Study of the Oscillations of a Nonlinearly Supported String Using Nonsmooth Transformations

Abstract

An analytical method for analyzing the oscillations of a linear infinite string supported by a periodic array of nonlinear stiffnesses is developed. The analysis is based on nonsmooth transformations of a spatial variable, which leads to the elimination of singular terms (generalized functions) from the governing partial differential equation of motion. The transformed set of equations of motion are solved by regular perturbation expansions, and the resulting set of modulation equations governing the amplitude of the motion is studied using techniques from the theory of smooth nonlinear dynamical systems. As an application of the general methodology, localized time-periodic oscillations of a string with supporting stiffnesses with cubic nonlinearities are computed, and leading-order discreteness effects in the spatial distribution of the slope of the motion are detected.