Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, China
Journal of Sound and Vibration (Impact Factor: 1.86). 06/2005; DOI: 10.1016/j.jsv.2004.07.024

ABSTRACT Stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated. The governing equation is derived from Newton's second law, the Kelvin constitution relation, and the geometrical relation. When the axial speed is a constant mean speed with small harmonic variations, the governing equation can be regarded as a continuous gyroscopic system under small periodically parametric excitations and a damping term. The method of multiple scales is applied directly to the governing equation without discretization. The stability conditions are obtained for combination and principal parametric resonance. Numerical examples are presented for beams with simple supports and fixed supports, respectively, to demonstrate the effect of viscoelasticity on the stability boundaries in both cases.

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    ABSTRACT: This paper investigates the steady-state periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam. For the first time, the nonlinear dynamic behaviors in the transverse parametric vibration of an axially moving Timoshenko beam are studied. The axial speed of the system is assumed as a harmonic variation over a constant mean speed. The transverse motion of the beam is governed by nonlinear integro-partial-differential equations, including the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation is applied to discretize the governing equations into a set of nonlinear ordinary differential equations. Based on the solutions obtained by the fourth-order Runge–Kutta algorithm, the stable steady-state periodic response is examined. Besides, the bifurcation diagrams of different bifurcation parameters are presented in the subcritical and supercritical regime. Furthermore, the nonlinear dynamical behaviors are identified in the forms of time histories, phase portraits, Poincaré maps, amplitude spectra, and sensitivity to initial conditions. Moreover, numerical examples reveal the effects of various terms Galerkin truncation on the amplitude–frequency responses, as well as bifurcation diagrams.
    Nonlinear Dynamics 01/2014; · 2.42 Impact Factor
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    ABSTRACT: The transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on simple supports are investigated. The supports are at the ends, and there is a support in between. The axial velocity is a sinusoidal function of time varying about a constant mean speed. Since the supports are immovable, the beam neutral axis is stretched during the motion, and hence, nonlinear terms are introduced to the equations of motion. Approximate analytical solutions are obtained using the method of multiple scales. Natural frequencies are obtained for different locations of the support other than end supports. The effect of nonlinear terms on natural frequency is calculated for different parameters. Principal parametric resonance occurs when the velocity fluctuation frequency is equal to approximately twice of natural frequency. By performing stability analysis of solutions, approximate stable and unstable regions were identified. Effects of axial velocity and location of intermediate support on the stability regions have been investigated.
    Journal of Vibration and Acoustics. 01/2011; 133:031013.
  • International Journal of Structural Stability and Dynamics 08/2014; 14(06):1450015. · 1.06 Impact Factor


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May 16, 2014