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 Bandgap Design of Metamaterial Structures by Varying Local Resonator Properties Hannes Wo¨hler and Sebastian Tatzko Abstract The concept of metamaterials can be used in structural dynamics to reduce vibrations. By attaching local resonators of given eigenfrequency to a host structure, a so-called bandgap is created which indicates significant vibration reduction in a broad frequency range. Although the resonators are basically represented by spring-mass systems the bandgap phenomenon goes beyond the well-known tuned mass damper behavior. This is due to the fact that the resonators are positioned at different locations of the host structure. It is well known that by increasing the mass of the resonators, the band gap width also increases. However, adding weight is not always a viable option when it comes to lightweight structures for example. In this study we investigate new ways to increase the bandgap width without adding mass to the metamaterial structure. We consider a set of resonators for which we change individual stiffness values and investigate different arrangements of the eigenfrequency distribution of the resonators while keeping the total mass constant. The resonators influence the host structure dynamics and can be seen as a band-pass filter for mechanical waves which offers various possible applications in the field of vibration mitigation. Keywords Metamaterial · Bandgap · Local resonators · Detuning· Mechanical filter Introduction In the field of structural dynamics, the use of the bandgap effect of metamaterials to attenuate vibrations is a current topic in research and development. Metamaterials consisting of a periodic e.g. geometric pattern generally offer a so-called phononic bandgap [1], which is in principle located in the higher frequency range. The origin of this effect can be traced back to research on phononic crystals [2]. Another approach is to add local resonators in a periodic pattern to a host structure, where traditionally resonators tuned to identical frequencies are distributed across the structure. This also creates a bandgap, which in this case is based on the so-called resonator effect. The big advantage for structural dynamics is that the usual bandgap is in a much lower frequency range. It is determined by the arrangement and properties of the resonators [3–5]. Recent research has extended this approach by exploring variations in resonator tuning to achieve multiple [6] and wider bandgaps [7, 8]. One such concept is a series of differently tuned resonators to cover broader frequency ranges. This study builds on by proposing systematic detuning patterns of resonators attached to a segment of a host structure. The main goal is to increase the bandgap width by employing different configurations of resonator detuning. To investigate the effectiveness of these configurations, we use a basic lumped-mass model that serves as a simplified framework for analyzing different metamaterial arrangements. The study examines several detuning patterns, including alternating and linearly in-/ decreasing triangular formations, each with three levels of detuning. In addition, we compare the calculated bandgap widths of these finite lumped-mass models with those obtained from the Bloch’s theorem model, which uses a unit cell with complex boundary conditions to calculate a dispersion diagram [9, 10]. Hannes Wo¨hler · Sebastian Tatzko Institute of Dynamics and Vibration Research, Leibniz University Hannover, An der Universita¨t 1, 30823 Garbsen, Germany e-mail: woehler@ids.uni-hannover.de; tatzko@ids.uni-hannover.de © The Author(s), under exclusive license to River Publishers 2025 71 Matthew Allen et al. (eds.), Special Topics in Structural Dynamics & Experimental Techniques, Vol. 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0150-4 9
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