Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations |
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| Institution: | 1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200444, China;2. Department of Mechanics, Shanghai University, Shanghai 200444, China;1. Department of Mechanics, Shanghai University, Shanghai 200444, China;2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China;3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China;1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China;2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China;3. Department of Mechanics, Shanghai University, Shanghai 200444, China;1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China;2. School of Engineering, University of California Merced, CA 95343, USA;3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China;4. Department of Mechanics, Shanghai University, Shanghai, 200072, China;1. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200444, China;2. School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China;1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China;2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China;3. Department of Mechanics, Shanghai University, Shanghai 2004444, China |
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| Abstract: | Stability is investigated for an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation, not simply the partial time derivative. The method of multiple scales is applied directly to the governing equation without discretization. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams constrained by simple supports with rotational springs in parametric resonance. The finite difference schemes are developed to solve numerically the equation of axially accelerating viscoelastic beams with fixed supports for the instability regions in the principal parametric resonance. The numerical calculations confirm the analytical results. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity. |
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