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陈立群, BY LI-QUN CHEN,
Proc. R. Soc. A (2005) 461, 2701-2720 Published online 25 July 2005,-0001,():
-1年11月30日
The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) non-trivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial-speed fluctuation amplitude, and the viscoelastic parameters.
nonlinear parametric vibration, method of multiple scales, stability, axially moving string, viscoelasticity
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陈立群, Li-Qun Chen, Xiao-Dong Yang
L. -Q. Chen, X. -D. Yang. Journal of Sound and Vibration 284 (2005) 879-891,-0001,():
-1年11月30日
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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陈立群, Li-Qun Chen , Xiao-Dong Yang
L. -Q. Chen, X. -D. Yang. International Journal of Solids and Structures 42 (2005) 37-50,-0001,():
-1年11月30日
Principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed. A nonlinear partial-differential equation governing the transverse vibration is derived from the dynamical, constitutive, and geometrical relations. Under certain assumption, the partial-differential reduces to an integro-partialdifferential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams. The method of multiple scales is applied to two equations to calculate the steady-state response. Closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelascity, and nonlinearity and to compare results obtained from two equations.
Axially accelerating beam, Principal parametric resonance, Nonlinearity, Viscoelascity
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陈立群, L.-Q. Chen, J. W. Zu, J. Wu
Acta Mechanica 162, 143-155 (2003),-0001,():
-1年11月30日
The steady-state transverse vibration of a parametrically excited axially moving string with geometric nonlinearity is investigated in this paper. The Boltzmann superposition principle is employed to characterize the material property of the string. The method of multiple scales is applied directly to the governing equation, which is a nonlinear partial-differential-integral equation. The solvability condition of eliminating the secular terms is established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the summation resonance are obtained. Some numerical examples showing effects of the viscoelastic parameter, the amplitude of excitation, the frequency of excitation, and the transport speed are presented.
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【期刊论文】The energetics and the stability of axially moving Kirchhoff strings (L)
陈立群, Li-Qun Chena, Wei-Jia Zhao
L. -Q. Chen and W. -J. Zhao: Letters to the Editor J. Acoust. Soc. Am., Vol. 117, No. 1, January 2005,-0001,():
-1年11月30日
The energetics and the stability of free transverse vibration of an axially moving string are investigated based on the Kirchhoff nonlinear model. The model is derived in a physically meaningful manner. The time-rate of the total mechanical energy associated with the vibration is calculated to show that the energy is not conserved. It is proved that there exists a conserved quantity that remains a constant during the nonlinear vibration. The conserved quantity is applied to verify the Lyapunov stability of the axially moving Kirchhoff string.
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