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.     

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.     

Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method

A new version of the differential quadrature method is presented in this paper to overcome the difficulty existing in the ordinary differential quadrature method for applying multi‐boundary conditions in two‐dimensional problems. Since the weighting coefficients of the first derivative are the same as for the ordinary differential quadrature method even with the introduction of multi‐degree‐of‐freedom at the boundary points, the method is easier to extend to two‐ or three‐dimensional problems. A new version of the differential quadrature plate element has been established for demonstration. The essential difference from the existing old version of the differential quadrature plate element is the way the weighting coefficients are determined. The methodology is worked out in detail and some numerical examples are given to show the efficiency of the present method. Copyright © 2004 John Wiley & Sons, Ltd.     

Comparative accuracy of DQ and HDQ methods for three‐dimensional vibration analysis of rectangular plates

An accuracy study between the Differential Quadrature (DQ) and Harmonic Differential Quadrature (HDQ) methods for three‐dimensional elasticity solutions of free vibration of rectangular plates is carried out. The solution capability of the DQ and HDQ methods is first studied. Then the numerical performance of both the methods is compared. It is found that the DQ method displays more superior convergence characteristics over the HDQ method for the lower modes of vibration. However, the HDQ method is advantageous over the DQ method for computing higher modes of vibration. It is also discovered that the DQ and HDQ methods produce better convergent solutions than the Finite Element Method (FEM) when a similar number of discrete points/nodes are used. Copyright © 1999 John Wiley & Sons, Ltd.     

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。     

Error estimates of local multiquadric‐based differential quadrature (LMQDQ) method through numerical experiments

In this article, we present an error estimate of the derivative approximation by the local multiquadric‐based differential quadrature (LMQDQ) method. Radial basis function is different from the polynomial approximation, in which Taylor series expansion is not applicable. So, the present analysis is performed through the numerical solution of Poisson equation. It is known that the approximation error of LMQDQ method depends on three factors, i.e. local density of knots h, free shape parameter c and number of supporting knots n_s). By numerical experiments, their contribution to the approximation error and correlation were studied and analysed in this paper. An error estimate ε ～ O((h/c)ⁿ) is thereafter proposed, in which n is a positive constant and determined by the number of supporting knots n_s. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Based on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem. The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices. By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.     

Semi‐analytical analysis for multi‐directional functionally graded plates: 3‐D elasticity solutions

Semi‐analytical 3‐D elasticity solutions are presented for orthotropic multi‐directional functionally graded plates using the differential quadrature method (DQM) based on the state‐space formalism. Material properties are assumed to vary not only through the thickness but also in the in‐plane directions following an exponential law. The graded in‐plane domain is solved numerically via the DQM, while exact solutions are sought for the thickness domain using the state‐space method. Convergence studies are performed, and the present hybrid semi‐analytical method is validated by comparing numerical results with the exact solutions for a conventional unidirectional functionally graded plate. Finally, effects of material gradient indices on the displacement and stress fields of the plates are investigated and discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

The static behaviour of spherical shallow shells bonded with piezoelectric actuators and subjected to electrical loading are studied in this paper by using the differential quadrature element method (DQEM). Geometrical non‐linear effects are considered. Detailed formulations for the DQ circular spherical shallow shell element and the DQ annular spherical shallow shell element are given for the first time. Numerical studies are performed to evaluate the effects of actuator size, thickness and boundary conditions. Very accurate results are obtained by the DQEM. Based on the results reported in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions for smart materials and structures exhibiting geometric non‐linear behaviours. Thickness effects cannot be neglected when the actuator thickness is comparable to that of the base material. Snap‐through may occur when the applied voltage reaches a critical value even without mechanical loading for certain geometric configurations. Copyright © 2001 John Wiley & Sons, Ltd.     

Free vibration analysis of double-bonded isotropic piezoelectric Timoshenko microbeam based on strain gradient and surface stress elasticity theories under initial stress using differential quadrature method

The size-dependent effect on free vibration of double-bonded isotropic piezoelectric Timoshenko microbeams using strain gradient and surface stress elasticity theories under initial stress is presented. This article is developed for isotropic piezoelectric material. Due to the high surface-to-volume ratio, surface stress has an important role with micro- and nanoscale materials. Thus, the Gurtin–Murdoch continuum mechanic approach is used. Governing equations of motion are derived by Hamilton's principle and solved by the differential quadrature method. The effects of pre-stress load, surface residual stress, surface mass density, surface piezoelectrics, Young's modulus of surface layers, three material length scale parameters, thickness to material length scale parameter ratios, various boundary conditions, and two elastic foundation coefficients are investigated. It is concluded that the effect of pre-stress load in greater modes is negligible for higher aspect ratios and this effect is similar to lower aspect ratios. Also, the size-dependent effect on the dimensionless natural frequency for strain gradient theory is higher than that for modified couple stress theory and classical theory, which is due to increasing stiffness of the Timoshenko microbeam model. Moreover, the results show that dimensionless natural frequency affects more by considering the material length scale parameters with respect to surface effect. The results are compared with the obtained results from the literature and show good agreement between them. It is concluded that the amplitude of the transverse displacements (w₀) for a microbeam (MB) is more than the transverse displacements (w₁) for a piezoelectric microbeam (PMB). On the other hand, using a piezoelectric layer for PMB, the amplitude of the transverse displacements (w₁) reduces considerably with respect to MB, in which this effect leads to increase the stiffness of the microbeam and stability of microstructures. With considering the piezoelectric layer, the obtained results can be used to control the amplitude and vibration of microstructures, prevent the resonance phenomenon, design smart structures, and can be employed for micro-electro-mechanical systems and nano-electro-mechanical systems.