Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 890813, 12 pages
Research Article

Accuracy Improvement of the Method of Multiple Scales for Nonlinear Vibration Analyses of Continuous Systems with Quadratic and Cubic Nonlinearities

Department of Information Systems Engineering, Asahikawa National College of Technology, 2-2-1-6 Syunkodai, Asahikawa, Hokkaido 071-8142, Japan

Received 26 August 2010; Revised 18 November 2010; Accepted 22 December 2010

Academic Editor: Cristian Toma

Copyright © 2010 Akira Abe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper proposes an accuracy improvement of the method of multiple scales (MMSs) for nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. As an example, we treat a shallow suspended cable subjected to a harmonic excitation, and investigate the primary resonance of the 𝑚 th in-plane mode ( Ω 𝜔 𝑚 ) in which Ω and 𝜔 𝑚 are the driving and natural frequencies, respectively. The application of Galerkin's procedure to the equation of motion yields nonlinear ordinary differential equations with quadratic and cubic nonlinear terms. The steady-state responses are obtained by using the discretization approach of the MMS in which the definition of the detuning parameter, expressing the relationship between the natural frequency and the driving frequency, is changed in an attempt to improve the accuracy of the solutions. The validity of the solutions is discussed by comparing them with solutions of the direct approach of the MMS and the finite difference method.