Article

# Dynamic stability of an axially accelerating viscoelastic beam

{ "0" : "Department of Mechanics, Shanghai University, Shanghai 200436, China" , "1" : "Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China" , "3" : "Axially accelerating beam" , "4" : "Viscoelasticity" , "5" : "Method of averaging"}

European Journal of Mechanics - A/Solids (Impact Factor: 1.9). 07/2004; DOI: 10.1016/j.euromechsol.2004.01.002 - [Show abstract] [Hide abstract]

**ABSTRACT:**The dynamic stability of an axially accelerating viscoelastic beam with two fixed supports is investigated. The Kelvin model is used for the constitutive law of the beam. A small simple harmonic is allowed to fluctuate about the constant mean speed applied to the beam, and the governing equation is truncated using the Galerkin method based on the eigenfunctions of the stationary beam. The averaged equations are derived for the cases of subharmonic and combination resonance. Finally, numerical examples are presented to demonstrate the effects of the viscosity coefficient, the mean axial speed and the beam bending stiffness on the stability boundaries.International Journal of Structural Stability and Dynamics 11/2011; 06(01). · 1.06 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Nonlinearly parametric resonances of axially accelerating moving viscoelastic sandwich beams with time-dependent tension are investigated in this paper. Based on the Kelvin differential constitutive equation, the controlling equation of the transverse vibration of a beam with large deflection is established. The system has been subjected to a time varying velocity and a harmonic axial tension. Here the governing equation of motion contains linear parametric terms and two frequencies, one is the frequency of axially moving velocity and the other one is the frequency of varying tension. The method of multiple scales is applied directly to the governing equation to obtain the complex eigenfunctions and natural frequencies of the system. The elimination of secular terms leads to the steady-state response and amplitude of vibrations. The influence of various parameters such as initial tension on natural frequencies and the amplitude of axial fluctuation, the phase angle between the two frequencies on response curves has been investigated for two different resonance conditions. With the help of numerical results, it has been shown that by using suitable initial tension, the amplitude of axial fluctuation, the phase angle, the vibration of the sandwich beam can be significantly controlled.Applied Mathematical Modelling 05/2014; 38(s 9–10):2558–2585. · 2.16 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This study focuses on the steady-state periodic response and the chaotic behavior in the transverse motion of an axially moving viscoelastic tensioned beam with two-frequency excitations. The two-frequency excitations come from the external harmonic excitation and the parametric excitation from harmonic fluctuations of the moving speed. A dynamic model is established to include the finite axial support rigidity, the material derivative in the viscoelastic constitution relation, and the longitudinally varying tension due to the axial acceleration. The derived nonlinear integro-partial-differential equation of motion possesses space-dependent coefficients. Applying the differential quadrature method (DQM) and the integral quadrature method (IQM) to the equation of the transverse motion, a set of nonlinear ordinary differential equations is obtained. Based on the Runge–Kutta time discretization, the time history of the axially moving beam is numerically solved for the case of the primary resonance, the super–harmonic resonance, and the principal parametric resonance. For the first time, the nonlinear dynamics is studied under various relations between the forcing frequency and the parametric frequency, such as equal, multiple, and incommensurable relationships. The stable periodic response and its sensitivity to initial conditions are determined using the bidirectional frequency sweep. Furthermore, chaotic motions are identified using different methods including the Poincaré map, the maximum Lyapunov exponent, the fast Fourier transforms, and the initial value sensitivity. Numerical simulations reveal the characteristics of the periodic, quasiperiodic, and chaotic motion of a nonlinear axially moving beam under two-frequency excitations.International Journal of Applied Mechanics 06/2013; 05(02). · 1.29 Impact Factor

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