Bifurcation And Chaos In Nonsmooth Mechanical Systems, Series A, Vol. 45
This book presents the theoretical frame for studying lumped nonsmooth dynamical systems, containing the mathematical methods, and adapted numerical methods.
TABLE OF CONTENTS Bifurcation And Chaos In Nonsmooth Mechanical Systems, Series A, Vol. 45
Chapter 14: Triple Pendulum with Impacts
Overview
In this chapter the triple pendulum with damping, external forcing and with impacts is investigated. We integrate the obtained system of governing equations between two successive impacts, and the discontinuity points are detected (by halving time step until obtaining required precision). At each impact time, the state of the system is transformed using the extended restitution coefficient rule. The theory of Aizerman and Gantmakher is used to calculate the fundamental solution matrices in the analysed system exhibiting discontinuities. The fundamental matrices are used during calculation of Lyapunov exponents, in stability analysis of periodic solutions (Floquet multipliers) and in shooting method for finding periodic orbits. Then some examples for three coupled identical rods with horizontal barrier are presented.
14.1 Introduction
A pendulum plays a very important role in mechanics since many interesting nonlinear dynamical behaviour can be illustrated and analysed using this simple system [Acheson and Mullin (1993)], [Bishop and Clifford (1996)], [Skeledon (1994)], [Yagasaki (1994)]. Even an externally excited simple pendulum can exhibit periodic, quasi-periodic and chaotic dynamics. It is obvious that coupled pendulums with obstacles can serve as models even for more complicated behaviour including that of energy pumped to the model, various resonances, jumps between different system states, various continuous and discontinuous bifurcations, symmetry breaking and crisis bifurcations, pools of attractions, oscillatory-rotational attractors, etc. We observe recently also an interest focused on experimental investigation of either simple or coupled mechanical and electronic setup of pendulums [Blackburn et. al. (1987)], [Doerner et. al. (1994)], [Heng et.
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