River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Structural Health Monitoring & Machine Learning, Vol. 12 Brian Damiano Babak Moaveni Antonio De Luca Keith Worden 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
Brian Damiano· Babak Moaveni · Antonio De Luca · Keith Worden Editors Structural Health Monitoring & Machine Learning, Vol. 12 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 97887-438-0157-3 (Hardback) ISBN 97887-438-0169-6 (eBook) https://doi.org/10.13052/97887-438-0157-3 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 Structural Health Monitoring & Machine Learning 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; Model Validation and Uncertainty Quantification; Dynamic Substructuring & Transfer Path Analysis; 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 and Data Science in Engineering. Each collection presents early findings from analytical, experimental and computational investigations on an important area within Structural Dynamics. Structural Health Monitoring is one of these areas which cover topics of interest of several disciplines in engineering and science. Structural Health Monitoring & Machine Learning are specific subject areas within the SEM umbrella of technical divisions, namely the Dynamics of Civil Structures including other technical activities devoted to structural analysis, testing, monitoring, and assessment. This volume covers a variety of topics including Bayesian Inference Methods, and Structural Health Monitoring with Digital Twinning. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Editor: Brian Damiano, Oak Ridge National Laboratory, Oak Ridge, TN, USA; Babak Moaveni, Tufts University, Medford, MA, USA; Antonio De Luca, Thornton Tomasetti, Ft. Lauderdale, FL, USA; Keith Worden, University of Sheffield, Sheffield, UK. v
Contents 1 Theoretical Foundations and Practical Applications of Damage Detection Using Autocovariance Functions 1 Jyrki Kullaa 2 On the Real Time Tightness Measurement of Complex Shaped Flanges 11 Wolfgang Witteveen, Simon Desch, and Lukas Koller 3 Parameter Rejection in Sensitivity-based Model Updating using Output Feedback Eigenstructure Assignment 17 Martin D. Ulriksen and Dionisio Bernal 4 Structural Health Monitoring of a Ferry Quay: Instrumentation and Impact of Tidal Levels on Modal Parameters 23 Luigi Sibille, Torodd Skjerve Nord, Ba Tung Le, Bartosz Siedziako, and Alice Cicirello 5 Outcomes from Field Measurements on the Magerholm Ferry Quay: System Identification, Finite Element Model Updating and Sensitivity Analysis 33 Ba Tung Le, Bartosz Siedziako, Torodd Skjerve Nord, and Luigi Sibille 6 A Robust Data-Driven Algorithm for Early Damage Detection in Structural Health Monitoring 43 Luigi Severa, Silvia Milana, Nicola Roveri, Eleonora Maria Tronci, Antonio Culla, and Antonio Carcaterra 7 Real-Time Structural Health Assessment of a Tension Rod Assembly Using Machine Learning 51 Ahmad Rababah, Osama Abdeljaber, Aniston Cumbie, Lauren Harpenau, and Onur Avci 8 Multi-Bridge Indirect Structural Health Monitoring: Leveraging Big Data and Drive-By Crowdsensing Techniques 59 Jiangyu Zeng, Qipei Mei, and Mustafa Gu¨l 9 A Comparative Study of Feature Selection Methods for Wind Turbine Gearbox Bearing Fault Prognosis 67 Feras Abla, Mohammad Hesam Soleimani-Babakamali, Sahabeddin Rifai, Ahmad Rababah, Shawn Sheng, Ertugrul Taciroglu, Serkan Kiranyaz, and Onur Avci 10 Damage Identification on Gear Drivetrains Using Neural Networks Trained by High-Fidelity Multibody Simulation Data 73 J. Koutsoupakis, D. Giagopoulos, G. Karyofyllas, P. Seventekidis, and S. Natsiavas 11 Advanced Condition Monitoring framework for CFRP Gear Drivetrains Using Machine Learning and Multibody Dynamics Simulations 85 G. Karyofyllas, J. Koutsoupakis, and P. Giagopoulos 12 On the use of Statistical Learning Theory for model selection in Structural Health Monitoring 95 C. A. Lindley, N. Dervilis, and K. Worden 13 Full-field Measurements for Anomaly Detection of Mechanical Systems using Convolutional Neural Networks and LSTM Networks 105 Carlos Quiterio Go´mez Mun˜oz, Mariano Alberto Garc´ıa Vellisca, Celso T. do Cabo, Yujie Xi, and Zhu Mao 14 A Generative Modeling Approach for the Translation of Operational Variables to Short-term Vibrations 113 vii
viii Contents Arthur Hatstatt and Konstantinos E. Tatsis 15 Effective Structural Health Monitoring of Rotating Propellers using Asynchronous Neuromorphic Tracking 123 Guillermo Toledo, Wyatt Saeger, Fernando Moreu, David Mascarenas, Christian Torres, and Jahsyel Rojas 16 Estimating Damage Detection of an Aircraft Component with Machine Learning Models 131 Brandon Jones, Milton Jones, Stephen Keyes, Eamon Mott, and Thomas J. Matarazzo 17 Physics-Informed Machine Learning for Advanced Structural Damage Detection and Localization 137 Zixin Wang, Mohammad R. Jahanshahi, and Shirley J. Dyke 18 Damage Detection Strategy Based on PCA/Mode-Shapes Developed on a Laboratory Truss Girder Subjected to Environmental Variations 143 M. Berardengo, F. Luca`, S. Manzoni, S. Pavoni, and M. Vanali
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 Theoretical Foundations and Practical Applications of Damage Detection Using Autocovariance Functions Jyrki Kullaa Autocovariance functions (ACFs) can be utilized as damage-sensitive features in structural health monitoring. They have several advantages: they include the necessary information about the dynamic characteristics of the structure for damage detection, they are easy to estimate, they contain many data points, their accuracy can be controlled, and they are spatiotemporally correlated. Damage detection is performed in the time domain. ACFs have the same form as a free decay of the system for a stationary random process. A major issue of ACFs is that the free decay at a natural frequency includes cross-terms also from other modes. Therefore, the data lie in a high-dimensional space. For spatial correlation, the required number of sensors therefore increases considerably with an increasing number of active modes. Fortunately, the number of sensors can be reduced if temporal correlation is also utilized. Using spatiotemporal correlation, it is possible to detect changes in both natural frequencies and mode shapes. The objective is to pursue data redundancy, so that damage can be detected in the noise space. A simple and useful formula is presented, relating the number of active modes, the number of sensors, and the model order for damage detection. The conditions are also given for the design parameters limiting detection to frequency changes only or mode shape changes only. It is also shown that the variability of the excitation characteristics between measurements has no effect on damage detection. Numerical experiments were performed to validate the theoretical considerations. Keywords Autocovariance function· Spatiotemporal correlation· Damage detection· Data redundancy · Model order Introduction Structural health monitoring (SHM) aims at receiving an early warning about structural damage or failure. Damage detection is based on vibration data measured from the structure during normal operation. Therefore, the excitation cannot be controlled or measured, and it may also vary between measurements. If the changes of the dynamic characteristics of the structure are statistically significant, it is an indication of damage. Training data are first acquired under different environmental or operational conditions to distinguish changes due to damage from changes due to environmental or operational variability. Damage detection in the time domain is an attractive alternative to parameter-based methods, because time domain data include more data points and system identification is not needed. Covariance functions have been used in damage detection [3–12]. In this study, autocovariance functions (ACFs) are used as damage-sensitive features. They have been shown to work with numerical and experimental data [11]. In this paper, the applicability of ACFs for damage detection is theoretically investigated. More specifically, a relationship between the number of active modes, the number of sensors, and the model order in the data analysis is explored. Additionally, the effect of the variability of the excitation characteristics is studied (operational variability). The objective of the proposed SHM system design is to make the measurement data redundant. Then, some variables can be computed using the remaining variables. For a redundant system, the data space can be divided into two subspaces, the signal space and the noise space. The signal space contains useful information of the vibration and the environmental or operational effects. The noise space consists mostly of measurement error. This was also discussed in [12]. Once damage appears, the new data are assumed to appear also in the noise space of the training data, which is easier to detect, because the variance is assumably much smaller in the noise space than in the signal space. It should be noted, however, that the Jyrki Kullaa Department of Automotive and Mechanical Engineering, Metropolia University of Applied Sciences, Leiritie 1, 01600 Vantaa, Finland e-mail: jyrki.kullaa@metropolia.fi © The Author(s), under exclusive license to River Publishers 2025 Brian Damiano et al. (eds.), Structural Health Monitoring & Machine Learning, Vol. 12, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0157-3 1
2 J. Kullaa existence of the noise space does not guarantee that the data from the damaged structure enters the noise space. Therefore, the degree of redundancy should be increased if possible so that the dimension of the noise space is greater than one. The paper is organized as follows. The autocovariance functions are introduced in the next section, in which important formulas are given for the SHM system design. The proposed damage detection and localization algorithm in the time domain follows. A numerical experiment is performed to validate the theoretical consideration. Finally, concluding remarks are given. Autocovariance Functions for Damage Detection If the excitation is stationary random, the autocovariance functions of the structural responses (e.g. acceleration, velocity, displacement, strain) have the same form as a free decay of the system, which is also the foundation of operational modal analysis (OMA) [13]. The autocovariance function of a zero-mean random process {y(t)} is Ryy(τ)=E[y(t)y(t+τ)] (1) where E[·] is the expectation operator andτ is the time lag. Using modal superposition of linear structures, the displacement response y(t) at time t can be written as y(t)= nX r=1 ϕrqr(t) (2) where n is the number of active modes, ϕr is the mode shape vector of mode r, and qr(t) is the modal coordinate of mode r. If we denote ϕir as the ith component of vector ϕr, the autocovariance function of the displacement degree-of-freedom (DOF) {yi(t)} is Rii(τ)=E" X r ϕirqr(t)X s ϕisqs(t+τ)#=X r Xs ϕirϕisE[qr(t)qs(t+τ)] (3) For a stationary random process with white noise excitation, the ACF can be derived [13, 14, 5]: Rii (τ)= nX s=1 [Aiscos(ωdsτ)+Bissin(ωdsτ) ] e−ζsωs (4) where Ais = nX r=1 Frsϕirϕisϕ T r BQBTϕs (5) and Bis = nX r=1 Grsϕirϕisϕ T r BQBTϕs (6) where Frs = 1 2mrωdrmsωds " −ζrωr−ζsωs (ζrωr+ζsωs) 2+(ωdr+ωds) 2+ ζrωr+ζsωs (ζrωr+ζsωs) 2+(ωdr−ωds) 2# (7) and Grs = 1 2mrωdrmsωds " ωdr+ωds (ζrωr+ζsωs) 2+(ωdr+ωds) 2+ ωdr−ωds (ζrωr+ζsωs) 2+(ωdr−ωds) 2# (8) where mr, ωr, ωdr, ζr are, respectively, the modal mass, the undamped natural circular frequency, the damped natural circular frequency, and damping ratio of the mode r. Matrix Bis the load distribution matrix, and Qis a diagonal matrix where the diagonal entries are the spectral densities of the independent loads. Bis assumed to be constant during a single measurement. An example of an ACF with 300 lags is shown in Fig. 1, when the number of modes is 7. From Equation (4), it can be seen that the ACF has the same form as a free decay of the system. However, the decay at natural frequency ωds includes also cross-terms from other modes, as seen in Equations (5) and (6). This can make damage detection challenging, because the modal amplitudes ratios between sensors (Ais/Ajs and Bis/Bjs) may be different in each measurement.
Theoretical Foundations and Practical Applications of Damage Detection Using Autocovariance Functions 3 Fig. 1 Autocovariance function with 7 active modes In the design of an SHM system, three parameters are important for damage detection: the number of active modes n, the number of sensors p, and the model order mused in the data analysis. The number of active modes cannot be controlled, but it affects the selection of the other two parameters. If the number of active modes is too large, filtering could be applied to remove the contribution of some modes. Oncenis known, the number of sensors can be determined. This decision influences the hardware and maintenance costs and should be kept to a minimum. Optimal sensor placement is beyond the scope of this study but should also be considered. If changes in both the mode shapes and frequencies must be detected, this number must be at least equal ton. Once the first two parameters are fixed, the model order can be determined. It only affects the duration of the data analysis and the memory usage without additional hardware costs. The selection of design parameters is based on redundancy of the ACF data. The redundancy can be spatial, temporal, or spatiotemporal. The results are only presented here without lengthy derivations. Spatial correlation (m=0) detects only changes in the mode shapes. The required number of sensors p is p> 1 2 n(n+1) (9) For many real structures, this condition may yield too large a sensor network and high costs. Temporal correlation (1 ≤p ≤n) detects only changes in the natural frequencies or modal damping. The model order used in the analysis must be m>2n (10) Spatiotemporal correlation (p≥n) can detect both changes in the frequencies, damping, and mode shapes. The model order mmust be m> 2n2 p − 1 (11) These formulas, especially Eq. (11) can be applied to the design of an SHM system. Note that environmental influences were not considered in the formulas. Their contribution can be taken into account by increasing the number of sensors pand the model order m. Damage Detection Algorithm The damage detection algorithm is briefly introduced. More details can be found in [15]. For each data set and for each sensor separately, the ACFs are estimated. The user has to determine the number of lags in the ACFs. The training data are comprised of the ACFs from the undamaged structure under different environmental or operational conditions. The data are copied and shifted mtimes for spatiotemporal correlation. The model order mis determined by the user with the help of
4 J. Kullaa Eqs. (10) or (11). A covariance matrix of the time-shifted training data is estimated. The covariance matrix is subjected to a whitening transformation for data normalization. When new data (ACFs) arrive, the same whitening transformation is applied to these data. If the dynamic characteristics of the structure have changed, the new data are assumed to appear in the noise space and thus outside the hypersphere (for Gaussian data). This can be detected by applying principal component analysis to all data. The first principal component is now dictated by the data points outside the hypersphere. The first principal component scores are only retained, reducing the data dimensionality to one. Extreme value statistics (EVS) distributions [16] are identified for the scores of the training data by dividing the data into equal sized subgroups. The extreme values of each subgroup are plotted on a control chart with control limits determined from the probability of a false alarm [17]. If the data points of the test data are outside the control limits, an alarm is raised. Numerical Experiment The structure being monitored was a bridge deck with a concrete slab and steel stiffeners (Fig. 2). A detailed description of the model can be found in [18]. The first 7 modes were active, having natural frequencies of 3.95 Hz, 5.35 Hz, 13.7 Hz, 15.4 Hz, 17.9 Hz, 24.1 Hz, and 24.8 Hz. The corresponding damping ratios were 0.01, 0.01, 0.02, 0.02, 0.03, 0.03, and 0.03. The sampling interval was ∆t =0.01 s and the measurement period was almost 44 min including 218 =262144 samples. A long measurement period is necessary to provide accurate ACF estimates [19]. Fig. 2 Finite element model of the bridge deck with labeled sensor positions. Damage location is close to sensor 11 Because environmental influences were not considered in this study, the excitation was the primary cause of varying the ACFs (operational variability). Some variability came from the measurement noise, which was also different in each measurement. The number of independent excitations varied randomly between 1 and 3. Furthermore, the excitation points were randomly selected from all nodes at the joints between the steel webs and flanges. The load histories were generated in the frequency domain between 0 and 25 Hz. Each measurement had different load functions with random amplitudes and phases at each frequency pin. In addition, it was randomly determined if a loading had zero contribution at a randomly located 5 Hz interval. This simulated measurements in which some modes were not excited. The FE analysis was performed in the frequency domain [20]. Measurement noise was simulated by adding Gaussian noise to each sensor. The average signal-to-noise ratio (SNR) was 30 dB. The ACFs were estimated from the long response time histories at lags τ between 0 and 3 s, resulting in a length of 300 data points in each measurement (Fig. 4a). In the data analysis for damage detection, the ACFs were concatenated to form a long data matrix (Fig. 4b). To remove discontinuities between measurements, mdata points had to be removed from each data set. The first 100 data sets were from the undamaged structure and the next 36 data sets from the damaged structure with an increasing crack length (Fig. 3). The number of different crack lengths was 6, and each damage level was monitored with 6 simulations. The first 70 data sets were used as training data.
Theoretical Foundations and Practical Applications of Damage Detection Using Autocovariance Functions 5 Fig. 3 A detail of the steel girder with a sequence of crack propagation [18] Fig. 4 a) Autocovariance functions of 28 accelerometers from measurement 1. b) Autocovariance functions of 9 accelerometers [2:3:28] from measurements 1 to 10 The whitening matrix was identified using the training data, but principal component analysis was applied to all data. The first principal component scores were only retained and divided into subgroups each consisting of all data from a single measurement. The minimum and maximum of each subgroup were used for damage detection by identifying extreme value distributions for the training data and determining the probability of a false alarm equal to 0.001 to compute the control limits. Finally, a control chart for the extreme values was plotted. Damage localization was based on the projection of the first principal component on the sensor coordinates and selecting the largest length, which was supposed to reveal the sensor closest to damage. SHM systems for damage detection for different sensor arrays were designed. The number of active modes was assumed known (n =7). In the first case, spatial correlation (m=0) was only considered. According to Eq. (9), the number of sensors had to be greater than 28. If all 28 accelerometers were used, the control chart in Fig. 5a resulted. Damage detection was successful even if the number of sensors was not sufficient. This is probably because distant modes had only a small contribution to the ACFs. Damage was localized to sensor 12 (Fig. 5b). Sensor 11 also had a high damage index. Because the crack was located between sensors 11 and 12, damage localization was considered successful. Note that with spatial correlation, changes in the mode shapes could only be detected.
6 J. Kullaa Fig. 5 a) EVS control chart for 28 accelerometers and spatial correlation (m=0). Data on the left side of the leftmost vertical line were used as the training data, and the other vertical lines indicate the six damage levels with increasing crack lengths. Logarithmic scaling was applied to the extreme values. b) Damage localization The second system had 14 accelerometers (sensors 2:2:28). If spatial correlation was used, damage detection failed (Fig. 6). Using Eq. (11), the condition for the model order was m>6. For the model order equal tom=10, the control chart in Fig. 7a was plotted. Damage was successfully detected with occasional false positives. Damage was incorrectly localized to sensor 16 (Fig. 7b). Fig. 6 EVS control chart for 14 accelerometers [2:2:28] and spatial correlation (m=0) For the third system, hardware costs were further reduced by installing only 9 accelerometers (sensors 2:3:28). Now, the model order in the data analysis had to be increased (m>9.9). If the model order equal to m=16 was used, the result was the control chart in Fig. 8a showing successful detection, but incorrect damage localization to sensor 8 (Fig. 8b). On the other hand, if too small a model order was used (m=4), damage could not be detected (Fig. 9a). The fourth system had only three accelerometers (sensors 2, 11, and 20), implying that temporal correlation was only possible. According to Eq. 10, the model order had to be greater than 14. Using m=20 resulted in the control chart in
Theoretical Foundations and Practical Applications of Damage Detection Using Autocovariance Functions 7 Fig. 7 a) EVS control chart for 14 accelerometers [2:2:28] and spatiotemporal correlation (m=10). b) Damage localization Fig. 8 a) EVS control chart for 9 accelerometers [2:3:28] and spatiotemporal correlation (m=16). b) Damage localization Fig. 9 a) EVS control chart for 9 accelerometers [2:3:28] and spatiotemporal correlation (m=4). b) EVS control chart for 3 accelerometers [2, 11, 20] and temporal correlation (m=20)
8 J. Kullaa Fig. 9b with reasonable damage detection capability. Detection of natural frequency changes was only possible. Damage was correctly localized to sensor 11 having the largest damage index (not shown). For systems 2–4, it was observed that the theoretical minimum for the model order mwas not optimal, but a slightly larger number yielded better results. The reason was probably an increased dimensionality of the noise space, increasing the probability of the test data to appear in that space. In addition, systems 2–3 with spatiotemporal correlation were able to detect changes in both frequencies, damping, and mode shapes. Damage localization was correct only when using spatial or temporal correlation. With spatiotemporal correlation, localization failed. On the other hand, the sensor closest to damage also had a high damage index in all cases. Conclusion Autocovariance functions are promising damage-sensitive features in the time domain. Their usage was theoretically and numerically explored. Practical formulas for the design parameters were given for the design of an SHM system. If the number of active modes is known, the minimum number of sensors can be computed. Once the number of sensors is determined, the required model order used in the data analysis can be computed. The main conclusions are: 1. To detect both frequency and mode shape changes, the number of sensors must be at least equal to the number of active modes. The required model order can then be computed. This spatiotemporal correlation model is suggested for most cases. 2. If the number of sensors is less than the number of modes, only changes in frequencies and damping can be detected but not changes in mode shapes. 3. For spatial correlation, changes of mode shapes can only be detected. A large number of sensors is necessary, which is impractical in many cases. 4. In spatiotemporal correlation analyses, a slightly greater model order than the required minimum was found to yield optimal results. 5. The excitation variability has no effect on damage detection, provided the process is stationary random and the design parameters fulfil the given conditions. 6. The proposed method yielded incorrect damage localization in some cases, but a large damage index resulted also for the sensor closest to damage. Acknowledgments This research has been supported by Metropolia University of Applied Sciences. References 1. Li, X.Y., Law, S.S., Matrix of the covariance of covariance of acceleration responses for damage detection from ambient vibration measurements. Mechanical Systems and Signal Processing, 24(4), 945–956, 2010. 2. Zhang, M., and Schmidt, R., Sensitivity analysis of an auto-correlation-function-based damage index and its application in structural damage detection. Journal of Sound and Vibration, 333(26), 7352–7363, 2014. 3. Zhang, M., and Schmidt, R., Study on an auto-correlation-function-based damage index: Sensitivity analysis and structural damage detection. Journal of Sound and Vibration, 359, 195–214, 2015. 4. Li, W., Huang, Y., A method for damage detection of a jacket platform under random wave excitations using cross correlation analysis and PCA-based method. Ocean Engineering, 214, 107734, 2020. 5. Yang, Z., Wang, L., Wang, H., Ding, Y., Dang, X., Damage detection in composite structures using vibration response under stochastic excitation. Journal of Sound and Vibration, 325(4–5), 755–768, 2009. 6. Wang, L., Yang, Z., Waters, T.P., Structural damage detection using cross correlation functions of vibration response. Journal of Sound and Vibration, 329(24), 5070–5086, 2010. 7. Trendafilova, I., A method for vibration-based structural interrogation and health monitoring based on signal cross-correlation. Journal of Physics Conference Series, 305(1), 012005, 2011. 8. Huo, L.-S., Li, X., Yang, Y.-B., Li, H.-N., Damage detection of structures for ambient loading based on cross correlation function amplitude and SVM. Shock and Vibration, 2016, 3989743, 2016. 9. Do¨hler, M., Mevel, L., Hille, F., Subspace-based damage detection under changes in the ambient excitation statistics. Mechanical Systems and Signal Processing, 45(1), 207–224, 2014. 10. Li, M., Huang, T., Structural damage detection based on quadratic correlation function of strain responses. Latin American Journal of Solids and Structures, 21(1), 2024.
Theoretical Foundations and Practical Applications of Damage Detection Using Autocovariance Functions 9 11. Kullaa, J., Jonscher, C., Liesecke, L., Rolfes, R., Damage detection with closely spaced modes using autocovariance functions. In: Proceedings of the 10th International Operational Modal Analysis Conference (IOMAC 2024), Vol. 2, Springer, 87–94, 2024. 12. Yan, A.-M., Golinval, J.-C., Null subspace-based damage detection of structures using vibration measurements. Mechanical Systems and Signal Processing, 20(3), 611–626, 2006. 13. Brincker R., Ventura, C., Introduction to operational modal analysis. Chichester, West Sussex: Wiley, 2015. 14. James III, G.H., Carne, T.G., Lauffer, J.P., The natural excitation technique (NExT) for modal parameter extraction from operating structures. Modal Analysis: The International Journal of Analytical and Experimental Modal Analysis, 10(4), 260–277, 1995. 15. Kullaa, J., Damage detection and localization under variable environmental conditions using compressed and reconstructed Bayesian virtual sensor data. Sensors, 22(1), 306, 2022. 16. Coles, S., An introduction to statistical modeling of extreme values. Bristol, UK: Springer, 2001. 17. Montgomery, D.C., Introduction to statistical quality control. 3rd edition. New York: Wiley, 1997. 18. Kullaa, J., Robust damage detection using Bayesian virtual sensors. Mechanical Systems and Signal Processing, 135, 106384, 2020. 19. Bendat, J.S., Piersol, A.G., Random data: Analysis and measurement procedures. 4th edition. Blackwell, Hoboken, N.J., USA: Wiley, 2010. 20. Clough, R.W., Penzien, J., Dynamics of structures. 2nd edition. New York: McGraw-Hill, 1993.
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 On the Real Time Tightness Measurement of Complex Shaped Flanges Wolfgang Witteveen, Simon Desch, and Lukas Koller Abstract The tightness of a flange is a time-varying variable and depends on the current state of the structure. For obvious reasons, this could be a critical and important information. It would be desirable to get such information in real-time. In this extended abstract, an example of a flange with 12 bores is used to give a numerical proof that this may be possible in principle. With the help of special trial vectors for a small sliding contact area, so-called contact modes, the tightness state of a joint can be reconstructed based on strain measurements. This is possible within a fraction of a second. This offers the possibility for corrective actions when a critical condition regarding tightness is detected. Keywords Flange tightness · State monitoring· Virtual sensor Introduction Loads and vibrations cause deformations in the contact area of flanges and these deformations obviously affect the tightness. In the worst case, the tightness is no longer guaranteed and the system leaks. The possibility of monitoring leak tightness of flanges in real-time can therefore be a significant improvement to such systems. Through numerical simulation and experimental studies, it is well documented in the literature that the gap and pressure distribution in a contact area are complex functions of the state, see for example [1] and [2]. The existing options for assessing tightness are only moderately suitable for dynamically loaded structures. Static pressure films can be clamped between the contacting surfaces. After the pressure has been applied, the bolts must be loosened again, the film is removed, and the color change is evaluated because it is a measure of the pressure. These films are suitable for assessing the tightness according to the pre-tensioned bolt, but not for permanent monitoring. In [2], a foil is used that must be clamped between the contact surfaces. The sensor is based on a material whose electrical properties are pressure dependent. The foil is divided into cells and the contact pressure can be determined for each individual cell. This technology seems suitable in principle for contact and leak monitoring, but there are some disadvantages: (1) Since the foil must be clamped, the behavior of the joint changes. (2) It is questionable how pressure and heat resistant the foil is - important questions for practical use. (3) Another question is whether such films can be used for very large flanges, such as those required for pipes with a diameter of 1m and more. In this project it was numerically investigated whether it might be possible to compute the gap distribution with a few strains outside the flange contact area. The magnitude of the strains must be within a measurable range and the result must not be too sensitive to noise. This extended abstract is organized as follows: The first section gives an idea about the theory which is applied in the following section to a flange contact with 12 bores. The results are investigated with respect to the strain magnitudes and its sensitivity with respect to noise. The full paper can be found in [3]. Wolfgang Witteveen University of Applied Sciences Upper Austria, Degree Program Mechanical Engineering, 4600 Wels, Austria e-mail: wolfgang.witteveen@fh-wels.at Simon Desch· Lukas Koller FH OOE Research and Development GmbH, 4600 Wels, Austria e-mail: simon.desch@fh-wels.at; lukas.koller@fh-wels.at © The Author(s), under exclusive license to River Publishers 2025 11 Brian Damiano et al. (eds.), Structural Health Monitoring & Machine Learning, Vol. 12, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0157-3 2
12 W. Witteveen et al. Brief Outline of the Theory Only the idea of the theory is outlined below. The detailed derivation can be found in [3]. It is assumed that a usable FE model of the structure including the flange area is available. With the use of proper trial vectors (commonly called “modes”), the normal displacement (=gap) between the two contact surfaces can be computed in the form of g =Φgq (1) where the (C×1) vector g holds the normal distance of all node-to-node contact pairs which form the contact area. The (C×Q) matrix Φg holds in its columns the portion of the modes which belong to the gap area and the (Q×1) vector q holds the scaling factors (commonly called “modal coordinates”) of the modes. It is obvious that the mode base must be able to capture global structural deformations like vibrations and very local ones like the deformations in the contact area. This can be ensured by extending proven mode bases with so-called contact modes. This allows both types of deformation to be represented with sufficient accuracy. For more information on contact modes, the interested reader is referred to [4]. Regarding linear strains, the relation ε =Φεq (2) also apply where the (E×1) vector ε holds all potential strain measurements at E positions on the structure. The (E×Q) matrix Φε maps the modal coordinates to the strains. By the use of the pseudo inverse equation (2) can be rearranged and plugged into (1) so that g =Gε (3) with the (C×E)matrix G=ΦgΦε† (4) is obtained. Relation (3) is not useful because it holds all potential strain gauges which is far too much for practical use. Training data is used to reduce the number of strain gauges. We assume that, for example through simulation, T training data ε1 to εT are available. They are arranged column by column in the (E×T) matrix ET. In a next step Proper Orthogonal Decomposition (POD, see exemplarily [5]) is used to extract a S dimensional subspace which optimally approximates all T training data in an Euclidian sense. The S vectors which span the subspace are arranged column wise in a (E×S) matrix ES. Now the Discrete Empirical Interpolation Method (DEIM, see exemplarily [6]) can be used for a somehow optimal extraction of S sensor positions out of E potential ones. The application of DEIM to ES gives a (S×E) matrix BT and a (E×S)matrixEDwith the properties εD =BTε (5) and ε =EDεD. (6) Note, that in relation (5) the (S×E) vector εDholds a Sdimensional subset of the entries inε. Consequently, BT selects the actual used sensors out of all potential ones. The matrix ED in relation (6) is used to computed all potential strains on the base of the selected subset stored inεD. When (6) is plugged into (3) the final relation g =GDεD (7) with the time invariant (C×S)matrix GD =GED (8) is obtained. Relation (7) can be used to compute the gap distributiong based on the strain measurements inεD. Numerical Example The proposed method is tested on a purely numerical example. The aim is to give a kind of numerical prove that the deformation state of a joint, and thus the tightness, can be determined using just a few strain gauge measurements. As shown in Fig. 1, two pipes are connected with a flange. The dimensions and all other relevant parameters can be found in [3]. The two flange rings are connected via 12 bores (M20 screws). The entire structure is rigidly mounted on one side and 3 forces are applied on the other side (Fx, Fy and Fz). The FE model consists of 131040 linear hexahedron elements and 179162 nodes. The contact surfaces of the flange are congruently meshed with 7944 (=C) node pairs. The contact pressures and the friction forces are computed by nonlinear penalty laws which can be found [3].
On the Real Time Tightness Measurement of Complex Shaped Flanges 13 Fig. 1 Pipe structure with flange Fig. 2 shows the positions of all potential strain measurements. The labeling of each position with two directions is intended to indicate that the strains in these two directions can be measured. The bolts are also fitted with a strain gauge so that the bolt strain can be measured in the longitudinal direction. In sum, this are 156 (=E) potential sensor positions. The potential of the presented method is numerically tested based on non-linear computations which serve as reference solutions. The results of these non-linear computations are the normal distances in the contact area (=gap) as well as the strains at the sensor positions. Based on these strains, the gap in the joint is then reconstructed using the proposed method and then compared with the actual gap from the reference computation. The details of the nonlinear computation can be found in [3]. For the sake of testing 120 load cases are constructed using random numbers for the forces Fx, Fy and Fz. A constant bolt load is acting in all cases. For the construction of ET, 600 training load cases have been created via the same procedure. For more details on the test and training sets, see [3]. Fig. 2 Potential strain measurement locations
14 W. Witteveen et al. Results The application of the former procedure leads to the selection of 10 strain measurements which are depicted in Fig. 3. Two of them are strain gauges in the bolts and 8 more on both flange rings measuring the strains in tangential direction. The result of all 120 test cases is of satisfying accuracy. The test case with the worst approximation is given in Fig. 4. The red color indicates areas of critical gapping. The reconstructed gap distribution is quite similar as the reference. In summary it can be stated that the gap distribution can be reconstructed based on a few (=10) strain measurements. Fig. 3 Selected strain measurements Fig. 4 Strain based gap (right) vs. reference computation (left). The red color indicates a critical opening in the contact area
On the Real Time Tightness Measurement of Complex Shaped Flanges 15 Comments on the Measurability of the Strains An important criterion for feasibility is the magnitude of the strains that occur. Very small strains do not lead to any problems in the simulation but are not measurable with real sensors. A look on the strain data reveals that in case of critical loads (like Fig. 4), the strains are in a good measurable range. This is not necessarily the case for low loads, but the flange is then not in a critical state. Comments on the Sensitivity to Noisy Data To get a first impression of the sensitivity to noise, the strains were artificially polluted by random numbers. It turns out that test sets with critical loads react very robustly to noise. Comments on the Real-Time Capability Real-time capability requires that in an actual realization, the transformation of the strain measurements into the gap distribution by the matrix-vector multiplication (7) is faster than the sampling time of the data acquisition system. Corresponding investigations have shown that this takes less than one millisecond with 10 sensors and almost 8000 entries in the gap vector g. Conclusion The numerical investigation presented suggests that the gap distribution inside a flange contact can be reconstructed via a model based virtual sensor with few strain measurements. This could be used to assess the tightness, for example. It turns out that the result has a certain robustness against noise and the strains are in a range that is good to measure, at least for critical load situations. The arithmetic operation that converts the strains into the gap distribution takes less than a millisecond. The reconstruction of the gap distribution presented here is actually a kind of worst case in terms of effort. Simpler questions such as ‘Is the gap uncritical or not?’ would probably be more relevant. It is conceivable, for example, to train a neural network for this question without reconstructing the entire gap distribution. References 1. Witteveen, W., Kuts, M., Koller, L., 2023, “Can transient simulation efficiently reproduce well known nonlinear effects of jointed structures?”, Mechanical Systems and Signal Processing, 190, doi:10.1016/j.ymssp.2023.110111. 2. Dreher, T., Brake, M., R., W., Seeger, B., Krack, M., 2021, “In situ, real-time measurements of contact pressure internal to jointed interfaces during dynamic excitation of an assembled structure”, Mechanical Systems and Signal Processing, 160, doi:10.1016/j.ymssp.2021.107859. 3. Witteveen, W., Desch, S., Koller, L., “Can real-time flange tightness monitoring be possible with strains? A numerical pre-study.”, Journal of Structural Dynamics [En ligne], Special issue on Tribomechadynamics, under review 4. Pichler, F., Witteveen, W., Fischer, P., 2017, “A complete strategy for efficient and accurate multibody dynamics of flexible structures with large lap joints considering contact and friction”, Multibody System Dynamics, 40(4):407–436, doi:10.1007/s11044-016-9555-2.370 5. Chatterjee, A., “An introduction to the proper orthogonal decomposition”, SIAM Journal on Scientific Computing, 78(7):808–817, URL http://www.jstor.org/stable/24103957. 6. Chaturantabut, S., Sorensen. D., C., 2010, “Nonlinear model reduction via discrete empirical interpolation”, SIAM Journal on Scientific Computing, 32(5):2737–2764, doi:10.1137/090766498.
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