Vibration Behavior and Simplified Design of Thick Rectangular Plates With Variable Thickness

2002 ◽  
Vol 124 (2) ◽  
pp. 302-309 ◽  
Author(s):  
M. Sasajima ◽  
T. Kakudate ◽  
Y. Narita
Keyword(s):  
Optimal Design ◽  
Boundary Conditions ◽  
Fundamental Frequency ◽  
Plate Theory ◽  
Free Vibration Analysis ◽  
Design Approach ◽  
Mindlin Plate ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Classical Plate Theory

A free vibration analysis and an optimal design approach have been presented for thick isotropic rectangular plates with varying thickness under general edge conditions. First, the analysis is developed for vibrating rectangular plates by using the Mindlin plate theory and an eigenvalue problem is formulated by extending a method of Ritz to arbitrary sets of standard boundary conditions. The classical plate theory is also used to derive the frequency equation for comparison purpose. Secondly, a simplified optimal design approach is proposed to maximize the fundamental frequency of the plates. In applying this approach, the thickness variation is assumed to be linear in one direction and a taper ratio is chosen to be a design variable that represents the whole plate design. This approach significantly reduces the process for obtaining optimal or nearly optimal design under constraint of the constant plate volume. Numerical results are presented for various sets of boundary conditions, thickness ratio and two different plate theories, and their effects on the optimal taper ratio and its corresponding maximized fundamental frequency are discussed.

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.

Buckling and Vibration of Orthotropic Nonhomogeneous Rectangular Plates With Bilinear Thickness Variation

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Yajuvindra Kumar ◽  
R. Lal
Keyword(s):  
Orthogonal Polynomials ◽  
Plate Theory ◽  
Ritz Method ◽  
Cartesian Product ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Two Dimensional ◽  
Classical Plate Theory ◽  
Simply Supported

An analysis and numerical results are presented for buckling and transverse vibration of orthotropic nonhomogeneous rectangular plates of variable thickness using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh–Ritz method on the basis of classical plate theory when uniformly distributed in-plane loading is acting at two opposite edges clamped/simply supported. The Gram–Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of various plate parameters such as nonhomogeneity parameters, aspect ratio together with thickness variation, and in-plane load on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported, and free edges correct to four decimal places. Three dimensional mode shapes for a specified plate for all the four boundary conditions have been plotted. By allowing the frequency to approach zero, the critical buckling loads in compression for various values of plate parameters have been computed correct to six significant digits. A comparison of results with those available in the literature has been presented.

scholarly journals Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 429 ◽  
Author(s):  
Krzysztof Żur ◽  
Piotr Jankowski
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Special Functions ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Circular Plates ◽  
Classical Plate Theory ◽  
Coupled Equations

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as a linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the neglected effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

On Free Vibration of Functionally Graded Mindlin Plate and Effect of In-Plane Displacements

2013 ◽  
Vol 29 (2) ◽  
pp. 373-384 ◽  
Author(s):  
A. Hasani Baferani ◽  
A.R. Saidi ◽  
H. Ehteshami
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Various Boundary Conditions ◽  
Aspect Ratios

AbstractIn this paper, free vibration analysis of functionally graded rectangular plate is investigated based on the first order shear deformation theory and the effect of in-plane displacements on the natural frequencies of such plate is studied. The governing equations of motion are obtained, which are five coupled partial differential equations, without any simplification. Some mathematical manipulation leads us to decouple the equations. The decoupled equations are solved by the Levy's method for various boundary conditions. As the results show, in some boundary conditions the in-plane displacements cause a drastic change of frequencies. In other words, neglecting the in-plane displacement, which is assumed in some papers, is not proper for these boundary conditions. However, in the other boundary conditions, the natural frequencies are not significantly affected by the in-plane displacements. The results for various boundary conditions are discussed in detail and some interpretations for these differences are provided. Besides to the comparisons, the accurate natural frequencies of the plate for six different boundary conditions with several aspect ratios, thickness-length ratios and power law indices are presented. The natural frequencies of Mindlin functionally graded rectangular plates with considering the in-plane displacements are reported for the first time and can be used as benchmark.

scholarly journals Exact Analytical Solution for Free Axisymmetric and Non-Axisymmetric Vibrations of FGM Porous Circular Plates

2018 ◽  
Author(s):  
Krzysztof Kamil Żur ◽  
Piotr Jankowski
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Special Functions ◽  
Plate Theory ◽  
Exact Analytical Solution ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Circular Plates ◽  
Classical Plate Theory ◽  
Coupled Equations

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the negligibled effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

A novel analytical approach for the buckling analysis of moderately thick functionally graded rectangular plates with two simply-supported opposite edges

2010 ◽  
Vol 224 (9) ◽  
pp. 1831-1841 ◽  
Author(s):  
M Mohammadi ◽  
A R Saidi ◽  
E Jomehzadeh
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Arbitrary Boundary ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Fg Plates

In this article, a novel analytical method for decoupling the coupled stability equations of functionally graded (FG) rectangular plates is introduced. Based on the Mindlin plate theory, the governing stability equations that are coupled in terms of displacement components are derived. Introducing four new functions, the coupled stability equations are converted into two independent equations. The obtained equations have been solved for buckling analysis of rectangular plates with simply-supported two edges and arbitrary boundary conditions along the other edges (Levy boundary conditions). The critical buckling loads are presented for different loading conditions, various thickness to side and aspect ratios, some powers of FG materials, and various boundary conditions. The presented results for buckling of moderately thick FG plates with two simply-supported edges are reported for the first time.

An Exact Levy Solution for Free Vibration Analysis of FGM Plates With Integrated Piezoelectric Layers

2011 ◽  
Author(s):  
M. A. Askari-Farsangi ◽  
A. R. Saidi
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Plate Theory ◽  
Free Vibration Analysis ◽  
Open Circuit ◽  
Classical Plate Theory ◽  
Arbitrary Boundary ◽  
Piezoelectric Layers ◽  
Coupled Plates

In the present research, an exact Levy solution for the free vibration analysis of FGM plates with integrated piezoelectric layers is established. The electrical boundary conditions and the Maxwell’s equation are satisfied by assuming the transverse distribution of electric potential to be a combination of a parabolic and a linear function. On the basis of classical plate theory, the coupled governing equations for both closed and open circuit piezoelectric coupled structures are reformulated into two new decoupled equations. The numerical results for the Levy type of boundary conditions, two opposite edges simply supported and arbitrary boundary conditions at the other edges, and both closed and open circuit piezoelectric coupled plates have been presented. The effects of piezoelectric layer thickness on the natural frequency for different power law indices, thickness-length ratios, modes, boundary conditions, and material properties are investigated.

Length scale-dependent natural frequencies of piezoelectric microplates

2017 ◽  
Vol 24 (13) ◽  
pp. 2749-2759 ◽  
Author(s):  
M Jafari ◽  
E Jomehzadeh ◽  
M Rezaeizadeh
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Free Vibration Analysis ◽  
Length Scale ◽  
Governing Equations ◽  
Classical Plate Theory ◽  
Wide Range ◽  
Piezoelectric Layers

The length-scale free vibration analysis of a rectangular microplate coupled with piezoelectric layers is presented. The modified couple stress theory is used to describe the size effect of the system. The governing equations of motion are obtained using Hamilton’s principle based on the classical plate theory. The transverse part of the electric potential for the piezoelectric layers is considered to satisfy the Maxwell’s equation and the electrical boundary conditions. A new procedure is introduced to decouple the governing equations and then an analytical Levy-type solution is obtained. The exact natural frequencies are established for a wide range of length scales, various plate dimensions, several piezoelectric layer thicknesses, and different boundary conditions. The results show that the effect of length scale parameter is decreased by the piezoelectric electrical field.

Dynamical Behaviors of Sinusoidal Nanocomposite Plates Subjected to Thermomechanical Loads

2021 ◽  
Author(s):  
Vu Ngoc Viet Hoang ◽  
Dinh Gia Ninh
Keyword(s):  
Thermal Environment ◽  
Plate Theory ◽  
Micromechanical Model ◽  
Functionally Graded ◽  
Geometrical Parameters ◽  
Rectangular Plates ◽  
Sine Function ◽  
Classical Plate Theory ◽  
Wolfram Mathematica ◽  
Number Of Cycles

In this paper, a new plate structure has been found with the change of profile according to the sine function which we temporarily call as the sinusoidal plate. The classical plate theory and Galerkin’s technique have been utilized in estimating the nonlinear vibration behavior of the new non-rectangular plates reinforced by functionally graded (FG) graphene nanoplatelets (GNPs) resting on the Kerr foundation. The FG-GNP plates were assumed to have two horizontal variable edges according to the sine function. Four different configurations of the FG-GNP plates based on the number of cycles of sine function were analyzed. The material characteristics of the GNPs were evaluated in terms of two models called the Halpin–Tsai micromechanical model and the rule of mixtures. First, to verify this method, the natural frequencies of new non-rectangular plates made of metal were compared with those obtained by the Finite Element Method (FEM). Then, the numerical outcomes are validated by comparing with the previous papers for rectangular FGM/GNP plates — a special case of this structure. Furthermore, the impacts of the thermal environment, geometrical parameters, and the elastic foundation on the dynamical responses are scrutinized by the 2D/3D graphical results and coded in Wolfram-Mathematica. The results of this work proved that the introduced approach has the advantages of being fast, having high accuracy, and involving uncomplicated calculation.

Vibration Analysis of Laminated Composite Rectangular Plates With General Boundary Conditions

2018 ◽  
Author(s):  
Yu Fu ◽  
Jianjun Yao ◽  
Zhenshuai Wan ◽  
Gang Zhao
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Geometric Parameters ◽  
General Boundary ◽  
Rectangular Plates ◽  
General Boundary Conditions ◽  
Benchmark Solutions

In this investigation, the free vibration analysis of laminated composite rectangular plates with general boundary conditions is performed with a modified Fourier series method. Vibration characteristics of the plates have been obtained via an energy function represented in the general coordinates, in which the displacement and rotation in each direction is described as an improved form of double Fourier cosine series and several closed-form auxiliary functions to eliminate any possible jumps and boundary discontinuities. All the expansion coefficients are then treated as the generalized coordinates and determined by Rayleigh-Ritz method. The convergence and reliability of the current method are verified by comparing with the results in the literature and those of Finite Element Analysis. The effects of boundary conditions and geometric parameters on the frequencies are discussed as well. Finally, numerous new results for laminated composite rectangular plates with different geometric parameters are presented for various boundary conditions, which may serve as benchmark solutions for future research.

Sign in / Sign up

Export Citation Format

Share Document