Periodic Motions to Chaos in a Spring-Pendulum System

Description

This book builds on the fundamental understandings, learned in undergraduate engineering and physics in principles of dynamics and control of mechanical systems.  The design of real-world mechanical systems and devices becomes far more complex than the spring-pendulum system to which most engineers have been exposed.  The authors provide one of the simplest models of nonlinear dynamical systems for learning complex nonlinear dynamical systems. The book addresses the complex challenges of the necessary modeling for the design of machines. The book addresses the methods to create a mechanical system with stable and unstable motions in environments influenced by an array of motion complexity including varied excitation frequencies ranging from periodic motions to chaos. Periodic motions to chaos, in a periodically forced nonlinear spring pendulum system, are presented through the discrete mapping method, and the corresponding stability and bifurcations of periodic motions on the bifurcation trees are presented. Developed semi-analytical solutions of periodical motions to chaos help the reader to understand complex nonlinear dynamical behaviors in nonlinear dynamical systems. Especially, one can use unstable motions rather than stable motions only. Preface Contents About the Authors 1 Introduction References 2 A Semi-analytical Method References 3 A Nonlinear Spring-Pendulum 3.1 Physical Description 3.2 Discretization References 4 Formulation for Periodic Motions 4.1 Period-1 Motions 4.2 Period-m Motions 4.3 Finite-Fourier Series References 5 Period-1 Motions to Chaos 5.1 Bifurcation Trees 5.2 Frequency-Amplitude Characteristics 5.3 Periodic Motions Illustrations References 6 Higher-Order Periodic Motions to Chaos 6.1 Analytical Bifurcation Trees 6.2 Harmonic Amplitudes 6.3 Complex Periodic Motion Illustrations Reference