Periodic and chaotic dynamics of composite laminated piezoelectric rectangular plate with one-to-two internal resonance

The existence and bifurcations of the subharmonic orbits for four-dimensional non-autonomous nonlinear systems are investigated in this paper. The improved subharmonic Melnikov method is presented by using the periodic transformations and Poincaré map. The theoretical results and the formulas are obtained, which can be used to analyze the subharmonic dynamic responses of four-dimensional nonautonomous nonlinear systems. The improved subharmonic Melnikov method is used to investigate the subharmonic orbits of a simply supported rectangular thin plate under combined parametric and external excitations for verifying the validity and applicability of the method. The theoretical results indicate that the subharmonic orbits can occur in the rectangular thin plate with 1:1 and 1:2 internal resonances. The results of numerical simulation also indicate the existence of the subharmonic orbits for the rectangular thin plate, which can verify the analytical predictions.

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“…(27) and (29)(30)(31), the subharmonic Melnikov function of Eq. (77) with 1:1 internal resonance is obtained as follows…”

Section: Subharmonic Orbit Of a Rectangular Thin Platementioning

confidence: 99%

“…Amabili et al [26] investigated the nonlinear vibrations of composite laminated plates with different boundary conditions. Zhang and Yao [27] studied the bifurcations and chaos of composite laminated piezoelectric rectangular plate with one-to-two internal resonance.…”

Section: Introductionmentioning

confidence: 99%

Subharmonic Melnikov method of four-dimensional non-autonomous systems and application to a rectangular thin plate

et al. 2015

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“…Axial and oblique impact crash tests on empty and honeycomb filled aluminum square tubes have been performed by Zarei and Kroger [17]. Zhang et al [18] established the nonlinear governing equations of motion for a simply supported laminated composite piezoelectric rectangular plate under combined parametric and transverse excitations and studied the periodic and chaotic dynamics in the case of one-to-two internal resonance. Yang and Lim [19] investigated nonlinear vibrations and stability of an axially moving beam.…”

Section: Introductionmentioning

confidence: 99%

Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a honeycomb sandwich plate

Zhang

2012

Acta Mech

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The global bifurcations and multi-pulse chaotic dynamics of a simply supported honeycomb sandwich rectangular plate under combined parametric and transverse excitations are investigated in this paper for the first time. The extended Melnikov method is generalized to investigate the multi-pulse chaotic dynamics of the non-autonomous nonlinear dynamical system. The main theoretical results and the formulas are obtained for the extended Melnikov method of the non-autonomous nonlinear dynamical system. The nonlinear governing equation of the honeycomb sandwich rectangular plate is derived by using the Hamilton's principle and the Galerkin's approach. A two-degree-of-freedom non-autonomous nonlinear equation of motion is obtained. It is known that the less simplification processes on the system will result in a better understanding of the behaviors of the multi-pulse chaotic dynamics for high-dimensional nonlinear systems. Therefore, the extended Melnikov method of the non-autonomous nonlinear dynamical system is directly utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the two-degree-of-freedom non-autonomous nonlinear system for the honeycomb sandwich rectangular plate. The theoretical results obtained here indicate that multi-pulse chaotic motions can occur in the honeycomb sandwich rectangular plate. Numerical simulation is also employed to find the multi-pulse chaotic motions of the honeycomb sandwich rectangular plate. It also demonstrates the validation of the theoretical prediction.

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“…That procedure has been verified in pursuing the periodic solution of the general SNOS [16,17]. For discussing the global bifurcation [18][19][20], mainly about the homoclinic bifurcation problem, we use the Melnikov method to investigate in detail the critical conditions for the associated homoclinic structures with induced saddle states basing on the improved averaged equations. At last, the numerical computations are employed to support the analytical results.…”

Section: Introductionmentioning

confidence: 99%

Global bifurcations of strongly nonlinear oscillator induced by parametric and external excitation

Wang¹,

Zhang²,

Feng³

2011

Sci. China Technol. Sci.

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The global bifurcation of strongly nonlinear oscillator induced by parametric and external excitation is researched. It is known that the parametric and external excitation may induce additional saddle states, and result in chaos in the phase space, which cannot be detected by applying the Melnikov method directly. A feasible solution for this problem is the combination of the averaged equations and Melnikov method. Therefore, we consider the averaged equations of the system subject to DuffingVan der Pol strong nonlinearity by introducing the undetermined fundamental frequency. Then the bifurcation values of homoclinic structure formation are detected through the combined application of the new averaged equations with Melnikov integration. Finally, the explicit application shows the analytical conditions coincide with the results of numerical simulation even disturbing parameter is of arbitrary magnitude. global bifurcation, strongly nonlinear, chaos, Melnikov method Citation:Wang W, Zhang Q C, Feng J J. Global bifurcations of strongly nonlinear oscillator induced by parametric and external excitation.

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Periodic and chaotic dynamics of composite laminated piezoelectric rectangular plate with one-to-two internal resonance

Cited by 64 publications

References 21 publications

Subharmonic Melnikov method of four-dimensional non-autonomous systems and application to a rectangular thin plate

Subharmonic Melnikov method of four-dimensional non-autonomous systems and application to a rectangular thin plate

Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a honeycomb sandwich plate

Global bifurcations of strongly nonlinear oscillator induced by parametric and external excitation

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