Chapter 7 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 An Undamped Dynamic Vibration Absorber on a Resonant Plate Shock Test David E. Soine, Adam J. Bouma, and Forrest J. Arnold Dynamic vibration absorbers are widely used for vibration reduction on vehicles, buildings, industrial equipment, and many other applications. If an undamped or very lightly damped absorber is tuned to suppress the resonant response of a structure, two new resonances are created at frequencies above and below the suppressed resonance peak. An experimental investigation was conducted to explore the effect of such a vibration absorber on shock response spectra obtained during resonant plate shock testing. The shock response spectrum at the original plate resonant frequency was reduced. The response of the two new resonant frequencies appeared equal in a Fourier spectrum, but only the response of the higher frequency resonance had a strong effect on the shock response spectrum, creating a new “knee frequency” at a higher frequency than that of the original unmodified plate. Keywords Resonant · Fixture · Plate · Pyroshock · Absorber Introduction Resonant fixture shock tests are a means to perform mid-field and far-field pyroshock simulation in the test laboratory. The method can be adapted to simulate aspects of stage separation, ejection shock, blast events, flight shock, and margin tests for other shock environments ordinarily performed on a vibration shaker but outside the shaker test capability. Resonant fixtures in use include square, rectangular, or round plates, beams, and bars of various desirable responding frequencies. Shock practitioners have developed means to adjust the natural frequency and damping of these resonant shock test systems and are on the lookout for new tools to adjust resonant shock test equipment to better meet test specifications. In the discussion after a presentation on resonant bar shock testing at IMAC-XL, it was mentioned that dynamic vibration absorbers (also called tuned mass dampers or tuned absorbers) can be applied to resonant shock test fixtures. For the resonant bar application, shock practitioners in some cases might desire to suppress or alter the bending modes of the resonant bar. The authors recognized that some existing experimental test items at Sandia National Laboratories could be fashioned into a dynamic vibration absorber and decided to perform initial work on a resonant plate. This decision allowed the team to use existing hardware and finite element models to get familiar with the application of dynamic absorbers as quickly as possible. A short investigation was conducted into the effects of such a dynamic absorber on a resonant plate bending mode. Background Resonant plate is a type of resonant fixture laboratory test that exploits the transverse bending mode of a plate, whose frequency is pre-determined by selecting appropriate plate dimensions. The test article is usually attached to the center of the plate surface, and the plate is struck with a projectile on the opposite side. The resulting pulse and ringdown event, measured with an accelerometer and analyzed via the Shock Response Spectrum (SRS), is very repeatable. To facilitate a wide range of shock event simulations, a test laboratory will modularize this technique, resulting in several resonant plates of different dimensions in the lab inventory, each designed to respond at different frequencies. To keep costs low, simple square plates of aluminum are most often procured and adapted for use [1]. David E. Soine · Adam J. Bouma · Forrest J. Arnold Sandia National LaboratoriesAlbuquerque, NM 87185 e-mail: desoine@sandia.gov; abouma@sandia.gov; fjarnol@sandia.gov © The Author(s), under exclusive license to River Publishers 2025 55 Alexandra Karlicek et al. (eds.), Dynamic Environments Testing, Vol. 7 of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0152-8 7
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