Modal interaction of resonantly forced oscillations of two-degree-of-freedom structure.
type: Item
instance of: scholarly article
title: Modal interaction of resonantly forced oscillations of two-degree-of-freedom structure.
zbMATH Open document ID: 1074.70013
DOI: 10.1016/S0096-3003(01)00278-8
author: A. F. El-Bassiouny
published in: Applied Mathematics and Computation
publication date: 2003-01-28T00:00:00Z
review text: The author studies the nonlinear response of internally resonant two-degree-of-freedom systems with quadratic and cubic nonlinearities to harmonic external excitations. More precisely, if \(X_{1}\) and \(X_{2}\) are two generalized coordinates describing the motion of the system, it is assumed that the kinetic energy can be expressed as a polynomial of the form \(T = \frac{1}{2}M_{1}(1+m_{1}X_{1}+m_{2}X_{2})\dot{X}_{1}^{2} + \frac{1}{2}M_{2}(1+m_{3}X_{1}+m_{4}X_{2})\dot{X}_{2}^{2}+ M_{3}X_{1}\dot{X}_{1}\dot{X}_{2} + M_{4}X_{2}\dot{X}_{1}\dot{X}_{2},\) and that the potential energy has the form \(V = d_{1}X_{1}^{2} + d_{2}X_{2}^{2} + \frac{1}{4} d_{3}X_{1}^{4} + d_{4} X_{1}^{3}X_{2} + \frac{3}{2}d_{5} X_{1}^{2}X_{2}^{2} + d_{6} X_{1}X_{2}^{3} + \frac{1}{3} d_{7} X_{2}^{4}.\) Moreover, the author considers the effect of weak damping described by the dissipation function \(D = \frac{1}{2} c_{1}\dot{X}_{1}^{2} + \frac{1}{2} c_{2}\dot{X}_{2}^{2}.\) In the proof, the method of multiple scales is used to determine four first-order ordinary differential equations which describe the modulation of amplitudes and phases caused by resonances. Also, numerical results are presented.
Mathematics Subject Classification ID: 70K30
Mathematics Subject Classification ID: 70K40
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 1861296
zbMATH Keywords: harmonic excitations
zbMATH Keywords: weak damping
zbMATH Keywords: method of multiple scales
reviewed by: Antonio Cañada
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1861296