Accurate higher-order analytical approximate solutions to nonconservative nonlinear oscillators and application to van der Pol damped oscillators

C. W. Lim, Siu Kai Lai

Research output: Journal article publicationJournal articleAcademic researchpeer-review

16 Citations (Scopus)


In general, this paper deals with general nonlinear oscillations of a nonconservative and single degree-of-freedom system with odd nonlinearity and, in particular, it presents accurate higher-order analytical approximate solutions to van der Pol damped nonlinear oscillators having odd nonlinearity and the Rayleigh equation. By combining the linearization of the governing equation with harmonic balancing and the method of averaging, we establish accurate analytical approximate solutions for the general weakly damped nonlinear systems. Unlike the classical harmonic balance method, simple linear algebraic equations instead of nonlinear algebraic equations are obtained upon linearization prior to harmonic balancing. The combination of these two methods results in very accurate transient response of the periodic solution. In addition and for the first time, this paper also presents a method for deducing fourth-, fifth- and higher-order linearized governing equations from the lower-order equations without the requirement of formulating the problem from the first principle. Three examples including the van der Pol damped nonlinear oscillator are presented to illustrate the excellent agreement with approximate solution using the exact frequency.
Original languageEnglish
Pages (from-to)483-492
Number of pages10
JournalInternational Journal of Mechanical Sciences
Issue number5
Publication statusPublished - 1 May 2006
Externally publishedYes


  • Harmonic balance
  • Linearization
  • Nonlinear oscillation
  • The method of averaging
  • Van der Pol equation

ASJC Scopus subject areas

  • Mechanical Engineering

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