Chapter 9 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 A Methodology for Feature Selection and Electrical Capability Prediction of a Coupled Shaker-DUT Model D. Scheg, J. Heinlen, C. Garcia, L. Redmond, T. Roberts, A. Ramirez, and C. Haynes Abstract In experimental pre-testing, it is often useful to predict the capability of an electrodynamic shaker to run an environmental test on a given device under test (DUT). In this work, a methodology is proposed for predicting shaker electrical capability through a coupled model and appropriate feature selection using Bayesian classification. The coupled model was formed through a reduced-order finite element model of a DUT and an experimentally-calibrated lumped parameter model of a shaker. A control problem was solved to calculate voltage spectral densities (VSDs) to meet the specified acceleration spectral density (ASD) for the test. Variance was given to model parameters and a Bayesian classifier was employed for feature selection. The Bayesian classifer was verified against three reduced-order model classes and the coupling methodology was validated experimentally. The paper culminates with an example problem which shows that the novel coupling methodology and the Bayesian classifier can be used to predict voltage requirements and select an unknown feature: joint stiffness. Keywords Environmental testing · Bayesian classification · Reduced-order modeling · Shaker model · Finite element analysis Introduction In practice, it is common to run an environmental test on a device under test (DUT). An environmental test is the excitation of a DUT, usually by a shaker, under similar conditions that it would be subjected to in its true application. Often a test operator will not know the dynamic responses of the DUT (mode shapes, modal frequencies, etc.) or know if the shaker has the capability of performing the environmental test due to electrical constraints. Thus, it becomes advantageous to create analytical models for pre-test investigation. Past work investigated modeling a shaker as a lumped parameter system with electrical degrees of freedom (DOFs) to predict shaker requirements in a vibration test [1]. However, this investigation modeled the DUT as mass coupled to the shaker and did not consider the dynamic properD. Scheg Department of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 e-mail: dscheg@purdue.edu J. Heinlen Mathematics & Computer Science Department, Whitworth University, Spokane, WA 992517 e-mail: jheinlen26@my.whitworth.edu C. Garcia Department of Mechanical Engineering, New Mexico Institute of Mining and Technology, Socorro, NM 87801 e-mail: chrisgarcia2003@gmail.com L. Redmond School of Civil, Environmental Engineering and Earth Sciences, School of Mechanical and Automotive Engineering, Clemson University, Clemson, SC 29634 e-mail: lmredmo@clemson.edu T. Roberts · A. Ramirez · C. Haynes Los Alamos National Laboratory, Los Alamos, NM 87545 e-mail: tproberts@lanl.gov; ajrami@lanl.gov; cmhaynes@lanl.gov © The Author(s), under exclusive license to River Publishers 2025 67 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 9
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