Differential Quadrature Technique Research Articles

Dynamic Analysis of Electrostatic Microactuators Using the Differential Quadrature Method

This work studies the dynamic behavior of electrostatic actuators using finite‐element package software (FEMLAB) and differential quadrature method. The differential quadrature technique is used to transform partial differential equations into a discrete eigenvalue problem. Numerical results indicate that length, width, and thickness significantly impact the frequencies of the electrostatic actuators. The thickness could not affect markedly the electrostatic actuator capacities. The effects of varying actuator length, width, and thickness on the dynamic behavior and actuator capacities in electrostatic actuator systems are investigated. The differential quadrature method is an efficient differential equation solver.

Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method

In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of laminated composite doubly-curved shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Examples of hyperbolic, catenary, cycloid, parabolic, elliptic and circular shell and panel structures are presented to illustrate the validity and the accuracy of the GDQ method. Furthermore, GDQ results are compared with those presented in literature and the ones obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed.

Magnetohydrodynamic Natural Convection Flow in an Enclosure with a Finite Length Heater Using the Differential Quadrature (DQ) Method

Analysis of heat and fluid flow transport due to natural convection and magnetohydrodynamic (MHD) flows in a square enclosure with a finite length heater has been performed using the differential quadrature (DQ) technique. The heater with constant heat flux is located on the bottom wall of the enclosure and isothermal boundary conditions are applied to the right vertical wall while the remaining walls are adiabatic. The effects of heater length (0.2 ≤ ϵ ≤ 0.8), heater location (0.1 ≤ c/L ≤ 0.9), and direction of magnetic force (0° ≤ φ ≤ 90°) for different values of Grashof (103 ≤ Gr ≤ 106) and Hartmann numbers (0 ≤ Ha ≤ 100) on the heat and fluid flow in the enclosure are studied. According to the results obtained, heat transfer reduces when increasing the Hartmann number. The rate of reduction is higher for high values of Grashof number. The heat transfer rate for the heater closer to the cold wall is considerably higher than the heaters far from the right wall.

Vibration analysis of orthotropic rectangular plates on elastic foundations

The vibration responses of orthotropic plates on nonlinear elastic foundations are numerically modeled using the differential quadrature method. The differential quadrature technique is utilized to transform partial differential equations into a discrete eigenvalue problem. Numerical results and those from literature closely correspond to each other. Numerical results demonstrate that elastically restrained stiffness, plate aspect ratio and foundation stiffness significantly impact the dynamic behavior of orthotropic plates.

On the Convergence of Random Differential Quadrature (RDQ) Method and Its Application in Solving Nonlinear Differential Equations in Mechanics

Differential Quadrature (DQ) is one of the efficient derivative approximation techniques but it requires a regular domain with all the points distributed only along straight lines. This severely restricts the DQ while solving the irregular domain problems discretized by the random field nodes. This limitation of the DQ method is overcome in a proposed novel strong-form meshless method, called the random differential quadrature (RDQ) method. The RDQ method extends the applicability of the DQ technique over the irregular or regular domains discretized using the random field nodes by approximating a function value with the fixed reproducing kernel particle method (fixed RUM), and discretizing a governing differential equation by the locally applied DQ method. A superconvergence condition is developed for the RDQ method, which gives more than O(h(p+1)) function value convergence for the uniform as well as random field nodes scattered in the domain. The RDQ method convergence analysis is carried out, and the superconvergence condition is verified by solving several 1D, 2D and elasticity problems. The applicability of the RDQ method to solve the nonlinear governing differential equations is successfully demonstrated by solving the fixed-fixed and cantilever beams for deflection due to the nonlinear electrostatic loading. It is concluded that the RDQ method effectively handles the irregular or regular domains discretized by the uniform or random field nodes, with good convergence rates.

Free vibration and buckling analysis of functionally graded deep beam-columns on two-parameter elastic foundations using the differential quadrature method

In this research, a differential quadrature method is applied for free vibration and buckling analysis of deep beam-columns composed of functionally graded materials on two-parameter elastic foundations. Derivation of equations is based on the unconstrained higher-order shear deformation theory taking into account the complete effects of shear deformation, depth change, and rotary inertia. It is assumed that the effective mechanical properties of functionally graded (FG) beam-columns are temperature dependent and vary continuously throughout the thickness direction according to volume fraction of the constituents defined by power-law function. The accuracy, convergence, and flexibility of the differential quadrature technique for simply supported FG deep beam-columns with complicated governing differential equations and boundary conditions are examined and verified with the known data in the literature.

3D thermoelasticity solutions for functionally graded thick plates

Thermal-mechanical behavior of functionally graded thick plates, with one pair of opposite edges simply supported, is investigated based on 3D thermoelasticity. As for the arbitrary boundary conditions, a semi-analytical solution is presented via a hybrid approach combining the state space method and the technique of differential quadrature. The temperature field in the plate is determined according to the steady-state 3D thermal conduction. The Mori-Tanaka method with a power-law volume fraction profile is used to predict the effective material properties including the bulk and shear moduli, while the effective coefficient of thermal expansion and the thermal conductivity are estimated using other micromechanics-based models. To facilitate the implementation of state space analysis through the thickness direction, the approximate laminate model is employed to reduce the inhomogeneous plate into a homogeneous laminate that delivers a state equation with constant coefficients. The present solutions are validated by comparisons with the exact ones for both thin and thick plates. Effects of gradient indices, volume fraction of ceramics, and boundary conditions on the thermomechanical behavior of functionally graded plates are discussed.

Dynamic behaviour of wind turbine blades

Wind power does not generate pollution and is a clean source. The dynamic problems associated with wind turbine blades are formulated using the differential quadrature method. The Euler—Bernoulli beam paradigm is used to characterize wind turbine blades. The differential quadrature technique is utilized to transform partial differential equations, which presents the dynamic behaviour of wind turbine blades, into a discrete eigenvalue problem. The effects of the number of sample points on the accuracy of the calculated natural frequencies are studied. Numerical results show that the rotational speed impacts significantly the first frequency of the wind turbine blade. The pitch angle could not markedly affect wind turbine blade frequencies.

A new method for free vibration analysis of piezoelectric-elastic laminated beams

Free vibration of the piezoelectric-elastic laminatedbeams is studied by combining the state-space method and differentialquadrature method. The state equations are dispersed at every sample pointby using the differential quadrature technique to obtain the solutions. Theconvergence of the present method is also studied by choosing differentnumber of sample points. The obtained results are validated to show that thedeveloped method is convenient for vibration analysis ofpiezoelectric-elastic laminated beams.

2-D solution for free vibrations of parabolic shells using generalized differential quadrature method

The Generalized Differential Quadrature (GDQ) procedure is developed for the free vibration analysis of complete parabolic shells of revolution and parabolic shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. The results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Several examples of parabolic shell elements are presented to illustrate the validity and the accuracy of GDQ method. Numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Femap/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. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions. Different typologies of non-uniform grid point distributions are considered. The effect of the distribution choice of sampling points on the accuracy of GDQ solution is investigated. New numerical results are presented.

Free vibration of a functionally graded piezoelectric beam via state-space based differential quadrature

Based on the theory of elasticity and piezoelectricity, the method called state-space based differential quadrature method (SSDQM) is extended to study the free vibration of a functionally graded piezoelectric material (FGPM) beam under different boundary conditions. The FGPM beam is approximated as a multi-layered cantilever. The state equations in terms of state variables of every discrete point are established using the differential quadrature technique. It is found that the results obtained in the present paper agree very well with both the finite element results and the results obtained by other investigations in special cases. Furthermore, at the end of this paper, the influence of some material parameters on the natural frequency of the FGPM beam is discussed.

Semi-analytical three-dimensional elasticity solutions for generally laminated composite plates

Semi-analytical solutions for bending and free vibration of composite laminated plates have been derived based on three-dimensional elasticity theory using a newly developed hybrid analysis, which perfectly combines the state space approach (SSA) and the technique of differential quadrature (DQ). The thickness direction of laminates is selected as the transfer direction in SSA, and the DQ technique is employed to discretize the in-plane domains. This actualizes the transformation of the original partial differential equations into a state equation consisting of first-order ordinary differential equations. In particular, the use of DQ technique makes ease of the treatment of various boundary conditions, which cannot be considered in the conventional exact SSA. To avoid numerical instabilities in the conventional transfer matrix method, artificial interfaces are introduced to divide each layer into several sub-layers to reduce the transfer distance and the joint coupling matrices are established according to the continuity conditions at actual and artificial interfaces to implement the global analysis. Comprehensive numerical examples are preformed to validate the present hybrid method. Effects of some parameters on mechanical properties of the laminates are discussed.

Stress analysis of anisotropic thick laminates in cylindrical bending using a semi-analytical approach

Semi-analytical elasticity solutions for bending of angle-ply laminates in cylindrical bending are presented using the state-space-based differential quadrature method (SSDQM). Partial differential state equation is derived from the basic equations of elasticity based on the state space concept. Then, the differential quadrature (DQ) technique is introduced to discretize the longitudinal domain of the plate so that a series of ordinary differential state equations are obtained at the discrete points. Meanwhile, the edge constrained conditions are handled directly using the stress and displacement components without the Saint-Venant principle. The thickness domain is solved analytically based on the state space formalism along with the continuity conditions at interfaces. The present method is validated by comparing the results to the exact solutions of Pagano’s problem. Numerical results for fully clamped thick laminates are presented, and the influences of ply angle on stress distributions are discussed.

Semi-analytical solutions for free vibration of anisotropic laminated plates in cylindrical bending

Elasticity solutions for free vibration of angle-ply laminates subjected to cylindrical bending are obtained using a newly developed semi-analytical approach. The thickness domain is solved analytically using the transfer matrix method based on the state space concept, while the in-plane domain is solved approximately via the technique of differential quadrature. The present method is applicable to arbitrarily thick laminates and for treating arbitrary edge conditions. The method is verified by comparisons with the exact solutions of Pagano's problem. Effects of variation of ply angle on the vibration properties of laminates are investigated; mode shape switching is observed when ply angle varies. Numerical results for fully clamped thick laminates are presented for future references.

Two-dimensional thermoelasticity solution for functionally graded thick beams

Two-dimensional thermoelasticity analysis of functionally graded thick beams is presented using the state space method coupled with the technique of differential quadrature. Material properties vary continuously and smoothly through the beam thickness, leading to variable coefficients in the state equation derived from the elasticity equations. Approximate laminate model is employed to translate the state equation into the one with constant coefficients in each layer. To avoid numerical instability, joint coupling matrices are introduced according to the continuity conditions at interfaces in the approximate model. The differential quadrature procedure is applied to discretizing the beam in the axial direction to make easy the treatment of arbitrary end conditions. A simply-supported beam with exponentially varying material properties is considered to validate the present method. Numerical examples are performed to investigate the influences of relative parameters.

Three-dimensional vibration analysis of multilayered piezoelectric composite plates

We investigate the free vibration of multilayered piezoelectric composite plates. A brief review of the theoretical developments in piezoelasticity that are relevant to smart composite structures is presented. An analysis is performed using the differential quadrature (DQ) technique to solve three-dimensional equations of piezoelasticity, and technical issues are thoroughly discussed. Solutions for piezoelastic laminated plates are made possible with the development and implementation of the DQ layerwise modeling technique, which allows different boundary conditions to be imposed at the edges of the plates. The numerical results of different plate problems are presented, and the effects of the piezoelasticity of these problems are investigated. The DQ model predictions are validated against existing results, with good agreement found.

A differential quadrature method solution for shear-deformable shells of revolution

This paper deals with the application of the differential quadrature method to the linear elastic static analysis of isotropic rotational shells. The governing equations of equilibrium, in terms of stress resultants and couples, are those from Reissner–Mindlin shear deformation shell theory. These equations, written in terms of the circular harmonic amplitudes of the stress resultants, are first put into generalized displacements form by the use of strain–displacement relationships and constitutive equations. The resulting systems are solved by means of the differential quadrature technique with favourable precision, leading to accurate stress patterns.

Analytical and differential quadrature results for vibration analysis of damaged circular arches

The present paper focuses on in-plane linear free vibrations of circular arches, in undamaged and damaged configurations. For the model herein utilized, the equations of motion, in terms of displacements and rotation, take into account shearing and axial deformations and rotary inertia. The cracked section of the arch is modeled with an elastic spring. An exact analytical method of solution and an approximate numerical one are presented. The first method solves the fundamental system in closed form, by means of a characteristic polynomial; the second one is based on a simple and efficient differential quadrature and domain decomposition technique. Natural frequencies and mode shapes are computed for some significant cases, showing very good agreement between the two approaches.

Three-dimensional analysis of the coupled thermo-piezoelectro-mechanical behaviour of multilayered plates using the differential quadrature technique

This paper investigates the behaviour of multilayered composite plates subject to thermo-piezoelectric-mechanical loading. The analysis is performed using the three-dimensional equations of thermo-piezoelasticity and the differential quadrature (DQ) numerical technique. Solutions to the thermo-piezoelectric laminated plates are made possible with the development and implementation of a DQ layerwise modelling technique. The formulation allows different boundary conditions to be imposed at the edges of the plate. Numerical results for different example plate problems are presented, and the effects of the thermo-piezoelasticity and boundary conditions of these problems are investigated. The DQ model predictions are validated with existing results as the comparison reveals good agreement between two.

Static Analysis of Thick Laminated Beams: Two-Dimensional Elasticity Solutions via Differential Quadrature

This paper intends to present two-dimensional elasticity solutions for static problem of thick laminated composite beams using a hybrid method of state-space-based differential quadrature. The technique of differential quadrature is employed to reduce the partial differential state equations into the ordinary differential ones at all arbitrary sampling points for each individual laminate. General solution to the assembled state equation is then obtained according to the matrix theory. Taking account of the continuity conditions at the interfaces of all the adjacent lamina, a relationship between state variables at the top and bottom surfaces of the beam is established through a global transfer matrix. After incorporating the boundary conditions at these two surfaces, an eigenvalue equation for static problem is then derived. Numerical examples are presented, through which the accuracy and convergence characteristics of the present method are investigated. It is shown that the present method is of excellent efficiency for laminated composite thick beams subjected to arbitrary end supporting conditions.