Modeling and Simulation of the Dynamics of Dissipative, Inelastic Spheres with Applications to Planetary Rovers and Gravitational Billiards
This dissertation provides a thorough treatment on the dynamic modeling and simulation of spherical objects, and its applications to planetary rovers and gravitational billiards. First, the equations governing the motion of a wind-driven spherical rover are developed, and a numerical procedure for their implementation is shown. Dynamic simulations (considering the Earth and Mars atmospheres) for several terrain types and conditions illustrate how a rover may maneuver across flat terrain, channels and craters. The effects of aerodynamic forces on the rover's motion is studied. The results show the wind force may both push and hinder the rover's motion while sliding, rolling and bouncing. The rover will periodically transition between these modes of movement when the rover impacts sloped surfaces. Combinations of rolling and bouncing may be a more effective means of transport for a rover traveling through a channel when compared to rolling alone. The aerodynamic effects, of drag and the Magnus force, are contributing factors to the possible capture of the rover by a crater. Next, a strategy is formulated for creating randomized Martian rock fields based on statistical models, where the rover's interactions with these fields are analyzed. Novel procedures for creating randomized Martian rock fields are presented, where optimization techniques allow terrain generation to coincide with the rover's motion. Efficient collision detection routines reduce the number of tests of potential collisions between the rover and the terrain while establishing new contact constraints. The procedures allow for the exploration of large regions of terrain while minimizing computational costs. Simulations demonstrate that bouncing is the rover's dominant mode of travel through the rock fields. Monte-Carlo simulations illustrate how the rover's down-range position depends on the rover design and atmospheric conditions. Moreover, the simulations verify the rover's capacity for long distance travel over Martian rock fields. Finally, a mathematical model that captures the essential dynamics required for describing the motion of a real world billiard for arbitrary boundaries is presented. The model considers the more realistic situation of an inelastic, rotating, gravitational billiard in which there are retarding forces due to air resistance and friction. The simulations demonstrate that the parabola has stable, periodic motion, while the wedge and hyperbola, at high driving frequencies, appear chaotic. The hyperbola, at low driving frequencies, behaves similarly to the parabola, and has regular motion. Direct comparisons are made between the model's predictions and previously published experimental data. The representation of the coefficient of restitution employed in the model resulted in good agreement with the experimental data for all boundary shapes investigated. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.
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- Applied Mathematics;Engineering, Aerospace;Engineering, Mechanical