Pulse Dynamics in a Chain of Granules With Friction

We study the dynamics of a pulse in a chain of granules with friction. We present theories for chains of cylindrical granules (Hertz potential with exponent $n=2$) and of granules with other geometries ($n>2$). Our results are supported via numerical simulations for cylindrical and for spherical granules ($n=5/2$).Comment: Submitted to PR

Non-Filippov dynamics arising from the smoothing of nonsmooth systems, and its robustness to noise

Switch-like behaviour in dynamical systems may be modelled by highly nonlinear functions, such as Hill functions or sigmoid functions, or alternatively by piecewise-smooth functions, such as step functions. Consistent modelling requires that piecewise-smooth and smooth dynamical systems have similar dynamics, but the conditions for such similarity are not well understood. Here we show that by smoothing out a piecewise-smooth system one may obtain dynamics that is inconsistent with the accepted wisdom --- so-called Filippov dynamics --- at a discontinuity, even in the piecewise-smooth limit. By subjecting the system to white noise, we show that these discrepancies can be understood in terms of potential wells that allow solutions to dwell at the discontinuity for long times. Moreover we show that spurious dynamics will revert to Filippov dynamics, with a small degree of stochasticity, when the noise magnitude is sufficiently large compared to the order of smoothing. We apply the results to a model of a dry-friction oscillator, where spurious dynamics (inconsistent with Filippov's convention or with Coulomb's model of friction) can account for different coefficients of static and kinetic friction, but under sufficient noise the system reverts to dynamics consistent with Filippov's convention (and with Coulomb-like friction).Comment: submitted to: Nonlinear Dynamic

Semi-classical generalized Langevin equation for equilibrium and nonequilibrium molecular dynamics simulation

Molecular dynamics (MD) simulation based on Langevin equation has been widely used in the study of structural, thermal properties of matters in difference phases. Normally, the atomic dynamics are described by classical equations of motion and the effect of the environment is taken into account through the fluctuating and frictional forces. Generally, the nuclear quantum effects and their coupling to other degrees of freedom are difficult to include in an efficient way. This could be a serious limitation on its application to the study of dynamical properties of materials made from light elements, in the presence of external driving electrical or thermal fields. One example of such system is single molecular dynamics on metal surface, an important system that has received intense study in surface science. In this review, we summarize recent effort in extending the Langevin MD to include nuclear quantum effect and their coupling to flowing electrical current. We discuss its applications in the study of adsorbate dynamics on metal surface, current-induced dynamics in molecular junctions, and quantum thermal transport between different reservoirs.Comment: 23 pages, 16 figur

Non-deterministic dynamics of a mechanical system

A mechanical system is presented exhibiting a non-deterministic singularity, that is, a point in an otherwise deterministic system where forward time trajectories become non-unique. A Coulomb friction force applies linear and angular forces to a wheel mounted on a turntable. In certain configurations the friction force is not uniquely determined. When the dynamics evolves past the singularity and the mechanism slips, the future state becomes uncertain up to a set of possible values. For certain parameters the system repeatedly returns to the singularity, giving recurrent yet unpredictable behaviour that constitutes non-deterministic chaotic dynamics. The robustness of the phenomenon is such that we expect it to persist with more sophisticated friction models, manifesting as extreme sensitivity to initial conditions, and complex global dynamics attributable to a local loss of determinism in the limit of discontinuous friction.Comment: 22 pages, 8 figure

Canards in stiction: on solutions of a friction oscillator by regularization

We study the solutions of a friction oscillator subject to stiction. This discontinuous model is non-Filippov, and the concept of Filippov solution cannot be used. Furthermore some Carath\'eodory solutions are unphysical. Therefore we introduce the concept of stiction solutions: these are the Carath\'eodory solutions that are physically relevant, i.e. the ones that follow the stiction law. However, we find that some of the stiction solutions are forward non-unique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further the understanding of the non-unique dynamics, we introduce a regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. We identify a repelling slow manifold that separates the forward slipping to forward sticking solutions, leading to a high sensitivity to the initial conditions. On this slow manifold we find canard trajectories, that have the physical interpretation of delaying the slip onset. We show with numerics that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, that were otherwise disconnected.Comment: Submitted to: SIADS. 28 pages, 12 figure

Fast Simulation of Vehicles with Non-deformable Tracks

This paper presents a novel technique that allows for both computationally fast and sufficiently plausible simulation of vehicles with non-deformable tracks. The method is based on an effect we have called Contact Surface Motion. A comparison with several other methods for simulation of tracked vehicle dynamics is presented with the aim to evaluate methods that are available off-the-shelf or with minimum effort in general-purpose robotics simulators. The proposed method is implemented as a plugin for the open-source physics-based simulator Gazebo using the Open Dynamics Engine.Comment: Submitted to IROS 201

Trajectory generation for road vehicle obstacle avoidance using convex optimization

This paper presents a method for trajectory generation using convex optimization to find a feasible, obstacle-free path for a road vehicle. Consideration of vehicle rotation is shown to be necessary if the trajectory is to avoid obstacles specified in a fixed Earth axis system. The paper establishes that, despite the presence of significant non-linearities, it is possible to articulate the obstacle avoidance problem in a tractable convex form using multiple optimization passes. Finally, it is shown by simulation that an optimal trajectory that accounts for the vehicle’s changing velocity throughout the manoeuvre is superior to a previous analytical method that assumes constant speed