This paper describes identification of spatial modes in a chaotic vibration involving dynam 孟 c snap −through fbr a buckled beam with aconcntrated mass , The beam is constrained by an axial e豆 astic support and both ends of the beam are fixed . Using KL ( Karhunen −Loev ¢ ) method , time histor 正 es are decomposed into components which have no corelation each other . E 童 genvalues of eevarriance matrix of the time histories correspond te contribution of the components to the original time history . The eigenvectors correspond to spatial modes ( KL modes
Experimental results are presented on chaotic vibrations of a rectangular plate with in-plane elastic constraint. The plate has initial imperfection. Opposite edges of the plate are clamped and the other edges are simply supported. One side of clamped edges is connected to elastic springs and is movable to in-plane direction. The simply-supported edges are connected to the boundaries with adhesive flexible films. Loading in-plane compressive force to the plate, the plate shows pre-buckled configuration with the type of a softening-and-hardening spring. Under periodic lateral excitation, chaotic responses are obtained in specific frequency regions. Predominant chaotic responses are examined with the Fourier spectra, the Poincaré projections and the maximum Lyapunov exponents. Furthermore, applying the Karhunen-Loéve method, contributions of vibration modes on the chaotic responses are confirmed. It is found that the chaotic responses are generated from the internal resonant vibrations with the first mode of vibration and higher modes of vibration. The chaotic responses are dominated by the lowest mode of vibration. The higher modes of vibration contribute from 11 to 20 percent to the chaos. As the exciting amplitude increases, the amplitude of the chaotic responses increases and frequency regions of the chaotic responses shift. In the larger amplitude of the response, the frequency region shifts to the higher range owing to the resonant response with the type of a hardening spring. In contrast, the frequency region shifts to the lower range when the amplitude of the chaotic response is smaller comparatively. The resonant response with the smaller amplitude corresponds to the type of a softening spring.
878vibrations of the plate are generated with large amplitude. In the typical regions of excitation frequency, chaotic vibrations are generated within small amplitude of the responses. Furthermore, multiple modes of vibration are excited simultaneously in the chaotic responses. It is necessary to reveal the chaotic vibrations and the contribution of vibration mode to the chaos.Nonlinear Vibration and chaotic phenomena of plates and shells were investigated by many researchers. Nonlinear vibrations of a simply-supported shell were studied by Amabili (1), (2) . Experimental and theoretical studies on chaotic vibrations of a shell-panel including the dynamic snap-through transition were conducted by Nagai et al. (3)-(5) . Nonlinear vibrations of a shell-panel with clamped boundaries were studied experimentally by Maruyama et al. (6) . Modal couplings due to internal resonances were examined in the experiment. Chaotic vibrations due to internal resonances of a plate were analyzed by Chang et al. (7) . However, it seems to authors that experimental results on chaotic vibrations of a plate under in-plane constraint at clamped edges are required.In the present paper, experimental results are shown on chaotic vibrations of the rectangular plate. The plate is clamped at opposite edges and simply-supported at the other edges....
This paper presents a new analytical procedure on bending vibrations of rectangular plates . Rectangular plate is divided 血 tO a few segmentS with rectangular elernents . ne mode shape 跏 ction is expressed with the product of t cated power series and 面 gonometdc fUnctions . The function is infinitely differentiable ction of class C °°with the oeoTdinate variable
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