Chapter 18 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 Finding the Limits of Beam Elements: Modeling Bolted Joints as an Effective Joint Region Nicholas Pomianek, Casey Whitworth, Enrique Gutierrez-Wing, Trevor Jerome, and J. Gregory McDaniel Abstract Interrupting a beam with a bolted joint introduces dynamic nonlinearity and time dependent variability arising from complex interfacial physics. High fidelity finite element simulation of large jointed structures is therefore expensive and has diminishing returns when considering only global behavior. We propose a simplified method to model bolted joints where a characteristic set of dynamic properties is determined for the jointed regions of a structure. Euler-Bernoulli beam elements are used to represent joints as continuous solid regions with tunable damping, stiffness, and density properties. These properties are updated by an optimization process based on experimental data using classical and recent frequency response similarity metrics. Assembly and reassembly effects are accounted for experimentally through variation of bolt tightening orders. The Brake-Reuß standardized beam is used as a test case for this work; results show robust agreement between the updated model and modal test data. This work develops a practical and effective framework for joint modeling that is readily applicable in any finite element software. Keywords Joint Mechanics· Model Updating· Similarity Metrics· Frequency Response· Finite Elements Introduction It is easy to find structural design cases where an estimation of global response is required at the modeling stage but including nonlinearity is not strictly necessary. These are typically large structures which often contain many instances of the same joint and loading scenarios significantly below expected service levels. If the common joints in these designs are sufficiently standardized and prevalent, then prototypes should be available and easy to test ex situ. Methods that leverage the available information under these circumstances to produce a high fidelity model that is inexpensive enough for structural design model updating are of increasing interest to the joint modeling community. Modeling nonlinear systems requires a choice between fidelity and computation time. The highest-fidelity techniques involve detailed scanning of the joint interface surfaces and modeling asperity interactions [1]. For modeling scenarios where joint surfaces are not available a priori, statistical descriptions of surface asperities are often employed alongside a combination of linear modal extraction and a quasi-static approach [2]. If the structure isn’t expected to encounter significant macroslip, then a Iwan or Masing contact element formulation is typically used [3]. More recent techniques for this application include the definition of Contact-Zone elements [4], or the replacement of bolts with springs [5]. Commercial finite element packages offer joint elements that entirely replace joints but they require empirical stiffness data. Finally, sufficiently linear systems are adequately modeled using friction contact conditions such as the penalty method that are available in commercial finite element software. All these modeling techniques require validation and calibration with experimental data for best results. If these data are available and joints are expected to be in the linear regime, then a simplified and calibrated linear model that can predict the most significant dynamics of joints is a powerful tool for structural design. Many joint modeling techniques have been developed to better predict the effects of friction across a wide range of loading scenarios but all require model updating and most require element types that are unavailable to finite element practitioners. Furthermore, the nonlinear approaches that many of these modeling techniques use prevent the use of frequency Nicholas Pomianek· Casey Whitworth· Enrique Gutierrez-Wing· J. Gregory McDaniel Department of Mechanical Engineering, Boston University, Boston, MA 02215 e-mail: pomianek@bu.edu; caseyw12@bu.edu; esgw@bu.edu; jgm@bu.edu Trevor Jerome Naval Surface Warfare Center, Carderock Division, West Bethesda, MD 20817 e-mail: trevor.w.jerome.civ@us.navy.mil © The Author(s), under exclusive license to River Publishers 2025 129 Ludovic Renson et al. (eds.), Nonlinear Structures & Systems, Vol. 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0146-7 18
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