A Mathematical Model and Numerical Solution of a Boundary Value Problem for a Multi-Structure Plate

Yazıcı, Vildan; Muradoğlu, Zahir

doi:10.3390/math7030244

Cited by 3 publications

(3 citation statements)

References 20 publications

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“…In this study, the transmission conditions obtained on the common border of the plates that constituted the system in our previous study [22] are extended for a system consisting of plates with different mechanical properties, and the finite difference expressions of these conditions are given. It is the aim of this study to determine how long the sizes of the plates should be for obtaining the desired amount of bending against the force affecting the system under different boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Determining the Geometrical Sizes of Plates with Internal Hinges by Using Additional Conditions

Yazıcı

Muradoğlu

2020

Advances in Mathematical Physics

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For a system obtained by placing more than two elastic plates side by side, the transmission conditions are obtained at the common boundaries. Finite difference equations are developed for the problem of plates with internal hinges and applied for determination of the response of a system assembled from three different plates with different mechanical constraints between adjacent plates in this study. An algorithm is written to find out how long the size of the plates should be in order to obtain the desired amount of bending against the force affecting the system under different boundary conditions. The bisection and multigrid methods are used for this. These two methods are compared based on the obtained data.

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Section: Introductionmentioning

confidence: 99%

Determining the Geometrical Sizes of Plates with Internal Hinges by Using Additional Conditions

Yazıcı

Muradoğlu

2020

Advances in Mathematical Physics

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“…Once Hopf bifurcation occurs, the panel will oscillate for ever, which is known as panel flutter. Zhang et al investigated both the local and global bifurcations of a rectangular, thin plate [10]. Ye studied the nonlinear aeroelastic flutter and stability of the panel [11].Yang et al also provided a study on the nonlinear thermal flutter of heated curved panels in supersonic air flow using the Newton iterative approach and Runge-Kutta method [12].…”

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation of Heated Panels Flutter in Supersonic Flow

Cao

Yao

2019

Mathematics

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A differential equation of panel vibration in supersonic flow is established on the basis of the thin-plate large deflection theory under the assumption of a quasi-steady temperature field. The equation is dimensionless, and the derivation of its second-order Galerkin discretization yields a four-dimensional system. The algebraic criterion of the Hopf bifurcation is applied to study the motion stability of heated panels in supersonic flow. We provide a supplementary explanation for the proof process of a theorem, and analytical expressions of flutter dynamic pressure and panel vibration frequencies are derived. The conclusion is that the algebraic criterion of Hopf bifurcation can be applied in high-dimensional problems with many parameters. Moreover, the computational intensity of the method established in this work is less than that of conventional eigenvalue computation methods using parameter variation.

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“…The governing equation thus obtained and the variation of material properties along certain directions, due to the incorporation of functionally-graded materials, make analytical solving more and more difficult. It has been found that even for a plate consisting of classical materials, the analytical solution is hard to obtain, and in most cases, numerical simulations have to be resorted to (for example, see recent studies [22,23]). Therefore, seeking an effective analytical technique, especially since the structural property of the analytical solution may clearly show the electromechanical coupling effect in the final results, seems to be valuable and urgent.…”

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confidence: 99%

Application of Multi-Parameter Perturbation Method to Functionally-Graded, Thin, Circular Piezoelectric Plates

Yang

et al. 2020

Mathematics

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In this study, a multi-parameter perturbation method is used for the solution of a functionally-graded, thin, circular piezoelectric plate. First, by assuming that elastic, piezoelectric, and dielectric coefficients of the functionally-graded materials vary in the form of the same exponential function, the basic equation expressed in terms of two stress functions and one electrical potential function are established in cylindrical coordinate system. Three piezoelectric coefficients are selected as perturbation parameters, and the established equations are solved by the multi-parameter perturbation method, thus obtaining up to first-order perturbation solutions. The validity of the perturbation solution obtained is verified by numerical simulations, based on layer-wise theory. The perturbation process indicates that adopting three piezoelectric coefficients as perturbation parameters follows the basic idea of perturbation theory—i.e., if the piezoelectricity may be regarded as a kind of introduced disturbance, the zero-order solution of the disturbance system corresponds exactly to the solution of functionally-graded plates without piezoelectricity. The result also indicates that the deformation magnitude of piezoelectric plates is smaller than that of plates without piezoelectricity, due to the well-known piezoelectric stiffening effect.

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A Mathematical Model and Numerical Solution of a Boundary Value Problem for a Multi-Structure Plate

Cited by 3 publications

References 20 publications

Determining the Geometrical Sizes of Plates with Internal Hinges by Using Additional Conditions

Determining the Geometrical Sizes of Plates with Internal Hinges by Using Additional Conditions

Hopf Bifurcation of Heated Panels Flutter in Supersonic Flow

Application of Multi-Parameter Perturbation Method to Functionally-Graded, Thin, Circular Piezoelectric Plates

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