Article

Dynamic stability of an axially accelerating viscoelastic beam

Shanghai University, Shanghai, Shanghai Shi, China
European Journal of Mechanics - A/Solids (Impact Factor: 1.68). 07/2004; 23(4):659-666. DOI: 10.1016/j.euromechsol.2004.01.002

ABSTRACT

This work investigates dynamic stability in transverse parametric vibration of an axially accelerating viscoelastic tensioned beam. The material of the beam is described by the Kelvin model. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The Galerkin method is applied to discretize the governing equation into a infinite set of ordinary-differential equations under the fixed–fixed boundary conditions. The method of averaging is employ to analyze the dynamic stability of the 2-term truncated system. The stability conditions are presented and confirmed by numerical simulations in the case of subharmonic and combination resonance. Numerical examples demonstrate the effects of the dynamic viscosity, the mean axial speed and the tension on the stability conditions.

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    • "¨ Oz [1] computed natural frequencies of an axially moving beam in contact with a small stationary mass under pinned-pinned or clamped-clamped boundary conditions. Chen et al. [2] studied the dynamic stability of an axially accelerating viscoelastic beam and analyzed the effects of the dynamic viscosity, the mean axial speed, and the tension on the stability conditions. Chen and Yang [3] developed two nonlinear models for transverse vibration of an axially accelerating viscoelastic beam and applied the method of multiple scales to compare the corresponding steady-state responses and their stability. "
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    ABSTRACT: The transverse free vibration of an axially moving beam made of functionally graded materials (FGM) is investigated using a Timoshenko beam theory. Natural frequencies, vibration modes, and critical speeds of such axially moving systems are determined and discussed in detail. The material properties are assumed to vary continuously through the thickness of the beam according to a power law distribution. Hamilton's principle is employed to derive the governing equation and a complex mode approach is utilized to obtain the transverse dynamical behaviors including the vibration modes and natural frequencies. Effects of the axially moving speed and the power-law exponent on the dynamic responses are examined. Some numerical examples are presented to reveal the differences of natural frequencies for Timoshenko beam model and Euler beam model. Moreover, the critical speed is determined numerically to indicate its variation with respect to the power-law exponent, axial initial stress, and length to thickness ratio.
    Full-text · Article · Jun 2015 · Mathematical Problems in Engineering
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    • "The dynamics of these systems is studied in two sub-and super-critical axial velocity regimes, the critical velocity is the limit between these two regimes, in which instability occurs. On the other hand, axial velocity fluctuations could cause instabilities even in a very low axial velocity [7] [8] [9] [10] [11] [12] [13]. Pakdemirli and Ulsoy [14] studied the parametric principal resonance and combination resonance for any two modes of axially moving string. "
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    ABSTRACT: The extraordinary properties of carbon nanotubes enable a variety of applications such as axially moving elements in nanoscale systems. For vibration analysis of axially moving nanoscale beams with time-dependent velocity, the small-scale effects could make considerable changes in the vibration behavior. In this research, by applying the nonlocal theory and considering small fluctuations in the axial velocity, the stability and non-linear vibrations of an axially moving nanoscale visco-elastic Rayleigh beam are studied. It is assumed that the non-linearity is geometric and is due to the axial stress changes. The energy loss in the system is considered by using the Kelvin-Voigt model. The governing higher order nonlocal equation of motion is derived by using the Hamilton's principle and is analyzed by applying the multiple scales and power series methods. Then the non-linear resonance frequencies and response of the system are obtained. Considering the solvability condition, the stability of the system is studied parametrically through the Lyapunov's first method. An interesting result is that, considering the small-scale effects changes the slope of the frequency response curves due to the fluctuations in the axial velocity, considerably.
    Full-text · Article · Mar 2015 · International Journal of Mechanical Sciences
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    • "In addition to elastic beams, axially accelerating viscoelastic beams have recently been investigated. Chen, Yang and Cheng (2004) applied the averaging method to a discretized system via the Galerkin method to present analytically the stability boundaries of axially accelerating viscoelastic beams with clamped-clamped ends. Chen and Yang (2005) applied the method of multiple scales without discretization to obtain analytically the stability boundaries of axially accelerating viscoelastic beams with pinned-pinned or clamped-clamped ends. "

    Full-text · Dataset · Jan 2014
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