MS08.4: Non-Smooth Dynamics

Rolling without slipping is often assumed during modeling unicycles resulting in kinematic constraints and simplified description of the system. However, in real cases, slipping may occur especially in case of low friction coefficients. In this study, the wheel-ground contact of the unicycle is modeled assuming Coulomb friction yielding a limitation on the friction force magnitude. The corresponding dynamical models for the rolling without slipping, and rolling with slipping cases are derived using the Newtonian approach. This limitation on the friction force restricts the feasible steady states and also the realizable maneuvers that can be carried out without the slipping of the wheel as also shown by simulations.

The mechanic vibration absorbers might feature nonlinear piece-wise linear phenomena, while acoustic absorbers usually will not. This research proposes introducing non-smooth like system in acoustic applications by the use of programmable electroacoustic resonators. An original discontinuous piece-wise linear restoring force is considered. The considered system is studied through analytical methods, and then experimentally implemented. Nonlinear phenomena such as quasi-periodic regimes and fast energy triggering are highlighted. The discontinuous restoring forcing function is compared to linear and cubic ones.

Dynamic models are widely used for simulating friction phenomena, however, the examination of the models is often limited to numerical simulations. In this paper, new models based on the Generalized Maxwell-Slip (GMS) model − the SGMS-Coulomb and the SGMS-Stribeck model − are examined. These models form a surprisingly interesting phase space, even with the use of only a single internal state variable. We apply these models through a simple mechanical example and perform phase space analyses. We present different types of discontinuities, special points, equilibria, and bifurcations, which may remain hidden if we rely solely on numerical simulations. The results of the study indicate further, deeper research work in the field of the family of GMS models.

Sigmoid functions contain a transition region between two constant values. As smoothed switches between quantities, these functions can be found in many fields of physics and engineering. The main motivation of the present work is the combined frictional effect between rigid bodies with the presence of different kinematic quantities. In several cases of modelling, it is not enough to consider a few parameters -- magnitude and steepness of the transition -- but the shape of the transition curve has to be known in more details. By choosing a special sigmoid function and applying appropriate transformations, a wide range of other sigmoid curves can be expanded by using either Fourier series or Legendre polynomials. It is demonstrated, that the resulting approximate models can be applied effectively for the qualitative analysis of dynamic problems of rigid bodies with coupled friction effect between slip and spin.