Transverse vibrations of an axially accelerating viscoelastic string with geometric nonlinearity |
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| Authors: | Li-Qun Chen Jean W Zu Jun Wu Xiao-Dong Yang |
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| Affiliation: | (1) Department of Mechanics, Shanghai University, Shanghai, 200436;(2) Department of Mechanical & Industrial Engineering, University of Toronto, Ontario, M5S 3G8, Canada;(3) Shanghai Institute of Applied Mathematics & Mechanics, Shanghai University, Shanghai, 200072, China;(4) Shanghai Institute of Applied Mathematics & Mechanics, Shanghai University, Shanghai, 200072, China |
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| Abstract: | Two-to-one parametric resonance in transverse vibration of an axially accelerating viscoelastic string with geometric nonlinearity is investigated. The transport speed is assumed to be a constant mean speed with small harmonic variations. The nonlinear partial differential equation that governs transverse vibration of the string is derived from Newton's second law. The method of multiple scales is applied directly to the equation, and the solvability condition of eliminating secular terms is established. Closed-form solutions for the amplitude of the vibration and the existence conditions of nontrivial steady-state response in two-to-one parametric resonance are obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation are presented. Lyapunov's linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions for two-to-one parametric resonance. Some numerical examples highlighting the effects of the related parameters on the stability conditions are presented. |
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| Keywords: | axially accelerating string geometric nonlinearity method of multiple scales stability viscoelasticity |
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