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Research Papers

# Steady-State Transverse Response in Coupled Planar Vibration of Axially Moving Viscoelastic Beams

[+] Author and Article Information
Li-Qun Chen1

Department of Mechanics, Shanghai University, Shanghai 200444, China; Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200077, Chinalqchen@staff.shu.edu.cn

Hu Ding

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200077, China

1

Corresponding author.

J. Vib. Acoust 132(1), 011009 (Jan 11, 2010) (9 pages) doi:10.1115/1.4000468 History: Received August 21, 2008; Revised May 29, 2009; Published January 11, 2010

## Abstract

Steady-state periodical response is investigated for planar vibration of axially moving viscoelastic beams subjected external transverse loads. A model of the coupled planar vibration is established by introducing a coordinate transform. The model can reduce to two nonlinear models of transverse vibration. The finite difference scheme is developed to calculate steady-state response numerically. Numerical results demonstrate there are steady-state periodic responses in transverse vibration, and resonance occurs if the external load frequency approaches the linear natural frequencies. The effect of material parameters and excitation parameters on the amplitude of the steady-state responses are examined. Numerical results also indicate that the model of coupled vibration and two models of transverse vibration predict qualitatively the same tendencies with the changing parameters, and the two models of transverse vibration yield satisfactory results.

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## Figures

Figure 8

The steady-state responses calculated from three models in the third resonance (dots for solution of Eq. 14, the dash-dot lines for solution of Eq. 22, and the solid lines for solution of Eq. 23); (a) b=1, α=0.0001, and k1=100; (b) b=1, α=0.00002, and k1=100; (c) b=1, α=0.0001, and k1=150; and (d) b=1.5, α=0.0001, and k1=100

Figure 9

The shape of beam calculated from three models in the first resonance (dots for solution of Eq. 14, the dash-dot lines for solution of Eq. 22, and the solid lines for solution of Eq. 23

Figure 10

The shape of beam calculated from three models in the third resonance (dots for solution of Eq. 14, the dash-dot lines for solution of Eq. 22, and the solid lines for solution of Eq. 23

Figure 7

The steady-state responses calculated from three models in the first resonance (dots for solution of Eq. 14, the dash-dot lines for solution of Eq. 22, and the solid lines for solution of Eq. 23); (a) b=0.5, α=0.0001, and k1=100; (b) b=0.5, α=0.0005, and k1=100; (c) b=0.5, α=0.0001, and k1=75; and (d) b=0.75, α=0.0001, and k1=100

Figure 6

Effects of viscosity coefficient α, (a) the first resonance (dash-dot line for α=0.0001 and solid line for α=0.0005), and (b) the third resonance (dash-dot line for α=0.00002 and solid lines for α=0.0005)

Figure 5

Effects of external excitation amplitude b, (a) the first resonance (dash-dot line for b=0.5 and solid line for b=0.75), and (b) the third resonance (dash-dot line for b=1 and solid lines for b=1.5)

Figure 4

Effects of nonlinear coefficient k1, (a) the first resonance (dash-dot line for k1=75 and solid line for k1=100), and (b) the third resonance (dash-dot line for k1=100 and solid lines for k1=150)

Figure 3

The response amplitude changing with the load frequency, (a) the amplitude-frequency relation, (b) local magnification of (a) at the first resonance, (c) local magnification of (a) at the second resonance, and (d) local magnification of (a) at the third resonance

Figure 2

Steady-state response in the third resonance, (a) time history, (b) local magnification of (a) at initial phase, (c) local magnification of (a) at transient phase, and (d) local magnification of (a) at steady-state phase

Figure 1

Steady-state response in the first resonance, (a) time history, (b) local magnification of (a) at initial phase, (c) local magnification of (a) at transient phase, and (d) local magnification of (a) at steady-state phase

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