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# Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, China
(Impact Factor: 1.81). 06/2005; 284(3-5):879-891. DOI: 10.1016/j.jsv.2004.07.024

ABSTRACT

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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• "recoverable nature of SMA and the adhesive property of Ti 2 AlC, the damping behavior of the GCMeC is largely frequencydependent viscoelastic. Even the viscoelastic materials have wide application in solving damping problems of many engineering systems [1] [2] [3] [4], such as aircraft, space structures, automobiles, buildings, bridges and so on, their damping models in most available commercial finite element software do not explicitly represent the environmentally effected behaviors of actual materials (such as excitation frequency, ambient temperature, dynamic loads, etc.). One of the effective viscoelastic damping models in engineering applications was developed by Golla and Hughes [5] and McTavish and Hughes [6], known as the Golla–Hughes–McTavish (GHM) method. "
##### Article: Inman, D.J.: Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping. J. Sound Vib. 332, 6177-6191
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ABSTRACT: This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla-Hughes-McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.
Journal of Sound and Vibration 08/2013; 332(23):6177-6191. DOI:10.1016/j.jsv.2013.06.016 · 1.81 Impact Factor
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• "Then they obtained the effects of moving speed and viscoelasticity on the variation of the lowest two eigenvalues. The Method of Multiple Scales (a perturbation method) was used by Chen and Yang [20]. They conducted a stability analysis for parametric resonances of axially moving viscoelastic beams. "
##### Article: Effect of Viscoelasticity on the Natural Frequencies of Axially Moving Continua
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ABSTRACT: Linear models of axially moving viscoelastic beams and viscoelastic pipes conveying fluid are considered. The natural frequencies of the models are calculated. For both models, viscoelasticity terms are assumed to be of order one. Natural frequencies corresponding to various beam and pipe parameters are presented. Effects of viscoelasticity on the natural frequencies are discussed.
Advances in Mechanical Engineering 04/2013; 2013. DOI:10.1155/2013/169598 · 0.58 Impact Factor
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• "The moving tensioned Euler-Bernoulli beam is one of the most common models of these systems. Chen and Yang [1] investigated the transverse vibration of viscoelastic beams with time-independent speed. "
##### Conference Paper: Active vibration control of an axially moving beam using varying velocity method
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ABSTRACT: In this paper, active vibration control method of a viscoelastic axially moving beam, using varying velocity approach, is investigated. The dynamic response of the moving beam is computed with a set of linear equations in state space, which are obtained by Galerkin finite-element method. Based on the approximated solution, the vibration energy is reformulated as a quadratic function. Lyapunov stability is employed to investigate the vibration energy. The varying velocity algorithm optimizing time of vibration suppression is achieved by using steepest descent method. Simulation results demonstrate the effectiveness of the control approach.
ICCAS-SICE, 2009; 09/2009