrometer 5 Fig. 1.6 MISO system ⎡ ⎢ ⎢ ⎣ X1 X2 X3 X4 ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎣ H11 · · · H14 . . . . . . . . . H41 · · · H44 ⎤ ⎥ ⎦· ⎡ ⎢ ⎢ ⎣ −F[q1· g1 (x1)] F2 −F[q2· g2 (x3 −x4)] F[q2· g2 (x3 −x4)] ⎤ ⎥ ⎥ ⎦ ⎧ ⎪ ⎨ ⎪ ⎩ X1 =−H11G1 +H12F2 −H13G2 +H14G2 X2 =−H21G1 +H22F2 −H23G2 +H24G2 X3 =−H31G1 +H32F2 −H33G2 +H34G2 X4 =−H41G1 +H42F2 −H43G2 +H44G2 Rewriting the equations in terms of the input force, one decomposes a single SIMO system with nonlinear feedback into a set of reverse MISO systems like in Fig. 1.6. ⎧ ⎪ ⎨ ⎪ ⎩ F2 =H−1 12 X1 +H−1 12 H11G1 +H−1 12 (H13 −H14)G2 F2 =H−1 22 X2 +H−1 22 H21G1 +H−1 22 (H13 −H14)G2 F2 =H−1 32 X3 +H−1 32 H31G1 +H−1 32 (H13 −H14)G2 F2 =H−1 42 X4 +H−1 42 H41G1 +H−1 42 (H13 −H14)G2 By feeding in any combination of location vectors wi, nonlinear operators gi(·, ·) and scaling factors qi, it is possible to assess the quality of the system by making use of standard spectral techniques. One of the best suitable metrics of causality is the multiple coherence between the one output and all the inputs, defined as γ 2 (ω) = SFX(ω)· S−1 XX(ω)· SH FX(ω) SFF (ω) Where S are the frequency-dependent averaged auto and cross-spectral density matrices. Finally, this can be turned into a coherence index by normalising its integral over the considered bandwidth: κ = 1 ω2 −ω1· ω2 ω1 γ 2 (ω) dω 1.4 Preliminary Analysis As stated in the previous section, accelerometers and laser data have different structure: the former consists of 30 different tests of 4 acceleration response channels acquired at the same time while the latter is a single collection of 30 velocity response channels acquired at different times. It is thus necessary to perform some preliminary data analysis to check the sanity of the data, in form of linearity and stationarity checks. Linearity checks are performed by super-imposing FRFs and coherences of the systems at different force levels. The plots in Figure 1.7 show severe curve degradations in the tests at 8 Nm torque.
6 A. delli Carri et al. (A) chan#1 (L) chan#30 10-10 10-5 10-10 1 500 1000 1500 2000 frequency [Hz] 3000 4000 5000 3500 0 γ2 xy γ 2 xy 1 0.5 0018mV 0180mV 1800mV 4500 2500 0 500 1000 1500 2000 frequency [Hz] 3000 4000 3500 0 1 0.5 4500 5000 2500 |H| |H| Fig. 1.7 FRFs and Coherences of LDV point 1 and LDV point 30 -4 -2 0 0 10 20 30 25 15 realisation# 5 0 10 20 30 25 15 realisation# 5 m s 2 (A) stationarity plot ×10-3 4 10 8 6 4 2 Fig. 1.8 Accelerometers stationarity plots If gaussian input voltage is provided, a gaussian output force is expected from the shaker. This is not generally the case due to shaker-structure interactions. A gaussian time history passing through a non-linear system always generates a nongaussian output [1]. Stationarity checks are performed to assess the quality of the data and determine if the system is changing over time (the structure might be settling on supports or getting to the operating temperature). The stationarity plots for both accelerometers and lasers are found in Figs. 1.8 and 1.9. By inspecting the mean and standard deviation of the channels, it can be observed that the accelerometric channels are quite stationary in mean, but the structure had to settle at around the 15th realisation for a stable standard deviation. The laser means and standard deviations are more scattered around an average. Since the laser channels were not captured at the same time it is complicated to discern if the ensemble data is ergodic or even stationary, as some channels might have been captured before the structure had any time to settle. Since the quality metric chosen is the multiple coherence, the data was sanitised by detrending. This operation has no impact on coherence, that only measures the input-output causality, but will hinder any calculation and retrieval of frequency response functions.
1 Detection of Sources of Nonlinearity in Multiple Bolted Joints by Use of Laser Vibrometer 7 -2 2 0.025 0.02 0.01 0.015 0.005 ×10-3 (L) stationarity plot 0 10 20 30 block# 40 50 60 70 80 90 100 0 10 20 30 block# 40 50 60 70 80 90 100 m s 0 Fig. 1.9 LDV stationarity plots 0.8 0.8 1 0.6 tanh(x) Taylor (7th) linear 0.6 0.4 0.4 0 0 0.2 0.2 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -1 -0.8 Fig. 1.10 Hyperbolic tangent function 1.5 Characterisation of the Nonlinearity The frequency response functions (Fig. 1.7) show a left-shift in natural frequencies over the force range, thus indicating a softening stiffness nonlinearity in play. This is to be expected, as loose joints have some degree of play and the prestress of the bolt is small enough that fails to provide stiffness after a given threshold. After feeding in several softening nonlinear operators into the system the most responsive (i.e. the one that gives the greatest improvement in multiple coherence) is a combination of alternating softening/hardening monomials and a taperingsaturation effect. This unfortunately makes the numerical analysis extremely cumbersome. The best approximation of this behaviour is the hyperbolic tangent function in Fig. 1.10. Its Taylor expansion being tanh(x) = ∞ n=0 22n 22n −1 B2n (2n)! x 2n−1 =x − 1 3 x 3 + 2 15 x 5 − 17 345 x 7 +. . . By using this nonlinear operator, it was possible to cut the processing time without incurring in any sensible loss of accuracy. Given a suitable nonlinear operator, the location of the nonlinearity is performed by reverse MISO analysis of every nonlinear term by rotating the location vectors. These can also be filtered to include or exclude locations based on any prior information available to the engineer. In this case it is known that the nonlinearity locations will always be between two consecutive channels, thus only three locations were tested for the accelerometric channels (i.e. between 1–2, 2–3,
8 A. delli Carri et al. Table 1.1 Results Location Coherence index Test3 Test4 Test5 LASER ACCEL LASER ACCEL LASER ACCEL LASER ACCEL 1–2 3–4 0.5622 0.9452 0.5228 0.9522 0.7747 0.9818 2–3 0.5575 0.5110 0.7957 3–4 0.5555 0.5113 0.8243 4–5 0.5556 0.5093 0.7912 5–6 2–3 0.5555 0.9603 0.5082 0.9716 0.7875 0.9592 6–7 0.5585 0.5151 0.7945 7–8 0.5634 0.5076 0.7962 8–9 0.5950 0.5132 0.7919 9–10 1–2 0.5783 0.9851 0.5243 0.9746 0.7858 0.9436 1 0.5 0 0 500 pcoh ACC#1 LOC#1 (1-2) 1000 2000 3000 4000 5000 2500 3500 4500 1500 1 0.5 0 0 500 pcoh ACC#1 LOC#2 (2-3) 1000 2000 3000 4000 5000 2500 coh tanh mcoh 3500 4500 1500 1 0.5 0 0 500 pcoh ACC#1 LOC#3 (3-4) 1000 2000 3000 4000 5000 2500 3500 4500 1500 Fig. 1.11 Test3 - accelerometers 3–4) and nine for the laser channels (i.e. 1–2, 2–3, 3–4, 4–5, 5–6, 6–7, 7–8, 8–9, 9–10). The multiple coherence index was then calculated for all these combinations. The highest multiple coherence index should indicate the correct location of the nonlinearity. Results can be found in Table 1.1 and all the relevant plots in Figs. 1.11 and 1.12. Accelerometric data features a much higher coherence index. This is expected, all channels being acquired at the same time and thus having direct causality relationship with the excitation. Test3 and Test5 are correctly identified while Test4 is not, with some degree of ambiguity. This might be due to the low spatial resolution of accelerometers, but it is more likely an issue with the test, as it is observed that laser data share the same degree of uncertainty. Laser data is more spatially dense and so more precise with respect to location. The main issue is the possible lack of causality due to the channels not being acquired at the same time (non-ergodicity or non-stationarity). Although the voltage excitation is the same, the force output from the shaker is different for every channel due to shaker-structure interactions and settling of the joints. This leads to generally poorer coherence and lack of causality that renders the method incapable of giving correct results in a consistent way, especially in non-lab conditions. As a result, laser results are much harder to interpret using the coherence index alone and reading the coherence plots (Figs. 1.11, 1.12, 1.13, 1.14, 1.15, and 1.16) proves challenging as well, as some locations perform better in certain frequency ranges than others, unlike what happens with accelerometric data.
1 Detection of Sources of Nonlinearity in Multiple Bolted Joints by Use of Laser Vibrometer 9 1 0.5 0 0 500 1000 2000 3000 4000 5000 3500 4500 2500 1500 pcoh ACC#1 LOC#1 (1-2) 1 0.5 0 0 500 1000 2000 3000 4000 coh tanh mcoh 5000 3500 4500 2500 1500 pcoh ACC#1 LOC#2 (2-3) 1 0.5 0 0 500 1000 2000 3000 4000 5000 3500 4500 2500 1500 pcoh ACC#1 LOC#3 (3-4) Fig. 1.12 Test4 – accelerometers 1 0.5 coh tanh mcoh 0 0 500 1000 pcoh ACC#1 LOC#1 (1-2) 2000 3000 4000 5000 2500 3500 4500 1500 1 0.5 0 0 500 1000 pcoh ACC#1 LOC#2 (2-3) 2000 3000 4000 5000 2500 3500 4500 1500 1 0.5 0 0 500 1000 pcoh ACC#1 LOC#3 (3-4) 2000 3000 4000 5000 2500 3500 4500 1500 Fig. 1.13 Test 5 – accelerometers
10 A. delli Carri et al. 0 1 pcoh for DOF#10 LOC#1 (1-2) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#2 (2-3) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#3 (3-4) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#4 (4-5) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#5 (5-6) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#6 (6-7) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#7 (7-8) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#9 (9-10) 0.5 coh tanh mcoh 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#8 (8-9) 0.5 0 1000 2000 3000 4000 5000 Fig. 1.14 Test3 - LDV
1 Detection of Sources of Nonlinearity in Multiple Bolted Joints by Use of Laser Vibrometer 11 0 1 pcoh for DOF#10 LOC#1 (1-2) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#2 (2-3) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#3 (3-4) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#4 (4-5) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#5 (5-6) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#6 (6-7) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#7 (7-8) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#9 (9-10) 0.5 coh tanh mcoh 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#8 (8-9) 0.5 0 1000 2000 3000 4000 5000 Fig. 1.15 Test4 - LDV
12 A. delli Carri et al. 0 1 pcoh for DOF#10 LOC#1 (1-2) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#2 (2-3) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#3 (3-4) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#4 (4-5) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#5 (5-6) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#6 (6-7) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#7 (7-8) 0.5 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#9 (9-10) 0.5 coh tanh mcoh 0 1000 2000 3000 4000 5000 0 1 pcoh for DOF#10 LOC#8 (8-9) 0.5 0 1000 2000 3000 4000 5000 Fig. 1.16 Test5 - LDV 1.6 Conclusions Random accelerometric data has successfully been used to locate the source of nonlinear effects. The main issue with accelerometers are the highly complex setup procedures, coupled with some inevitable mass loading effects. One of the clear advantages of LDV measurements is the possibility to easily acquire many more points than accelerometers can achieve, without any mass loading drawbacks. Having a finer point-mesh makes the location procedure ideally more accurate, albeit at the cost of much more expensive processing power. However, the biggest drawback of LDV measurements lies in the fact that the points are not excited and acquired all at the same time - unlike with accelerometers – but over the span of several minutes. This means that any settling of the structure, small relaxation in boundary conditions or any time-variant properties of the environment like temperature or exogenous inputs causes the random process to become non-stationary and therefore non-ergodic, resulting in a loss of causality between excitations and responses. All these drawbacks make the application of random LDV measurements for location of nonlinearities yet unsuitable for unexperienced operators and industrial settings but confined to extremely well-equipped laboratory environments with trained personnel.
1 Detection of Sources of Nonlinearity in Multiple Bolted Joints by Use of Laser Vibrometer 13 References 1. delli Carri, A., Weekes, B., Di Maio, D., Ewins, D.J.: Extending modal testing technology for model validation of engineering structures with sparse nonlinearities: a first case study. Mech. Syst. Signal Process. 84, 97–115 (2017) 2. Di Maio, D., Bozzo, A., Peyret, N.: Response phase mapping of nonlinear joint dynamics using continuous scanning LDV measurement method. In: AIP Conference Proceedings, vol. 1740, (2016) 3. Di Maio, D., Bennett, P., Schwingshackl, C., Ewins, D.: Experimental non-linear modal testing of an aircraft engine casing assembly. In: Kerschen, G., Adams, D., Carrella, A. (eds.) Topics in Nonlinear Dynamics, Volume 1 SE - 2, vol. 35, pp. 15–36. Springer, New York (2013) 4. Di Maio, D., Ramakrishnan, G., Pascalis, S., Rajasagaran, Y., Ghambir, S.: A study on detection of nonlinearity using an aero-engine casing assembly. In: ISMA (2016) 5. Bendat, J.S., Piersol, A.G.: Random Data: Analysis and Measurement Procedures, vol. 729. John Wiley & Sons (2011) 6. Muhamad, P., Sims, N.D., Worden, K.: On the orthogonalised reverse path method for nonlinear system identification. J. Sound Vib. 331(20), 4488–4503 (2012) 7. Richards, C.M., Singh, R.: Identification of multi-degree-of-freedom non-linear systems under random excitations by the ‘reverse path’ spectral method. J. Sound Vib. 213(4), 673–708 (1998) 8. Bendat, J.S.: New techniques for nonlinear system analysis and identification from random data. ASA. 102, 3075 (1997) 9. Ewins, D.J., Weekes, B., Carri, A.D.: Modal testing for model validation of structures with discrete nonlinearities. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 373(2051), 20140410 (2015)
Chapter2 Single High-Speed Camera Based 3D Deflection Reconstruction in Frequency Domain Janko Slavicˇ, Domen Gorjup, and Miha Boltežar Abstract High-speed camera systems are a well-established alternative to traditional vibration measurement techniques, particularly in cases when the region of interest on the observed object is near-planar. With the introduction of 3D digital image correlation 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. In this paper the possibility of extending the use of high-speed camera systems to vibration measurement of arbitrarily shaped structures by applying methods, commonly used in multi-view computer vision is explored. A single high speed camera is used to record the vibrating structure from multiple points of view. By utilizing properties of linear, time-invariant mechanical systems, multi-view triangulation is then performed in frequency domain on displacement data, extracted from these image sequences using optical flow or digital image correlation. The acquired 3D spectra are finally used in full-field deflection reconstruction. Keywords High-speed camera · Vibration measurement · Frequency domain · Multiview geometry · Optical flow 2.1 Introduction The use of high-speed cameras in vibration measurement is best suited to near-planar structures due to an inherent limitation of 2D imaging systems. Depth information, lost in the imaging process, can be recovered by using the well-established 3D DIC technique [1]. Its field of view is, however, usually limited to a single face of the object, observed by the stereo pair. 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 [3– 5]. Multiple digital cameras used in the measurement process can be arranged in various configurations around [6]. Data acquired by a moving stereo-pair of high-speed cameras can also be used to extend the field of view of 3D DIC in a process called surface stitching [7, 8]. 2.2 Measurement Setup A concave steel object, composed of three 1 mm thick 200 ×200 mm sheet metal planes, bent and welded at one seam (Fig. 2.1) was placed on a LDS V555 electrodynamic shaker and excited with a constant profile of 3 g in the 25 Hz–2000 Hz frequency range. The object was mounted to the shaker in such a way that the excitation force vector formed an equal angle with all of its three planes. Six separate video sequences were acquired using a single Photron Fastcam SA-Z monochrome high-speed camera operating at 20,000 fps and a resolution of 640×640 pixels. Each video contained a sequence of 20,000 images for a sampling period of 1 s. The camera was stationary during the measurement process, but the object was rotated by 60 degrees around the vertical axis between each consecutive video acquisition session. The multiview system was calibrated using the Perspective n-point algorithm [2] and a set of markers with known positions on the surface of the observed object. J. Slavicˇ ( ) · D. Gorjup · M. Boltežar Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia e-mail: janko.slavic@fs.uni-lj.si © Society for Experimental Mechanics, Inc. 2019 C. Niezrecki et al. (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12935-4_2 15
16 J. Slavicˇ et al. Fig. 2.1 Multiview measurement setup 2.3 Image Based 2D Displacement Measurement Acquired images were processes by the Simplified gradient-based optical flow method [9] to identify the 2D displacements for each video sequence. The points to be analyzed in each image were selected by projecting a rectangular grid of 30×30 points onto each of the three object planes, totaling 2700 points per image. A rectangular region of interest of 11×11pixels with the grid node in the middle was analyzed for each of the selected points. 2.4 Multiview Geometry and Triangulation Each camera position in a multiview imaging setup (Fig. 2.1) can be defined by a transformation matrix that projects the coordinates of a point Xin space into the image plane: x =PX (2.1) wherexdenotes the coordinates of a point in an image andP=K[R| t] is a projective transform matrix, composed of a 3×3 matrix of intrinsic camera parameters K, 3×3 rotation matrixRanda 3×1 translation vector t [2]. By matching the position of a point in an imagexto the position of the same real-world point in another image, x , the3D position of the original point in a chosen global coordinate frame is triangulated by solving the following system of equations for three unknown coordinates inX[10]: x =P1X x =P2X (2.2) Each camera view adds another matrix equation to the already overdetermined system of algebraic equations, which was solved in a least-squares sense using singular value decomposition in our case.
2 Single High-Speed Camera Based 3D Deflection Reconstruction in Frequency Domain 17 Fig. 2.2 Examples of measured spatial deflection shapes 2.5 Results The measured displacements were first transformed into frequency domain using Fast Fourier transform (Fig. 2.1). Deflection magnitude peaks were identified in the resulting amplitude spectra (Fig. 2.2). Multiview triangulation was then performed for the deflection shapes at selected frequencies in each of the 6 video sequences, assuming linearity of the observed response. Examples of resulting 3D deflection shapes are visualized in Fig. 2.2. In this experiment, spatial deflection shapes of a vibrating 3D object at frequencies up to 1500 Hz were successfully reconstructed using a single moving high-speed camera. References 1. Chu, T.C., Ranson, W.F., Sutton, M.A.: Applications of digital-image-correlation techniques to experimental mechanics. Exp. Mech. 25(3), 232–244 (1985) 2. Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, New York (2003) 3. Harvent, J., Bugarin, F., Orteu, J.-J., Devy, M., Barbeau, P., Marin, G. Inspection of aeronautics parts for shape defect detection using a multi-camera system. In: Proc. SEM XI Int. Congr. Exp. Appl. Mech. Orlando, FL, USA, pp. 2–5, 2008 4. Orteu, J.-J., Bugarin, F., Harvent, J., Robert, L., Velay, V.: Multiple-camera instrumentation of a single point incremental forming process pilot for shape and 3D displacement measurements: methodology and results. Exp. Mech. 51(4), 625–639 (2011) 5. Wang, Y., Lava, P., Coppieters, S., Houtte, P.V., Debruyne, D.: Application of a multi-camera stereo DIC set-up to assess strain fields in an Erichsen test: methodology and validation. Strain. 49(2), 190–198 (2013) 6. Pan, B.: Digital image correlation for surface deformation measurement: historical developments, recent advances and future goals. Meas. Sci. Technol. 29(8), 82001 (2018) 7. LeBlanc, B., Niezrecki, C., Avitabile, P., Sherwood, J., Chen, J.: Surface stitching of a wind turbine blade using digital image correlation. In: Topics in Modal Analysis II, vol. 6, pp. 277–284. Springer, New York (2012) 8. Patil, K., Baqersad, J., Sheidaei, A.: A multi-view digital image correlation for extracting mode shapes of a tire. In: Shock & Vibration, Aircraft/Aerospace, Energy Harvesting, Acoustics & Optics, vol. 9, pp. 211–217. Springer, Cham (2017) 9. Javh, J., Slavicˇ, J., Boltežar, M.: The subpixel resolution of optical-flow-based modal analysis. Mech. Syst. Signal Process. 88, 89–99 (2017) 10. Hartley, R.I., Sturm, P.: Triangulation. Comput. Vis. Image Underst. 68(2), 146–157 (1997)
Chapter3 Operational Modal Analysis of a Thin-Walled Rocket Nozzle Using Phase-Based Image Processing and Complexity Pursuit Marc A. Eitner, Benjamin G. Miller, Jayant Sirohi, and Charles E. Tinney Abstract In this work, the modal parameters of a reduced scale, thin-walled, metallic rocket nozzle are extracted through Operational Modal Analysis (OMA). The specimen is excited using pressurized gas from a rocket nozzle test stand. Deformation of the nozzle lip is measured using a non-contact optical technique consisting of two-dimensional marker tracking in conjunction with phase-based motion amplification. OMA methods that use digital image data suffer from low signal to noise ratios (SNR), especially in higher modes with small amplitude vibrations. The structural displacements are often on a subpixel scale and therefore difficult to analyze without additional image processing. Phase-based motion amplification (PMA) offers a possible solution to this problem by magnifying subpixel motion. This work focuses on the implementation of the technique in a marker-tracking algorithm, which serves to extract the time-history of high-contrast markers placed on a large area ratio nozzle with a 5.3 inch exit diameter. Grayscale images taken with a high-speed camera are first processed with the phase-based algorithm to increase the marker motion in a certain broad frequency band. This results in a set of modified images, which are then analyzed with a tracking algorithm that identifies centroid positions of fluorescent markers. The time-history of these markers is then used as input for an OMA algorithm, namely the Complexity Pursuit algorithm, which leads to estimates of eigenfrequencies, damping ratios and mode shapes. A quantitative comparison between the modal parameters obtained with and without additional motion magnification is provided. Results of a numerical simulation are provided that demonstrate the improvement of estimated modal parameters. The modal parameters of the first six modes of the nozzle are found using this method. The motion was amplified in the range of 0–1400 Hz which includes the six eigenfrequencies. Without application of the broad band PMA, the highest mode cannot clearly be identified and the quality of the other modes decreases. Keywords Operational modal analysis · Phase based motion estimation · Complexity pursuit · Blind source separation · Optical vibration measurement 3.1 Introduction This study was motivated by the problem of unsteady side loads in large area-ratio rocket nozzles. Unsteady flow phenomena consisting of oscillating compression shocks occurring in the supersonic part of a converging-diverging rocket nozzle can lead to large pressure fluctuations [1, 2]. The resulting forces acting on the nozzle induce structural deformation and potentially large bending moments acting in the support structure of the nozzle [3]. If the nozzle is sufficiently compliant, the unsteady pressure fluctuations induce vibrations of the nozzle wall which are large enough to be measured using optical techniques [4]. In this study, a metallic large area-ratio nozzle with a thin wall (0.03 inches) was excited by these unsteady flow phenomena. The vibration of the structure was captured using a digital camera in combination with 2-dimensional point tracking. In theory, knowledge of the displacement history of discrete points of the nozzle in combination with an adequate reduced order model of the nozzle, should allow reconstruction of the aerodynamic forces. The solution of such an inverse problem depends on the order and quality of the model. For the construction of such a model, knowledge of the modal parameters (eigenfrequencies, damping ratios, mode shapes) is crucial. The quality of the modal parameters extracted from experimental data depends, among other things, on the algorithm used and on the signal to noise ratio (SNR) in the data. In this study the Complexity Pursuit (CP) algorithm [5] was used to M. A. Eitner ( ) · B. G. Miller · J. Sirohi The University of Texas at Austin, Department of Aerospace Engineering and Engineering Mechanics, Austin, TX, USA e-mail: marceitner@utexas.edu C. E. Tinney The University of Texas at Austin, Applied Research Laboratories, Austin, TX, USA © Society for Experimental Mechanics, Inc. 2019 C. Niezrecki et al. (eds.), Rotating Machinery, Optical Methods & Scanning LDV Methods, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12935-4_3 19
20 M. A. Eitner et al. estimate modal parameters from output only (camera images) data. The goal of this process is to find as many vibrational modes as possible. Finding accurate modal parameters becomes more and more difficult as the eigenfrequency increases. This is generally due to the reduction in SNR, meaning that for higher modes, the low amplitude and high frequency oscillations of the test specimen eventually fall below the level of noise present in any given measurement. Therefore it is necessary to utilize signal processing techniques to increase the SNR of the higher modes. A potentially promising algorithm that preprocesses the set of high-speed images is the Phase-based Motion Amplification algorithm (PMA). PMA, or motion microscope was developed by Wadhwa et al. [6]. This image processing technique utilizes complexsteerable pyramids to decompose, filter and reconstruct a set of images in order to amplify motion in a certain frequency range. Since this algorithm is fairly new, few studies have been published on its application to structural dynamics and modal parameter identification. Poozesh et al. [7] demonstrated experimentally that by amplifying the motion in the frequency range of a noisy mode, the SNR of that mode can be significantly improved. Using OMA and comparing modal parameters before and after motion amplification it was shown that a clear improvement could be observed in terms of the modal assurance criterion (MAC). In this paper, motion is amplified in a broad frequency range that includes all known eigenfrequencies. The earlier work of Yang et al. [8] used PMA in combination with Complexity Pursuit to extract and visualize full-field mode shapes of a scaled multi-story building. PMA was used after the CP algorithm, like a post-processing step. This is significantly different from the approach in this study, where PMA is used in a preprocessing step. The current study uses PMA as a preprocessing step to increase the SNR of images from high-speed video from vibration tests. The images contain several discrete markers that are painted on the surface of a structure. Using PMA, the motion of the markers is amplified within a certain broad frequency band. The images are then analyzed by a two-dimensional point tracking (2DPT) algorithm that extracts the marker positions in each image frame, which results in the displacement history of each marker. These amplified displacements are used to estimate the modal parameters of the structure using the CP algorithm. By also estimating the modal parameters of the unprocessed images directly, the effect of motion amplification can be quantified. This paper is structured in the following way. The first section covers a summary of the algorithms and methods used in this study. The 2DPT, PMA, and CP algorithms are briefly described. The entire process is summarized and outlined. In the second part of this paper, two experiments were performed. First, the vibration of an eight degree of freedom system (8DOF) was simulated in a numerical study. Images showing oscillating markers were artificially created and processed using PMA, 2DPT, and CP to obtain estimates of the system’s modal parameters. Since the true modal parameters were known, a detailed error analysis was performed. Then, in a laboratory experiment, vibration tests were performed on a large area-ratio nozzle with a 5 inch exit diameter. Actual images from high-speed video taken of the vibrating nozzle were used to obtain the nozzle’s modal parameters. As in the numerical experiment, the modal parameters are estimated from the original images as well as from the motion amplified images. The last chapter contains the discussion and evaluation of the test results. 3.2 Methods The broad-band PMA methodology for modal parameter estimation described in this paper is based on a two-dimensional point tracking method, the Phase-based Motion Amplification algorithm, and the Complexity Pursuit algorithm. These three methods are outlined in this section. 3.2.1 2-Dimensional Point Tracking (2DPT) When performing vibration measurements, the goal is to obtain the displacement history of discrete points on the structure. A common way of doing this is to use accelerometers. Attaching these to the surface of a structure allows the measurement of acceleration at these points, which can then be numerically integrated to obtain the displacement history. Numerical errors such as drift and errors resulting from the added mass of the sensor are disadvantages of this method. By using cameras to detect the vibration of the structure, non-contact measurements can be performed. The 2DPT method consist of painting a number of discrete markers on the surface of a structure and tracking their locations using a high-speed camera and an image processing algorithm. The markers are painted on the structure using fluorescent paint. High contrast between the markers and the surface of the structure is obtained. A marker tracking algorithm then detects the position of each marker in each camera image. The algorithm used in this paper first converts each greyscale camera image into a binary image (pixels either pure black or pure white) based on a user-defined pixel intensity threshold. The algorithm then looks for accumulations of
3 Operational Modal Analysis of a Thin-Walled Rocket Nozzle Using Phase-Based Image Processing and Complexity Pursuit 21 Fig. 3.1 Outline of point tracking method. Grayscale image taken with high-speed camera (left); identified markers (center); displacement history of centroid of single marker (right) high intensity pixels to find all pixels that make up a marker. In a final step, the centroid of each marker is calculated. This is done by weighing the location of each pixel that is part of the marker with the pixel intensity and then dividing by the number of pixels in the marker. This process is performed independently for each image, leading to the displacement history of all markers throughout the entire measurement. An example of this process is shown in Fig. 3.1, where the position of markers on the lip of a nozzle are extracted from grayscale test images. 3.2.2 Phase-Based Motion Amplification (PMA) The original PMA algorithm by Wadhwa et al. [6] reads an input video, amplifies motions in a certain frequency band, and outputs a reconstructed video. Motions are processed by first breaking the video into a set of images, and then breaking each image into constitutive layers through spatial filtering. Spatial filtering is performed through the Complex Steerable Pyramid Decomposition (CSPD) process developed by Simoncelli et al. [9, 10]. In CSPD, a 2D transfer function library is constructed using a set of Gabor filters. These are linear filters that have a direction, position, and frequency. Convolving an image by this library decomposes the original image into several layers or levels containing filtered images differing in spatial resolution. Summing these back together results in the reconstructed image, whereby lower-ordered levels of the decomposition (containing low-resolution decompositions) contribute more to the overall behavior – a structure similar to a Taylor expansion. The total number of levels is dictated by the image size and number of filter orientations. Localized phase changes in CSPD are proportionally related to displacements through Fourier transformation [6]. The local motion in an image can therefore be computed by subtracting the phase angles of each frame from the phase angles of the first frame. A band-pass filter is then applied to these motions and they are multiplied by an amplification factor (α+1). When the pyramid is ‘collapsed’, meaning the decomposition into different layers is reversed, a reconstructed video is obtained that contains the amplified motion. 3.2.3 Modal Parameter Estimation Using Complexity Pursuit The Blind Source Separation algorithm called Complexity Pursuit (CP) is used to extract modal parameters from the displacement history of the markers, identified by the previously described 2DPT algorithm. CP was proposed by Stone [5] and uses a temporal predictability function that acts as a measure of complexity of an observed signal. Stone’s theorem states that source signals making up any measured signal have higher predictability than the measured signal itself. The algorithm assumes that measured observations are made up of a linear mixture of source signals according to: −→x =[A]−→s (3.1)
22 M. A. Eitner et al. Images from structural vibration tests Motion-Amplified Images Displacement History of Markers Modal parameters and R2 2D Fourier Amplify Phases Reconstruct Images within chosen frequency band Transform Generate Gabor Filters CSPD: Convolve Identify Estimate modal Curve fit modal coordinates to impuls response function matrix and modal coordinates with Complexity Pursuit Markers Calculate weighted centroids Images and Filters Fig. 3.2 Outline of the modal parameter identification process Here [A] is the N×Nmixing matrix, −→s is the N×1 vector of source signals with maximum predictability and−→x is the N×1 vector of measurements. The CP algorithm estimates the mixing matrix in such a way that the resulting source signals exhibit maximum temporal predictability. When applied to structural dynamics the matrix [A] is the modal matrix containing the mode shapes, −→x is the vector of instantaneous displacement measurements at Nlocations and−→s is the vector of modal coordinates. If the structure was excited with an impulsive force, the extracted modal coordinates will be impulse response functions of damped single degree of freedom oscillators. The modal coordinates are therefore fit to an impulse response function of the form si =Ce−ζ2πfit sin 2πfi 1−ζ 2 i t −φ , (3.2) where fi and ζi are the eigenfrequency and damping ratio of the i th mode respectively and Cand φ are constants. The curve fit results in values for the eigenfrequency and damping ratio associated with that mode. If the system was excited with random force the response of the modal coordinates will also be random. Computing their autocorrelation function recovers the impulse response function [11] and the modal parameters can be estimated once again via curve fitting. Using a goodness of fit value (e.g. R2) gives an indication of the quality of the separated modal coordinates. A summary of the entire process is shown in Fig. 3.2. 3.3 Experiments Two sets of experiments were performed to test and quantify the effect of PMA on modal parameter estimation. First the structural vibration of an 8DOF system was numerically simulated. Images of markers placed on the simulated system were artificially generated, as if they had been taken during a real vibration test. These images were used in conjunction with PMA, 2DPT and CP to obtain estimates of the system’s modal parameters. Since the true modal parameters of the simulated system were known, errors in modal parameters could be computed. Next, two vibration tests of a small, thin-walled, large area-ratio nozzle were performed. The nozzle was attached to a test rig and excited by pressurized air from a plenum. The induced vibration was captured using a high-speed camera. The test was performed twice, once with the camera placed close to the nozzle and once with the camera placed further away to decrease the SNR. The resulting images were then used to estimate the modal parameters of the nozzle, using the previously discussed process of PMA, 2DPT and CP. 3.3.1 Numerical Simulation A simulation was performed to compare the modal parameters estimated from motion amplified images to true values, thus allowing detailed error analysis. The simulated structure consisted of four point masses arranged in a square connected to
RkJQdWJsaXNoZXIy MTMzNzEzMQ==