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永井, 健一 ; 丸山, 真一 ; 長谷川, 光貴 ; 山口, 誉夫
概要:
application/pdf<br />Journal Article<br />This study presents experimental results on chaotic vibrations of a shallow cylindrial shell-panel with clamped and simply-supported boundaries. Both opposite sides of rectangular boundaries
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are clamped by blocks and the other opposite sides are simply supported by flexible films. First, to find fundamental properties of the shell-panel, the linear natural frequencies and the characteristics of restoring force of the shell-panel are measured. The characteristic of restoring force of the shell-panel shows a type of a soften-hardening spring. Then, the shell-panel is excited with lateral periodic acceleration. In typical ranges of the exciting frequency, non-periodic responses are generated. These responses are examined with the Fourier spectra, the principal component analysis, the maximum Lyapunov exponents and the Poincare projections. The responses are found to be the chaotic responses. The chaotic responses accompany with sub-harmonic resonance responses of the order 1/2. The chaotic responses are generated from internal resonances dominated by the lowest mode and higher modes of vibration.
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柳澤, 大 ; 永井, 健一 ; 丸山, 真一
概要:
application/pdf<br />Journal Article<br />Experimental results and analytical results are presented on the effects of axial compressive load on chaotic vibrations of a clamped-supported beam subjected to periodic lateral
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acceleration. The beam is elastically compressed by an axial spring at the simply supported end. In the experiment, nonlinear and chaotic responses are detected under various compressions. In the analysis, the governing equation is reduced to nonlinear differential equations of a multiple-degree-of-freedom system by the Galerkin procedure. The nonlinear periodic responses are calculated by the harmonic balance method. The chaotic responses are numerically integrated by the Runge-Kutta-Gill method. The chaotic responses of the beam under axial compression are examined with the Fourier spectra, the Poincare projections, the maximum Lyapunov exponents and the principal components by the Karhunen-Loeve transformation. Both results of the experiment and the analysis coincide fairly well in detail. The frequency region of the chaos spread with the increment of the axial compressive force. The chaotic responses are generated within the axial compression range of the same ratio to the each buckling load. Distinct fractal pattern can be observed in the attractors of Poincare projections. The number of modes generated in the chaos is counted as three by the maximum Lyapunov exponent and the Lyapunov dimension
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丸山, 真一 ; 永井, 健一 ; 山口, 誉夫 ; 加藤, 考行
概要:
application/pdf<br />Journal Article<br />Numerical results are presented on chaotic vibrations of a cantilevered beam u
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nder vibroimpact. The analytical model consists of a cantilevered beam, subjected to periodic excitation, and a bar that restrains the amplitude of the beam. Equation of the motion of the beam is discretised by the finite element method and impacts of the beam are computed by using the coefficient of restitution rule. Time responses of the beam are calculated with direct integration by the Newmark-β method. Then the responses are inspected with the frequency response curves, the Fourier spectra, the Lyapunov exponents and the principal component analysis. The numerical results are compared with the experimental results that are previously presented by the authors, which verify our numerical results. Effects of the location of the bar and of the clearance between the bar and the beam on the chaotic responses are examined by the numerical results. As the location of the bar becomes farther from the clamped end and the clearance becomes smaller, the frequency region of the impact vibration is enlarged. The chaotic responses of the beam generally have contribution ratio of higher vibration modes less than 5%. However, when the location of the impact is close to a node of a higher mode or super-harmonic resonance of a higher mode is excited, contribution of the higher mode increases up to 10% to 30%.
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論文
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丸山, 真一 ; 加藤, 考行 ; 永井, 健一 ; 山口, 誉夫
概要:
application/pdf<br />Journal Article<br />Experimental results are presented on chaotic vibrations of a cantilevered beam under vibroimpact. A rigid bar that is located close to the free end of the beam limits the amplitude of the beam under
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lateral periodic acceleration, and then asymmetric vibroimpacts are induced. In the frequency region of the impact vibration near the fundamental resonance of the lowest mode, two regions of chaotic responses are observed. There is one impact in an excitation cycle in the higher frequency region of the chaos, while occurrence of impacts in the lower frequency region is less frequent than the former one. The maximum Lyapunov exponent of the chaotic response in the higher frequency region takes higher value than that of the chaos in the lower frequency. Mode contributions to the chaos are inspected by the principal component analysis. The lowest mode of vibration prevails in the vibroimpacting response. In the regions of the chaos, contribution of the second mode of vibration to the chaotic responses increases up to 5%. As the exciting frequency is increased, the contribution of the second vibration mode to the chaos becomes larger owing to the increase in the amplitude of impact vibration. Furthermore, contribution of the second vibration mode drastically increases as the super-harmonic resonance of the second mode is generated. The maximum Lyapunov exponent increases as the contribution of the second mode increases, which implies the close relation between the complexity of the chaotic responses and the participation of the higher modes of vibration.
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