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| 1.
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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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| 2.
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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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| 3.
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丸山, 真一 ; 永井, 健一 ; 服部, 伯慕 ; 山口, 誉夫
概要:
application/pdf<br />Journal Article<br />Analytical results are presented on chaotic vibrations of a shallow arch with initial configuration of catenary. The shallow arch is simply supported at both ends and is subjected to periodic
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lateral force. Equations of motion are reduced to ordinary differential equations of multiple-degree-of-freedom system by the Galerkin procedure. To clarify effects of sag-to-span ratio on chaotic motions, first, steady-state responses are calculated by the harmonic balance method. The chaotic responses are obtained by numerical integration and are examined by Poincare projection, the maximum Lyapunov exponent and bifurcation diagrams. For the arch with comparatively small sag-to-span ratio, chaotic vibrations are generated mainly in the regions of subharmonic resonance of 1/3 order and of ultra-subharmonic resonance of 3/2 order, accompanied by the lowest-symmetric-mode of vibration. As the sag-to-span ratio increases, chaotic responses are generated from the subharmonic resonances both of 1/2 and 1/3 orders. Furthermore, chaotic motion appears within a wider range of exciting frequency and the number of modes that contribute to the chaos increases. Period doubling bifurcation was mainly observed in the bifurcation from periodic responses to chaotic ones. Hopf bifurcation was also found for the arch that satisfies the condition of internal resonance.
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| 4.
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山口, 誉夫 ; 永井, 健一 ; 丸山, 真一
概要:
application/pdf<br />Journal Article<br />It's difficult to clarify deformation patterns in chaotic vibration of a beam, because multiple modes are generated simultaneously. This paper describes identification of spatial modes in a
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chaotic vibration for a buckled beam. KL (Karhunen-Loeve) method was applied to the identification. Using this method, time histories are decomposed into components which have no correlation each other. Contribution of the components to the original time histories can be estimated as eigenvalues of covarriance matrix of the time histories. Moreover, we used the corresponding eigenvectors to identify spatial modes in the chaos. We focused on chaotic motion of the beam involving a dynamic snap-through phenomena. The time histories for the identification were given from numerical analysis. The identified eigenvectors were compared with the natural modes of vibration. As a result, effectiveness of KL method was revealed.
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| 5.
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永井, 健一 ; 鈴木, 央 ; 山口, 誉夫 ; 丸山, 真一
概要:
application/pdf<br />Journal Article<br />Analytical result is presented on chaotic oscillations of a buckled beam under an axial spring. The beam with a concentrated mass is clamped at both ends. The beam is compressed to the
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post-buckled configuration by the axial spring. The beam is subjected to periodic lateral acceleration. Introducing a mode shape function to a basic equation, nonlinear ordinary differential equations of multiple-degree-of-freedom system is reduced to by the Galerkin procedure. Changing a stiffness of the axial spring, an internal resonance condition of one to two is selected. First, steady-state resonance responses are calculated by the harmonic balance method. The chaotic responses are obtained by a numerical integration. The chaotic responses are examined by the Poincare projection, the maximum Lyapunov exponent and bifurcation diagrams. The chaos due to the internal resonance is easily generated by a small amplitude of excitation. As the exciting frequency decreased, transition to the chaos from a periodic response needs larger amplitude of excitation. In a lower range of frequency, the chaotic oscillations are mixed with the internal resonance and the dynamic snap buckling. Two modes of vibration contribute to the chaos related to the internal resonance. Number of the modes increases more than three for the chaos involved the dynamic snap buckling.
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山口, 誉夫 ; 永井, 健一
概要:
application/pdf<br />Journal Article<br />This paper presents numerical results on chaotic vibrations of a shallow cylindrical shell-panel under harmonic lateral excitation. The shell with a rectangular boundary is simply supported
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for deflection and the shell is constrained elastically in both in-plane directions. The effects of bi-axial in-plane elastic constraint on the chaotic behaviors of the shell are focused on. Using the Donnell-Mushtari-Vlasov type equation modified with an inertia force, the basic equation is reduced to nonlinear differential equation of a multiple-degree-of-freedom system by the Galerkin procedure. To estimate regions of the chaos, first, nonlinear responses of steady state vibration are calculated by the harmonic balance method. Next, time progresses of the chaotic response are obtained numerically by the Runge-Kutta-Gill method. The chaotic responses due to dynamic snap-through are examined by Fourier spectrum, Poincare projection and Lyapunov exponent. Contribution of multiple modes of vibration to the chaos is examined carefully by the Lyapunov dimension. The main results can be summarized as follows. Loosening the in-plane constraint perpendicularly along the curved edges and tightening the in-plane constraint along the straight edges, chaotic motion is restricted with less number of modes of vibration.
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| 7.
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山口, 誉夫 ; 永井, 健一 ; 鈴木, 央
概要:
application/pdf<br />Journal Article<br />Chaotic responses are investigated for a post-buckled beam under the interaction between internal resonance and dynamic snap-through. The beam with variable cross section is clamped at both ends and
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the beam is axially compressed to a post-buckled configuration. The buckled beam is excited by a periodic acceleration. Applying the Galerkin method to the governing equation of the beam, nonlinear differential equations of a multi-degree-of-freedom system are reduced. Periodic solutions of steady-state responses are calculated by the harmonic balance method. In a typical frequency region, chaotic response bifurcates from the periodic response. Time progress of the chaotic motion is calculated by the numerical integration. The chaotic response is examined in detail by the Fourier spectrum, the Poincare section, the Lyapunov exponent and the Lyapunov dimension, respectively. Under the condition of the one-to-two internal resonance, the chaos is generated by a small amplitude of excitation. Two modes of the vibration induced in the chaos. Increasing the amplitude of excitation, the chaotic response transits to the complicated response coupled with the dynamic snap-through and the internal resonance. Induced modes in the chaos are counted as nearly four.
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| 8.
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永井, 健一 ; 大山, 貞夫 ; 山口, 誉夫
概要:
application/pdf<br />Journal Article<br />Analytical results are presented on chaotic vibrations of arches with axial elastic constraints. The arch with variable-cross-section is subjected to a periodic lateral force. The mode shape
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function proposed by the senior author-satisfies four sets of boundary conditions combined with a simply supported end and a clamped end. Basic equation is reduced to an ordinary differential equation of multiple-degrees-of-freedom system by the Galerkin procedure. Steady-state responses of the arches are calculated by the harmonic balance method. When the arch is contained simply supported condition, chaotic responses are excited easily in a specific frequency region. The chaotic responses are examined by the Runge-kutta-gill method. Following conclusion was obtained. In the chaos of the arch with the rigid axial constraint, many natural modes of vibration of both symmetric and asymmetric modes are excited simultaneously. The Lyapunov dimension increases as the rigid axial constraint of the arch becomes more tight. Furthermore, the chaotic attractors in the Poincare section show fractal geometries.
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| 9.
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永井, 健一 ; 春日, 邦夫 ; 鎌田, 昌樹 ; 山口, 誉夫 ; 谷藤, 克也
概要:
application/pdf<br />Journal Article<br />Experimental results are presented on chaotic oscillations of a reinforced beam subjected to lateral excitations. The beam is partially reinforced with boxed-type stringers. The beam is clamped at
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both ends on a base frame. One end of the beam is arranged to move to an axial displacement by attachment to an elastic spring. The beam is deformed to a post-buckled configuration by the axial constraint. Under the post-buckled condition of the beam, chaotic responses are generated in specified regions of exciting frequency. A response is expected from a system with a lower degree of freedom. The chaotic responses are analyzed by the Fourier spectrum, the Poincare section and the maximum Lyapunov exponent. It is found that the chaos of the beam is generated with the fundamental mode of vibration. Chaotic response includes the resonance modes both of a higher lateral vibration and of an axial vibration. The Poincare projections of the chaos show clearly the stretching-and-folding mechanism of the chaos attractor. The instability boundary of the chaos is obtained in the plane of exciting frequency and amplitudes of excitation.
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| 10.
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永井, 健一 ; 山口, 誉夫
概要:
application/pdf<br />Journal Article<br />This paper presents the numerical simulation for the chaotic vibrations of a post-buckled beam with a variable cross section. The beam is clamped at both ends and is compressed in an axial
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direction. When the beam is deformed to a post-buckled configuration, chaotic vibration will be excited easily under periodic lateral acceleration. Applying Galerkin method, the basic equation of the beam is reduced to the ordinary differential equation of multiple-degrees-of-freedom systems. To get the steady-state response of the beam, the harmonic balance method is used. In a typical region of frequency, the chaotic response is excited. The chaotic time progress is calculated by the numerical integration. The chaotic response is examined carefully by the Poincare projection, the Lyapunov exponent and the Lyapunov dimension. The results of tapered beams are compared with that of the straight beam. The chaotic response with the asymmetric mode of vibration appears remarkably. The response shows the complicated projection to the chaotic attracter, moreover, the Lyapunov dimension takes higher value. The chaos of the beam with variable cross section shows the more complicated behavior to that of straight beam.
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