Non-linear phenomena in forced vibrations of a nearly square plate: Antisymmetric case

Yang, Xing; Sethna, P. R.

doi:10.1016/0022-460x(92)90710-f

Cited by 45 publications

(16 citation statements)

References 21 publications

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“…The averaged equations depend on seven parameters including the frequency of external excitation, the damping, the two linear natural frequencies, the amplitudes of excitation, and the mass of each pendulum. The averaged equations are very similar to those that have arisen in the previous studies with different physical systems with Z 2 O Z2 symmetry; namely, the motion of a nearly square plate (Yang and Sethna [8]) and the dynamics of surface waves in a nearly square container (Feng and Sethna [6]). Constant solutions of the averaged equations are studied using the local bifurcation theory.…”

Section: Introductionsupporting

confidence: 63%

“…One-to-one internal resonances occur in several physical systems, such as the spherical pendulum (Miles [3]), the stretched string (Miles [4], Johnson and Bajaj [5]), surface waves in a nearly square container (Feng and Sethna [6]), forced response of a simply supported beam (Maewal [7]), forced response of a nearly square plate (Yang and Sethna [8]) and orthogonal planar pendulum (Bridges [9]). Yang and Sethna [8], and Bridges [9] have included the effect of the detuning of the natural frequencies in their work.…”

Section: Introductionmentioning

confidence: 99%

“…Yang and Sethna [8], and Bridges [9] have included the effect of the detuning of the natural frequencies in their work.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcations in the dynamics of an orthogonal double pendulum

Samaranayake

Bajaj

1993

Nonlinear Dyn

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The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1 : 1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.

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Section: Introductionsupporting

confidence: 63%

Section: Introductionmentioning

confidence: 99%

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Bifurcations in the dynamics of an orthogonal double pendulum

Samaranayake

Bajaj

1993

Nonlinear Dyn

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“…For example, in square plates, antisymmetric modes appear in pairs with one linear natural frequency of oscillation. Among the studies in continuous systems accounting for internal resonance, and thus involving more than one mode, are the works of Sridhar et al [8], Maewal [9], Yasuda and Torii [10], Johnson and Bajaj [11], and Yang and Sethna [12].…”

Section: Introductionmentioning

confidence: 99%

“…modal equations to four first-order differential equations representing the slow-time evolution of amplitudes of harmonic motion of the two modes. These amplitude or averaged equations are a generalization of those that describe the motion of square plates [12] and membranes [10] and, when the additional restriction of circular symmetry is imposed, they have arisen in the study of resonant motions of a spherical pendulum [4], a stretched string [11], and forced response of axisymmetric shells [15] and beams [9]. The amplitude equations for the rectangular plate depend on three nonlinear coefficients, in contrast to the two independent nonlinear coefficients found in the above mentioned studies.…”

Section: Introductionmentioning

confidence: 99%

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

1993

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Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. It is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. Numerical results are presented specifically for 1:1 resonance in the (1, 2) and (3, 1) plate modes.

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Bibliography

2004

Linear and Nonlinear Structural Mechanics

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Non-linear phenomena in forced vibrations of a nearly square plate: Antisymmetric case

Cited by 45 publications

References 21 publications

Bifurcations in the dynamics of an orthogonal double pendulum

Bifurcations in the dynamics of an orthogonal double pendulum

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

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