The multi-mode response of a square membrane.

Yasuda, Kimihiko; Torii, Takao

doi:10.1299/kikaic.53.286

Cited by 1 publication

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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%

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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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