Steady state response of nonlinear vibration of axially accelerating viscoelastic rayleigh beam
DING Hu1，CHEN Li-qun1,2
1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China;2. Department of Mechanics, Shanghai University, Shanghai 200444, China;3. School of Electronic Science & Applied Physics, Hefei University of Technology, Hefei, 230009, China
Abstract：The sub-harmonic resonance of axially accelerating nonlinear viscoelastic Rayleigh beams are investigated via an approximate analytical method with differential quadrature method verification. The mathematical model of transverse vibration of deformations infinitesimal beams is established by calculus of variations. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The solvable conditions of parametric vibration for sub-harmonic resonance are established via the methods of multiple scales. Therefore the steady state periodic response is presented for accelerating viscoelastic Rayleigh beams with simple supports boundary conditions. Numerical examples show the effects of the nonlinear coefficient and the viscoelastic coefficient on the steady state response are studied individually. The differential governing equation for transverse vibration of an axially moving slender Rayleigh beam is numerically solved via differential quadrature method. The numerical calculations confirm the analytical results. Numerical examples demonstrate that the approximate analysis results are with rather high precision.
2012, Vol. 31 
