Application of a modified lindstedt-poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

C. W. Lim, Siu Kai Lai, B. S. Wu, W. P. Sun, Y. Yang, C. Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

11 Citations (Scopus)


An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear-nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt-Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge-Kutta solutions.
Original languageEnglish
Pages (from-to)411-431
Number of pages21
JournalArchive of Applied Mechanics
Issue number5
Publication statusPublished - 1 May 2009
Externally publishedYes


  • Modified Lindstedt-Poincaré (L-P) method
  • Nonlinear oscillation
  • Quadratic nonlinear oscillator
  • Two-degree-of-freedom
  • Two-mass system

ASJC Scopus subject areas

  • Mechanical Engineering


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