On Nonlinear Dynamics of the Pendulum with Periodically Varying Length

TL;DR

This paper analyzes the nonlinear dynamics of a pendulum with a periodically varying length, deriving instability boundaries, exploring different motion domains, and investigating routes to chaos through numerical methods.

Contribution

It provides analytical expressions for instability boundaries and characterizes motion domains in a pendulum with variable length, including transitions to chaos.

Findings

Derived asymptotic expressions for instability boundaries near resonance frequencies.

Identified domains of oscillation, rotation, and mixed motions analytically.

Numerically investigated two types of chaos transition depending on parameters.

Abstract

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of child's swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with numerical study. Two types of transitions to chaos of the pendulum depending on problem parameters are investigated numerically.