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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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| 2.
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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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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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山口, 誉夫 ; 堺本, 和也 ; 永井, 健一 ; 丸山, 真一
概要:
application/pdf<br />Journal Article<br />This paper describes that eigen frequencies for two dimensional beam structures under gravity. Because the beam structures were consisted of extremely thin flexible components, the shapes of
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the structures were changed due to their self-weight. And both ends of the beam structures were assumed to be clamped. To analyze these phenomena, discrete equations using finite element in consideration with geometrical nonlinearity were derived as cubic simultaneous nonlinear differential equations. First, large static deformations of the structures due to gravity were calculated using the proposed FEM. Next, linear natural frequencies for the deformed structures were investigated. The calculated results for straight beams and shallow arches using the FEM were consistent with the theoretical results carried by authors previously. Further, the influences of self-weight on eigen frequencies of the structure which was comprised of three straight beams were clarified.
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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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| 6.
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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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| 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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NAGAI, Ken-ichi ; KASUGA, Kunio ; KAMADA, Masaki ; YAMAGUCHI, Takao ; TANIFUJI, Katsuya
概要:
application/pdf<br />Journal Article<br />Experimental results are presented on chaotic oscillations of a post-buckled reinforced beam subjected to lateral excitations.The beam is partially reinforced and is clamped at both ends.One end
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of the beam is arranged to move to an axial displacement by 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.The chaotic responses are examined by the Fourier spectrum, the Poincare projection and the maximum Lyapunov exponent.It is found among other tings that the chaos of the reinforced beam is mainly generated with the lower modes of vibration.The instability regions of chaos as well as the effect of inertia in the axial direction are also examined.
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| 9.
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井開, 重男 ; 樫本, 弘 ; 大沢, 恵一 ; 長屋, 幸助
概要:
application/pdf<br />Journal Article<br />This paper presents an actuator with a sensor for controlling vibrations of machines. It consists of a voice coil type electromagnetic actuator connected to piezoelectric sensor and a coil spring.
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The actuator is used for controlling a motor laid on the beam. The disturbance force due to the centrifugal force of the motor is detected by the sensor, and the force which is equivalent to the disturbance force is applied to the motor and the beam by the actuator. This implies that the disturbance force is canceled. The control technique as just mentioned has advantages because it reduces the vibration in the low frequency region, in addition, the actuator becomes compact, and the control force becomes small. But the control is insufficient for controlling the resonance peak. The PD control is available for reducing the resonance peak, but the displacement and velocity sensors are required. In the present article, a method of vibration control of disturbance cancellation combining the PD control is presented in which both the displacement and velocity sensors are not required. The analytical results for obtaining the control current and the response of both the motor and beam have been obtained. To validate the present actuator sensor system and the control method, experimental tests are carried out.
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| 10.
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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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