Abstract
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 th in-plane mode () in which and 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.
Highlights
It is well known that when thin structures are subjected to time variable loads and vibrate with finite amplitudes that are of the order of their thicknesses, nonlinear dynamic behavior occurs frequently
When the MMS is applied to ordinary differential equations having cubic nonlinear terms, accurate results are given for the primary, subharmonic, and internal resonances 38–40 by defining the detuning parameter, which expresses the relationship between linear natural frequency and a driving frequency in the quadratic form
Nonlinear vibrations of a suspended cable with quadratic and cubic nonlinearities were investigated using the discretization approach of the MMS, where we focused on the primary resonance of the first antisymmetric mode the second in-plane mode
Summary
It is well known that when thin structures are subjected to time variable loads and vibrate with finite amplitudes that are of the order of their thicknesses, nonlinear dynamic behavior occurs frequently. A considerable number of publications have studied the vibration problems of nonlinear partial differential equations e.g., 1–12. To solve the nonlinear vibration problems of continuous systems, many researchers have used the method of multiple scales MMSs developed by Nayfeh and Mook and Nayfeh. For the analyses of nonlinear continuous systems, the application of MMS can be divided into two categories: direct and discretization approaches. The MMS is applied directly to the governing partial differential equations, whereas, in the discretization approach, the MMS is applied to the ordinary differential equations derived from Galerkin’s procedure. By comparing the results obtained from both these approaches in the vibration analysis for the nonlinear continuous systems, the superiority of the direct approach to the discretization approach was reported 19–32
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