Chapter 8 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 Investigating the Propulsive Efficiency of Bio-Inspired Fish-like Elastic Caudal Fin through Dynamic Analysis and Experimental Validation E. Paifelman, S. Milana, A. Culla, G. Costantino Muniz, F. Passacantilli, and E. Ciappi Abstract The present work focuses on the dynamic analysis of bio-inspired fish-like robotic fins in order to evaluate their propulsive efficiency. This study relates the effect of the main factors on which this efficiency depends, particularly materials and stiffness in a typical biomimetic range. Numerical and experimental modal analysis techniques are employed to examine the vibrational properties of different fins. This analysis is essential to characterize the elastic materials and understand how the respective stiffness levels affect the dynamic behavior of the fin. A numerical analysis is then conducted using an analytical model that simulates the coupled dynamics between the fluid and the elastic fin. This model takes into account the two-ways interaction between the hydrodynamic forces and elastic deformations of the fin, allowing a detailed evaluation of propulsive efficiency parameters. Key performance indicators, such as elastic deformation, thrust, and efficiency, are calculated and analyzed in relation to changes in materials and stiffness. Hydrodynamic tank experiments are ongoing to directly measure the performance of the robotic fins, comparing the experimental data with the predictions of the analytical model. This validation phase is critical to confirm the accuracy of the model and to further refine the predictions regarding propulsive efficiency. The results of this paper provide valuable insights for the future design of underwater robots, with potential applications in areas such as marine exploration, environmental monitoring, and bioinspired robotics. Keywords Fish-like · Bionspired robot · FSImodel · Modal analysis · Biomimetic propulsion Introduction The use of biomimetic thrusters in the marine environment offers significant advantages for marine transportation, particularly through reduced noise compared to conventional thrusters and the ability to provide a wide range of propulsive efficiency. Although the applicability of such systems is currently limited to small marine vehicles, such as AUVs (Autonomous Underwater Vehicles) or ROVs (Remotely Operated Vehicles), interest in their performance is growing. Numerous studies in the literature point out that these thrusters are often designed on a small scale, with dimensions on the order of a centimeter, mainly for educational purposes or laboratory testing [1, 2]. The objective of this study is to overcome these limitations by proposing a systematic analysis on elastic propulsion system intended for larger robotic prototypes, with dimensions of the order of a meter. This allows them to cope with more complex operating conditions, such as higher cruising speeds and greater operational depths, than the micro-robots currently in use. Specifically, the paper presents an analysis of the propulsive behavior of different types of elastic thrusters designed to move an AUV with fish-inspired morphology. The analysis is based both on numerical results and on a preliminary experimental campaign, with the aim of identifying the most suitable materials and the relative frequencies peculiar to the systems under consideration. E. Paifelman· G. Costantino Muniz · F. Passacantilli · E. Ciappi National Research Council, Institute of Marine Engineering, Via di Vallerano 139, 00128 Rome, Italy e-mail: elena.paifelman@cnr.it; costantinomuniz.1793090@studenti.uniroma1.it; fabio.passacantilli@cnr.it; elena.ciappi@cnr.it S.Milana · G. Costantino Muniz Sapienza University of Rome, Department of Aerospace and Mechanical Engineering, Via Eudossiana 18, 00184 Rome, Italy e-mail: silvia.milana@uniroma1.it; costantinomuniz.1793090@studenti.uniroma1.it © The Author(s), under exclusive license to River Publishers 2025 65 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 8
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