River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Dynamic Environments Testing, Vol. 7 Alexandra Karlicek Dagny Beale Cora Taylor 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
Alexandra Karlicek· Dagny Beale · Cora Taylor Editors Dynamic Environments Testing, Vol. 7 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 97887-438-0152-8 (Hardback) ISBN 97887-438-0164-1 (eBook) https://doi.org/10.13052/97887-438-0152-8 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 Environments Testing 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; Dynamic Substructuring & Transfer Path Analysis; Model Validation and Uncertainty Quantification; Dynamic Substructures; Special Topics in Structural Dynamics & Experimental Techniques; Computer Vision & Laser Vibrometry; 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 Dynamic Environments Testing & other Structural Dynamics areas. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Editor: Alexandra Karlicek–Anduril Industries, MA, USA; Dagny Beale–Sandia National Laboratories, NM, USA; Cora Taylor, Michigan Technological University, MI, USA. v
Contents 1 Comparison of Accelerometer Selection Algorithms for a Modal Filter Application 1 Tyler F. Schoenherr 2 An Improved Fatigue Damage Spectrum for MIMO Random Testing 9 Enrico Proner and Emiliano Mucchi 3 Characterization of Load Uncertainty on Simulated Dynamic Responses 17 Beale Dagny and Yarberry Kieran 4 Toward Generalized MIMO Random Vibration Specifications 25 Ryan Schultz and Garrett Nelson 5 A Method for Response Replication at Component-Level in MIMO Random Testing 37 Enrico Proner and Emiliano Mucchi 6 Making Modal Analysis Easy and More Reliable – Challenging Ai-Based Algorithms with the Barc Example 45 Tim Kamper, Denis Beljan, Patrick Hu¨skens, and Haiko Bru¨cher 7 An Undamped Dynamic Vibration Absorber on a Resonant Plate Shock Test 55 David E. Soine, Adam J. Bouma, and Forrest J. Arnold 8 Analysis of a Tuned Vibration Absorber for Resonant Plate Shock Testing 61 Adam J. Bouma and David E. Soine 9 A Methodology for Feature Selection and Electrical Capability Prediction of a Coupled Shaker-DUT Model 67 D. Scheg, J. Heinlen, C. Garcia, L. Redmond, T. Roberts, A. Ramirez, and C. Haynes 10 Motivating Multivariate Specifications for Multiple-Input, Multiple-Output Vibration Testing 77 Colin Haynes, Thomas Thompson, and Adam Watts 11 Understanding Changes in Global Behavior Due to Control Location 83 Sarah Johnson, John Schultze, and Shannon Danforth 12 Estimating Component Level Environments Using Next Assembly Measurements 93 Marcus Behling, Matthew S. Allen, Randall L. Mayes, and Washington J. DeLima 13 Dynamic Analysis of a Modular Test Stand for Multi-Axis Vibration Testing 107 H. R. Kramer, J. F. Schultze, S. M. Danforth, and B. P. Mann 14 Evaluating the Effect of Shaker Placement Optimization Priorities on Multi-Axis Test Results 121 Risto Djishev, Kieran Elrod, Connor Tasik, Jim DeClerck, Brittany Ouellette, John Schultze, and Shannon Danforth vii
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 Comparison of Accelerometer Selection Algorithms for a Modal Filter Application Tyler F. Schoenherr Abstract A modal filter is a useful tool to characterize the full-field dynamic response of a structure. It provides the amplitudes of the inherent modes of the structure, which can be used to understand its full-field response. As useful as the modal filter is, the execution of the technique is challenging. Forming a modal filter requires representative modes observed at appropriate measurement locations. Choosing those locations is critical to reducing fitting and over-fitting errors in the modal filter process. One means of choosing the accelerometer locations recently developed is to minimize the Modal Projection Error. This paper examines using the Modal Projection Error and two other means of choosing measurement locations in order to compare their efficacy with respect to the modal filter. This comparison of the different methods are evaluated using an experimental data set and a novel means of comparing the effectiveness of the modal filter. Keywords Modal · Projection · Error · Filter · Accelerometer Introduction One challenging aspect in dynamic environment characterization is the ability to measure the full-field response of a structure. The most direct means of acquiring the response of a structure is to take direct discrete measurements of acceleration using accelerometers. Although data acquisition systems and accelerometers have greatly improved in the past decades, measuring the approximate full-field response of a continuous structure is impractical in most applications. Strain is caused by relative motion within a structure. Knowledge of the full-field deflection of the structure at any moment of time is imperative if strain and damage is to be quantified in a dynamic environment. Direct discrete measurements are not adequate on their own to calculate the full-field strain in the structure. Acquiring the full-field response requires additional knowledge of the structure in addition to response measurements. A modal filter is the combination of the structural dynamic properties of a structure in the form of mode shapes and discrete measurements. The mode shapes are curve fit to the measured data through a least-squares process. By combining the mode shapes and measured data, the full-field response of a structure can be calculated. There are errors generated during the curve fitting process when combining measured acceleration and mode shapes to calculate the full-field response. Such errors have unique challenges in providing meaningful modal filter results and are described in this paper as fitting and over-fitting errors. Intelligent selections of where the discrete measurements are during the dynamic environment can mitigate some of the curve fitting errors. Several methods with different modal-based objective functions have been previously proposed to improve the modal filter calculation. This paper attempts to compare the efficacy of three existing degree of freedom selection methods: EffecThis article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DENA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.energy.gov/downloads/doe-public-access-plan. Tyler F. Schoenherr Sandia National Laboratories, P.O. Box 5800 - MS0346, Albuquerque, NM, 87185 e-mail: tfschoe@sandia.gov © The Author(s), under exclusive license to River Publishers 2025 Alexandra Karlicek et al. (eds.), Dynamic Environments Testing, Vol. 7 of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0152-8 1
2 T. F. Schoenherr tive Independence [1], condition number, and Modal Projection Error [2]. All three of them are examined in this paper to determine if there are benefits of one method over the others and to evaluate how robust the calculations are to changes and errors in the mode shape basis set. Theory The focus of this paper is on the modal filter process. The modal filter is rooted in modal superposition theory that states that a structural response is comprised of a linear combination of the structure’s mode shapes ¨¯x≈ϕ¨¯q, (1) where ¨¯xis the vector of physical accelerations at a set of degrees-of-freedom, ϕis the mode shape matrix at the same set of degrees-of-freedom and ¨¯q is the modal acceleration or amplitude for each corresponding mode. The linear combination of modes in Eq 1 is only an approximation of the acceleration due to modal truncation. The modal filter is the inverse of Eq. 1 such that the modal accelerations can be calculated with knowledge of the structure’s mode shapes and physical accelerations. In order to transform the physical accelerations into modal accelerations, the physical accelerations are projected on to the mode shapes by using the Moore-Penrose pseudo-inverse of ϕ, ϕ+¨¯x≈¨¯q, (2) where the + superscript denotes the pseudo-inverse of a matrix. The pseudo-inverse in the modal filter is a least squares fitting process for over-determined systems [3]. Even though this paper focuses on the structural dynamics aspect of this process, the mode shape matrix can also be referred to as a basis set of vectors as the vectors do not need to be structural mode shapes [4]. For the remainder of this paper, this matrix is referred to as a basis set. The least squares fitting process introduces errors in the calculation of the modal coordinates. If the mode shapes in Eq. 2 do not span the space of the physical accelerations, there is a fitting error that is minimized in a least squares sense and the modal accelerations will not combine with the mode shapes to perfectly reproduce the physical accelerations. The second error is an over-fitting of the shapes to the data. This error occurs when two or more shapes linearly combine to fit the physical acceleration that is not fit by the other shapes. This error can end up being detrimental as these subset of shapes will greatly increase in amplitude to either fit noise or fitting error in the basis set. The over-fitting error is caused when two or more vectors in the basis set can closely reproduce any other vector in the basis set. This allows the least-squares fitting process to inflate all these vectors to fit measurement noise or fitting errors. If the vectors cannot combine to fit errors, then the resultant error from the projection will be a fitting error. It is advantageous to have the error be from the fit as it provides evidence that is easier to interpret and is quantified by the Modal Projection Error (MPE) [5]. One way to reduce the overfitting error is to select a set of degrees of freedom to measure that makes the basis vectors unable to combine to form another basis vector. Many methods of selecting degrees of freedom for ultimate usage in a modal filter application exist. Three of those methods are examined in this paper. The first method is a minimization of the MPE. The modal projection error is a quantity of how well a set of vectors can linearly combine to create a given vector calculated MPE=Ψ2 n =1− ¯ϕ+ TnϕSϕ + S ¯ϕTn, (3) where Ψn is the modal projection error of the n th target vector, ϕTn. The set of vectors, ϕS, is fit to the target vector. The other two objective functions are the maximizing of the Effective Independence (EFI) of the basis set [1] and the minimization of the condition number of the basis set. These quantities do not directly compute the basis vectors ability to linearly combine to recreate a separate vector in the basis set, but these methods have been used in other studies of modal filter applications with success. In order to optimize the MPE, EFI, or condition number, one degree of freedom is removed at a time and the objective function is recalculated. This process is repeated over and over in a Greedy algorithm [6] until a desired number of degrees of freedom are selected. Test Hardware and Calibration to the Model In order to demonstrate the efficacy of different instrumentation selection methods, experimental data is ideal. Use of experimental data is ideal because it contains realistic errors in the basis set from the finite element model (FEM) shapes in their ability to perfectly fit the measured experimental data.
Comparison of Accelerometer Selection Algorithms for a Modal Filter Application 3 The hardware used to generate data for this paper is the frame-plate-wing (FPW) test bed [7]. This hardware is chosen because development of an adequate model is quick and a large experimental data set already exists. A large channel count test provides channel reduction options for the optimization methods. In addition, a large channel count also provides more validation for the basis set of mode shapes. The FPW was tested in the laboratory in a free boundary condition and impacted with a modal hammer. A picture of the test setup can be seen in Figure 1. Fig. 1 Photograph of the experimental setup and node locations for the Frame Plate Wing structure The FEM is compared to experimentally derived mode shapes and natural frequencies. The Modal Assurance Criteria (MAC) with associated natural frequency comparisons can be seen in Figure 2. From the comparison in Figure 2, it can be seen that the model does a good job fitting to the experimental shapes below 800 Hz even though the model natural frequencies are 7-10% higher than the experimentally measured natural frequencies. Above 800 Hz, the FEM shapes begin to represent the physical hardware less. Having modes that match well and match poorly is leveraged in this paper to demonstrate how well the modal filter works with a basis set expected to fit the data well under 800Hz and a basis set with that will incur fitting error under 1800Hz. Experimental Results In order to fit a basis set to test data, test data is needed. The response data used in this study is a transient set of data caused by a modal hammer strike of the FPW structure at node 305 in the direction normal to the flat surface of the wing as shown in Figure 1. For all modal filters implemented in this study, the modes are generated by the FEM of the hardware. The test data is ideal for comparing different means of selecting accelerometer degrees of freedom because the test is highly instrumented at 95 channels. This channel count provides a large pool from which the different algorithms can select degrees of freedom. Evaluating the efficacy and accuracy of the modal filter is challenging. The output of the modal filter is modal accelerations if the data measured is from accelerometers. Since there is no direct measurement for modal acceleration, success is inferred. With imperfect mode shapes, the modal filter will always be a least squares fit to the test data. Only modes and accelerations measurements from a finite element model would provide a perfect fit and produce exact modal accelerations. Using experimental data, there is always some error in the estimations of the modal accelerations.
4 T. F. Schoenherr Fig. 2 Modal Assurance Criteria and associated natural frequency error of the Finite Element Model when compared to experimental modal results It is of interest to alter parameters of the modal filter that have the largest effect on the calculation of the modal accelerations. When analyzing the modal accelerations output by the modal filter, four parameters are considered: • Method of selecting the degrees of freedom in the filter • Number of degrees of freedom selected • Number of modes in the basis set • Accuracy of the basis set The changes in modal accelerations when altering the aforementioned parameters are compared to ‘truth’ modal accelerations. The truth modal accelerations are calculated from using all the degrees of freedom and modes 1 through 32 with modes 20 and 25 omitted in the modal filter. Modes 20 and 25 are omitted because they individually closely resemble other modes in the basis set. Using this set of shapes will be referred to as the full set of shapes. The frequency content of the truth modal accelerations calculated for modes 1 through 18 can be seen in Figure 3. Because the input force is an impulse from a hammer strike, the rigid body modal acceleration for modes 1 through 6 should be flat at low frequencies with translation in the Z direction (mode 3), rotation about X (mode 4), and rotation about Y (mode 5) being of largest magnitude. These rigid body modes should be highest in magnitude of the rigid body modes due to the location and direction of the force vector. The modal acceleration of the elastic modes 7 through 15 present themselves as single degree-of-freedom oscillators which is expected from modal analysis theory. The modal accelerations in Figure 3 provide evidence that the modal filter worked well enough to use these modal accelerations as ‘truth’ data. To increase our confidence in the truth modal accelerations calculated from a modal filter, the modal accelerations are multiplied by the basis set to calculate the response at the measured DOFs . The set of measured DOFs in Figure 4 are not included in the modal filter process so they are considered a blind check of the validity of the modal acceleration calculation. Comparing the response of the measured DOFs to the resynthesized DOFs below 600 Hz, the comparison provides evidence of the modal accelerations being an excellent representation of the response of the system. To compare the efficacy of the three different DOF selection methods, four modal filters are applied to the test data with varying modal filter parameters. A summary of the modal parameters can be found in Table 1. The first filter uses only modes 1 through 19 and 20 degrees of freedom. Modes 1 through 19 are chosen because they all have high MAC values when compared to the experimental modal data. The second run also uses modes 1 through 19 in the modal filter, but uses 34 degrees of freedom in the modal filter. The third run uses the full set of shapes and 31 degrees of freedom. The ratio in the third run basis set provides one more degree of freedom than basis vectors. The final run uses the full set of shapes and 45 degrees of freedom.
Comparison of Accelerometer Selection Algorithms for a Modal Filter Application 5 Fig. 3 Modal accelerations of modes 1 through 18 taken to be the ‘truth’ values Fig. 4 Selection of degrees of freedoms not included in the modal filter with the modal filter having 19 modes and 20 degrees of freedom in the filter For all four runs, the different basis sets are used to calculate the modal accelerations. For all comparisons, the peak of the modal acceleration of the Power Spectrum Density is compared to the modal accelerations from the truth calculation.
6 T. F. Schoenherr Table 1 Summary of the four modal filter parameters compared Run # Modes Included (m) # of DOFs in Filter run1 1:19 m+1 run2 1:19 m+15 run 3 1:19, 21:24, 26:32 m+15 run 4 1:19, 21:24, 26:32 m+1 The percent difference to the truth modal peak acceleration is documented and is shown in Figure 5 for comparison. It is important to acknowledge that the real modal acceleration value is unknown, but observing how much the modal acceleration changes when the degrees of freedom of the basis set changes is the focus of this research. Fig. 5 Error of the modal acceleration at its resonance for each of the FEM modes used in the inverse Run 1: 19 Modes with 20 DOFs in inverse Run 2: 19 Modes with 34 DOFs in inverse Run 3: 30 Modes with 45 DOFs in inverse Run 4: 30 Modes with 31 DOFs in inverse There are three notable observations of the percent differences of the different selection methods. The first observation is that the largest error occurs for the Effective Independence selection method for mode 21 as the magnitude changes by a factor of 3. However, this result is misleading because the magnitude of the modal acceleration for mode 21 is small
Comparison of Accelerometer Selection Algorithms for a Modal Filter Application 7 enough that it is fitting only electronic noise and not structure response. Small changes to the basis set affecting the modal acceleration calculation is expected when the modal acceleration is in the noise floor. The second observation is that the methods of selecting DOFs did not appear to have an effect on the stability of the modal filter. Each picking method has modes that caused the result to be the most consistent and the least consistent. Not shown in Figure 5 is the result when a random set of DOFs are chosen. When selecting a random set of DOFs for the basis set identical to run1, the inverse had an average different of around 800%. It is noteworthy that the DOFs that are selected by the three techniques are different from each other. Figure 6 shows the DOFs selected for run 1 by the three methods. Fig. 6 Degrees of freedom chosen by 3 separate algorithms The third observation is that the modal accelerations calculated are fairly robust to changes in the basis set given that the basis vectors do a good job of projecting to the measured data. Changes to number of modes included or number of DOFs included generally only altered the magnitude of the modal acceleration by less than 5%. This level of robustness is encouraging for usage of modal accelerations as quantities for characterizing the full-field response in dynamic environments characterization. Conclusion This research intended to provide insight into the benefits of using either the EFI, condition number, or MPE DOF selection method over another with respect to a modal filter calculation. In using a relatively accurate model to provide the basis set to fit experimental acceleration measurements, the results show that there is very little benefit of using one selection method over another. However, it is important to use one of the methods as random selection resulted in serious degradation of the modal filter. The manner in assessing the success of the modal filter in this paper is novel in comparison to other papers on modal filter techniques. This paper uses the modal accelerations to compare techniques instead of the measured and resynthesized acceleration data. Direct comparison of the modal accelerations instead of a subset of measured accelerations provides a full-field response comparison that otherwise is implicitly estimated by comparing selected degrees of freedom. A benefit to comparing modal accelerations directly is that changes in the magnitude of the modal acceleration is explicit instead of implied when comparing physical coordinates. Additionally, examination of the modal accelerations while altering the parameters of the modal filter provides some information regarding its stability. This is important because an unstable modal filter would be an indication of a poor modal filter. In addition, it appears that most of the modal accelerations change by less than 5%. This conclusion provides a confidence level of using modal accelerations in environment response characterization. References 1. Daniel C Kammer and L Yao. Enhancement of on-orbit modal identification of large space structures through sensor placement. Journal of Sound and Vibration, 171(1):119–139, 1994. 2. Tyler F Schoenherr and Jerry W Rouse. Characterizing dynamic test fixtures through the modal projection error. Mechanical Systems and Signal Processing, 204:110746, 2023. 3. Roger Penrose. A generalized inverse for matrices. Mathematical proceedings of the Cambridge philosophical society, 51(3):406–413, 1955. 4. Yuanchang Chen. A non-model based expansion methodology for dynamic characterization. Thesis, 2019.
8 T. F. Schoenherr 5. Tyler F Schoenherr and Jelena Paripovic. Using Modal Projection Error to Evaluate SEREP Modal Expansion, pages 111–138. Springer, 2022. 6. Andrew Vince. A framework for the greedy algorithm. Discrete Applied Mathematics, 121(1-3):247–260, 2002. 7. D. Roettgen, G. Lopp, A. Jaramillo, and B. Moldenhauer. Experimental substructuring of the dynamic substructures round-robin testbed. In Matthew Allen, Walter D’Ambrogio, and Dan Roettgen, editors, Dynamic Substructures, Volume 4, pages 119–123. Springer International Publishing.
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 An Improved Fatigue Damage Spectrum for MIMO Random Testing Enrico Proner and Emiliano Mucchi This study focuses on Multiple-Input Multiple-Output (MIMO) vibration control tests, designed to accurately mimic a structure’s operational vibration environment. In MIMO random testing, the Cross Spectral Density (CSD) between each pair of control channel plays a key role in the outcome of the test. In fact, the same Power Spectral Densities (PSDs) but with different CSDs may determine a different response on the Device Under Test (DUT) and different drives required to conduct the test. In this scenario, this paper proposes a methodology that aim to replicate a random vibration environment at the component level with minimal excitation. The approach pre-defines the phase and correlation of the reference Spectral Density Matrix (SDM) that maximize the response of the DUT. Then, the reference SDM is completed with the minimum PSD values ensuring that the structure’s response in the laboratory aligns with the target vibration environment. This method enhances the consistency of laboratory tests with actual operational environments while minimizing the required energy, ultimately preserving the excitation system’s integrity. The paper provides both a theoretical and an experimental analysis of the proposed methodology. The experiments are carried on specifically designed specimen excited by a multi-axis vibration environment, provided by a three-axis shaker. Keywords Multi-axis test tailoring· Random testing · Fatigue · Fatigue Damage Spectrum· Environmental testing Introduction One way to assess the durability of a component is to conduct random vibration tests, which aim to mimic the operational environment by means of random excitation. In general, it is unfeasible to conduct life-long tests, therefore, the vibration environment is usually exaggerated in order to accelerate the test, overcoming time and costs limitations. Typically, the test profiles are derived from Standards [1, 2]. However, standard profiles are generally much more severe than real vibration environments and may lead to over-testing. The best way to conduct the test is to adopt tailoring techniques [3]. These techniques require to gather operational data to synthesise vibration profiles that are more representative of the operational environment. Nowadays, the most accredited tailoring procedure is the one proposed by Lalanne [4] . Lalanne formalized two types of frequency spectra: the Extreme Response Spectrum (ERS) and the Fatigue Damage Specteum (FDS) [5]. According to Lalanne, the test article is represented by a of Single-Degree-of Freedom (SDOF) systems, with varying natural frequency in the bandwidth of interest. he vibration environment is applied as excitation to the base of the SDOF system. The FDS for a random excitation can be computed as: FDS(fn)= n+ 0 T C Kb √2zrms(fn) b Γ 1+ b 2 (1) where n+ 0 is the mean number of zero up-crossing, zrms is the root mean square of the SDOF relative displacement and Γ is the gamma function. In particular, the FDS represents the damage potential of a vibration environment. Procedures based on the equivalence of the FDS are widely studied and researched [6–11]. The formulation of the FDS is limited to uni-axial excitations, therefore, it neglects the possibility of multi-axis tests. However, real vibration environments are, in general, multi-axial. Moreover, in the recent past, multi-axis tests have been proven to be more realistic and more severe than traditional single-axis testing procedures [12, 13]. In this context, this paper investigates the possibility of a novel Enrico Proner · Emiliano Mucchi Department of Mechanical Engineering, University of Ferrara, Italy, Via Saragat 1 e-mail: enrico.proner@unife.it; emiliano.mucchi@unife.it © The Author(s), under exclusive license to River Publishers 2025 9 Alexandra Karlicek et al. (eds.), Dynamic Environments Testing, Vol. 7 of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0152-8 2
10 E. Proner and E. Mucchi formulation of theFDS, suitable for multi-axis random excitations. In particular theMulti-Input FDS (MI-FDS) is presented. The MI-FDS takes into account the CSDs between multiple excitations, allowing to identify the real damage potential of a multi-axis random vibration. Moreover, this work provides the experimental verification of the MI-FDS. The experimental verification demonstrates the ability of the MI-FDS to evaluate the damage potential of multi-axis vibration environments, which enables the possibility of more accurate multi-axis tests. The novel proposal: Multi-Input Fatigue Damage Spectrum In this section the novel MI-FDS is presented. The general concept of the MI-FDS, compared to the traditional FDS, is described in Figure 1 . Fig. 1 Comparison between the calculation of the MI-FDS andFDS for a random vibration environment. The MI-FDS shares most of the assumptions of the traditional FDS. However, the MI-FDS considers multiple simultaneous excitations applied to the SDOF system. As a consequence, the total response PSD of the SDOF systemGz subjected to multiple random excitations can be expressed as: Gz =H1×LSinHH 1×L (2) where H1 ×L is the 1×Ltransfer function of the system , Sin is the L×Lexcitation Spectral Density Matrix (SDM) and the superscript H indicates the complex conjugate transpose operator. Each element of H1×L is computed as a frequency response function with amplitude: |H¨xz(f,fn)| = 1 (2πfn) 2 s 1− f fn 2 2 + 2ζ f fn 2 (3) andphase: ∠H¨xz(f,fn)=atan 2ζffn f2 n −f2 (4) Equation (2) can be also expressed in terms of phases and coherences of Sin: Gz =ΣL j=1Sin,jjWjj +2ΣL−1 j=1 ΣL k=j+1qγ 2 jkSin,jjSin,kk|Wjk|cos(ϕjk −ξjk) (5) where W=HH 1×L H1×L , ξjk is the phase of Wjk, γjk is the coherence and ϕjk is the phase of Sin,jk. In Equation (5) the phase and coherence of the excitation SDM are explicit. In particular, γjk and cos(ϕjk −ξjk) defines how much the jth andkt excitation are combined constructively or destructively to increase or decrease the response of the SDOF system.
An Improved Fatigue Damage Spectrum for MIMO Random Testing 11 Moreover, it should be noted that the phase difference between Sin,jk and Wjk is specific of the SDOF system. Therefore, the value ϕjk −ξjk must be coherent between the test and the SDOF system to guarantee the consistency of the calculated damage potential. The resulting response of Equation (5) can be used to compute the MI-FDS: MI-FDS = n+ 0 T √2b C Kb Z ∞ 0 Gzdf b Γ 1+ b 2 (6) Experimental Verification In this section the experimental verification of the MI-FDSis presented. The objective of the test campaign is to demonstrate the capability of the MI-FDSto take into account the phase and coherence between multiple simultaneous random excitation and provide an accurate evaluation of the damage potential of multi-axis vibration environments. The first part of this section is dedicated to the presentation of the experimental set up and the results of the testing campaign. In the second part, the results of the experimental campaign will be discussed. Test set up and results The specimen chosen for the test campaign is a cantilever aluminum beam with rectangular section and U-shaped notches near the fixed end. A lumped mass of 0.47 kg is placed on the free-end of the beam, to tune its natural frequencies in a convenient frequency range. The detailed geometry of the beam is depicted in Figure 2. The excitation system used for the test campaign is the three-axial electro-dynamic shaker available at University of Ferrara. This excitation system is composed of three actuators (10 kN each) that allow to excite the specimen mounted on the head expander along three orthogonal translational DOFs. Figure 3. SCADAS Mobile SCM202V and MIMO Random of Simcenter Testlab are used as data acquisition hardware and vibration control software, respectively. The control accelerometer (model 356B21 by PCB Piezotronics) is placed on the fixture, near the fixed end of the cantilever beam. In addition, an accelerometer is placed on the tip of the beam to monitor the response of the specimen. Fig. 2 Detailed geometry of the specimen used in the testing campaign (all measures are inmm). The objective of the experimental campaign is to obtain the Time-to-Failure (TTF) of the specimen under different types of vibration environments. In particular, a reference vibration environment is defined in the [25−200]Hz bandwidth. Then, the traditional FDS and the novel MI-FDS are used to generate equivalent tests. Four kinds of test are carried out, each one repeated on three specimens and conducted until total failure: 1. Test Environment 0 (TE-0): three-axial reference vibration environment. 2. Test Environment 1 (TE-1): synthesise three separate PSDs to have the sameFDS, computed according to Equation (1), and the same TTF of TE-0. The obtained PSDs are applied separately and sequentially to the specimen for a duration
12 E. Proner and E. Mucchi Fig. 3 Specimen mounted on the shaker. equal to the mean TTF of TE-0. If the specimen survives the first sequence of excitations, the test is performed again until failure. 3. Test Environment 2 (TE-2): the PSDs obtained for TE-1 are applied simultaneously to the specimen. The phase and coherence between each axis is defined according to the Extreme Dynamic Response Method (EDRM) [14]. 4. Test Environment 3 (TE-3): The vibration environment of TE-2 is scaled to have the same MI-FDS, computed according to (6), and the same TTF of TE-0. The SDM of TE-3 is obtained according to the following equation: SinTE−3 = MI-FDSTE−0 MI-FDSTE−2 SinTE−2 (7) whereSinTE−3 andSinTE−2 are the excitation SDM of TE-3 and TE-2 andMI-FDSTE−0 andMI-FDSTE−2 are theMI-FDS of TE-0 and TE-2. The resulting vibration environment is a SDM with the same phase and coherence of TE-2 but with the PSDs that match the MI-FDS of TE-0. Figure 4 shows the SDM of TE-0 (solid blue), TE-2 (dashed red) and TE-3 (dashed green). The plots on the diagonal show the excitation PSD along each-axis. The upper triangular plots show the coherence, the lower triangular plots the phase between each pair of excitations. Figure 5 shows the FDS (left plot) and MI-FDS (right plot) computed for each test environment. The parameters used for the computation of the FDS and the MI-FDS are b = 5.4 , C = 5.1 · 1017 and a damping ratio of ζ = 0.1%. In the plot on the left, the total FDS of each test environment is depicted. TE-0, TE-1 and TE-2 are represented by the same curve (dashed gray). The blue curve depicts the FDS computed for TE-3. The total FDSs are obtained by summing the FDSs of each single-axis excitation. The plot on the right shows the MI-FDS computed for TE-0 (blue curve), TE-1 (gray curve), TE-2 (red curve) and TE-3 (green curve), computed according to Equation (6). Finally, Table 1 shows the TTFs obtained for each specimen in the testing campaign. The TTFs from Table 1 are also depicted in the box plot of Figure 6. In Figure 6, each box describes the first quartile of the TTFs obtained from each test environment, the red line and the black cross are the median and the mean of the TTFs, respectively. Discussion of the results The first result that needs to be addressed is the difference between the traditional FDS and the novel MI-FDS. In Figure 5, in the left plot, the total FDS computed for TE-0, TE-1 and TE-2 is the same. In fact, the traditional FDS considers only the PSD of each excitation. TE-0, TE-1 and TE-2 excite the system with the same PSDs, therefore, the total FDS is the same for these test environments. Moreover, the total FDS of TE-3 resulted lower than the the total FDS of TE-0, as the PSDs of TE-3 are lower. On the other hand, the MI-FDS produces three distinct curves for TE-0, TE-1 and TE-2. In particular TE-1
An Improved Fatigue Damage Spectrum for MIMO Random Testing 13 50 100 150 200 Hz 10-2 10-1 g2/Hz TE-0 TE-2 TE-3 50 100 150 200 Hz 0 0.2 0.4 0.6 0.8 1 / TE-0 TE-2 TE-3 50 100 150 200 Hz 0 0.2 0.4 0.6 0.8 1 / TE-0 TE-2 TE-3 50 100 150 200 Hz -100 0 100 deg TE-0 TE-2 TE-3 50 100 150 200 Hz 10-3 10-2 g2/Hz TE-0 TE-2 TE-3 50 100 150 200 Hz 0 0.2 0.4 0.6 0.8 1 / TE-0 TE-2 TE-3 50 100 150 200 Hz -100 0 100 deg TE-0 TE-2 TE-3 50 100 150 200 Hz -100 0 100 deg TE-0 TE-2 TE-3 50 100 150 200 Hz 10-3 10-2 10-1 g2/Hz TE-0 TE-2 TE-3 Fig. 4 SDM of the TE-0 (solid blue) ,TE-2 (dashed red) and TE-3 (dashed green) 50 100 150 200 Frequnecy - [Hz] 10-31 10-30 10-29 10-28 10-27 10-26 10-25 10-24 10-23 FDS - [damage potential] Total FDS (X+Y+Z) - TE-0; TE-1; TE-2 Total FDS (X+Y+Z) - TE -2 50 100 150 200 Frequnecy - [Hz] 10-31 10-30 10-29 10-28 10-27 10-26 10-25 10-24 10-23 MI-FDS - [damage potential] MI-FDS (X+Y+Z) - TE-1 MI-FDS - TE-0 MI-FDS - TE-2 MI-FDS - TE-3 Fig. 5 FDS (left) and MI-FDS (right) of the tests
14 E. Proner and E. Mucchi Table 1 TTFs obtained in the testing campaign Time-to-Failure - [s] spec #1 spec #2 spec #3 Mean Standard dev. TE-0 1537 1620 1920 1692,3 201,4 TE-1 6036 5265 5522 5607,6 392,5 TE-2 1380 1292 1008 1226,6 194,4 TE-3 1648 1818 1472 1646,0 173,0 TE-0 TE-1 TE-2 TE-3 1000 1500 2500 5000 Time to failure - [s] Fig. 6 Box plot of the TTFs obtained in the test campaign has the lowest damage potential. This is expected, as the combination of multiple simultaneous excitations determines a greater response of the tested system, resulting in a higher damage output [15–17]. The highest damage potential, according to the MI-FDS, is given by TE-2. In Equation (6), cos(ϕjk −ξjk) and γjk are both equal to 1 for TE-2. In fact, the phases and coherences of the SDM of TE-2 are set according to the EDRM, which maximize the response of the system and, therefore, achieves the maximum damage potential. Moreover, the MI-FDS of TE-0 is lower than the one of TE-2 as the correlation between the excitations determines a lower response of the system. Finally, TE-3 has been obtained to match the MI-FDS of TE-0, with the phases and coherences of TE-2. The PSDs of TE-3 are lower than the ones of TE-0, however, the contribution of the CSDs determines an increased response of the system that allows to match the MI-FDS of TE-0. The previous consideration imply the following predictions regarding the outcome of the tests: 1. According to the traditional FDS: TE-0, TE-1 and TE-2 will be equally damaging to the specimen while TE-3 will be the least damaging environment. 2. According to the novel MI-FDS: TE-1 will be the least damaging test, TE-2 will be the most damaging test, TE-0 and TE-3 will deliver the same damage to the specimen. These predictions are now compared to the actual outcome of the tests, described by Table 1 and Figure 6. The results of the testing campaign demonstrate that the predictions made according to the MI-FDS are correct. In fact, in comparison to the mean TTF of TE-0, TE-1 determined a 231%increase in the TTF, TE-2 caused a 27%reduction of the mean TTF and, finally, the mean TTF of TE-3 deviates form the mean TTF of TE-0 by only 2.7%. These results are depicted in Figure 6. It
An Improved Fatigue Damage Spectrum for MIMO Random Testing 15 can be seen that TE-1 determined the highest TTFs (least damage), while TE-2 determined the lowest TTFs (most damage), in accordance to the prediction made by the MI-FDS. Moreover, TE-0 and TE-3 determined very similar TTFs distributions. This result indicates that, although the PSDs of the TE-3 are lower than the PSDs of TE-0, the damage output of TE-3 is the same of TE-0. This is to be attributed to the contribution of the CSDs, which allow TE-3 to have the same MI-FDS of TE-0 but with lower PSDs. The analysis of the obtained TTFs confirms that the predictions made according to the MI-FDS reflect the experimental results of the testing campaign. Therefore, the MI-FDS correctly takes into account the contribution of the CSDs of multiaxis random vibration environments to the fatigue damage inflicted to the specimen. Furthermore, the MI-FDS has been able to modify TE-2 to generate a new vibration environment that deals the same damage as TE-0 to the specimen in the same amount of time. Finally, the test campaign demonstrates the incapability of the traditional FDS approach to evaluate multi-axis vibration environments. In fact, the FDS could not distinguish TE-0, TE-1 and TE-2, which have the same PSDs but dealt different damage to the specimen. Conclusion In this paper a new approach to the Fatigue Damage Spectrum has been investigated. The Multi-Input FDS is proposed, to consider the correlation of multiple simultaneous random excitations in the computation of the damage potential of a multiaxis vibration environment. The validity of the proposed approach has been tested in a dedicated test campaign. In particular, ad-hoc designed specimens have been tested until failure under different vibration environments and the MI-FDS has been used to evaluate the damage potential of each vibration environment. The proposed approach allowed to make predictions on the outcome of the tests coherent with the experimental results. In fact, the MI-FDS was able to correctly identify the most and least damaging environments and recognised the environments that determine the same damage output. Moreover, the MI-FDS allowed to synthesise a new vibration environment equally damaging to the tested article as the reference vibration environment. The experimental results proved that the MI-FDS offers a significant improvement over the traditional FDS in evaluating the damage potential of multi-axis random vibrations. References 1. NSA. “Environmental conditions”. AECTP 200, category 240(2006) 2. de la Defense Delegation Generale pour l’armement, M. “Essais generaux en environment des materials (general tests of materials in environment)”. GAMEG-13. 3. of Defence, U.S.D. “Environmental engineering considerations and laboratory tests. method 527.1: Multi-exciter test”. MIL-STD 810G w/CHANGE1. 4. Lalanne, C. “Specification development, third ed.”. John Wiley & Sons, Inc - ISTE, London(2014) 5. Lalanne, C. “Maximum response and fatigue damage spectra – part i & part ii”. The journal of environmental sciences, July/Augst (1984) 6. Xu, F., Ahlin, K., and Wang, B. “Optimization of damage equivalent accelerated test spectrum derivation using multiple non-gaussian vibration data”. Journal of Sensors, 2021(1):3668726 (2021) 7. Coutinho, I.A. and Landre Jr, J. “Enhanced tailored durability test specification for multiaxial dynamically excited structures attached to a single input base under off-road loading environment”. Engineering Structures, 219:110768 (2020) 8. Decker, M., Kinscherf, S., Bauer, N., David, P., and Serifsoy, M. “Deriving fatigue equivalent power spectral density spectra for the vibration testing of engine components”. Materialwissenschaft und Werkstofftechnik, 49(3):392–405 (2018) 9. Cianetti, F., Alvino, A., Bolognini, A., Palmieri, M., and Braccesi, C. “The design of durability tests by fatigue damage spectrum approach”. Fatigue & Fracture of Engineering Materials & Structures, 41(4):787–796 (2018) 10. Angeli, A., Cornelis, B., and Troncossi, M. “Synthesis of sine-on-random vibration profiles for accelerated life tests based on fatigue damage spectrum equivalence”. Mechanical Systems and Signal Processing, 103:340–351 (2018) 11. Aime´, M., Banvillet, A., Khalij, L., Pagnacco, E., Chatelet, E., and Dufour, R. “A framework proposal for new multiaxial fatigue damage and extreme response spectra in random vibrations frequency analysis”. 213 (2024) 12. Proner, E., Mucchi, E., and Tovo, R. “A relationship between fatigue damage estimation under multi-axis and single-axis random vibration”. Mechanical Systems and Signal Processing, 215:111402 (2024) 13. Nath, N. and Aglietti, G.S. “Study the effect of tri-axis vibration testing over single-axis vibration testing on a satellite”. In 2022 IEEE Aerospace Conference (AERO), pages 1–10 (2022) 14. D’Elia, G., Mucchi, E., and Dalpiaz, G. “A novel methodology for dynamic response maximisation in multi-axis accelerated random fatigue testing”. 181 (2022) 15. Gregory, D., Bitsie, F., and Smallwood, D. “Comparison of the response of a simple structure to single axis and multiple axis random vibration inputs” (2008) 16. Whiteman, W. and Berman, M. “Inadequacies in uniaxial stress screen vibration testing”. Journal of the IEST, 44:20–23 (2001) 17. French, R., Handy, R., and Cooper, H. “A comparison of simultaneous and sequential single-axis durability testing” (2006)
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