Chapter 6 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 Making Modal Analysis Easy and More Reliable – Challenging Ai-Based Algorithms with the Barc Example Tim Kamper, Denis Beljan, Patrick Hu¨skens, and Haiko Bru¨cher Though modal analysis is a common tool to evaluate the dynamic properties of a structure, there are still many individual decisions to be made during the process which are often based on experience and make it difficult for occasional users to gain reliable and correct results. The paper presents how on different steps of the process experience-based decisions can be supported and replaced by automated evaluations to make modal analysis accessible to less experienced users. In addition to traditional methods such as the Least-Squares Complex Frequency-domain (LSCF) estimator the presented approach takes advantage of more innovative methods such as a neural network to reduce experience-based decisions and increase reliability. The advantage of the presented approach is shown based on the example of a Box Assembly with Removable Component (BARC). BARC is known as a challenge structure for approaches for the evaluation of dynamic properties. In the study two groups of users are exposed to perform a modal analysis of the BARC. The groups are selected as heterogeneous groups with different levels of experience in modal analysis. The first group performs the modal analysis based on classic algorithms and methods whereas the second group is supported by AI based technology. Keywords Modal Analysis · Optimal Driving Point · AI Introduction Modal analysis is a widely used method to analyze the dynamic properties of structures. It is used for a variety of tasks such as troubleshooting, validation of numerical models and the parametrization of numerical models. Due to its various use cases the user group of modal analysis is very heterogeneous with different background-knowledge and experience. Unfortunately, the process of modal analysis, that starts the planning of the measurements and ends with the postprocessing of the results requires a lot of decisions based on the experience of the user. Thus, it is difficult to judge on the reliability of the results of modal analysis and the whole process is hardly reproducible. Furthermore, wrong decisions at an early step of the process can often only be discovered in the very end of the process, if at all. This is particularly painful, as the process of modal analysis can be a quite time-consuming task. This paper focuses on the evaluation of different methods that are designed to contribute to an easier process of modal analysis that delivers more reliable results. Figure 1 gives an overview of the process steps of modal analysis with the corresponding decisions that are necessary to be made by the individual user. All these decisions have an impact on the completeness and the quality of the results, such as number of extracted mode shapes, accuracy of the fitted frequency response functions (FRF) and of the detected mode shapes. Before the user can start the experimental data acquisition, the test must be designed properly. During the test design phase, the used data acquisition method and hardware must be chosen (shaker excitation, impact testing with roving hammer/accelerometer, laser vibrometer, etc.). Choosing the right geometric resolution of measurement points enables capturing local shape characteristics. The selection of correct number and positions of good reference degrees of freedom (DOFs) is crucial, as it results in satisfactory signal-to-noise-ratio (SNR) and deflection magnitudes for all modes within the desired frequency range, enabling the detection of all corresponding eigenfrequencies during postprocessing. Furthermore, the number of reference DOFs may be kept down by a good selection. This leads to less equipment needed and less data to TimKamper · Denis Beljan· Patrick Hu¨skens · Haiko Bru¨cher HEAD acoustics GmbH, Ebertstr. 30 a, D-52134 Herzogenrath, Germany e-mail: Tim.Kamper@head-acoustics.com; Denis.Beljan@head-acoustics.com; patrick.hueskens@head-acoustics.com; haiko.bruecher@head-acoustics.com © The Author(s), under exclusive license to River Publishers 2025 45 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 6
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