scholarly journals Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates

Materials ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 3820
Author(s):  
Chiung-Shiann Huang ◽  
S. H. Huang
Keyword(s):  
Boundary Conditions ◽  
Material Properties ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Series Solutions ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Graded Material

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.

Dynamic Analysis of Functionally Graded Euler Beam with Elastically Restrained Edges

2014 ◽  
Vol 684 ◽  
pp. 182-190 ◽  
Author(s):  
Jun Feng Zhao ◽  
Jing Fang ◽  
Yao Li
Keyword(s):  
Boundary Conditions ◽  
Material Properties ◽  
Natural Frequencies ◽  
Functionally Graded ◽  
Spring Stiffness ◽  
Euler Beam ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

Free vibration of functionally graded materials (FGMs) Euler beam with elastically restrained edges is investigated. The material properties of the FGMs beam vary continuously in the thickness direction according to the power law form. The neutral axis site of the FGMs beam is determined by the static equilibrium condition. The governing equation and boundary conditions are found by applying the Hamilton’s principle. The linear combination of a Fourier cosine series and auxiliary Legendre polynomial function is used to obtain the natural frequencies of the FGMs beam. The effects of the rotational spring stiffness, the translational spring stiffness and the gradient index on the natural frequencies are discussed and analyzed for different material properties and different boundary conditions, indicating that the frequencies are sensitive to the gradient variation of material properties and the spring stiffness.

scholarly journals Free and Forced Vibration of the Moderately Thick Laminated Composite Rectangular Plate on Various Elastic Winkler and Pasternak Foundations

2017 ◽  
Vol 2017 ◽  
pp. 1-23 ◽  
Author(s):  
Dongyan Shi ◽  
Hong Zhang ◽  
Qingshan Wang ◽  
Shuai Zha
Keyword(s):  
Boundary Conditions ◽  
Forced Vibration ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Future Research ◽  
Rectangular Plates ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Foundation Model

An improved Fourier series method (IFSM) is applied to study the free and forced vibration characteristics of the moderately thick laminated composite rectangular plates on the elastic Winkler or Pasternak foundations which have elastic uniform supports and multipoints supports. The formulation is based on the first-order shear deformation theory (FSDT) and combined with artificial virtual spring technology and the plate-foundation interaction by establishing the two-parameter foundation model. Under the framework of this paper, the displacement and rotation functions are expressed as a double Fourier cosine series and two supplementary functions which have no relations to boundary conditions. The Rayleigh-Ritz technique is applied to solve all the series expansion coefficients. The accuracy of the results obtained by the present method is validated by being compared with the results of literatures and Finite Element Method (FEM). In this paper, some results are obtained by analyzing the varying parameters, such as different boundary conditions, the number of layers and points, the spring stiffness parameters, and foundation parameters, which can provide a benchmark for the future research.

scholarly journals An Exact Series Solution for the Vibration of Mindlin Rectangular Plates with Elastically Restrained Edges

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Xue Kai ◽  
Wang Jiufa ◽  
Li Qiuhong ◽  
Wang Weiyuan ◽  
Wang Ping
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Cross Sectional ◽  
Fourier Cosine ◽  
Cosine Series ◽  
The Matrix ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

An analysis method is proposed for the vibration analysis of the Mindlin rectangular plates with general elastically restrained edges, in which the vibration displacements and the cross-sectional rotations of the mid-plane are expressed as the linear combination of a double Fourier cosine series and four one-dimensional Fourier series. The use of these supplementary functions is to solve the possible discontinuities with first derivatives at each edge. So this method can be applied to get the exact solution for vibration of plates with general elastic boundary conditions. The matrix eigenvalue equation which is equivalent to governing differential equations of the plate can be derived through using the boundary conditions and the governing equations based on Mindlin plate theory. The natural frequencies can be got through solving the matrix equation. Finally the numerical results are presented to validate the accuracy of the method.

Sound Radiation Analysis of Functionally Graded Porous Plates with Arbitrary Boundary Conditions and Resting on Elastic Foundation

2020 ◽  
Vol 20 (05) ◽  
pp. 2050068 ◽  
Author(s):  
Zhengmin Hu ◽  
Kai Zhou ◽  
Yong Chen
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Sound Radiation ◽  
Functionally Graded ◽  
Harmonic Load ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Arbitrary Boundary Conditions ◽  
Porosity Coefficient

In this paper, the sound radiation behaviors of the functionally graded porous (FGP) plate with arbitrary boundary conditions and resting on elastic foundation are studied by means of the modified Fourier series method. It is assumed that a total of three types of porosity distributions are considered in the present study. The material parameters are determined according to the porosity coefficient used to denote the size of pores in the plate. The governing equations of the FGP plate are derived by utilizing the Hamilton’s principle on the basis of the first-order deformation theory (FSDT). Each displacement component of the FGP plate is expanded as the Fourier cosine series combined with auxiliary polynomial functions introduced to enhance the convergence rate of the series expansions. The acoustic response of the FGP plate due to a concentrated harmonic load is calculated by evaluating the Rayleigh integral. Good agreements are attained by comparing the present results with those in available literatures, which show the accuracy and versatility of the developed method in this paper. Finally, the influences of the porosity distribution type, porosity coefficient, boundary condition and elastic foundation on the sound radiation of the FGP plate are analyzed in detail.

An Exact Series Solution for the Vibration of Mindlin Rectangular Plates with Elastically Restrained Edges

2013 ◽  
Vol 572 ◽  
pp. 489-493 ◽  
Author(s):  
Kai Xue ◽  
Jiu Fa Wang ◽  
Qiu Hong Li ◽  
Wei Yuan Wang ◽  
Ping Wang
Keyword(s):  
Series Solution ◽  
Rectangular Plates ◽  
Analysis Method ◽  
Original Function ◽  
Cross Sectional ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

An analysis method has been proposed for the vibration analysis of the Mindlin rectangular plates with general elastically boundary supports, in which the vibration displacements and the cross-sectional rotations of the mid-plane are sought as the linear combination of a double Fourier cosine series and auxiliary series functions. The use of these supplementary functions is to solve the potential discontinuity associated with the x-derivative and y-derivative of the original function along the four edges, so this method can be applied to get the exact solution. Finally the numerical results are presented to validate the correct of the method.

scholarly journals Boundary Conditions in 2D Numerical and 3D Exact Models for Cylindrical Bending Analysis of Functionally Graded Structures

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
F. Tornabene ◽  
S. Brischetto ◽  
N. Fantuzzi ◽  
M. Bacciocchi
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Numerical Models ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Cylindrical Bending ◽  
Exact Model ◽  
Graded Material ◽  
Generalized Differential ◽  
Graded Structures

The cylindrical bending condition for structural models is very common in the literature because it allows an incisive and simple verification of the proposed plate and shell models. In the present paper, 2D numerical approaches (the Generalized Differential Quadrature (GDQ) and the finite element (FE) methods) are compared with an exact 3D shell solution in the case of free vibrations of functionally graded material (FGM) plates and shells. The first 18 vibration modes carried out through the 3D exact model are compared with the frequencies obtained via the 2D numerical models. All the 18 frequencies obtained via the 3D exact model are computed when the structures have simply supported boundary conditions for all the edges. If the same boundary conditions are used in the 2D numerical models, some modes are missed. Some of these missed modes can be obtained modifying the boundary conditions imposing free edges through the direction perpendicular to the direction of cylindrical bending. However, some modes cannot be calculated via the 2D numerical models even when the boundary conditions are modified because the cylindrical bending requirements cannot be imposed for numerical solutions in the curvilinear edges by definition. These features are investigated in the present paper for different geometries (plates, cylinders, and cylindrical shells), types of FGM law, lamination sequences, and thickness ratios.

scholarly journals The Modified Fourier-Ritz Approach for the Free Vibration of Functionally Graded Cylindrical, Conical, Spherical Panels and Shells of Revolution with General Boundary Condition

2017 ◽  
Vol 2017 ◽  
pp. 1-32 ◽  
Author(s):  
Lijie Li ◽  
Haichao Li ◽  
Fuzhen Pang ◽  
Xueren Wang ◽  
Yuan Du ◽  
...  
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
General Boundary ◽  
General Boundary Condition ◽  
Shells Of Revolution ◽  
Fourier Cosine ◽  
Cosine Series

The aim of this paper is to extend the modified Fourier-Ritz approach to evaluate the free vibration of four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions. The first-order shear deformation theory is employed to formulate the theoretical model. In the modified Fourier-Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solve the natural frequencies by means of the Ritz method. As one merit of this paper, the functionally graded cylindrical, conical, spherical shells are, respectively, regarded as a special functionally graded cylindrical, conical, spherical panels, and the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.

LARGE DEFLECTION FINITE STRIP ANALYSIS OF FUNCTIONALLY GRADED PLATES UNDER PRESSURE LOADS

2007 ◽  
Vol 07 (02) ◽  
pp. 193-211 ◽  
Author(s):  
H. R. OVESY ◽  
S. A. M. GHANNADPOUR
Keyword(s):  
Functionally Graded Material ◽  
Material Properties ◽  
Large Deflection ◽  
Normal Pressure ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Pressure Loading ◽  
Finite Strip ◽  
Graded Material

Description is given for a finite strip method for analyzing the large deflection response of simply supported rectangular functionally graded plates under normal pressure loading. The material properties of the functionally graded plates are assumed to vary continuously through the thickness of the plate, according to the simple power law and exponential law distribution. Both distributions of material properties are used to examine the stress variations. The fundamental equations for rectangular plates of functionally graded material (FGM) are obtained by discretizing the plate into some finite strips, which are developed by combining the Von–Karman theory for large transverse deflection and the concept of functionally graded material. The solution is obtained by the minimization of the total potential energy. The Newton–Raphson method is used to solve the non-linear equilibrium equations. Numerical results for square functionally graded plates are given in dimensionless graphical forms, and compared to the available results, wherever possible. The effects of material properties on the stress field through the thickness and on the variation of the central deflection at a given value of normal pressure loading are determined and discussed.

Supersonic Aeroelasticity and Dynamic Instability of Functionally Graded Porous Cylindrical Shells Using a Unified Solution Formulation

2020 ◽  
Vol 20 (12) ◽  
pp. 2050132
Author(s):  
Mohammad Rahmanian ◽  
Masoud Javadi
Keyword(s):  
Boundary Conditions ◽  
Power Law ◽  
Dynamic Instability ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Computational Effort ◽  
Functionally Graded ◽  
Clear Understanding ◽  
Stability Margins ◽  
Graded Material

This study provides an overview on the effect of porosity on free vibrations and more importantly aeroelastic stability margins of cylindrical shells. A general formulation for cylindrical shells is first developed including the effects of shear deformation and rotary inertia along with Sander’s rigid body rotation modification. Two porosity distributions of even and uneven are considered for functionally graded shells. The most general form of power-law model which is known as four-parameter power-law is utilized to provide a clear understanding for the qualification of functionally graded material. A Ritz-based solution algorithm being capable of representing all combinations of symmetric and asymmetric boundary conditions by a penalty method is also introduced. In addition to the capability of satisfying all boundary conditions, the presented solution method is very fast in terms of convergence and computational effort. Various parametric studies are provided and practical results are reported.

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