Chapter 10 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 Investigation of the Use of Commercial Robotic Arms for Real-Time Hybrid Substructuring Arian Kist, David Stadler, Rok Belsˇak, Vasja Plesec, Timi Karner, Gregor Harih and Daniel Rixen Abstract Real-Time Hybrid Substructuring (RTHS) allows components of dynamic systems to be tested under realistic boundary conditions from an early stage of development by coupling the prototype of the component in real-time with a cosimulation of the remaining structure of the dynamic system. Real-time synchronization of the interfaces using actuators and sensors is a prerequisite for stable experiments and high-fidelity results. Since these hardware components suffer from imperfect transfer behavior and delays, additional controllers are required in most RTHS setups. This paper presents an RTHS framework specifically designed to use off-the-shelf robotic arms as actuators. A combination of Iterative Leaning Control and Normalized Passivity Control acts as a pure outer-loop controller in the robot’s task-space, making it independent of the robot used and avoiding integration into the robot’s low-level controllers. We demonstrate the ability of the control framework to provide high-fidelity results in virtual, i.e. fully simulated, RTHS experiments. We also present a first experimental realization of the framework using a KUKA® KR16 robot arm as an actuator. Although a stable RTHS experiment could be performed with this hardware, the fidelity improvement through iterative learning could not be experimentally validated yet. Keywords Real-time hybrid substructuring · Robotic arm· Iterative learning control · Passivity control · Delay compensation Introduction Dynamic Substructuring is a well-established method in the field of structural dynamics [1]. In this approach, the dynamics of an overall system is examined by independently analyzing the dynamic behavior of several substructures that compose the overall system. This can be done both experimentally and numerically. The dynamics of the overall system is then emulated by coupling the individual subcomponents. To do this, the compatibility of the displacements and the equilibriumof the forces at each of the interfaces between the substructures must be satisfied. This concept is visualized in fig. 1a. The overall system is decomposed into two substructures (A and B). To preserve the overall dynamics when investigating the dynamics of the substructures individually, the compatibility condition requires the interface displacements to be equal, i.e. zA =zB, and the equilibrium condition requires the resulting interface force to be zero, i.e. FA+FB =0. Based on these conditions, the dynamics of the whole system can be reconstructed from the individual results of the substructures. Real-Time Hybrid Substructuring (RTHS) is an efficient and economical testing approach to investigate the dynamic behavior of systems under realistic conditions by coupling a numerical co-simulation to physical components in realtime [2, 3, 4]. The approach therefore makes use of the concept of Dynamic Substructuring and the system under test is ArianKist · David Stadler · Daniel Rixen Chair of Applied Mechanics, TUM School of Engineering and Design, Munich Institute of Robotics and Machine Intelligence (MIRMI), Technical University of Munich, Boltzmannstr. 15, 85748 Garching, Germany e-mail: arian.kist@tum.de; david.stadler@tum.de; rixen@tum.de Rok Belsˇak · Timi Karner Laboratory for Robotics, Faculty of Mechanical Engineering, University of Maribor, Smetanova Ulica 17, 2000 Maribor, Slovenia e-mail: rok.belsak@um.si; timi.karner@um.si Vasja Plesec · Gregor Harih Laboratory for Integrated Product Development and CAD, Faculty of Mechanical Engineering, University of Maribor, Smetanova Ulica 17, 2000 Maribor, Slovenia e-mail: vasja.plesec@um.si; gregor.harih@um.si © The Author(s), under exclusive license to River Publishers 2025 101 Walter D’Ambrogio, et al. (eds.), Dynamic Substructuring & Transfer Path Analysis, Vol. 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0149-8 10
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