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永井, 健一
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小野里, 直樹 ; 丸山, 真一 ; 永井, 健一 ; 山口, 誉夫 ; 黒澤, 正樹
概要:
application/pdf<br />Journal Article<br />Experimental results are presented on chaotic vibrations of a rectangular plate with an in-plane elastic constraint. Opposite edges of the plate are clamped and the other edges are simply
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supported. One of the clamped edges is connected with elastic springs and is movable to an in-plane direction. The simply-supported edges are composed with adhesive elastic thin films. Under the specified in-plane compressive force, the plate shows the type of a softening-and-hardening spring. Moreover, the plate has the condition of the two to three internal resonance between the second and third modes. Under periodic lateral excitation, non-periodic responses are obtained in specific frequency regions. The non-periodic responses are examined with the Fourier spectra, the Poincare projections and the maximum Lyapunov exponents. It is found that two types of chaotic responses are generated from the internal resonances dominated by the first mode and higher modes of vibration. Appying the Karhunen Loeve method, contributions of vibration modes on the chaotic responses are confirmed. The first mode contributes to the chaotic responses dominantly. The higher three modes including the second, third and fourth modes have 10 to 20 percent of contribution. As exciting amplitude increases, the lower frequency region of chaotic response with large amplitude shifts to the higher frequency range owing to the hardening nonlinearity of the plate, while the higher region of chaotic response with small amplitude shifts to the lower frequency range.
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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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山口, 誉夫 ; 永井, 健一 ; 丸山, 真一 ; 斉藤, 友明
概要:
application/pdf<br />Journal Article<br />This paper describes vibration analysis using finite element method for two viscoelastic blocks connected by a nonlinear concentrated spring. One of the blocks is supported by another linear
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concentrated spring. The restoring force of the nonlinear concentrated spring has cubic nonlinearity and linear hysteresis damping. Finite element for the nonlinear spring is expressed and is connected to the viscoelastic blocks modeled by linear solid finite elements in consideration with complex modulus of elasticity. Further, the discretized equations in physical coordinate are transformed into the nonlinear ordinary coupled equations using normal coordinate corresponding to linear natural modes. This transformation yields computation efficiency. Influences of share of dissipated energy on nonlinear frequency responses for the viscoelastic blocks are clarifi
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永井, 健一 ; 岡田, 賢二 ; 丸山, 真一 ; 山口, 誉夫
概要:
application/pdf<br />Journal Article<br />This paper presents experimental results on chaotic vibrations of a thin circular plate with a circular center hole. The plate is clamped around the outer edge by rigid rings. Asymmetric
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defletion of the plate is induced by initial imperfection and in-plane compressive stress due to thermal elongation of the plate. Two natural modes of vibration with one nodal diameter are generated in each natural frequency. These nodal diameters are perpendicular to each other. Under periodic excitation, a dominant chaotic response generated due to the type of one-to-two internal resonance in a specific frequency range. The chaotic response is inspected by the Fourier spectrum, the Poincare projection and the maximum Lyapunov exponent. The principal component analysis is adapted on the chaotic response to confirm modal contributions. The multiple time responses of the plate are measured at four positions simultaneously for long-time interval. The modes of vibration without nodal diameter and the two modes with one nodal diameter contribute to the chaotic response, predominantly. Furthermore, changing the calculation of principal component with shorttime interval, the chaotic response of the plate shows the motion of irregular traveling waves.
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丸山, 真一 ; 小野里, 直樹 ; 永井, 健一 ; 山口, 誉夫
概要:
application/pdf<br />Journal Article<br />Experimental results are presented on nonlinear vibrations of a cylindrical shallow shell-panel with a rectangular platform. Opposite cylindrical edges of the shell-panel are clamped and the
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other opposite straight edges are simply supported. One side of the clamped edges is connected with elastic springs and is movable to an in-plane direction. The simply supported edges are supported by adhesive elastic films. Under the in-plane compressive force over the buckling load, the post-buckled configuration is obtained. In the specific compressive force, the shell-panel has the conditions of internal resonances. By exciting the shell-panel with periodic lateral acceleration, nonlinear periodic and non-periodic responses are observed in specific regions of the excitation frequency. These responses are examined with the Fourier spectra and the principal component analysis. Non-periodic responses are evaluated by the maximum Lyapunov exponent. Under the smaller magnitude of the compressive force, the principal resonance and the sub-harmonic resonance of 1/2 order corresponding to the lowest mode are obtained. Furthermore, the combination resonance dominated by the higher modes of vibration is also obtained. Under the larger magnitude of the compressive force, the chaotic response is generated from internal resonances close to the lowest natural frequency. Applying the principal component analysis, dominant modes of vibration and their contributions are confirmed on the combination resonance and the chaotic response.
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永井, 健一 ; 丸山, 真一 ; 武藤, 康太 ; 山口, 誉夫
概要:
application/pdf<br />Journal Article<br />Analytical results are presented on nonlinear vibrations of a cantilevered beam constrained by a stretched string at the tip end. The beam is subjected to periodic lateral acceleration. Governing
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equations of motion and boundary condition, which have nonlinear geometrical coupling with an axial force, are derived by the Hamilton's principle. The mode shape function proposed by the senior author is introduced to reduce the governing equations to the nonlinear differential equation by the modified Galerkin method. The mode shape function is expressed with the product of the trigonometrical function and the finite power series truncated with the fourth order. The coefficients of the power series are determined by solving the homogeneous equation for the boundary condition at the buckling state of the beam. The trigonometrical function can express the form of higher mode of vibration. Nonlinear periodic responses are calculated by the harmonic balance method. Nonlinear non-periodic responses are obtained by the numerical integration. Comparing the nonlinear responses by the analyses with the approaches of single-degree-of-freedom and of multiple-degree-of-freedom, higher mode of vibration play important roles both in the periodic response and in the chaotic response. It is shown that the analysis based on the approach of the mode shape function makes fairly good prediction to the nonlinear phenomena of the cantilevered beam constrained by the stretched string.
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| 8.
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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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図書
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永井健一著
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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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