Stability in parametric resonance of axially accelerating beams constituted by Boltzmann's superposition principle |
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| Institution: | 1. Department of Engineering Mechanics, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, China;2. Department of Mechanics, Shanghai University, Shanghai 200436, China;1. Tianjin Key Laboratory of Nonlinear Dynamics and Control, Department of Mechanics, Tianjin University, Tianjin 300072, China;2. School of Mechanical Engineering and Automation, Northeastern University, Shenyang 100819, China;3. Department of Mechanics, School of Science, Harbin Institute of Technology, Shenzhen 518055, China;1. Univ Lyon, ENTPE, LTDS UMR CNRS 5513, Rue Maurice Audin, F-69518, Vaulx-en-Velin Cedex, France;2. CEREMA - Direction territoriale Sud-Ouest, Département Laboratoire de Bordeaux, 24 rue carton - CS 41635, Bordeaux Cedex 33073, France;1. Department of Ammunition Engineering, Mechanical Engineering College, 97 West Heping Rd., Xinhua District, Shijiazhuang, Hebei Province 050003, PR China;2. Department of Missile Engineering, Mechanical Engineering College, 97 West Heping Rd., Xinhua District, Shijiazhuang, Hebei Province 050003, PR China |
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| Abstract: | Stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated. The governing equation is derived from Newton's second law, Boltzmann's superposition principle, and the geometrical relation. When the axial speed is a constant mean speed with small harmonic variations, the governing equation can be treated as a continuous gyroscopic system with 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 demonstrate that the increase of the viscosity coefficient causes the lager instability threshold of speed fluctuation amplitude for given detuning parameter and smaller instability range of the detuning parameter for given speed fluctuation amplitude. The instability region is much bigger in lower order principal resonance than that in the higher order. |
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