In this paper, the parametric stability of axially accelerating viscoelastic beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was approximately assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The governing equations are linearized into the governing equation in the transverse direction and the expression of the longitudinally varying tension. The method of multiple scales is employed to analyze the parametric stability of transverse motion. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances. In terms of stability boundaries, the governing equations with or without the longitudinal variance of tension are compared and the effects of the finite support rigidity are also examined. Some numerical examples are presented to demonstrate the effects of the stiffness, the viscosity, and the mean axial speed on the stability boundaries. The differential quadrature scheme is developed to numerically solve the governing equation, and the computational results confirm the outcomes of the method of multiple scales.

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# Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions

Li-Qun Chen

,
Li-Qun Chen

Department of Mechanics, Shanghai University, Shanghai 200444, China; Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China; Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China; Modern Mechanics Division,

lqchen@staff.shu.edu.cn
E-Institutes of Shanghai Universities

, Shanghai 200072, China

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You-Qi Tang

You-Qi Tang

School of Mechanical Engineering,Shanghai Institute of Technology,Shanghai 201418, China;Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072,

China

Search for other works by this author on:

Li-Qun Chen

Department of Mechanics, Shanghai University, Shanghai 200444, China; Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China; Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China; Modern Mechanics Division,

E-Institutes of Shanghai Universities

, Shanghai 200072, China

lqchen@staff.shu.edu.cn

You-Qi Tang

China

*J. Vib. Acoust*. Feb 2012, 134(1): 011008 (11 pages)

**Published Online:**December 28, 2011

Article history

Received:

August 25, 2010

Revised:

March 15, 2011

Online:

December 28, 2011

Published:

December 28, 2011

Citation

Chen, L., and Tang, Y. (December 28, 2011). "Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions." ASME. *J. Vib. Acoust*. February 2012; 134(1): 011008. https://doi.org/10.1115/1.4004672

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