Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed |
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Affiliation: | 1. Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street W., Montreal, QC H3A 2K6, Canada;2. Department of Bioresource Engineering, McGill University, 21111 Lakeshore Road, Montreal, QC H9X 3V9, Canada;1. Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35 (Agora), FI-40014, Finland;2. Institute for Problems in Mechanics RAS, Prospect Vernadskogo 101, Bld. 1, 119526 Moscow, Russian Federation;1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, People׳s Republic of China;2. MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, 100081 Beijing, People׳s Republic of China;1. IMB, CNRS-UMR 5584, Université Bourgogne Franche-Comté, 9 avenue Alain Savary, 21078 Dijon Cedex, France;2. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France |
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Abstract: | In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs). |
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