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Research Papers

Forced Vibrations of Supercritically Transporting Viscoelastic Beams

[+] Author and Article Information
Hu Ding1

 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, P. R. C.dinghu3@shu.edu.cn

Guo-Ce Zhang

 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, P. R. C.

Li-Qun Chen

 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, P. R. C.; Department of Mechanics, Shanghai University, Shanghai 200444, P. R. C.

Shao-Pu Yang

 Shijiazhuang Tiedao University, Shijiazhuang 050043, P. R. C.

1

Corresponding author. Present Address: 149 Yan Chang Road, Shanghai 200072, P. R. C.

J. Vib. Acoust 134(5), 051007 (Jun 05, 2012) (11 pages) doi:10.1115/1.4006184 History: Received May 10, 2011; Revised January 29, 2012; Published June 04, 2012; Online June 05, 2012

This study focuses on the steady-state periodic response of supercritically transporting viscoelastic beams. In the supercritical speed range, forced vibrations are investigated for traveling beams via the multiscale analysis with a numerical confirmation. The forced vibration is excited by the spatially uniform and temporally harmonic vibration of the supporting foundation. A nonlinear integro-partial-differential equation is used to determine steady responses. The straight equilibrium configuration bifurcates in multiple equilibrium positions at supercritical translating speeds. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for nontrivial equilibrium configuration. The natural frequencies and modes of the supercritically traveling beams are analyzed via the Galerkin method for the linear standard form with space-dependent coefficients under the simply supported boundary conditions. Based on the natural frequencies and modes, the method of multiple scales is applied to the governing equation to determine steady-state responses. To confirm results via the method of multiple scales, a finite difference scheme is developed to calculate steady-state response numerically. Quantitative comparisons demonstrate that the approximate analytical results have rather high precision. Numerical results are also presented to show the contributions of foundation vibration amplitude, viscoelastic damping, and nonlinearity to the response amplitude for the first and the second mode.

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Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Comparisons among the two-term, four-term, and eight-term Galerkin. (a) The first resonance and (b) the second resonance.

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Figure 5

The instability boundaries of the first two resonances versus the detuning parameter. (a) The first resonance and (b) the second resonance.

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Figure 6

The effects of the viscoelastic coefficient on instability boundaries of the first two resonances. (a) The first resonance and (b) the second resonance.

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Figure 7

The effects of the nonlinear coefficient on instability boundaries of the first two resonances. (a) The first resonance and (b) the second resonance.

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Figure 8

The first two resonances versus the detuning parameter with b = 0.001, α = 0.0001, k1  = 100, kf  = 0.8, and γ = 4.0. (a) The first resonance and (b) the second resonance.

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Figure 9

The first two resonances versus the detuning parameter with b = 0.002, α = 0.0001, k1  = 100, kf  = 0.8, and γ = 4.0. (a) The first resonance and (b) the second resonance.

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Figure 10

The first two resonances versus the detuning parameter with b = 0.001, α = 0.0002, k1  = 100, kf  = 0.8, and γ = 4.0. (a) The first resonance and (b) the second resonance.

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Figure 11

The first two resonances versus the detuning parameter with b = 0.001, α = 0.0001, k1  = 75, kf  = 0.8, and γ = 4.0. (a) The first resonance and (b) the second resonance.

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Figure 12

The first two resonances versus the detuning parameter with b = 0.001, α = 0.0001, k1  = 100, kf  = 0.7, and γ = 4.0. (a) The first resonance and (b) the second resonance.

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Figure 13

The first two resonances versus the detuning parameter with b = 0.001, α = 0.0001, k1  = 100, kf  = 0.8, and γ = 3.5. (a) The first resonance and (b) the second resonance.

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Figure 2

The effects of excitation amplitude on the first two resonances versus the detuning parameter. (a) The first resonance and (b) the second resonance.

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Figure 3

The effects of viscoelastic coefficient on the first two resonances versus the detuning parameter. (a) The first resonance and (b) the second resonance.

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Figure 4

The effects of nonlinear coefficient on the first two resonances versus the detuning parameter. (a) The first resonance and (b) the second resonance.

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