Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate

Yang, Hao; Chen, L.H.; Zhang, W.; Lei, Jian

doi:10.1016/j.jsv.2007.11.033

Cited by 173 publications

(67 citation statements)

References 31 publications

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“…The edge width, length and thickness of the FGM rectangular plate in the x, y and z directions are a, b and h respectively. Especially the width-to-length ratio of the FGM rectangular  where 1  and 2  are the amplitudes of two modes and all the coefficients can be found in paper [1].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Hao et al [1] transformed the aforementioned equations into the averaged equations in subsequent form. The Jacobi matrix of (1) evaluated at the initial equilibrium solution 1 2 3 4 ( , , , ) x x x x  (0,0,0,0) is as follows:…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…In order to use the functionally graded materials effectively and efficiently, researchers have done various studies on stability, nonlinear vibration, bifurcation and chaos of the functionally graded materials [1][2][3][4][5][6]. In addition, much work related to complex dynamic bifurcation phenomena exhibited by a nonlinear autonomous system in the vicinity of compound critical points have been discussed by a few researchers in detail.…”

Section: Introductionmentioning

confidence: 99%

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Local Stability and Bifurcation Analysis of A Functionally Graded Material Plate

Zhang¹,

Han²

2017

Proceedings of the 3rd Annual International Conference on Mechanics and Mechanical Engineering (MME 2016)

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Abstract. In this paper, we investigated the local stability of a simply supported functionally graded material (FGM) rectangular plate subjected to the transversal and in-plate excitations in the uniform thermal environment both analytical and numerical approaches. Three kinds of critical points of the bifurcation response equations are considered, which are characterized by a double zero eigenvalues, a simple zero and a pair of pure imaginary eigenvalues as well as two pairs of pure imaginary eigenvalues in nonresonant case, respectively. With the aid of a symbolic computation language Maple and normal form theory, the explicit expressions of critical bifurcation lines are obtained. Finally, the numerical solutions obtained by using fourth-order Runge-Kutta method agree with the analytic predictions.

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Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Local Stability and Bifurcation Analysis of A Functionally Graded Material Plate

Zhang¹,

Han²

2017

Proceedings of the 3rd Annual International Conference on Mechanics and Mechanical Engineering (MME 2016)

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“…28 Hao et al observed chaos in nonlinear dynamics of a rectangular FGM plate in a thermal environment. 29 However, none of the referred papers have considered the thermo-mechanical coupling for nonlinear forced vibrations of functionally graded plates. The novelty of this paper is the full derivation of the governing equations in the threedimensional case while accounting for geometrical nonlinearity, non-homogeneity, coupled fields of temperature, and displacement in the extension of our preliminary research, which only includes the general framework for this purpose.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Vibrations and Chaos in Rectangular Functionally Graded Plates with Thermo-mechanical Coupling

Naghibi¹,

Mahzoon²

2016

IJAV

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We analyze the nonlinear dynamics of a simply supported, rectangular, and functionally graded plate in terms of a newly derived coupled system of thermo-elasticity and energy equations, which is then expanded here in derivations and explored for chaotic responses through a parameter study in the state space.1 The plate properties vary linearly in thickness. Three-dimensional stress-strain relations are considered in general case and nonlinear strain-displacement relations are deployed to account for the plate's large deflection. A lateral harmonic force is applied on the plate, and there is a heat generation source within it and the surfaces are exposed to free convection. By integrating over the thickness, four new thermal parameters are introduced, which together with the midplane displacements constitute a system of seven partial differential equations. These equations are changed into ordinary differential equations in time using Galerkin's approximation and solved by using the 4 th order RungeKutta method. Finally, a parameter study is performed and the appropriate conditions resulting in chaotic solutions are determined by using numerical features such as the Lyapunov exponent and power spectrum.

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“…Gunes and Reddy [7] investigated the geometrically nonlinear analysis of functionally graded circular plates subjected to mechanical and thermal loads. Hao [8] investigated the nonlinear dynamics of a simply-supported FGM rectangular plate with the parametric and forcing excitations.…”

Section: Introductionmentioning

confidence: 99%

Stability and Bifurcation Analysis of Functionally Graded Materials Plate Under Different Loads and Boundary Conditions

2016

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Abstract. This paper presents the stability and bifurcation analysis of a simply-supported functionally graded materials (FGMs) rectangular plate subject to the transversal and in-plane excitations. A two-degree-of-freedom nonlinear system of the FGM plate is obtained via the Hamilton's principle and the Galerkin approach. The case of primary parametric resonance and 1:2 internal resonance is considered. The asymptotic perturbation method is utilized to obtain four-dimensional nonlinear averaged equation. With the aid of Matlab and normal form theory, the various types of dynamical behavior in the neighborhood of a kind of degenerated equilibrium point are investigated. It was found that static bifurcation and Hopf bifurcation exist for the FGM rectangular plate under certain conditions.

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Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate

Cited by 173 publications

References 31 publications

Local Stability and Bifurcation Analysis of A Functionally Graded Material Plate

Local Stability and Bifurcation Analysis of A Functionally Graded Material Plate

Nonlinear Vibrations and Chaos in Rectangular Functionally Graded Plates with Thermo-mechanical Coupling

Stability and Bifurcation Analysis of Functionally Graded Materials Plate Under Different Loads and Boundary Conditions

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