Discreteness effects in the forced dynamics of a string on a periodic array of non-linear supports

Gary Salenger, Alexander F. Vakakis

Research output: Contribution to journalArticlepeer-review


We analyze forced oscillations of an infinite string supported by a periodic array of non-linear elasticities. The envelope of the excitation possesses 'slow' and 'fast' scales and is periodic with respect to the 'fast' scale. The 'fast' spatial scale is defined by the distance between adjacent non-linear supports. To eliminate the singularities from the governing equations of motion that arise due to the discrete nature of the supports, we employ the non-smooth transformations of the spatial variable first introduced by Pilipchuck [Prikladneya Matematika i Mekhanika, Vol. 49, pp. 572-578 (1985)]. Thus, we convert the problem to a set of two non-homogeneous non-linear boundary value problems which we solve by means of perturbation theory. The boundary conditions of these problems arise from 'smoothness conditions' that are imposed to guarantee sufficient differentiability of the results. As an application, we study forced localized vibrations of the string, and analytically compute discreteness effects in the forced dynamics. In addition, we present an application of the method to the analysis of localized motions of a string on a periodic array of clearance, or vibro-impact non-linearities.

Original languageEnglish (US)
Pages (from-to)659-673
Number of pages15
JournalInternational Journal of Non-Linear Mechanics
Issue number4
StatePublished - Jul 1998


  • Non-linear oscillations
  • Non-smooth transformation

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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