River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Dynamic Substructuring & Transfer Path Analysis, Vol. 4 Walter D’Ambrogio Dan Roettgen Maarten van der Seijs Proceedings of the 43rd IMAC, A Conference and Exposition on Structural Dynamics 2025 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman Society for Experimental Mechanics, Inc., Bethel, USA i
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. ii
Walter D’Ambrogio· Dan Roettgen· Maarten van der Seijs Editors Dynamic Substructuring & Transfer Path Analysis, Vol. 4 Proceedings of the 43rd IMAC, A Conference and Exposition on Structural Dynamics 2025 River Publishers
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 97887-438-0149-8 (Hardback) ISBN 97887-438-0161-0 (eBook) https://doi.org/10.13052/97887-438-0149-8 Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2025 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 Dynamic Substructuring & Transfer Path Analysis represents one of twelve volumes of technical papers presented at the 43rd IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held February 10-13, 2025. The full proceedings also include volumes on Nonlinear Structures & Systems; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Special Topics in Structural Dynamics & Experimental Techniques; Computer Vision & Laser Vibrometry; Dynamic Environments Testing; Sensors & Instrumentation and Aircraft/Aerospace Testing Techniques; Topics in Modal Analysis & Parameter Identification Iⅈ Data Science in Engineering; and Structural Health Monitoring & Machine Learning. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. Coupled structures or, substructuring, is one of these areas. Substructuring is a general paradigm in engineering dynamics where a complicated system is analyzed by considering the dynamic interactions between subcomponents. In numerical simulations, substructuring allows one to reduce the complexity of parts of the system in order to construct a computationally efficient model of the assembled system. A subcomponent model can also be derived experimentally, allowing one to predict the dynamic behavior of an assembly by combining experimentally and/or analytically derived models. This can be advantageous for subcomponents that are expensive or difficult to model analytically. Substructuring can also be used to couple numerical simulation with real-time testing of components. Such approaches are known as hardware-in-the-loop or hybrid testing. Whether experimental or numerical, all substructuring approaches have a common basis, namely the equilibrium of the substructures under the action of the applied and interface forces and the compatibility of displacements at the interfaces of the subcomponents. Experimental substructuring requires special care in the way the measurements are obtained and processed in order to assure that measurement inaccuracies and noise do not invalidate the results. In numerical approaches, the fundamental quest is the efficient computation of reduced order models describing the substructure’s dynamic motion. For hardware-in-the-loop applications difficulties include the fast computation of the numerical components and the proper sensing and actuation of the hardware component. Recent advances in experimental techniques, sensor/actuator technologies, novel numerical methods, and parallel computing have rekindled interest in substructuring in recent years leading to new insights and improved experimental and analytical techniques. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Editors: Walter D’Ambrogio–University of L’Aquila, L’Aquila, Italy; Dan Roettgen–Sandia National Laboratories, NM, USA; Maarten van der Seijs–VIBES technology, Netherlands v
Contents 1 Modal Vibration Testing of a Frame and Wing Structure 1 Joshua E. Blackham and Matthew S. Allen 2 Toward 3D Experimental Impulse-Based Substructuring Using the Virtual Point Transformation 11 Oliver M. Zobel, Francesco Trainotti, and Daniel J. Rixen 3 A Look at Overfitting in Source Estimation Problems 35 Steven Carter 4 A Numerical Decoupling of a Wing Together with Two Nonlinear Radius-pylons, from the Technical Division’s Substructuring Four-unit Frame Structure 55 A. Linderholt, B. Moldenhauer and D. Roettgen 5 Finite Element Modeling and Substructuring to Simulate Shock Plate Testing 59 Harrison C. Denning, Scott Tuley, Matthew S. Allen, Jeffrey R. Hill, and Daniel R. Roettgen 6 Condition Monitoring for the Mounts of a Simple Supported Beam Using Transfer Path Analysis 69 Michael Kreutz and Daniel J. Rixen 7 Frequency Response Function Expansion using a Symmetry Preserving Gysin Expansion Technique 79 Christopher Page and Peter Avitabile 8 Cutting Force Estimation in Sensor-Equipped Metal Cutting Tools Using Strain-Force Transfer Function 85 Wu Peng, Anders Liljerehn, Martin Magnevall, and Dan O¨ stling 9 Enhanced Mode Selection in Modal Domain Substructuring on the Round Robin Structure 95 Jure Korbar, Miha Pogacˇar, and Gregor Cˇ epon 10 Investigation of the Use of Commercial Robotic Arms for Real-Time Hybrid Substructuring 101 Arian Kist, David Stadler, Rok Belsˇak, Vasja Plesec, Timi Karner, Gregor Harih and Daniel Rixen 11 Model Initialization in Real Time Hybrid Testing for Experimental Detection of Isolated Branches 113 A. Mario Puhwein and Markus J. Hochrainer 12 Extrapolating Dynamic Transfer Functions from Multi-Input Multi-Output Vibration Testing and Simulation 123 E. J. Perez, E. E. Regula, C. J. Wynn, J. D. Blessinger, J. L. Davis, E. P. Dawson, and S. J. Zimmerman 13 Revisiting the Dual Admittance-Based Quasi-Static Formulation for the Identification of Linear Joint Dynamics with Dynamic Substructuring 135 Francesco Trainotti, M. Brons, S. Klaassen, D. J. Rixen 14 On the Use of Frequency-Dependent Modal Basis for Interface Modeling in Frequency-Based Substructuring 143 Domen Ocepek, Miha Pogacˇar, Jure Korbar, and Gregor Cˇ epon vii
viii Contents 15 Assessing the Importance of Contact Joints Relative to Other Sources of Uncertainty in Dynamic Substructuring 147 Samuel Choi, Manuel Vega, Tyler Alvis, and Teresa Portone 16 High-frequency Dynamic Characterization of Rubber Mounts through an Enhanced Virtual Point Transformation 155 Bart Forrier, Fabio Bianciardi, and Karl Janssens
Chapter 1 Chapter 1 On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model Georgios Tsialiamanis and Charles R. Farrar Abstrac t Detection and identification of nonlinearity is a task of high importance for structural dynamics. On the one hand, identifying nonlinearity in a structure would allow one to build more accurate models of the structure. On the other hand, detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage within the structure. Common damage cases which cause nonlinear behaviour are breathing cracks and points where some material may have reached its plastic region. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model selected for the current application is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as input accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be unimodal and quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some of which are nonlinear. More specifically, the nonlinearity is introduced as a column-bumper nonlinearity, aimed at simulating the effects of a breathing crack and at different levels, i.e. different values of the initial gap between the bumper and the column. Following the proposed method, the statistics of the distributions of the gradients for the different scenarios can indeed be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for lower values of the initial column-bumper gap, i.e. for “more nonlinear” scenarios. Keyword s Structural health monitoring (SHM) · Structural dynamics · Nonlinear dynamics · Machine learning · Neural networks 1.1 Introduction In the pursuit of making everyday life safer, humans have extensively tried to model the environment around them. Structures are an important part of the environment, in which humans live. They are man-made and should be safe throughout their lifetime. Structures are exposed to numerous environmental factors, which may cause them to fail. Moreover, during operation, structures are subjected to dynamic loads, which, in time, may cause failure. Such failures will most probably result in economic damage to society and may even result in loss of human lives. Therefore, for the purpose of maintaining structures safe, the field of structural health monitoring (SHM) [1] has emerged. G. Tsialiamanis ( ) Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: g.tsialiamanis@sheffield.ac.uk C. R. Farrar Engineering Institute, MS T-001, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: farrar@lanl.gov © The Society for Experimental Mechanics, Inc. 2024 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-36999-5_1 1 Modal Vibration Testing of a Frame and Wing Structure Joshua E. Blackham and Matthew S. Allen Large, multi-component bolted structures can exhibit complicated vibration responses due to the complexity of frictional interactions at the bolt-plate interface. The effect of nonlinearity due to the joints can be characterized by performing a test to quantify how the effective natural frequency and damping ratio of each mode change with vibration amplitude. In this paper, modal hammer impact testing was performed on the Round Robin Frame and Wing Structure. Various amplitude impacts were used to excite any nonlinearities in the structure. Experimental modal analysis was performed to separate the modal responses of the system. Mode shapes and frequencies are examined to understand any nonlinearities in the system. Keywords Nonlinear Vibration· Experimental Modal Analysis · Frame and Wing Structure Introduction The goal of modeling and testing complicated bolted structures is typically to create a model that can accurately replicate the effective natural frequency and damping of a test structure at the vibration amplitudes of interest. In order to make further progress in experimental-analytical substructuring, the Frame and Wing Structure (FWS) was introduced as a testbed structure that is relatively easy to model and test, with the ultimate goal of comparing and learning about different substructuring methods that are commonly used among different researchers and engineers [1]. Since the introduction of the Frame and Wing Structure, groups at Sandia have performed tests and modeled the structure. Moldenhauer and Roettgen [2] performed substructuring on the original FWS and were able to model the structure’s mode frequencies and damping successfully. They also explored certain modes for large nonlinearities in the damping and found that most of the modes only exhibited weak nonlinearities. Linderholt et al. [3] also used substructuring to model the FWS, however, they modeled a new ”swept” wing that represents a standard plane wing more accurately than the rectangular wing used in earlier models. Linderholt et al. [4] again modeled the FWS using substructuring methods. This time more complicated elements were added to the structure, such as vibration damping studs placed between the frame and the wing, or small payloads added on the wings. These elements were included in an attempt to introduce more nonlinearity into the system and to be able to model these changes through the developed substructuring processes. While the original purpose of the frame wing project was to develop and compare different substructuring methods, this paper focuses on testing and characterizing the FWS. A roving impact test was performed to excite vibration modes from multiple locations and at various amplitudes. The frequency response functions (FRFs) obtained from both low-amplitude (i.e. assumed to be linear) responses and higher amplitude (i.e. nonlinear) responses were analyzed using the Algorithm of Mode Isolation to quantify the frequency and damping of each vibration mode. Mode shapes were also found and used to help explain any nonlinearities that might occur in each mode. The Hilbert transform [5, 6] was then applied to some of the high-amplitude impact data to quantify nonlinearities in the frequency and damping of certain modes. Joshua E. Blackham Department of Mechanical Engineering, Brigham Young University, Provo, UT 84601 e-mail: joshblackham1925@gmail.com Matthew S. Allen Department of Mechanical Engineering, Brigham Young University, Provo, UT 84601 e-mail: matt.allen@byu.edu © The Author(s), under exclusive license to River Publishers 2025 Walter D’Ambrogio, et al. (eds.), Dynamic Substructuring & Transfer Path Analysis, Vol. 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0149-8 1
2 J. E. Blackham and M. S. Allen Testing Procedure The Frame and Wing Structure that was tested included both the frame component SN013 and the thin wing component Wing013A. For reference, the weights of the frame and wing components are 629.7 g and 527.7 g, respectively [7]. In this setup, the frame and wing components were attached using 4 10-32 bolts and #10 washers on each side of the frame as seen in Figure 1. The washers were placed only between the head of the bolt and the contact surface of the wing part. No nuts were used since the frame part was manufactured with threaded inserts placed in all bolt holes. All bolts were pretensioned to 3.4 Nm and the structure was suspended with bungees to approximate free-free boundary conditions. All rigid body modes of the suspended structure were found to be below 3 Hz, which was well below the first elastic mode of the structure. Fig. 1 Frame and Wing structure suspended by bungees in preparation for modal impact testing The goal of this testing was to characterize the mode frequencies and shapes of the frame and wing structure and to find modes that exhibit large nonlinearities due to the bolted joints. An initial round of testing focused on both low- and highamplitude impact responses that were measured up to 1280 Hz. Low-amplitude impacts were used to find a baseline of linear mode frequencies and to find the mode shapes of the structure. High-amplitude impacts exercised the joint nonlinearity and hence could change the frequency and damping observed in a linearized FRF. The locations of the impact and accelerometers are shown in Figure 2, with 4 accelerometers positioned at the corners of the wing and one accelerometer on the frame. Note Fig. 2 Locations of hammer impact points during testing. Impact locations are marked with red dots with a 2 or 3 digit numbering system. Unless specified, all impacts are in the Z direction. Accelerometers are located at points marked with orange circles and numbered 2-6 with directions specified.
Modal Vibration Testing of a Frame and Wing Structure 3 that Accelerometer 6 was located on the side of the frame and oriented in the x direction in order to capture the response when impacts were performed in the X direction at Point 123. The impact locations extensively cover the length of the frame and wing parts and were chosen to allow the mode shapes to be found on both the frame and the wing components. After reviewing the initial testing, additional high-amplitude impacts were performed, which excited the structure up to about 700 Hz, in order to investigate nonlinear effects further. Frequency Response The H1 FRF was computed from the impacts at low and high amplitudes and is shown in Figure 3. Figure 4 shows an expanded view for specific modes of interest. While investigating the corresponding mode shapes, the peak at 230 Hz exhibited unusual responses (see Figure 4c) and so two modes were suspected to exist at this frequency. The Complex Mode Indicator Function (CMIF) [8] was computed for the 24×5matrix of FRFs from the 5 accelerometers and 24 impact points, and the result is seen in Figure 5. The CMIF confirms that there are two modes that occur at 230 Hz as well as another pair near 650 Hz. 0 200 600 800 1000 1200 Frequencies (Hz) 10 -1 10 0 10 1 10 2 Non-Linear Data Linear Data 400 Fig. 3 Comparison of average FRFs from low and high amplitude impact tests. Mode Frequencies and Shapes After verifying the number of modes located at each frequency, the Algorithm of Mode Isolation (AMI) [9] was used to identify the modal parameters. In the cases where two modes occurred at about the same frequency, AMI used a two-mode fit to extract both modes from the measurements [10]. The mode frequencies and damping are summarized in Table 1, and the first eight mode shapes are seen in Figure 7. Modes 1, 2 and 7 (see Figs. 7a, 7b, and 7g) are predominantly wing bending modes. Modes 4 and 5 (see Figs. 7d and 7e) are frame bending modes with the wings undergoing torsion with Mode 4 in phase and Mode 5 out of phase. Modes 3 and 6 (see Figs. 7c and 7f) involve more complicated wing torsion modes and certain interactions between the wing and the frame. Mode 8 in Fig. 7h is a higher order wing bending mode that interacts with a frame torsion mode.
4 J. E. Blackham and M. S. Allen 65 65.5 66 66.5 67 67.5 68 Frequencies (Hz) 10 1 10 2 Comparisons of Compiled FRFs Non-Linear Data Linear Data (a) Mode 1 - 66.55 Hz 114.5 115 115.5 116 116.5 117 117.5 118 Frequencies (Hz) 10 1 10 2 Comparisons of Compiled FRFs Non-Linear Data Linear Data (b) Mode 2 - 116.26 Hz 222 222.5 223 223.5 224 224.5 225 Frequencies (Hz) 10 1 10 2 Comparisons of Compiled FRFs Non-Linear Data Linear Data (c) Mode 3 - 223.14 Hz and Mode 4 - 223.79 Hz Fig. 4 Zoom in on Frequency Response Function for the first 4 modes of the frame and wing structure A few of the modes included in-plane motion. To identify these modes, the average of the FRFs for each of the accelerometers (i.e. the drive points) was plotted and is shown in Figure 6. Most of the modes are dominant in the accelerometers located on the wing and in the Z-direction (i.e. Acc. 2 - 5). The two exceptions are modes 9 and 14 at about 511 and 876 Hz, which are far more visible in the X-direction accelerometer 6 than in the Z-direction. The plots of the mode shapes in Figure 7 can be used to ascertain which might be most sensitive to nonlinearity due to the bolted joints. For example, Mode 1 would exert a load that would want to open and close the bolted joints. This type of motion typically does not change the effective frequency or damping much [11], and the FRFs in Fig. 8 seem to support this. Mode 2 also shows little change in frequency or damping, and this seems to be corroborated by the fact that the mode shape
Modal Vibration Testing of a Frame and Wing Structure 5 0 200 400 600 800 1000 1200 1400 Frequency 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 Magnitude of Singular Values Complex Mode Indicator Function 1 2 3 4 5 Fig. 5 Complex Mode Indicator Function for frame and wing low-amplitude impact test. 0 200 400 600 800 1000 1200 1400 Frequency (Hz) 10 -2 10 -1 10 0 10 1 10 2 10 3 Magnitude of Accelerometer Readings Composites of Drive Points Acc. 2 Acc. 4 Acc. 3 Acc. 5 Acc. 6 Fig. 6 Composite of drive points found using AMI.
6 J. E. Blackham and M. S. Allen (a) Mode 1 - 66.55 Hz (b) Mode 2 - 116.26 Hz (c) Mode 3 - 223.14 Hz (d) Mode 4 - 223.79 Hz (e) Mode 5 - 267.88 Hz (f) Mode 6 - 307.58 Hz (g) Mode 7 - 387.83 Hz (h) Mode 8 - 430.17 Hz Fig. 7 First eight Mode Shapes of Frame and Wing Structure
Modal Vibration Testing of a Frame and Wing Structure 7 Table 1 Mode frequencies and damping ratios of the frame and wing structure for low and high amplitude testing. LowAmp. HighAmp. Difference Mode Description Freq. (Hz) Damp. (%) Freq. (Hz) Damp. (%) Freq. (%) Damp. (%) 1 1st Wing Bending 66.55 0.24 66.46 0.22 -0.14 -8.3 2 2nd Wing Bending 116.26 0.22 116.33 0.33 0.06 50 3 1st Wing Torsion 223.14 0.10 223.12 0.23 -0.009 130 4 1st Frame Bending 223.79 0.15 223.67 0.23 -0.054 53 5 1st Frame Bending 267.88 0.22 267.66 0.23 -0.082 4.5 6 Torsion of Frame and Wing 307.58 0.24 307.16 0.22 -0.14 -8.3 7 3rd Wing Bending 387.83 0.19 387.39 0.21 -0.11 10 8 1st Frame Torsion 430.17 0.13 429.53 0.16 -0.15 23 9 In-Plane Mode 1 511.28 0.20 510.77 0.12 -0.10 -40 10 - 642.63 0.32 641.30 0.38 -0.21 19 11 - 644.18 0.16 643.73 0.17 -0.070 6.2 12 - 786.68 0.12 786.23 0.15 -0.057 25 13 - 797.01 0.09 796.45 0.14 -0.070 55 14 In-Plane Mode 2 876.25 0.14 874.50 0.21 -0.20 50 15 - 1065.4 0.10 1064.4 0.12 -0.094 20 16 - 1217.4 0.18 1215.6 0.20 -0.15 11 10 -3 10 -2 10 -1 10 0 1 1.5 2 Damping Ratio 10 -3 Mode 1 Mode 2 Mode 3 Mode 4 10 -3 10 -2 10 -1 10 0 Acceleration Amplitude (g) -0.4 -0.2 0 Delta Nat. Freq. (Hz) Fig. 8 Amplitude dependent damping and frequency of the first 4 modes.
8 J. E. Blackham and M. S. Allen would not put much stress on the joints. This trend is similar for many of the bending and torsion modes of the structure. On the other hand, Modes 8 and 14 involve shearing of the joint, suggesting significant nonlinearity, and the results show that they exhibit somewhat larger than normal changes in frequency and damping. It should be noted, however, that these inferences based on the mode shapes are not infallible. For example, Modes 3 and 4 exhibit a relatively large increase in damping, even though their shapes mostly involve opening and closing of the joints as with most of the other modes. On the other hand, these modes are close in frequency so that may introduce error in estimating their damping. To ascertain which modes have nonlinear frequencies and damping, the Hilbert transform was used and will be discussed later. Hilbert Transform and Nonlinear Analysis After mode frequencies and damping were found using AMI, the next step was to investigate any nonlinear effects that might occur due to the bolted joints and friction interfaces in the structure. First, a modal filter was used to separate the responses of the closely spaced pairs of modes. A band-pass filter was also used to obtain only the frequencies for specific modes of interest. After both filters were applied to the high amplitude impact data, the Hilbert transform was then used as described in [6] to find the amplitude dependent frequencies and damping for the first four modes and for Mode 9 of the frame and wing structure The results are shown in Figs. 8 and 9. 10 -2 10 -1 1 1.5 2 2.5 3 Damping Ratio 10 -3 10 -2 10 -1 Acceleration Amplitude (g) 509 509.5 510 510.5 511 Natural Frequency (Hz) Fig. 9 Amplitude dependent damping and frequency of Mode 9. Mode 1 has the cleanest data in both damping and natural frequency and shows a small change across the amplitude range of interest. Such an increase in damping and decrease in stiffness is characteristic of structures with bolted-joint nonlinearities [12]. Mode 2 shows almost no change in natural frequency and a much noisier damping curve, however, the
Modal Vibration Testing of a Frame and Wing Structure 9 general trend is that the damping is relatively unchanged as well. Modes 3 and 4 are the two modes that occur at the same frequency and so the assumptions that modal coupling does not occur may not have been met. This could be a possible explanation for the large amounts of noise in the Mode 4 curves. Overall, it seems that these four modes are predominantly linear across range of amplitudes that were accessible in these tests. Noisy results such as those shown in Fig. 8 are typical when applying this technique to data when the response is linear. Figure 9 shows the nonlinear damping and natural frequency for Mode 9. Although slightly noisy, the damping increases by a factor of 3 as amplitude increases. The natural frequency has a less clear trend although the frequency does seem to drop by almost 2 Hz as amplitude increases. This type of response is typical for a nonlinear mode. Conclusion In this paper, the Sandia Frame and Wing structure was tested to classify its vibration modes in order to understand the effect of bolted joints on the structure. Low amplitude hammer impact testing was performed to find the linear frequencies, damping and mode shapes. A second set of FRFs was obtained at higher force levels, to quickly screen for nonlinearity. Mode shapes were also presented to visualize the interaction between the wing and frame through the bolted joints. High amplitude hammer impacts yielded a greater range of amplitude dependent data. By using the Hilbert transform, the amplitude dependent frequency and damping of several modes was investigated. For this structure, nonlinearity was found have only a minor effect on the frequency and damping of the majority of modes in the structure. The fact that many of these modes are linear is notable since the purpose of this structure is to develop and compare different linear substructuring methods, and hence the frame seems well suited for this purpose. Though the modes in this structure tend to predominantly linear, two pairs of modes were discovered that occur at almost the same frequency. Hence, care must be taken when treating those modes. References 1. Roettgen, D., Lopp, G., Jaramillo, A., and Moldenhauer, B. “Experimental Substructuring of the Dynamic Substructures Round-Robin Testbed”. In Allen, M., D’Ambrogio, W., and Roettgen, D., editors, Dynamic Substructures, Volume 4, pages 119–123, Cham (2023) Springer International Publishing. 2. Moldenhauer, B. and Roettgen, D. “Nonlinear Substructuring of the Dynamic Substructures Technical Division Structure”. In 41st International Modal Analysis Conference (IMAC XLI), Texas, USA (2023) Society for Experimental Mechanics. 3. Linderholt, A., Roettgen, D., and Moldenhauer, B. “Combining steel and aluminum components of the Benchmark Structure for the Technical Division on Dynamic Substructuring”. In 41st International Modal Analysis Conference (IMAC XLI), Texas, USA (2023) Society for Experimental Mechanics. 4. Linderholt, A., Roettgen, D., and Moldenhauer, B. “Nonlinear Subcomponent Attachments for the Technical Division on Dynamic Substructuring Benchmark Structure”. In42nd International Modal Analysis Conference (IMAC XLII), Florida, USA (2024) Society for Experimental Mechanics. 5. Feldman, M. “Non-linear system vibration analysis using Hilbert transform–I. Free vibration analysis method ’Freevib”’. Mechanical Systems and Signal Processing, 8(2):119–127 (1994) 6. Roettgen, D.R. and Allen, M.S. “Nonlinear characterization of a bolted, industrial structure using a modal framework”. Mechanical Systems and Signal Processing, 84:152–170 (2017) 7. Dynamic Substructuring Focus Group. “SEM 4UF Measured Properties”. SEM Wikis.Accessed: 2024-06-10. [Online.] Available: https: //wiki.sem.org/wiki/SEM 4UF Measured Properties. 8. Shih, C.Y., Tsuei, Y.G., Allemang, R.J., and Brown, D.L. “Complex mode indication function and its applications to spatial domain parameter estimation”. Mechanical Systems and Signal Processing, 2(4):367–377 (1988) 9. Allen, M.S. and Ginsberg, J.H. “A Global, Single-Input-Multi-Output (SIMO) Implementation of The Algorithm of Mode Isolation and Applications to Analytical and Experimental Data”. Mechanical Systems and Signal Processing, 20:1090–1111 (2006) 10. Allen, M.S. and Ginsberg, J.H. “Global, Hybrid, MIMO Implementation of the Algorithm of Mode Isolation”. In 23rd International Modal Analysis Conference (IMAC XXIII), Florida, USA (2005) Society for Experimental Mechanics. 11. Clark, Brennen, Allen, Matthew S., and Pacini, Benjamin. “Nonlinear Normal Modes and Response to Random Inputs of Systems with Bilinear Stiffness”. Journal of Sound and Vibration, (Submitted April 2024) (2024) 12. Segalman, D.J., Gregory, D.L., Starr, M.J., Resor, B.R., Jew, M.D., Lauffer, J.P., and Ames, N.M. “Handbook on Dynamics of Jointed Structures”. Technical report, Sandia National Laboratories, Albuquerque, NM 87185 (2009)
Chapter 2 Chapter 1 On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model Georgios Tsialiamanis and Charles R. Farrar Abstrac t Detection and identification of nonlinearity is a task of high importance for structural dynamics. On the one hand, identifying nonlinearity in a structure would allow one to build more accurate models of the structure. On the other hand, detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage within the structure. Common damage cases which cause nonlinear behaviour are breathing cracks and points where some material may have reached its plastic region. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model selected for the current application is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as input accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be unimodal and quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some of which are nonlinear. More specifically, the nonlinearity is introduced as a column-bumper nonlinearity, aimed at simulating the effects of a breathing crack and at different levels, i.e. different values of the initial gap between the bumper and the column. Following the proposed method, the statistics of the distributions of the gradients for the different scenarios can indeed be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for lower values of the initial column-bumper gap, i.e. for “more nonlinear” scenarios. Keyword s Structural health monitoring (SHM) · Structural dynamics · Nonlinear dynamics · Machine learning · Neural networks 1.1 Introduction In the pursuit of making everyday life safer, humans have extensively tried to model the environment around them. Structures are an important part of the environment, in which humans live. They are man-made and should be safe throughout their lifetime. Structures are exposed to numerous environmental factors, which may cause them to fail. Moreover, during operation, structures are subjected to dynamic loads, which, in time, may cause failure. Such failures will most probably result in economic damage to society and may even result in loss of human lives. Therefore, for the purpose of maintaining structures safe, the field of structural health monitoring (SHM) [1] has emerged. G. Tsialiamanis ( ) Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: g.tsialiamanis@sheffield.ac.uk C. R. Farrar Engineering Institute, MS T-001, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: farrar@lanl.gov © The Society for Experimental Mechanics, Inc. 2024 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-36999-5_1 1 Toward 3D Experimental Impulse-Based Substructuring Using the Virtual Point Transformation Oliver M. Zobel, Francesco Trainotti, and Daniel J. Rixen Abstract The application of Impulse-Based Substructuring (IBS) might be advantageous compared to the classical FrequencyBased Substructuring (FBS) approach because it is a time domain method, e.g., allowing for the addition of non-linear elements. In order to be viable for experimental applications, the method must be able to handle real-world systems and interfaces as well as all the experimental issues commonly encountered with substructuring. For this, the Virtual Point Transformation (VPT), commonly used for FBS to model a rigid interface connection, is adapted for use with IBS. With the shown approach, it is possible to correctly reconstruct some of the initial acceleration response peaks of an experimental test-case. The results show that combining a time domain deconvolution, downsampling with low-pass filter, and the VPT fundamentally enables experimental substructuring of three-dimensional structures in the time domain using the IBS method. Keywords Impulse-based substructuring · Experimental substructuring · Virtual point transformation, · Time domain substructuring· Experimental techniques Introduction Impulse-based substructuring (IBS) is the time-domain counterpart of the well-established frequency-based substructuring method (FBS). The IBS method is especially suited for determining shock responses, where the main goal might be to correctly predict the maximum amplitudes, e.g., of the accelerations. Its advantages originate from it being a time-domain method, i.e., high-frequency content, potentially consisting of many modes, can be represented in a short time series. Additionally, since there is no transformation into the frequency domain, there is no forced prioritization, i.e., leakage does not occur. While this method has been successfully applied for numerical models in the past, see for instance [1–3], experimental applications are just now starting to be conducted. In the authors’ previous work [4], it was shown that the IBS method can be used experimentally to determine the initial response peaks of aluminum and polyoxymethylene (POM) rods considered one-dimensional. This was enabled by using a time domain deconvolution procedure to experimentally estimate the impulse response functions (IRF), and additionally, a downsampling approach was proposed for test cases with limited excitation bandwidth, like the POM rods. Whereas this showed that an experimental application of IBS is generally possible, the practical use-cases are limited when only one-dimensional systems or interfaces can be used. Therefore, in this paper, the fundamentals are laid out to apply the IBS method experimentally to three-dimensional systems. For this, the Virtual Point Transformation (VPT), as proposed in [5], commonly used within FBS to realize experimental six degree of freedom interface coupling, is adapted for the IBS method. The adapted IBS scheme using VPT is then tested experimentally on a substructuring benchmark structure and compared to reference measurements. Oliver M. Zobel · Francesco Trainotti · Daniel J. Rixen Chair of Applied Mechanics, TUM School of Engineering and Design, Technical University of Munich, Boltzmannstr. 15, 85748 Garching, Germany e-mail: oliver.zobel@tum.de; francesco.trainotti@tum.de; rixen@tum.de © The Author(s), under exclusive license to River Publishers 2025 11 Walter D’Ambrogio, et al. (eds.), Dynamic Substructuring & Transfer Path Analysis, Vol. 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0149-8 2
12 O. M. Zobel et al. Theory of Impulse-Based Substructuring Here, only a short summary of the dually assembled Impulse-Based Substructuring (IBS) theory proposed by RIXEN in 2010 in [2] is given. Further details can be found there or in the authors’ previous work, ‘Enabling Experimental Impulse-Based Substructuring through Time Domain Deconvolution and Downsampling’ [4]. The time-continuous IBS method is given by: Definition Impulse-Based Substructuring Method[2] y(s)(t)=Z t 0 H(s)(t −τ) f(s)(τ)+B (s) f T λf(τ) dτ Equations of Motion NSX s=1 B(s) y y (s)(t)=0 Compatibility (1) where: y(t) System response H(t) Matrix of impulse response functions (s) Substructure index f(t) Externally applied forces B Signed Boolean constraint matrix ∗f Force mapping NS Number of substructures λ(t) Global vector of Lagrange multipliers ∗y Response mapping These equations must be discretized in time, and then a time-marching scheme over the discrete time steps k has to be evaluated. The aforementioned scheme first calculates a prediction of the system responses using a convolution product between the impulse response functions (IRFs) and the applied forces while neglecting the compatibility condition for the current time stepk. Then, based on the existing interface gap, Lagrange multipliers λare calculated that, for a linear system, exactly close the gap when applied to the system in the corrector step [2, 4]. This is summarized in fig. 1. Predictor Step Without λ[k−1] y[k] = ˜y(s)[k] Calculate λ[k−1] to Close Gaps g[k] ! =0 Corrector Step for λ[k−1] y[k]+=y (s) λ [k] Calculation of Gaps at Time k g[k] = NSX s=1 B(s) ˜y(s)[k] Increment Discrete Time Stepk Fig. 1 Flowchart of IBS scheme calculations within each time step The IRFs are determined from impact measurements using the time domain convolution procedure proposed by [6]. This procedure allows averaging of multiple impacts and essentially is the time domain counterpart of the H1 FRF estimator. Depending on the excitation bandwidth compared to the full measurement bandwidth, the system might not be properly excited at all frequencies, see fig. 2. Contrary to FBS, where the higher frequencies can even be omitted after the substructuring procedure, for IBS, the improperly excited frequency content must be removed before the IRF calculation. The authors proposed in [4] to do this with downsampling after applying an appropriate low-pass filter, limiting the considered bandwidth. Virtual Point Transformation for IBS For three-dimensional structures, coupling two triaxial sensors on the interface, one on each substructure, is not sufficient as this does not account for the rotational dofs and only couples the three measured translations, while in reality, the interface connection is a line or surface contact, rather than a point contact. Measuring additional rotational dofs is very difficult because the required torque excitation and rotational sensors are not standard measurement equipment [7]. Instead, it was proposed by DE KLERK ET AL. in [7] to determine the rotational dofs by combining multiple triaxial sensors. Nevertheless, this approach was found to be non-ideal, as the interface problem could be overdetermined, and measurement errors can lead to unwanted stiffening and spurious peaks in the coupled FRFs.
Toward 3D Experimental Impulse-Based Substructuring Using the Virtual Point Transformation 13 0 10 20 30 40 50 10−4 10−2 100 102 104 2x 3x 15x 10x 7x 5x 4x Frequency / kHz Amplitude (logarithmic) Transfer function Acceleration response Excitation force Fig. 2 Example of transfer function calculated from measured acceleration response and excitation signal of the experimental test case; all quantities transformed into the frequency domain. Vertical lines indicate the cut-off point for a downsampling factor as indicated at the top of the line To solve this problem, the coupling using a virtual point was introduced in [5] by VANDERSEIJS ETAL.. The interface is assumed to be rigid, i.e., only six rigid body modes, three translations and three rotations, are considered, and the remainder or residual µis neglected. The measured interface translations (at least nine) are then projected onto the virtual point with six dofs. This projection weakens the interface coupling and is solved for in a least-square sense, averaging out measurement errors [5]. Projection of Responses For the virtual point transformation, the triaxial sensors are assumed to be rigidly connected to the interface, i.e., the virtual point. The distance from the virtual point to the sensor i is given by the position vector ri, as shown on the left in fig. 3. ri,z ri,y ri,x x z y yi,z yi,y yi,x Virtual Point Triaxial Accelerometer ri x z y x z y x y z Translation inx-direction Rotation around y-axis Rotation around z-axis qx ri,z ri,y ri,z · qθy −ri,y · qθz q θy qx qθz Fig. 3 Projection of triaxial sensor dofs on virtual point exemplarily shown for the measured x-direction yi,x of sensor i, figure adapted frompyFBS documentation [8] Using this vector, the responses yi measured by the sensor i can be projected onto the virtual point dofs q, by deriving the required relations, as exemplary shown for the measured response in the x-direction in fig. 3: yi,x yi,y yi,z = 1 0 0 0 ri,z −ri,y 0 1 0 −ri,z 0 ri,x 0 0 1 ri,y −ri,x 0 qx qy qz qθx qθy qθz (2) Equation (2), yi = Ry,iq in short notation, describes how the three sensor dofs yi can be described using the matrix Ry,i containing the rigid body modes of sensor i and the reduced virtual point dofs q. In case the orientation of the sensor and the virtual point coordinate system does not line up, a rotation matrix has to be added to the equation. Writing this for all nS sensors and adding a residual µto account for the neglected flexible interface motion, the response yi of all interface sensors is given by [5]: y1 .. . ynS = Ry,1 .. . Ry,nS q+µ (3)
14 O. M. Zobel et al. An optimal solution for the desired virtual point dofs q can then be found by minimizing the residual µ, which yields, without weighting matrices, the desired transformation matrix Ty, projecting the measured responses y on the virtual point dofs q [9]: Ty = RT y Ry −1 RT y with q =Tyy (4)
Toward 3D Experimental Impulse-Based Substructuring Using the Virtual Point Transformation 15 Projection of Forces Similar to the projection of the measured responses yi of sensor i, the forces or impacts fi applied to the system can be projected onto the virtual point [5]: mx my mz mθx mθy mθz = 1 0 0 0 1 0 0 0 1 0 −ri,z ri,y ri,z 0 −ri,x −ri,y ri,x 0 ei,x ei,y ei,z fi (5) where the vector ri describes the distance between the virtual point and the point where the force is acting and the vector ei the direction of the scalar force value fi. Again, all nI impacts can be combined in one equation m=RT f f. An optimal solution for the transformation of the virtual point forces minto the physical space of the measured forces f is given by [9]: TT f =Rf RT f Rf −1 with f =TT f m (6) Adaption of IBS Equations - Interface Compatibility First, the interface compatibility is no longer written for the physical responses y(s), but for the virtual point dofs q(s) for each virtual point of each substructure s: NSX s=1 B(s) q q (s) =0 with q(s) =hq (s) x q (s) y q (s) z q (s) θx q (s) θy q (s) θz i T (7) For this, the boolean constrain matrixB (s) f has to be rewritten for the virtual point dofs q as B (s) q . Inserting the transformation matrix from eq. (4), i.e., q =Tyy, gives the interface compatibility condition on the virtual point w.r.t. y(s): Interface Compatibility on Virtual Point NSX s=1 B(s) q (nV·6)× n (s) aS +nV·6 T(s) y n (s) aS +nV·6 ×n (s) S y(s) n (s) S ×1 = 0 n (s) S ×1 (8) where: nS Total number of sensor dofs nV Number of Virtual Points (s) Substructure index naS Number of additional sensor dofs NS Total number of substructures Adaption of IBS Equations - Interface Equilibrium Next, the interface equilibrium has to be written for the virtual point coordinates q(s), i.e., the Lagrange multipliers λmare applied to the virtual point as virtual forces: m(s) λ =B(s) m T λm (9) To calculate a response to these forces as required for the IBS scheme, they must be transformed into the basis of the impulse response functions H(s), namely the physical space. This is done by inserting the transformation eq. (6), i.e., TT f m, into eq. (9): Interface Equilibrium on Virtual Point f (s) λ n (s) I ×1 = T (s) f T n (s) I × n (s) aI +nV·6 B(s) m T n (s) aI +nV·6 ×(nV·6) λm ((nV·6)×1) (10) where: nI Total number of impact locations nV Number of Virtual Points (s) Substructure index naI Number of additional impact locations NS Total number of substructures
16 O. M. Zobel et al. Adaption of IBS Equations - Lagrange Multiplier Calculation Lastly, the calculation of the Lagrange multipliers must be adjusted. Using the discretization used in [4], this is given as: NSX s=1 B(s) y H(s)[0]B (s) f T! λf[k−1] =− NSX s=1 B(s) y ˜y (s)[k] · 1 ∆t (11) On the right-hand side, the compatibility condition is exchanged with the one written for VPT coupling (eq. (8)). On the left-hand side, the interface forces B (s) f T λf = f (s) λ are exchanged with the expression in eq. (10). As the left-hand side would then yield a response due to the Lagrange multipliers in the real space, a premultiplication by the transformation from eq. (4) is necessary to be consistent with the right-hand side, representing the gap in the virtual point coordinates. This yields the new Lagrange multiplier calculation with VPT coupling: Calculation of Lagrange Multipliers on VPT Interface NSX s=1 B(s) q T (s) y H(s)[0]T (s) f T B(s) m T! λm[k−1] =− NSX s=1 B(s) q T (s) y ˜y (s)[k] · 1 ∆t (12) Discretized IBS Equations with Virtual Point Coupling The only remaining step is to substitute the projection of the VPT interface forces (eq. (10)) into the predictor and corrector step of the IBS scheme. To summarize, the adapted IBS equations with virtual point coupling are given by: IBS with VPT Predictor Step ˜y(s)[k] = k−2X i=0 H(s)[k−(i +1)] f(s)[i]+T (s) f T B(s) m T λm[i] ∆t!+H(s)[0] f(s)[k−1] ∆t (13) Calculation of Lagrange Multipliers on VPT Interface NSX s=1 B(s) q T (s) y H(s)[0]T (s) f T B(s) m T! λm[k−1] =− NSX s=1 B(s) q T (s) y ˜y (s)[k] · 1 ∆t (14) IBS with VPT Corrector Step y (s) λ [k] =H(s)[0]T (s) f T B(s) m T λm[k−1] ∆t (15) Sensor and Impact Consistency Criterion When applying the VPT for measurements, various issues could exist, e.g., some impacts are too soft (bad signal-to-noise ratio), or sensor positions, orientations, or calibrations are wrong. Also, the rigid interface assumption will not hold for higher frequencies. Such issues within FBS applications can be detected using the established performance indicators of sensor and impact consistency [5]. Therefore, an adaption for IBS is highly motivated. The sensor consistency is defined using the response to a unit load case at any excitation dof, which in the frequency domain is represented by a magnitude of one at all frequencies. This essentially extracts the respective column of the FRF matrix [5]. In the time domain, the equivalent of the unit load case is given by a Dirac impulse. Then, the convolution of the IRF matrix with a Dirac impulse at dof i also extracts the i-th columnHi of the IRF matrixH. The filtered (time) response ¯y, i.e., transformed onto the virtual point and back to the physical space, can then be found by substituting eq. (4) in eq. (3): ¯y =RyTyy =RyTyHi (16) Similarly, the impact consistency uses the response at one sensor channel to all performed impacts, equivalent to one row of the FRF matrix [5], respectivelyH1i of the IRF matrix. The filtered force ¯f is found by substituting eq. (5) in eq. (6): ¯f =TT f RT f f =T T f RT f H1i (17)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==