Bifurcation and chaos of an axially accelerating viscoelastic beam |
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| Institution: | 1. “Vin?a” Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia;2. ”Vin?a” Institute of Nuclear Sciences, Laboratory for Nuclear and Plasma Physics, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia;3. Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovi?a 4, 21000 Novi Sad, Serbia;4. Department of Physics, Crete Center for Quantum Complexity and Nanotechnology, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece;5. National University of Science and Technology MISiS, Leninsky prosp. 4, Moscow 119049, Russia |
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| Abstract: | This paper investigates bifurcation and chaos of an axially accelerating viscoelastic beam. The Kelvin–Voigt model is adopted to constitute the material of the beam. Lagrangian strain is used to account for the beam's geometric nonlinearity. The nonlinear partial–differential equation governing transverse motion of the beam is derived from the Newton second law. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. By use of the Poincaré map, the dynamical behavior is identified based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented in the case that the mean axial speed, the amplitude of speed fluctuation and the dynamic viscoelasticity is respectively varied while other parameters are fixed. The Lyapunov exponent is calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially accelerating viscoelastic beam. |
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