Techno Press
Techno Press

Structural Engineering and Mechanics
  Volume 30, Number 5, November30 2008 , pages 593-602
DOI: https://doi.org/10.12989/sem.2008.30.5.593
 

Periodic solutions of the Duffing equation
Jale Tezcan and J. Kent Hsiao

 
Abstract
    This paper presents a new linearization algorithm to find the periodic solutions of the Duffing equation, under harmonic loads. Since the Duffing equation models a single degree of freedom system with a cubic nonlinear term in the restoring force, finding its periodic solutions using classical harmonic balance (HB) approach requires numerical integration. The algorithm developed in this paper replaces the integrals appearing in the classical HB method with triangular matrices that are evaluated algebraically. The computational cost of using increased number of frequency components in the matrixbased linearization approach is much smaller than its integration-based counterpart. The algorithm is computationally efficient; it only takes a few iterations within the region of convergence. An example comparing the results of the linearization algorithm with the ?exact? solutions from a 4th order Runge-Kutta method are presented. The accuracy and speed of the algorithm is compared to the classical HB method, and the limitations of the algorithm are discussed.
 
Key Words
    duffing equation; harmonic balance method; nonlinear oscillator; linearization.
 
Address
Jale Tezcan and J. Kent Hsiao: Dept. of Civil & Environmental Engineering, Southern Illinois University, Carbondale, IL 62901, USA
 
Techno-Press: Publishers of international journals and conference proceedings.       Copyright © 2026 Techno Press
P.O. Box 33, Yuseong, Daejeon 305-600 Korea, Tel: +82-42-828-7996, Fax : +82-42-828-7997, Email: admin@techno-press.com