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 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.

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.

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 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.

scholarly journals Free vibration of functionally graded carbon nanotube-reinforced conical panels integrated with piezoelectric layers subjected to elastically restrained boundary conditions

2017 ◽  
Vol 9 (7) ◽  
pp. 168781401771181 ◽  
Author(s):  
Jianyu Fan ◽  
Jin Huang ◽  
Junbo Ding ◽  
Jie Zhang
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Geometrical Parameters ◽  
Arbitrary Boundary ◽  
Piezoelectric Layers ◽  
Elastically Restrained

This article presents the free vibration of piezoelectric functionally graded carbon nanotube-reinforced composite conical panels with elastically restrained boundary conditions. The material properties of carbon nanotube-reinforced composites are assumed to be temperature-dependent and are obtained using the extended rule of mixture. First-order shear deformation theory is adopted to obtain the kinematics of the hybrid panels, and the boundary spring technique is used to implement arbitrary boundary conditions. Meanwhile, two types of electrical boundary conditions, closed circuit and open circuit, are considered for the free surfaces of the piezoelectric layers. The complete sets of electro-mechanically coupled governing equations are obtained using the Rayleigh–Ritz procedure with the Chebyshev polynomial basis functions. The resultant eigenvalue problem is solved to obtain natural frequencies and mode shapes of the hybrid panels. Convergence and comparison studies have been conducted to verify the stability and accuracy of the proposed method. Several numerical examples are examined to reveal the influences of the carbon nanotube volume fractions, carbon nanotube distribution types, boundary conditions, geometrical parameters, and temperatures on the natural frequencies of the hybrid panel. Moreover, the mode shapes of the hybrid panels under various boundary conditions are also presented.

Analysis of vibration in rotating pretwisted functionally graded graphene platelets reinforced nanocomposite laminated blades with an attached point mass

2021 ◽  
pp. 095440622110084
Author(s):  
Amin Ghorbani Shenas ◽  
Parviz Malekzadeh ◽  
Sima Ziaee
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Polymer Matrix ◽  
Natural Frequencies ◽  
Weight Fraction ◽  
Functionally Graded ◽  
Point Mass ◽  
Attached Mass ◽  
Graphene Platelets ◽  
Twisted Beam

This work presents an investigation on the free vibration behavior of rotating pre-twisted functionally graded graphene platelets reinforced composite (FG-GPLRC) laminated blades/beams with an attached point mass. The considered beams are constituted of [Formula: see text] layers which are bonded perfectly and made of a mixture of isotropic polymer matrix and graphene platelets (GPLs). The weight fraction of GPLs changes in a layer-wise manner. The effective material properties of FG-GPLRC layers are computed by using the modified Halpin-Tsai model together with rule of mixture. The free vibration eigenvalue equations are developed based on the Reddy’s third-order shear deformation theory (TSDT) using the Chebyshev–Ritz method under different boundary conditions. After validating the approach, the influences of the GPLs distribution pattern, GPLs weight fraction, angular velocity, the variation of the angle of twist along the beam axis, the ratio of attached mass to the beam mass, boundary conditions, position of attached mass, and geometry on the vibration behavior are investigated. The findings demonstrate that the natural frequencies of the rotating pre-twisted FG-GPLRC laminated beams significantly increases by adding a very small amount of GPLs into polymer matrix. It is shown that placing more GPLs near the top and bottom surfaces of the pre-twisted beam is an effective way to strengthen the pre-twisted beam stiffness and increase the natural frequencies.

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels integrated with piezoelectric layers

2014 ◽  
Vol 21 (4) ◽  
pp. 571-587 ◽  
Author(s):  
Hamid Reza Saeidi Marzangoo ◽  
Mostafa Jalal
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Simply Supported ◽  
Piezoelectric Layers ◽  
Curved Panels

AbstractFree vibration analysis of functionally graded (FG) curved panels integrated with piezoelectric layers under various boundary conditions is studied. A panel with two opposite edges is simply supported, and arbitrary boundary conditions at the other edges are considered. Two different models of material property variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential law distribution of the material properties through the thickness are considered. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. For the simply supported boundary conditions, closed-form solution is given by making use of the Fourier series expansion, and applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free-end conditions. Natural frequencies of the hybrid curved panels are presented by solving the eigenfrequency equation, which can be obtained by using edges boundary conditions in this state equation. The results obtained for only FGM shell is verified by comparing the natural frequencies with the results obtained in the literature.

Magneto-electro-elastic vibration of moderately thick FG annular plates subjected to multi physical loads in thermal environment using GDQ method by considering neutral surface

2019 ◽  
Vol 233 (10) ◽  
pp. 2140-2159 ◽  
Author(s):  
Ehsan Arshid ◽  
Ali Kiani ◽  
Saeed Amir
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Thermal Environment ◽  
Differential Quadrature Method ◽  
Shear Deformation Theory ◽  
Elastic Vibration ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Neutral Surface ◽  
Sensors And Actuators

The vibration analysis of an annular plate made up of functionally graded magneto-electro-elastic materials subjected to multi physical loads is presented. The plate is in thermal environment and temperature is distributed non-uniformly in its thickness direction. In addition, the plate is assumed moderately thick, the material properties vary through the thickness, and the exact neutral surface position is determined and took into account. According to Hamilton’s principle and the first-order shear deformation theory, the governing motion equations are extracted. Numerical results for various boundary conditions are obtained via the generalized differential quadrature method and are validated in simpler states with those of the literature. The effects of different parameters such as material property gradient index, multi physical loads, temperature variations, boundary conditions and geometric specifications of the plate on the natural frequencies and mode shapes are investigated. Temperature changes have little effect on the natural frequencies and the effect of electric potential on them is opposite of magnetic one. In other words, by increasing the magnetic potential, the rigidity of the plate increases too, and the frequency increases. The results of this study are useful to design more efficient sensors and actuators used in the smart or intelligent structures.

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