Studies in non-linear multi-mode responses of harmonically excited rectangular plates with internal resonances
Abstract
Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. Through the Galerkin procedure, the von Karman equations of motion are reduced to a coupled set of nonlinear ordinary differential equations governing the motions of the N-modes. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. When the two modes are in primary or secondary resonance with the external excitation, the response of the plate can be approximated by two coupled modal equations governing the motions of the two modes in internal resonance. The method of averaging is utilized in the two resonance cases to transform the two nonlinear modal equations into sets of four dimensional dynamical systems which govern the amplitudes and phases of the two modes. The averaged systems are studied in detail by using local bifurcation theory and it is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. The mixed-mode motion can be in the form of, either a standing wave or a traveling wave. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period-doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. The averaged equations for the case of subharmonic motions of order-three are also found to exhibit complicated dynamics including periodic and chaotic solutions. The conservative limit of the two-mode approximation of the plate equations is then studied by using the canonical averaging technique. The integrable autonomous system is found to have homoclinic and heteroclinic orbits that are destroyed by external forcing into chaotic motions.
Degree
Ph.D.
Advisors
Krousgrill, Purdue University.
Subject Area
Mechanical engineering
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