Computerized implementation of the multiple scales perturbation method using Mathematica

Raya Khanin, Matthew Cartmell, Anthony Gilbert

Research output: Contribution to journalArticlepeer-review

31 Citations (Scopus)


One of the well-established analytical techniques for solving engineering vibration problems, which are represented by ordinary differential equations, is the method of multiple scales (MS). This method can be applied to find approximate solutions to a wide range of nonlinear problems. The main idea of the MS method is to split up the single independent variable into several new independent variables. The method allows the construction of a set of perturbation equations that can be solved under the condition of removal of secular terms.

The main emphasis in this paper is on how to generalise a computer implementation of the MS method and its application to nonlinear vibration problems. The necessary macro-steps that are used for the development of the computational system are formulated and the practical ways of encoding these steps using Mathematica are discussed. The Mathematica package “MultipleScale.m” has been developed as a deliverable in this research. This package is capable of performing perturbation analysis on a wide class of multi-degree-of-freedom vibration systems. An example is given to illustrate the concepts discussed.
Original languageEnglish
Pages (from-to)565-575
Number of pages11
JournalComputers and Structures
Issue number5
Publication statusPublished - 2000


  • Mathematica
  • nonlinear vibrations
  • multiple scale method
  • perturbation methods


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