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

[+] Author and Article Information
V. N. Pilipchuk

Department of Applied Mathematics, State Chemical and Technological University of Ukraine, Dniepropetrovsk, Ukraine

A. F. Vakakis

Department of Mechanical and Industrial Engineering, University of Illinois at Urbana—Champaign, Urbana, IL 61801

J. Vib. Acoust 120(2), 434-440 (Apr 01, 1998) (7 pages) doi:10.1115/1.2893848 History: Received June 01, 1995; Online February 26, 2008


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.

Copyright © 1998 by The American Society of Mechanical Engineers
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