Lossless collisions and complex behaviour in simple impacting systems

Abstract

Humans have always tried to take cues from the animal kingdom, incorporating ideas into their creations, either w ith an intention o f devising a new one or making the existing ones better. Animal locomotion not only interests persons who study biomechanics but also engineers, who are constantly trying to improve upon mobile equipments. Animals interact with the forces of the environment around them to produce well co-ordinated and graceful movements. In the present work, we create and study simple mathematical models for two different kinds o f animal gait, namely, hopping and galloping. The nervous and the muscular systems o f an animal interact to produce controlled and sustained gait. There are studies in biomechanics which indicate lower levels o f muscular activity during human walking. This implies that it may be possible to generate sustained gait using simple passive models - models which do not need external control. In passive dynamic models that model a form o f animal gait, energy loss is the most im portant factor which governs sustained motion. Animal gait, like hopping and galloping, invariably involves impacts with the ground and consequently a loss o f energy. The energy lost during these collisions can be considerable if no clever handling o f collisions exists. The aim o f the present work is to study simple mathematical models which mimic animal locomotion in order to explore the existence o f such solutions which give dissipationless impacts! An inelastic impact is generally associated with a loss o f energy. We, however exploit special conditions tha t lead to plastic collisions w ith no dissipation. Also, an impact is a source o f strong nonlinearity and makes a dynamical system non-smooth. Study o f passive impacting systems is a step towards understanding the dynamics o f driven impacting systems and addressing issues like stability o f motions, bifurcations and possibility o f chaotic motions. Moreover, inelastic collisionsproduce stresses in the material and often result in a premature failure o f the equipment. The energy lost during impacts also affects the efficiency o f impacting systems. Hopping and galloping are two o f the simplest kinds of animal gait. Many animals, especially the smaller animals, use hopping as a means of locomotion while galloping is the mode of locomotion in large-sized animals. We study two simple mathematical models for hopping and galloping. The model used for hopping is a two mass-spring system. The model for galloping consists o f a rigid disk with an eccentric centre of mass. In both models, the only loss o f energy is due to the plastic collisions with the frictionless ground. There are no forces or initial momenta in the horizontal direction, which means th a t the centre o f mass in each the systems moves in a vertical line. Since we assume no other form of dissipation except plastic collisions, the energy in the system is conserved if the collisions are lossless. We show that, in both the systems, there exist solutions (i.e., initial conditions) which lead to lossless inelastic collisions. There are infinite number o f such solutions, each representing a different energy of the system. The solutions in the hopping model are sine-waves with fixed frequency but adjustable amplitude. In the galloping model, the solutions have fixed amplitude but variable frequency. The system dynamics can be understood by studying the dynamics o f one system variable - the non-dimensional velocity o f the upper mass in case o f the hopping model, and the angular displacement in case o f the galloping model. We construct a one-dimensional map to understand the global dynamics o f both the systems. The solutions which give lossless collisions become the fixed points o f the map. We show tha t the fixed points are stable only from one side, i.e., they exhibit one-way stability. Since the total energy o f the system has to decrease with each impact, this peculiar stability is plausible. The interval o f attraction around each fixed point decreases with an increase in the system parameters - the mass ratio in the hopping model, and the eccentricity o f the centre o f mass in case o f the gallop model. We define the basin o f attraction as the set of all initial conditions which eventually settle down to a fixed point for a range o f values o f the system parameters. The presence o f infinite fixed points, with each fixed point exhibiting one-way stability, has im portant consequences. We show tha t the basin o f attraction for both the models is extremely complex. We also showthat the basin o f attraction has structure at finer scales and is self-simiiar. The highly complex nature of the basin of attraction renders predicting the fate o f any arbitrary initial condition very difficult. The hopping model and the galloping model show common features, namely, infinite solutions which give lossless collisions. The solutions have one-way stability. Though we have studied two simple passive models, we believe tha t the characteristics o f these models described in this work are perhaps generic to a class o f passive impacting systems