Non-linear vibrations of a clamped beam with initial deflection and initial axial displacement, Part I: Theory

Yamaki, N.; Mori, Atsunobu

doi:10.1016/0022-460x(80)90417-4

Cited by 62 publications

(25 citation statements)

References 14 publications

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“…In (29), b (1) i (τ ) and b (2) i (τ ) correspond to the deflection and the velocity in the state space, respectively. New dissipation terms of linear damping are introduced.…”

Section: Problem Formulationmentioning

confidence: 99%

“…Transforming the equation of the variableb j (τ ) to the normal coordinates b (1) i , τ = b (2) i , b (2) i , τ = −2ε i ω i b (2) …”

Section: Problem Formulationmentioning

confidence: 99%

“…Nonlinear and chaotic vibrations of a beam have been investigated by many researchers including the authors [1][2][3][4][5][6][7][8][9][10]. Pizeshki and Dowell [6] studied chaotic vibrations of a post-buckled beam using the Lyapunov dimension to estimate the number of vibration modes in the chaos.…”

Section: Introductionmentioning

confidence: 99%

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Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint

et al. 2011

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Analytical results are presented on chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. The L-shaped beam is composed of two beams which are a horizontal beam and a vertical beam. The two beams are firmly connected with a right angle at each end. The beams joint with the right angle is attached to a linear spring. The other ends are firmly clamped for displacement. The L-shaped beam is compressed horizontally via the spring at the beams joint. The L-shaped beam deforms to a post-buckled configuration. Boundary conditions are required with geometrical continuity of displacements and dynamical equilibrium with axial force, bending moment, and share force, respectively. In the analysis, the mode shape function proposed by the senior author is introduced. The coefficients of the mode shape function are fixed to satisfy boundary conditions of displacements and linearized equilibrium conditions of force and moment. Assuming responses of the beam with the sum of the mode shape function, then applying the modified Galerkin procedure to the governing equations, a set of nonlinear ordinary differential equations is obtained in a multiple-degree-offreedom system. Nonlinear responses of the beam are calculated under periodic lateral acceleration. Nonlinear frequency response curves are computed with the harmonic balance method in a wide range of excitation frequency. Chaotic vibrations are obtained with the numerical integration in a specific frequency region. The chaotic responses are investigated with the Fourier spectra, the Poincaré projections, the maximum Lyapunov exponents and the Lyapunov dimension. Applying the procedure of the proper orthogonal decomposition to the chaotic responses, contribution of vibration modes to the chaotic responses is confirmed. The following results have been found: The chaotic responses are generated with the ultra-subharmonic resonant response of the two-third order corresponding to the lowest mode of vibration. The Lyapunov dimension shows that three modes of vibration contribute to the chaotic vibrations predominantly. The results of proper orthogonal decomposition confirm that the three modes contribute to the chaos, which are the first, second, and third modes of vibration. Moreover, the results of the proper orthogonal decomposition are evaluated with velocity which is equivalent to kinetic energy. Higher modes of vibration show larger contribution to the chaotic responses, even though the first mode of vibration has the largest contribution ratio.

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“…In (29), b (1) i (τ ) and b (2) i (τ ) correspond to the deflection and the velocity in the state space, respectively. New dissipation terms of linear damping are introduced.…”

Section: Problem Formulationmentioning

confidence: 99%

“…Transforming the equation of the variableb j (τ ) to the normal coordinates b (1) i , τ = b (2) i , b (2) i , τ = −2ε i ω i b (2) …”

Section: Problem Formulationmentioning

confidence: 99%

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Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint

et al. 2011

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“…The beam, initially straight with no axial force, gave a Duffing's equation response along with superharmonic and subharmonic motions. Yamaki and Mori [6] and Yamaki et al [7] analytically and experimentally studied a clamped-clamped beam response for the first three symmetric modes using the method of harmonic balance. The results, referenced to the initial equilibrium position, were obtained in terms of initial axial displacements but the effects of damping, asymmetric modes and internal resonance were not studied.…”

Section: Introductionmentioning

confidence: 99%

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

Kovacs

Ibrahim

1996

Nonlinear Dyn

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The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N -1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.

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“…Singh and Ali [13] studied a moderately thick clamped beam with a sinusoidal rise function by adding the effects of transverse shear and rotary inertia. Yamaki and Mori [14] analyzed a clamped buckled beam by considering the first three symmetric modes. Krishnan and Suresh [15] carried out free vibration studies using two beam elements, which one of these had three degree-of-freedom and the other four.…”

mentioning

confidence: 99%

Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

2009

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In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlinear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of multiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequencies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.

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Non-linear vibrations of a clamped beam with initial deflection and initial axial displacement, Part I: Theory

Cited by 62 publications

References 14 publications

Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint

Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

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