Thermo-mechanical vibration of orthotropic cantilever and propped cantilever nanoplate using generalized differential quadrature method

In this article, the vibration frequency of an orthotropic nanoplate under the effect of temperature change is investigated. Using nonlocal elasticity theory, governing equations are derived. Based on the generalized differential quadrature method for cantilever and propped cantilever boundary conditions, the frequencies of orthotropic nanoplates are considered and the obtained results are compared with valid reported results in the literature. The effects of temperature variation, small scale, different boundary conditions, aspect ratio, and length on natural nondimensional frequencies are studied. The present analysis is applicable for the design of rotating and nonrotating nano-devices that make use of thermo-mechanical vibration characteristics of nanoplates.     

The generalized differential quadrature method for frequency analysis of a rotating conical shell with initial pressure

Using the generalized differential quadrature (GDQ) method which is an improved version of the differential quadrature (DQ) method, this paper examines the influence of initial pressure load on the free vibration of a rotating thin truncated circular isotropic conical shell with different boundary conditions. The present governing equations of motion include the influence of initial stress field and the effects of initial hoop tension and also the centrifugal and coriolis accelerations due to rotation. The influence of initial pressure on the frequency characteristics is discussed in detail for various conditions. To validate the present analysis, frequency comparisons are made with those available in published works, and very good agreements are obtained. Copyright © 2000 John Wiley & Sons, Ltd.     

Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method

The transverse vibration of a rotary tapered microbeam is studied based on a modified couple stress theory and Euler–Bernoulli beam model. The governing differential equation and boundary conditions are derived according to Hamilton's principle. The generalized differential quadrature element method is then used to solve the governing equation for cantilever and propped cantilever boundary conditions. The effect of the small-scale parameter, beam length, rate of cross-section change, hub radius, and nondimensional angular velocity on the vibration behavior of the microbeam is presented.     

Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method

The small scale effect on the vibration analysis of orthotropic single layered graphene sheets (SLGS) is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived for the graphene sheets. Differential quadrature method (DQM) is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) material properties and (iv) boundary conditions on nondimensional vibration frequencies are investigated.     

Moving least squares differential quadrature method and its application to the analysis of shear deformable plates

A moving least squares differential quadrature (MLSDQ) method is developed and employed for the analysis of moderately thick plates based on the first‐order shear deformation theory (FSDT). To carry out the analysis, the governing equations in terms of the generalized displacements (transverse deflection and two rotations) of the plate are formulated by employing the moving least squares approximation. The weighting coefficients used in the MLSDQ approximation are computed through a fast computation of shape functions and their derivatives. Numerical examples illustrating the accuracy, stability and convergence of the MLSDQ method are presented. Effects of support size, order of completeness and node irregularity on the numerical accuracy are investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

Extended three-dimensional generalized differential quadrature method: The basic equations and thermal vibration analysis of functionally graded fiber orientation rectangular plates

The present article deals with free vibration of functionally graded fiber orientation rectangular plates considering temperature effect. Three different types of fiber orientation distributions through the thickness of the plate are proposed. The properties of the plate are assumed to be temperature-dependent. Equations of motions are derived based on a three-dimensional theory of elasticity. General differential quadrature method is used to discretize these equations. Effects of temperature, fiber orientation, and boundary conditions besides some geometric parameters are presented. Also, some interesting conclusions are obtained since temperature and functionality of a functionally graded plate have a significant effect on the natural frequency of the plate.     

Application of a new differential quadrature methodology for free vibration analysis of plates

A new methodology is introduced in the differential quadrature (DQ) analysis of plate problems. The proposed approach is distinct from other DQ methods by employing the multiple boundary conditions in a different manner. For structural and plate problems, the methodology employs the displacement within the domain as the only degree of freedom, whereas along the boundaries the displacements as well as the second derivatives of the displacements with respect to the co‐ordinate variable normal to the boundary in the computational domain are considered as the degrees of freedom for the problem. Employing such a procedure would facilitate the boundary conditions to be implemented exactly and conveniently. In order to demonstrate the capability of the new methodology, all cases of free vibration analysis of rectangular isotropic plates, in which the conventional DQ methods have had some sort of difficulty to arrive at a converged or accurate solution, are carried out. Excellent convergence behaviour and accuracy in comparison with exact results and/or results obtained by other approximate methods were obtained. The analogous DQ formulation for a general rectangular plate is derived and for each individual boundary condition the general format for imposing the given conditions is devised. It must be emphasized that the computational efforts of this new methodology are not more than for the conventional differential quadrature methods. Copyright © 2002 John Wiley & Sons, Ltd.     

Differential quadrature method for nonlocal nonlinear vibration analysis of a boron nitride nanotube using sinusoidal shear deformation theory

This article presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single-walled boron nitride nanotubes (SWBNNTs). The surrounding elastic medium is simulated based on nonlinear Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the SWBNNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the nonlinear frequency is presented, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, vibrational modes, length, and elastic medium on the nonlinear frequency of SWBNNTs are considered.     

Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory

ABSTRACT

This article investigates the nonlinear vibration of piezoelectric nanoplate with combined thermo-electric loads under various boundary conditions. The piezoelectric nanoplate model is developed by using the Mindlin plate theory and nonlocal theory. The von Karman type nonlinearity and nonlocal constitutive relationships are employed to derive governing equations through Hamilton's principle. The differential quadrature method is used to discretize the governing equations, which are then solved through a direct iterative method. A detailed parametric study is conducted to examine the effects of the nonlocal parameter, external electric voltage, and temperature rise on the nonlinear vibration characteristics of piezoelectric nanoplates.     

Free vibration analysis of generally laminated beams via state-space-based differential quadrature

A new method of state-space-based differential quadrature is presented for free vibration of generally laminated beams. By discretizing the state space formulations along the axial direction using the technique of differential quadrature, new state equations at discrete points are established. Applying end conditions and using matrix theory, the general solution is derived. Taking account of the boundary conditions at the top and bottom planes, frequency equation governing the free vibration of generally laminated beams is then formulated. The method is validated by comparing numerical results with that available in the literature.     

截锥壳颤振分析的微分求积法

引入微分求积法(Differential Quadrature Method,简称DQM)对截锥壳气动弹性方程离散,采用一阶活塞理论气动力,运用特征值分析方法求解系统的颤振临界动压。研究了半顶角、径厚比、长径比等几何参数对颤振临界动压的影响。结果表明,DQM求解截锥壳气动弹性方程具有良好的精度和计算效率,结构产生1阶~2阶耦合型颤振的最低临界动压对应的周向波数较大,并因几何参数而异;颤振临界动压参数随半顶角的增大而减小,随着径厚比的增大而增大,随长径比的增大而减小。     

Three‐dimensional theory of elasticity for free vibration analysis of composite laminates via layerwise differential quadrature modelling

In this paper, the free vibration analysis of simply‐supported and clamped composite laminates, especially thick laminates, is carried out. The three‐dimensional theory of elasticity is integrated into a layerwise model via differential quadrature discretization. All physical governing equations are satisfied, including the additional constraints of the characteristics of continuity and discontinuity of interfacial transverse and in‐plane strains and stresses along the interfaces of composite laminates. Effects of plate aspect and thickness ratios on the free vibration of these laminates are examined in detail. This study demonstrates the applicability, accuracy, and stability of the present methodology, for vibration analyses of composite structures of thick laminated constitution. Copyright © 2003 John Wiley & Sons, Ltd.     

Vibration analysis of size-dependent flexoelectric nanoplates incorporating surface and thermal effects

This article is concerned with the thermo-mechanical vibration behavior of flexoelectric nanoplates under uniform and linear temperature distributions. Flexoelectric nanoplates have higher natural frequencies than conventional piezoelectric nanoplates, especially at lower thicknesses. Both nonlocal and surface effects are considered in the analysis of flexoelectric nanoplates for the first time. Hamilton's principle is employed to derive the governing equations and the related boundary conditions, which are solved applying the Galerkin-based solution. A comparison study is also performed to verify the present formulation with those of previous data. Numerical results are presented to investigate the influences of the flexoelectricity, nonlocal parameters, surface elasticity, temperature rise, plate thickness, and various boundary conditions on the vibration frequencies of thermally affected flexoelectric nanoplate.     

基于微分求积法的轴向运动板横向振动分析

研究受面内载荷轴向运动薄板横向振动的运动微分方程,采用微分求积法计算四边简支轴向运动薄板的固有频率和临界速度。分析轴向运动速度、板材料刚度及长宽比对板横向振动固有频率及临界速度的影响。结果发现,随着轴向速度增大,各阶固有频率减小;随着刚度的增大,各阶固有频率增大;当长宽比较小时,轴向运动板可以用梁模型分析。     

The generalized differential quadrature rule for fourth‐order differential equations

The generalized differential quadrature rule (GDQR) proposed here is aimed at solving high‐order differential equations. The improved approach is completely exempted from the use of the existing δ‐point technique by applying multiple conditions in a rigorous manner. The GDQR is used here to static and dynamic analyses of Bernoulli–Euler beams and classical rectangular plates. Numerical error analysis caused by the method itself is carried out in the beam analysis. Independent variables for the plate are first defined. The explicit weighting coefficients are derived for a fourth‐order differential equation with two conditions at two different points. It is quite evident that the GDQR expressions and weighting coefficients for two‐dimensional problems are not a direct application of those for one‐dimensional problems. The GDQR are implemented through a number of examples. Good results are obtained in this work. Copyright © 2001 John Wiley & Sons, Ltd.     

Buckling analysis of composite plates using moving least squares differential quadrature method

This work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Young's modulus gradation ratio, shear modulus gradation ratio, Poisson's ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions.     

On the random differential quadrature (RDQ) method: consistency analysis and application in elasticity problems

Differential Quadrature (DQ) is an efficient derivative approximation technique but it requires a regular domain with uniformly arranged nodes. This restricts its application for a regular domain only discretized by the field nodes in a fixed pattern. In the presented random differential quadrature (RDQ) method however this restriction of the DQ method is removed and its applicability is extended for a regular domain discretized by randomly distributed field nodes and for an irregular domain discretized by uniform or randomly distributed field nodes. The consistency analysis of the locally applied DQ method is carried out, based on it approaches are suggested to obtain the fast convergence of function value by the RDQ method. The convergence studies are carried out by solving 1D, 2D and elasticity problems and it is concluded that the RDQ method can effectively handle regular as well as irregular domains discretized by random or uniformly distributed field nodes.     

Buckling analysis of orthotropic thin rectangular plates subjected to nonlinear in-plane distributed loads using generalized differential quadrature method

In the present work, buckling analysis of orthotropic thin rectangular plates with uniform thickness resting on Pasternak foundation are investigated for eight types of boundary conditions: SSSS, CCCC, SCSC, SSSC, SSCC, CCCF, SSFC, and CFCF. Based on classical plate theory, governing differential equation in buckling are solved numerically using generalized differential quadrature method (GDQM) to obtain critical buckling loads and corresponding modes. The kinds of nonlinear loading are presented in six cases including symmetrical and unsymmetrical distribution. In addition, the effects of aspect ratio, orthotropic moduli ratio and coefficients of foundation on the buckling load are illustrated. The present work is the first attempt to consider the influence of the nonlinearity of distributed in-plane bi-directional loading in determination of buckling load and representation of the corresponding shape modes. Some numerical examples are provided to demonstrate good accuracy of the GDQ method to evaluate the critical buckling load in case of nonlinear distributed bi-directional compressive loads. As shown, profile of distributed in-plane loading plays an important role on buckling behavior of the rectangular plate.     

Application of generalized differential quadrature rule in Blasius and Onsager equations

The generalized differential quadrature rule (GDQR) proposed recently by the authors is applied here to third‐order non‐linear differential equations of the Blasius type and to sixth‐order linear Onsager differential equations. High (?3rd)‐order differential equations in fluid mechanics are dealt with without using δ‐point techniques. The half‐space domain is simplified in a practical way. Accurate results are obtained for both kinds of problems. The wide applicability of the GDQR in high‐order differential equations is manifested further through this work. Copyright © 2001 John Wiley & Sons, Ltd.     

A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains

A differential quadrature hierarchical finite element method (DQHFEM) is proposed by expressing the hierarchical finite element method matrices in similar form as in the differential quadrature finite element method and introducing interpolation basis on the boundary of hierarchical finite element method elements. The DQHFEM is similar as the fixed interface mode synthesis method but the DQHFEM does not need modal analysis. The DQHFEM with non‐uniform rational B‐splines elements were shown to accomplish similar destination as the isogeometric analysis. Three key points that determine the accuracy, efficiency and convergence of DQHFEM were addressed, namely, (1) the Gauss–Lobatto–Legendre points should be used as nodes, (2) the recursion formula should be used to compute high‐order orthogonal polynomials, and (3) the separation variable feature of the basis should be used to save computational cost. Numerical comparison and convergence studies of the DQHFEM were carried out by comparing the DQHFEM results for vibration and bending of Mindlin plates with available exact or highly accurate approximate results in literatures. The DQHFEM can present highly accurate results using only a few sampling points. Meanwhile, the order of the DQHFEM can be as high as needed for high‐frequency vibration analysis. Copyright © 2016 John Wiley & Sons, Ltd.