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.

Spatially Localized and Chaotic Motions of a Discretely Supported Elastic Continuum

“…Finally, the transverse displacement and the external excitation are expressed in terms of the nonsmooth variables as [4]: Following the aforementioned steps and using the chain rule of differentiation, the equation of motion is transformed to the following fonn, …”

“…Noting that t't" = 0 [4] eliminating the remaining singular terms, and then setting separately terms prop0rlional and not proportional to T' equal to zero, equation (5) is decomposed into the following set of coupled nonlinear nonhomogeneous boundary value problems (NBVPs) [4J:…”

“…To overcome the difficulty encountered in dealing with the singularities inherent in this system, a nonsmooth transformation, first developed by Pilipchuk [1,2], is applied to the spatial variable. The method, also detailed in [3], was first applied to this problem in [4] for the unforced case, and in [5], to compute solitary waves for a system with hardening cubic and vibro-impact supports and forcing.…”

Damping-induced interplay between vibrations and waves in a forced non-dispersive elastic continuum with asymmetrically placed local attachments

“…The second set of solutions that leads to the realization of travelling waves on the left side of the string is 10) where the normalized mass μ can be chosen independently from σ and κ. From these equations we see that, because σ is always positive, it must be satisfied that the forcing frequency ω is either above or below the undamped natural frequency of the absorber, whereas the right part is dominated by pure standing waves.…”

“…In particular, at frequencies and wavenumbers corresponding to stop bands, no transmission of waves to the far field of the periodic medium is possible, and an applied disturbance is spatially confined close to the point of its application through the excitation of near-field standing or spatially decaying waves. These results have been extended to weakly [7,8] and strongly nonlinear [9,10] periodic media, recently including acoustic granular metamaterials [11][12][13][14][15].…”