A Study Of Four Nonlinear Systems With Parametric Forcing

Abstract

This thesis considers four nonlinear systems with parametric forcing. The first problem involves an inverted pendulum with asymmetric elastic restraints subjected to harmonic vertical base excitation. On linearizing trigonometric terms the pendulum is governed by an asymmetric Mathieu equation. Solutions to this equation are scaleable. The stability regions in the parameter plane are studied numerically. Periodic solutions at the boundaries of stable regions in the parameter plane are found numerically and then their existence is proved theoretically. The second problem involves use of the method of multiple scales to elucidate the dynamics associated with early and delayed ejection of ions from Paul traps. A slow flow equation is developed to approximate the solution of a weakly nonlinear Mathieu equation to describe ion dynamics in the neighborhood of the nominal stability boundary of ideal traps. Since the solution to the unperturbed equation involves linearly growing terms, some care in identification and elimination of secular terms is needed. Due to analytical difficulties, harmonic balance approximations are used within the formal implementation of the method. The third problem involves the attenuation, caused by weak damping, of harmonic waves through a discrete, periodic structure with wave frequency nominally within the Propagation Zone. Adapting the transfer matrix method and using the harmonic balance for nonlinear terms, a four-dimensional map governing the dynamics is obtained. This map is analyzed by applying the method of multiple scales upto first order. The resulting slow evolution equations give the amplitude decay rate in the structure. The fourth problem involves the dynamic response of a strongly nonlinear single-degree-of-freedom oscillator under a constant amplitude, parametric, periodic, impulsive forcing, e.g., a pendulum with strongly nonlinear torsional spring that is periodically struck in the axial direction. Single-term harmonic balance gives an approximate, but explicit, 2-dimensional map governing the dynamics. The map exhibits many fixed points (both stable and unstable), higher period orbits, transverse intersections of stable and unstable manifolds of unstable fixed points, and chaos.