Chapter 13 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 Full-field Measurements for Anomaly Detection of Mechanical Systems using Convolutional Neural Networks and LSTM Networks Carlos Quiterio Go´mez Mun˜oz, Mariano Alberto Garc´ıa Vellisca, Celso T. do Cabo, Yujie Xi, and Zhu Mao Abstract Structural health monitoring (SHM) of mechanical systems is of great importance for a variety of industries, and an early detection and localization of damage in such situations are often desired. One of the features that can be used for SHM is its dynamic characteristics, which are traditionally obtained by using contact and sparse sensing. In nowadays, technologies that can collect full-field information and perform noncontact sensing have become more advantageous. Among such equipment, laser Doppler vibrometers (LDV) can provide high-quality vibrational data with full-field information. However, damage detection based on vibrational data can also be difficult in complex systems. With the recent advancement of computational power, data-driven approaches have become increasingly popular for complex systems. One of the most known approaches is the convolutional neural network (CNN), which has great capabilities for image processing and classification. Since the LDV measurements can provide full-field information, these data can be converted into 2D matrices and employed in CNN as images. However, CNNs commonly do not keep any temporal dependency between the frames, making it difficult to be employed in vibration analysis where data are all time series. This work proposes combining CNNs with long short-term memory (LSTM) networks, to process video data instead of frames, allowing time-dependent image processing for damage detection. In addition, the results are expected to predict not only if there is damage to the system, but also to classify different types of damage. Keywords Machine Learning· Convolutional Neural Network · Long Short-Term Memory· Laser Doppler Vibrometer Introduction The classification of normal and abnormal vibrations in structure has gained substantial attention due to its implications in Structural Health Monitoring (SHM) and fault detection. Vibrational analysis is critical for identifying potential issues in industrial machinery, infrastructures, and mechanical systems [1–3]. By detecting anomalies early, it is possible to prevent costly repairs and ensure the safety of operations. With the increase interest in machine learning techniques, many studies have been performed in the field of structural vibration for anomaly detection [4, 5]. Specifically, Convolutional Neural Networks (CNNs) have great capabilities for image processing and classification, been employed previously to dynamic systems [6–8]. However, CNN architectures only capture the dependance between pixels in an image. For time-series datasets representing the time dependency of the signal is challenging, leading to the usage of Long Short-Term Memory (LSTM) networks. Such architectures can be combined with CNN to obtain time dependency between frames for video-based sensing [9, 10], or applied into Remaining Useful Life (RUL) applications [11]. This study focuses on analyzing vibrations of a metallic plate subjected to different frequency excitations in both linear and non-linear states. The linear state is characterized by the absence of external disruptions, where the plate vibrates freely. In contrast, the non-linear state introduces rubber bumpers placed at various positions on the plate, altering its vibrational Carlos Quiterio Go´mez Mun˜oz · Mariano Alberto Garc´ıa Vellisca HCTLab Research Group, Universidad Auto´noma de Madrid, Madrid, Spain e-mail: carlosq.gomez@uam.es; marianoa.garcia@uam.es Celso T. do Cabo· YujieXi · ZhuMao Department of Mechanical and Materials Engineering, Worcester Polytechnic Institute, Worcester, MA, United States e-mail: ctdocabo@wpi.edu; yxi@wpi.edu; zmao2@wpi.edu © The Author(s), under exclusive license to River Publishers 2025 105 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 13
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