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 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.

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.

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.

On the Buckling of Isotropic, Transversely Isotropic and Laminated Composite Rectangular Plates

2014 ◽  
Vol 14 (07) ◽  
pp. 1450020 ◽  
Author(s):  
Atteshamuddin S. Sayyad ◽  
Yuwaraj M. Ghugal
Keyword(s):  
Boundary Conditions ◽  
Elasticity Theory ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Transversely Isotropic ◽  
Buckling Analysis ◽  
Transverse Shear Strain ◽  
Rectangular Plates

This paper presents the uniaxial and biaxial buckling analysis of rectangular plates based on new trigonometric shear and normal deformation theory. The theory accounts for the cosine distribution of the transverse shear strain through the plate thickness and on the free boundary conditions on the plate surfaces without using the shear correction factor. Governing equations and boundary conditions of the theory are derived by the principle of virtual work. The Navier type solutions for the buckling analysis of simply supported isotropic, transversely isotropic, orthotropic and symmetric cross-ply laminated composite rectangular plates subjected to uniaxial and biaxial compressions are presented. The effects of variations in the degree of orthotropy of the individual layers, side-to-thickness ratio and aspect ratio of the plate are examined on the buckling characteristics of composite plates. The present results are compared with those of the classical plate theory (CPT), first order shear deformation theory (FSDT) and exact three-dimensional (3D) elasticity theory wherever applicable. Good agreement is achieved of the present results with those of higher order shear deformation theory (HSDT) and elasticity theory.

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.

Vibration of a Rectangular Plate Carrying a Massive Machine with Elastic Supports

2016 ◽  
Vol 16 (10) ◽  
pp. 1550069 ◽  
Author(s):  
Lingzhi Wang ◽  
Zhitao Yan ◽  
Zhengliang Li ◽  
Zhimiao Yan
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Engineering Practice ◽  
Element Analysis ◽  
Optimal Position ◽  
Elastic Supports ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Plate Size

In engineering practice, a massive machine may be placed on a plate supported by beams considered as elastic boundary conditions. The vibration of the plate due to the periodic excitation of the massive machine will cause noises or damages to the building in which the machine is housed. An analytical approach for the vibration analysis of a rectangular plate carrying a massive machine with uniform elastic supports is presented. The machine is simplified as a distributed mass. The transverse plate displacement is determined by the superposition of a two-dimensional (2D) Fourier cosine series and several supplementary functions. All the unknown Fourier coefficients are calculated directly from the Rayleigh–Ritz formulation. To validate the present approach, several numerical examples with classical boundary conditions are presented. The results reveal good agreement between the analytical results and those based on the finite element analysis (ANSYS). The effects of the plate size, location of the machine, and support stiffness on the modal, and transient response of the plate are investigated. From the results it is found that the transient displacement amplitude of the plate decreases almost linearly as the thickness increases, it increases nonlinearly along with the increase in the support stiffness, and that the optimal position for deploying the transformer is the center of the plate.

scholarly journals A Series Solution for the Vibration of Mindlin Rectangular Plates with Elastic Point Supports around the Edges

2018 ◽  
Vol 2018 ◽  
pp. 1-21 ◽  
Author(s):  
Fuzhen Pang ◽  
Haichao Li ◽  
Yuan Du ◽  
Shuo Li ◽  
Hailong Chen ◽  
...  
Keyword(s):  
Series Solution ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Cross Sectional ◽  
Numerical Examples ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Fourier Series Method ◽  
Edge Conditions

A series solution for the transverse vibration of Mindlin rectangular plates with elastic point supports around the edges is studied. The series solution for the problem is obtained using improved Fourier series method, in which the vibration displacements and the cross-sectional rotations of the midplane are represented by a double Fourier cosine series and four supplementary functions. The supplementary functions are expressed as the combination of trigonometric functions and a single cosine series expansion and are introduced to remove the potential discontinuities associated with the original admissible functions along the edges when they are viewed as periodic functions defined over the entire x-y plane. This series solution is approximately accurate in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. The convergence, accuracy, stability, and efficiency of the proposed method have been examined through a series of numerical examples. Some numerical examples about the nondimensional frequency and mode shapes of Mindlin rectangular plates with different point-supported edge conditions are given.

scholarly journals Forced Vibration Analysis of Laminated Composite Shells Reinforced with Graphene Nanoplatelets Using Finite Element Method

2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Trung Thanh Tran ◽  
Van Ke Tran ◽  
Pham Binh Le ◽  
Van Minh Phung ◽  
Van Thom Do ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Forced Vibration ◽  
Vibration Analysis ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Thermal Environments ◽  
Forced Vibration Analysis ◽  
Laminated Composite Shells ◽  
Element Method

This paper carries out forced vibration analysis of graphene nanoplatelet-reinforced composite laminated shells in thermal environments by employing the finite element method (FEM). Material properties including elastic modulus, specific gravity, and Poisson’s ratio are determined according to the Halpin–Tsai model. The first-order shear deformation theory (FSDT), which is based on the 8-node isoparametric element to establish the oscillation equation of shell structure, is employed in this work. We then code the computing program in the MATLAB application and examine the verification of convergence rate and reliability of the program by comparing the data of present work with those of other exact solutions. The effects of both geometric parameters and mechanical properties of materials on the forced vibration of the structure are investigated.

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