An efficient differential quadrature methodology for free vibration analysis of arbitrary straight-sided quadrilateral thin plates

In this paper, a new differential quadrature (DQ) methodology is employed to study free vibration of irregular quadrilateral straight-sided thin plates. A four-nodded super element is used to map the irregular physical domain into a square domain in the computational domain. Second order transformation schemes with relative ease and less computation are employed to transform the fourth order governing equation of thin plates between the two domains. The only degree of freedom within the domain is the displacement, whereas along the boundaries, the displacement as well as the second order derivative of the displacement with respect to associated normal co-ordinate variable in computational domain are the two degrees of freedom. Implementing the method, the formulation for the DQ method for the free vibration analysis of plates of straight-sided shapes was presented together with the implementation procedure for the different boundary conditions. To demonstrate the accuracy, convergency and stability of the new methodology, detail studies are made on isotropic plates at acute angles with different geometries, boundary and loading conditions including DQ free-edge boundary condition implementations. Accurate results even with fewer degrees of freedom than for those of comparable numerical algorithms were achieved.     

A semi-analytic solution for free vibration of annular sector plates

The paper describes a semi-analytical method in which the basic function in the circumferential direction satisfying the boundary conditions of the radial edges is substituted into the free vibration equation of the curved plate. By a suitable transformation, an ordinary differential equation is obtained. The resulting equation is solved by a finite difference technique. Tabulated results have been presented for annular sector plates possessing different boundary conditions. Excellent accuracy has been obtained wherever comparisons have been possible.     

Three-dimensional free vibration of thick functionally graded annular plates in thermal environment

The free vibration analysis of functionally graded (FG) thick annular plates subjected to thermal environment is studied based on the 3D elasticity theory. The material properties are assumed to be temperature dependent and graded in the thickness direction. Considering the thermal environment effects and using Hamilton's principle, the equations of motion are derived. The effects of the initial thermal stresses are considered accurately by obtaining them from the 3D thermoelastic equilibrium equations. The differential quadrature method (DQM) as an efficient and accurate numerical tool is used to solve both the thermoelastic equilibrium and free vibration equations. Very fast rate of convergence of the method is demonstrated. Also, the formulation is validated by comparing the results with those obtained based on the first-order shear deformation theory and also with those available in the literature for the limit cases, i.e. annular plates without thermal effects. The effects of temperature rise, material and geometrical parameters on the natural frequencies are investigated. The new results can be used as benchmark solutions for future researches.     

求解声辐射问题的微分求积法与微分求积单元法

基于无限域中的Helmholtz波动方程,将微分求积法与微分求积单元法应用于二维及三维声辐射问题的求解,对最外层节点施加不同阶数的人工边界条件,区域内使用均匀及非均匀的节点分布方式,分析了节点分布方式及人工边界条件对计算结果的影响,比较了两种数值方法的计算精度。研究结果表明:微分求积法与微分求积单元法,前者精度更高,而后者耗时更少,在频率较低时,具备较高的效费比。人工边界条件对计算结果的影响主要体现在低频段,而节点分布方式的影响主要体现在高频段。非均匀的节点分布方式在不同频段都具备更好的计算精度。     

Closed-form solution for free vibration of piezoelectric coupled annular plates using Levinson plate theory

Free vibration analysis of annular moderately thick plates integrated with piezoelectric layers is investigated in this study for different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Levinson plate theory (LPT). The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study the closed-form solution for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. To demonstrate the accuracy of the present solution, comparison studies is first carried out with the available data in the literature and then natural frequencies of the piezoelectric coupled annular plate are presented for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric and boundary conditions. Present analytical model provides design reference for piezoelectric material application, such as sensors, actuators and ultrasonic motors.     

Free vibration of annular sector plates by an integral equation technique

A numerical method is presented for the free vibration analysis of polar orthotropic clamped annular sector plates. The results are compared with analytical and experimental values of other investigators. A parametric study has been done by varying the sector angle and radii ratio. The frequencies for isotropic and orthotropic cases are presented in the form of graphs.     

Natural vibration of circular and annular thin plates by Hamiltonian approach

The present paper deals with the natural vibration of thin circular and annular plates using Hamiltonian approach. It is based on the conservation principle of mixed energy and is constructed in a new symplectic space. A set of Hamiltonian dual equations with derivatives with respect to the radial coordinate on one side of the equations and to the angular coordinate on the other side are obtained by using the variational principle of mixed energy. The separation of variables is employed to solve Hamiltonian dual equations of eigenvalue problem. Analytical frequency equations are obtained based on different cases of boundary conditions. The natural frequencies are the roots of the frequency equations and corresponding mode functions are in terms of the dual variables q₁(r, θ). Three basic edge-constraint cases for circular plates and nine edge-constraint cases for annular plates are calculated and the results are compared well with existing ones.     

An extended transfer matrix-finite element method for free vibration of plates

A combination of extended transfer matrix and finite element methods is proposed for obtaining vibration frequencies of structures. This method yields the value of the frequency once a trial value is assumed. By using this technique, the number of nodes required in the regular finite element method is reduced and therefore a smaller computer can be used. Besides, no plotting of the values of the determinants corresponding to each assumed frequency is necessary. A worked example is given for the case of vibration of a cantilever plate. The results show fast convergence from the assumed value to the true natural frequency.     

变厚度圆板、环板振动分析的传递矩阵法*

变厚度圆板和环板是在工程设计中经常遇到的一类构件,与等厚度板相比,通过适当的沿径向厚度的变化,这种变厚度板在振动、失稳、弯曲等方面能起到更好的效果。将沿径向任意变厚度圆板、环板划分为一系列等厚度环板单元,基于Mindlin中厚板理论采用逆向推导的方式推导了其传递矩阵,建立起变厚度圆板、环板的频率方程。通过计算线性变厚度环板自由振动时的频率,并与ANSYS模态分析结果相比较,验证了计算模型的精确性。逆向推导法避免了高阶数传递矩阵推导复杂的问题,是对传递矩阵法的很好推广。     

Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method

This paper presents a mesh-free Galerkin method for the free vibration and stability analyses of stiffened plates via the first-order shear deformable theory (FSDT). The model of a stiffened plate is formed by (1) regarding the plate and the stiffener separately, (2) imposing displacement compatible conditions between the plate and the stiffener so that displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate, and (3) superimposing the strain energy of plate and stiffener. Because there are no meshes used in this method, the stiffeners can be placed anywhere on the plate and need not be placed along the mesh lines. Several numerical examples are computed by this method to show its accuracy and convergence. The present results demonstrate good agreement with the existing solutions given by other researchers and the ANSYS. Influences of support size and order of the complete basis functions on the numerical accuracy are also investigated.     

Three-dimensional free vibration analysis of isotropic rectangular plates using the B-spline Ritz method

This paper presents three-dimensional free vibration analysis of isotropic rectangular plates with any thicknesses and arbitrary boundary conditions using the B-spline Ritz method based on the theory of elasticity. The proposed method is formulated by the Ritz procedure with a triplicate series of B-spline functions as amplitude displacement components. The geometric boundary conditions are numerically satisfied by the method of artificial spring. To demonstrate the convergence and accuracy of the present method, several examples with various boundary conditions are solved, and the results are compared with other published solutions by exact and other numerical methods based on the theory of elasticity and various plate theories. Rapid, stable convergences as well as high accuracy are obtained by the present method. The effects of geometric parameters on the vibrational behavior of cantilevered rectangular plates are also investigated. The results reported here may serve as benchmark data for finite element solutions and future developments in numerical methods.     

Nonlinear free vibrations of curved double walled carbon nanotubes using differential quadrature method

Nonlinear free vibration analysis of curved double-walled carbon nanotubes (DWNTs) embedded in an elastic medium is studied in this study. Nonlinearities considered are due to large deflection of carbon nanotubes (geometric nonlinearity) and nonlinear interlayer van der Waals forces between inner and outer tubes. The differential quadrature method (DQM) is utilized to discretize the partial differential equations of motion in spatial domain, which resulted in a nonlinear set of algebraic equations of motion. The effect of nonlinearities, different end conditions, initial curvature, and stiffness of the surrounding elastic medium, and vibrational modes on the nonlinear free vibration of DWCNTs is studied. Results show that it is possible to detect different vibration modes occurring at a single vibration frequency when CNTs vibrate in the out-of-phase vibration mode. Moreover, it is observed that boundary conditions have significant effect on the nonlinear natural frequencies of the DWCNT including multiple solutions.     

2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures

This paper focuses on the dynamic behavior of functionally graded conical, cylindrical shells and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The first-order shear deformation theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. The homogeneous isotropic material is inferred as a special case of functionally graded materials (FGM). The governing equations of motion, expressed as functions of five kinematic parameters, are discretized by means of the generalized differential quadrature (GDQ) method. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. For the homogeneous isotropic special case, numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Nastran, Straus, Pro/Mechanica. Very good agreement is observed. Furthermore, the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. Different typologies of non-uniform grid point distributions are considered. Finally, for the functionally graded material case numerical results illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behavior of shell structures.     

Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method

The main aim of this paper is to provide a simple yet efficient solution for the free vibration analysis of functionally graded (FG) conical shells and annular plates. A solution approach based on Haar wavelet is introduced and the first-order shear deformation shell theory is adopted to formulate the theoretical model. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. The separation of variables is first performed; then Haar wavelet discretization is applied with respect to the axial direction and Fourier series is assumed with respect to the circumferential direction. The constants appearing from the integrating process are determined by boundary conditions, and thus the partial differential equations are transformed into algebraic equations. Then natural frequencies of the FG shells are obtained by solving algebraic equations. Accuracy and reliability of the current method are validated by comparing the present results with the existing solutions. Effects of some geometrical and material parameters on the natural frequencies of shells are discussed and some selected mode shapes are given for illustrative purposes. It’s found that accurate frequencies can be obtained by using a small number of collocation points and boundary conditions can be easily achieved. The advantages of this current solution method consist in its simplicity, fast convergence and excellent accuracy.