Maximization of Natural Frequencies for Functionally Graded Rectangular Plates

2021 ◽  
Vol 891 ◽  
pp. 116-121
Author(s):  
Aleksander Muc
Keyword(s):  
Analytical Methods ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Optimization Techniques ◽  
Point Of View ◽  
Rectangular Plates ◽  
Classical Plate Theory ◽  
Arbitrary Boundary

In this paper optimal design of free vibrations for functionally graded plates is studied using the analytical methods. The analytical methods can be employed for the solution of six of 21 arbitrary boundary conditions (the combinations of clamped, simply supported and free). The influence of various models of porosity and forms of different reinforcements with nanoplatelets and carbon nanotubes are investigated, including variations of stiffness/density along the thickness of a plate. The analysis is carried out for the classical plate theory. Parametric studies illustrate the possibility of increasing natural frequencies and the necessity of implementing the optimization techniques to find the best solutions from the engineering point of view.

Homogenization Technique for Non-Linear Free Vibrations Analysis of FGM Rectangular Plates

2014 ◽  
Vol 971-973 ◽  
pp. 516-533 ◽  
Author(s):  
A. Abdenbi Boukhzer ◽  
Khalid El Bikri ◽  
Benamar Rhali
Keyword(s):  
Plate Theory ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Geometrically Nonlinear ◽  
Nonlinear Frequency ◽  
Rectangular Plates ◽  
Classical Plate Theory ◽  
Homogenization Technique ◽  
Non Linear ◽  
Good Agreement

In the present study, the problem of geometrically nonlinear free vibrations of functionally graded rectangular plates (FGRP) is studied. A homogenization technique has been developed to reduce the FGRP problem under consideration to that of isotropic homogeneous rectangular plate. The material properties of the functionally graded composites examined herein are assumed to be graded in the thickness direction of the plate and estimated through the rule of mixture. The proposed theoretical model is based on the classical plate theory and the Von Karman relationships, and the amplitude equation is derived in the form of a set of non-linear algebraic equation using Hamilton’s principle and a multimode approach. The fundamental nonlinear frequency parameters and the bending stress are then calculated using the iterative and explicit methods of solution to show the effect of the vibration amplitudes and the material distributions. The results obtained in this study are found to be in a good agreement with the published ones dealing with the problem of large vibration of functionally graded plates.

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.

Analytical Modeling on Vibration Analysis of Cracked Functionally Graded Plate Submerged in Fluid

2019 ◽  
Vol 12 (3) ◽  
pp. 240-247
Author(s):  
Shashank Soni ◽  
Nitin K. Jain ◽  
Prasad V. Joshi
Keyword(s):  
Fundamental Frequency ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Functionally Graded ◽  
Classical Plate Theory ◽  
Fluid Medium ◽  
Fluid Forces ◽  
Bernoulli’S Equation ◽  
Submerged Plate ◽  
Line Spring Model

Background: It is established that the vibration response of submerged structures is quite different than that calculated in vacuum. Therefore, the study of vibration characteristics of submerged plate structures is important for safety and its designing purpose. Objective: To investigate the fundamental frequency of partially cracked Functionally Graded (FG) submerged plate based on analytical approach. Methods: The governing differential equation of the cracked-submerged plate is derived based on Kirchhoff’s thin classical plate theory in conjunction with the potential flow theory. The line spring model is used to incorporate the effect of crack in the form of additional bending whereas the effect of fluid medium is incorporated in form fluids forces associated with inertial effects of its surrounding fluids. The Bernoulli’s equation and velocity potential function are used to define the fluid forces acting on plate surface. Results: An approximate solution for governing equation of coupled fluid-plate system is obtained by using the Galerkin’s method. For validation of the present results, they are compared with the existing results of the previous published work, which are in good agreements. New results for natural frequencies as affected by gradient index, crack length, level of submergence and immersed depth of plate are presented for Simply Supported (SSSS) boundary condition. Conclusion: It has been concluded that the presence of crack and fluidic medium significantly affect the natural frequencies of the plate. It is observed that the increase in the length of crack and level of submergence decreases the fundamental frequency. In this paper, few patents have been discussed.

Buckling Analysis of Thin Functionally Graded Rectangular Plates Resting on Elastic Foundation

2010 ◽  
Author(s):  
Meisam Mohammadi ◽  
A. R. Saidi ◽  
Mehdi Mohammadi
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Plate Theory ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Arbitrary Boundary ◽  
Coupled Equations ◽  
Aspect Ratios ◽  
Special Cases

In the present article, the buckling analysis of thin functionally graded rectangular plates resting on elastic foundation is presented. According to the classical plate theory, (Kirchhoff plate theory) and using the principle of minimum total potential energy, the equilibrium equations are obtained for a functionally graded rectangular plate. It is assumed that the plate is rested on elastic foundation, Winkler and Pasternak elastic foundations, and is subjected to in-plane loads. Since the plate is made of functionally graded materials (FGMs), there is a coupling between the equations. In order to remove the existing coupling, a new analytical method is introduced where the coupled equations are converted to decoupled equations. Therefore, it is possible to solve the stability equations analytically for special cases of boundary conditions. It is assumed that the plate is simply supported along two opposite edges in x direction and has arbitrary boundary conditions along the other edges (Levy boundary conditions). Finally, the critical buckling loads for a functionally graded plate with different boundary conditions, some aspect ratios and thickness to side ratios, various power of FGM and foundation parameter are presented in tables and figures. It is concluded that increasing the power of FGM decreases the critical buckling load and the load carrying capacity of plate increases where the plate is rested on Pasternak in comparison with the Winkler type.

scholarly journals Vibration characteristics of functionally graded carbon nanotube reinforced composite rectangular plates on Pasternak foundation with arbitrary boundary conditions and internal line supports

2018 ◽  
Vol 5 (1) ◽  
pp. 10-34 ◽  
Author(s):  
Rui Zhong ◽  
Qingshan Wang ◽  
Jinyuan Tang ◽  
Cijun Shuai ◽  
Qian Liang
Keyword(s):  
Carbon Nanotubes ◽  
Natural Frequencies ◽  
Functionally Graded ◽  
Internal Line ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Pasternak Foundation ◽  
Arbitrary Boundary ◽  
Reinforced Composite ◽  
Internal Line Supports

Abstract This paper presents the first known vibration characteristics of moderately thick functionally graded carbon nanotube reinforced composite rectangular plates on Pasternak foundation with arbitrary boundary conditions and internal line supports on the basis of the firstorder shear deformation theory. Different distributions of single walled carbon nanotubes (SWCNTs) along the thickness are considered. Uniform and other three kinds of functionally graded distributions of carbon nanotubes along the thickness direction of plates are studied. The solutions carried out using an enhanced Ritz method mainly include the following three points: Firstly, create the Lagrange energy function by the energy principle; Secondly, as the main innovation point, the modified Fourier series are chosen as the basic functions of the admissible functions of the plates to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges; Lastly, solve the natural frequencies as well as the associated mode shapes by means of the Ritz-variational energy method. In this study, the influences of the volume fraction of CNTs, distribution type of CNTs, boundary restrain parameters, location of the internal line supports, foundation coefficients on the natural frequencies and mode shapes of the FG-CNT reinforced composite rectangular plates are presented.

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.

A Semi-Analytical Study of Geometrically Nonlinear Free Axisymmetric Vibrations of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

2014 ◽  
Vol 704 ◽  
pp. 131-136
Author(s):  
El Kaak Rachid ◽  
Khalid El Bikri ◽  
Benamar Rhali
Keyword(s):  
Plate Theory ◽  
Mass Density ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Algebraic Equations ◽  
Circular Plates ◽  
Classical Plate Theory ◽  
Explicit Analytical Solution ◽  
Linear Algebraic Equations ◽  
Non Linear

. This paper deals with nonlinear free axisymmetric vibrations of functionally graded thin circular plates (FGCP) whose properties vary through its thickness. The inhomogeneity of the plate is characterized by a power law variation of the Young’s modulus and mass density of the material along the thickness direction, whereas Poisson’s ratio is assumed to be constant. The theoretical model is based on Hamilton’s principle and spectral analysis using a basis of admissible Bessel’s functions to yield the frequencies of the circular plates under clamped boundary conditions on the basis of the classical plate theory. The large vibration amplitudes problem, reduced to a set of non-linear algebraic equations, is solved numerically. The non-linear to linear frequency ratios are presented. Then, explicit analytical solutions are presented, based on the semi-analytical model previously developed by EL Kadiri et al. [1-2] for beams and rectangular plates, which allow direct and easy calculation for the first non-linear axisymmetric mode shape, with their associated non-linear frequencies of FG circular plates and which are expected to be very useful in engineering applications and in further analytical developments. An excellent agreement is found with the results obtained by the iterative method.

scholarly journals Nonlinear Dynamic Analysis for Rectangular FGM Plates with Variable Thickness Subjected to Mechanical Load

2019 ◽  
Vol 35 (3) ◽  
Author(s):  
Khuc Van Phu ◽  
Le Xuan Doan ◽  
Nguyen Van Thanh
Keyword(s):  
Variable Thickness ◽  
Nonlinear Dynamic ◽  
Volume Fraction ◽  
Mechanical Load ◽  
Plate Theory ◽  
Geometrical Nonlinearity ◽  
Nonlinear Dynamic Analysis ◽  
Dynamic Responses ◽  
Rectangular Plates ◽  
Classical Plate Theory

 In this paper, the governing equations of rectangular plates with variable thickness subjected to mechanical load are established by using the classical plate theory, the geometrical nonlinearity in von Karman-Donnell sense. Solutions of the problem are derived according to Galerkin method. Nonlinear dynamic responses, critical dynamic loads are obtained by using Runge-Kutta method and the Budiansky–Roth criterion. Effect of volume-fraction index k and some geometric factors are considered and presented in numerical results.

On the High-Temperature Free Vibration Analysis of Elastically Supported Functionally Graded Material Plates Under Mechanical In-PlaneForce Via GDQR

2019 ◽  
Vol 141 (10) ◽  
Author(s):  
Roshan Lal ◽  
Rahul Saini
Keyword(s):  
Mechanical Properties ◽  
Plate Theory ◽  
Elastic Equilibrium ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Effect Of Temperature ◽  
Axisymmetric Vibration ◽  
Thickness Direction ◽  
Classical Plate Theory ◽  
Linear Temperature

In this article, the effect of Pasternak foundation on free axisymmetric vibration of functionally graded circular plates subjected to mechanical in-plane force and a nonlinear temperature distribution (NTD) along the thickness direction has been investigated on the basis of classical plate theory. The plate material is graded in thickness direction according to a power-law distribution and its mechanical properties are assumed to be temperature-dependent (TD). At first, the equation for thermo-elastic equilibrium and then equation of motion for such a plate model have been derived by Hamilton's principle. Employing generalized differential quadrature rule (GDQR), the numerical values of thermal displacements and frequencies for clamped and simply supported plates vibrating in the first three modes have been computed. Values of in-plane force parameter for which the plate ceases to vibrate have been reported as critical buckling loads. The effect of temperature difference, material graded index, in-plane force, and foundation parameters on the frequencies has been analyzed. The benchmark results for uniform and linear temperature distributions (LTDs) have been computed. A study for plates made with the material having temperature-independent (TI) mechanical properties has also been performed as a special case. Comparison of results with the published work has been presented.

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