MS08.2: Non-Smooth Dynamics

In this paper, we propose formulations for several elastoplastic models, using nonsmooth differential algebraic systems of equations (DAE) which are suitable for numerical evaluation in an implicit time-stepping scheme. We will extend the Jenkins element, i.e. ˙a spring in series with a friction element, and demonstrate its use for modeling elastoplastic behavior in rockfall ring nets, where components deform plastically under impact loading.

This work examines the instability mechanism of a two-degree-of-freedom mass-on-moving belt system. The frictional interaction between the mass and the belt is governed by the Amontons-Coulomb law and a nonlinear normal contact force represented by the Hertz-Damp model. Nonlinear numerical analysis and a linearised stability analysis are conducted for both the unforced and the forced system. It is shown that, for certain model parameters, the excited system becomes unstable. The influence of damping on the stability is assessed as well. This study also demonstrates how the presence of instability affects vibration-induced friction modulation, leading to a reduced but largely scattered average friction force for specific belt velocities.

This work investigates the dynamic responses of a fluid-structure interaction (FSI) system executing vortex-induced vibrations (VIVs) when subjected to soft impacts with a deformable barrier. Discontinuity-induced bifurcations (DIBs) like grazing, followed period-added cascades interspersed with chaotic orbits, are observed on varying stiffness of the barrier. A comprehensive study of altering stiffnesses of the impacting barrier is carried out. Additionally, the effect of a pre-stressed barrier is also explored. Transverse discontinuity mappings (TDMs) associated with these interactions are studied with an intent to analyze the dynamics in the vicinity of the barrier. Further analysis, utilizing both Floquet multipliers and Lyapunov exponents, serves the crucial purpose of evaluating the system’s stability and predicting its response to perturbations, thereby offering valuable insights into the overall dynamical behavior.

We discuss analogies between first-order optimization algorithms and smooth/non-smooth dynamical systems. These analogies are not only used to understand the phenomenon of acceleration in both convex and nonconvex settings, but allow us to derive a new class of algorithms for constrained optimization. An important property of these algorithms is that constraints are expressed in terms of velocities instead of positions, which naturally leads to sparse, local, and convex approximations of the feasible set (even if the feasible set is nonconvex). Thus, the complexity tends to grow mildly in the number of decision variables and in the number of constraints, which makes the algorithms suitable for solving large-scale constrained optimization problems. We illustrate our algorithms on a compressed sensing and a sparse regression problem, showing that we can treat nonconvex $\ell^p$ constraints ($p< 1$) efficiently, while recovering state-of-the-art performance for $p=1$.

This study examines the non-smooth dynamics resulting from impacts between two elastic Euler-Bernoulli (EB) beams oscillating in the same plane separated by a certain constant distance along the span. We herein spatial discretization and the coefficient of restitution (CoR) based approach to estimate the post-impact velocities. The time integration of the discretized equations is based on the Moreau’s midpoint time-stepping algorithm and estimates the displacements at every time step. Numerical investigations demonstrate the method’s efficacy in providing responses that exhibit energy conservation (in the case of unity CoR) and is a potent and efficient tool to investigate the impact dynamics of continuous systems such as beams and strings.