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Research Papers

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

[+] Author and Article Information
Li-Qun Chen1

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, Chinalqchen@staff.shu.edu.cn

You-Qi Tang

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

1

Corresponding author.

J. Vib. Acoust 134(1), 011008 (Dec 28, 2011) (11 pages) doi:10.1115/1.4004672 History: Received August 25, 2010; Revised March 15, 2011; Published December 28, 2011; Online December 28, 2011

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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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 3

The natural frequencies changing with the axial speeds for different beam stiffnesses: (a) the first mode, (b) the second mode

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Figure 4

The effect of the viscosity coefficients on the stability boundaries for the summation parametric resonance of the first and second modes

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Figure 5

The effect of the mean axial speeds on the stability boundaries for the summation parametric resonance of the first and second modes

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Figure 6

The effect of the stiffness on the stability boundaries for the summation parametric resonance of the first and second modes

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Figure 7

The stability boundaries for the summation parametric resonances of the first and second modes for different models

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Figure 8

The effects of the viscosity coefficients on the stability boundaries for the first two principal parametric resonances: (a) the first mode, (b) the second mode

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Figure 9

The effects of the mean axial speeds on the stability boundaries for the first two principal parametric resonances: (a) the first mode, (b) the second mode

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Figure 10

The effects of the stiffnesses on the stability boundaries for the first two principal parametric resonances: (a) the first mode, (b) the second mode

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Figure 11

The stability boundaries for the first two principal parametric resonances for different models: (a) the first mode, (b) the second mode

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Figure 12

The comparisons of the first two natural frequencies of the linear generating system: (a) the first mode, (b) the second mode

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Figure 13

The comparison of the analytical and numerical stability boundaries for different parametric resonances in plane σ-γ1 : (a) summation parametric resonance of the first and second modes, (b) first principal parametric resonance, and (c) second principal parametric resonance

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Figure 1

The physical model of an axially moving viscoelastic beam

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Figure 2

The natural frequencies changing with the axial speeds for different support rigidity parameters: (a) the first mode, (b) the second mode

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