Chapter 12 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 Extrapolating Dynamic Transfer Functions from Multi-Input Multi-Output Vibration Testing and Simulation E. J. Perez, E. E. Regula, C. J. Wynn, J. D. Blessinger, J. L. Davis, E. P. Dawson, and S. J. Zimmerman Abstract Finding transfer functions between an applied force and a measured output is well defined for dynamic systems. These transfer functions, which assume a force input, can have many names depending on the output they measure, such as receptance for displacement, mobility for velocity, or accelerance for acceleration. Similarly, the transfer path is a type of transfer function, which is simply the ratio of two outputs from a system in the frequency domain. For single-input linear systems, finding the transfer path between two outputs is a well-understood calculation. These techniques become more complex when dealing with multi-input multi-output tests. While these multi-input vibration tests are more complicated, they provide a more complete understanding of the system being analyzed. These complex systems have well-defined methods for obtaining transfer functions if a forcing function is known. However, when the input force is not known, the methods to find the transfer path between two locations is less understood. This work explores methods for finding unknown transfer paths using accelerations. The tests and validations are done using the Box Assembly, a common structure used in dynamics testing. Experimentally measured and finite-element approximated transfer functions are used to create a transmissibility matrix. This matrix allows for estimation of unknown accelerations based on measured accelerations at different locations on the box assembly. Keywords Transfer Paths · Transmissibility· FRF· MIMO· FEA Introduction Frequency response functions (FRFs) are well understood in dynamic systems and are useful for system identification and structural health monitoring. Finding the FRF for a single-input single-output (SISO) case is straightforward, as only scalar values are used. However, these calculations become more complex with multi-input multi-output (MIMO) testing because the scalar values are replaced with matrices. Not only are the FRFs of MIMO harder to obtain, but so are the transfer paths. Similar to FRFs, which are ratios of output to input, transfer paths are ratios of two system outputs. This relationship E. J. Perez Department of Engineering, University of California Riverside, Riverside, CA 92524 e-mail: epere194@ucr.edu E. E. Regula Department of Engineering, Messiah University, Mechanicsburg, PA 17055 e-mail: er1324@messiah.edu C. J.Wynn Department of Engineering, University of Utah, Salt Lake City, UT 84112 e-mail: cjwynn7@outlook.com J. D. Blessinger Los Alamos National Laboratory, Los Alamos, NM 87545 e-mail: jbless@lanl.gov S. J. Zimmerman Department of Engineering, University of Southern Indiana, Evansville, IN 47712 e-mail: edawson@lanl.gov © The Author(s), under exclusive license to River Publishers 2025 123 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 12
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