Chapter 17 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 Efficient Simulation of Soft Tissue for Human, Robotics, and Exoskeleton Applications Stefan Holzinger, Andreas Zwo¨lfer, and Daniel Rixen Abstract This paper explores the simulation of human lower limb dynamics using the floating frame of reference formulation to model bones and soft tissues as flexible bodies discretized with 3D volume finite elements. The study compares joint forces during landing on one leg using different modeling approaches: a purely rigid body model, a rigid body model with lumped rigid wobbling masses, and the aforementioned floating frame of reference formulation-based model. Results demonstrate that wobbling masses reduce joint forces compared to purely rigid body models. Although more computationally demanding than the rigid body models, the floating frame of reference formulation approach offers insight into soft tissue behavior, making it a promising tool for human body dynamics and related applications. Keywords Multibody · FEM· Human · Robotics · Exoskeletons Introduction Human pain and injuries are often caused by loads imposed during movements of the human body especially when high accelerations are present [1]. Loads within the human body are difficult to measure directly, but can be calculated using multibody and finite element simulations [2]. Multibody dynamics refers to the computational methods employed for the dynamic analysis of systems that consist of interconnected rigid and deformable components. Multibody human models conventionally represent human segments as rigid bodies actuated by lumped muscle (e.g. Hill type) force elements [3]. However, bones are covered with muscles and other soft tissues. This so-called wobbling mass represents more than 80% of the total mass of a segment [4], and it has been shown that it cannot be neglected due to its essential contribution to human body dynamics [4]. Modellers interested in the behaviour of wobbling mass often employ the aforementioned rigid/lumped musculoskeletal multibody models to calculate skeletal motion and muscle activation, in order to use this information as an input for computationally expensive non-linear finite element analyses, see e.g., [5]. This one-way coupled simulation approach is tedious from a modelling perspective, computationally intricate, and physically inaccurate when the effects of wobbling masses (inertia and damping) has a significant impact on the overall human motion. The floating frame of reference formulation is a standard method in multibody dynamics and used to model bodies with complex shapes that can undergo large rigid body motions but exhibit only small deformations [6]. An advantage of the floating frame of reference formulation is that it accounts for the nonlinear coupling of small deformations superimposed to arbitrarily large rigid body motion [7] and modal reduction techniques can be applied, reducing the system size to just the included modes [8]. A promising approach, both from the computational and the modelers perspective is the so-called nodal-based floating floating frame of reference formulation, as it eliminates inertia shape integrals entirely, simplifying both the derivation and implementation without needing lumped mass approximations [6, 8]. Although the floating frame of reference formulation is conventionally employed for small deformation problems and linear elastic materials, it still can be used to analyze the dynamics of a human lower limb. This is because bone and muscle make up about 70% of the total weight of a human lower limb [9] and it has been shown that a linear material law is often sufficient to model their mechanical behavior [10]. While the floating frame of reference formulation has already been used to model the human skeletal system [11], to the best of the authors knowledge, it has not been applied for the dynamic analysis of a human lower limb with wobbling masses. In this paper, we demonstrate Stefan Holzinger · Andreas Zwo¨lfer · Daniel Rixen Chair for Applied Mechanics, Technical University of Munich, Garching b. Mu¨nchen 85748, Germany email: stefan.holzinger@tum.de; andreas.zwoelfer@tum.de; rixen@tum.de © The Author(s), under exclusive license to River Publishers 2025 131 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 17
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