Differential Quadrature Solution Research Articles

Investigation of moving load-induced vibrations in layered functionally graded Terfenol-D beams: a differential quadrature method and analytical approach

The paper investigates the potential of the full-sinusoidal Fourier series as a solution form for the deflection of layered functionally graded beams associated with smart actuators, drawing on the fundamental principles of classical Fourier series theory. The current method simplifies the difficult beam problem to a set of linear algebraic equations by using the Duhamel integration technique and the orthogonality of the trial function. The study takes into account generalized boundary conditions and moving forces, which are seldom discussed in previous research. Under the action of a moving load, the boundary value problem for a functionally graded beam integrated with Terfenol-D is efficiently addressed using the approach of combined differential and integral quadrature. The generalized boundary conditions may be easily achieved by adjusting the stiffness of the restraining springs. The significant agreement between the differential quadrature solution and the Fourier series solution underlines the efficiency and accuracy of both methods. Furthermore, the influences of various crucial physical characteristics on the natural frequencies and the essential flow velocities are explored, including boundary stiffness, foundation parameters, and geometric parameters.

Transversely isotropic effects on the coupled thermo‐hydro‐mechanical performance for layered saturated media

AbstractIn this manuscript, a novel transformed differential quadrature solution to the coupled thermo‐hydro‐mechanical (THM) problem of layered transversely isotropic (TI) saturated media is proposed, accompanied by a sensitivity analysis of pertinent parameters. Initially, the THM governing equations that encompass the transverse isotropy characteristics of thermal, permeable, and mechanical properties are introduced. Subsequently, utilizing the transformed differential quadrature method (TDQM), the discretized algebraic equations in the transformed domain are derived. Boundary conditions related to stresses, displacements, and temperatures are defined leveraging integral transform techniques and the stress‐strain relationship, while algebraic equations for continuity conditions across adjacent layers are incorporated. The global matrix equation is assembled and solved. After verifying the proposed TDQM solution, a sensitivity analysis on the complex transversely isotropic parameters is conducted to identify the principal factors influencing THM coupling behaviors, thereby simplifying the parameters for discussion in practical engineering applications.

Postcritical Behavior of Nonlocal Strain Gradient Arches: Formulation and Differential Quadrature Solution

Arches are important components in nanostructures and systems. Because of the remarkable strength of nanomaterials, nanoarches are slender, thus enhancing their vulnerability to geometric instability, such as buckling. Although molecular simulations are often employed to analyze nanostructures, their use in routine analysis/design is formidable due to prohibitively exhaustive computation. Equivalent continuum models were developed as alternatives. The buckling and postcritical phenomena of the classical arch also form an important benchmark problem in nonlinear mechanics. This study investigates the buckling and the postcritical behavior of nanoarch subjected to inward pressure using nonlocal (NL), strain gradient (SG) continuum theory. The governing equations are derived in the form of a sixth-order nonlinear integrodifferential equation, unlike the fourth-order for a classical arch. The equation is then solved numerically using a differential quadrature (DQ) with appropriate boundary conditions. An incremental-iterative arc-length continuation is employed for the solution of the resulting algebraic system of equations. The equilibrium paths are traced for the possible instability modes, such as symmetric/antisymmetric bifurcations, snap-through and limit point instability, ascertained with the internal-external force diagrams. A particular instability mode is triggered at a certain threshold/range of the slenderness ratio of the arch, which is significantly influenced by the NL and SG interactions. These interactions not only cause quantitative changes in the instability behavior but also lead to qualitative changes, such as cessation, shift, and conversion of modes, more prominently for SG arches. Similar to classical arches, the prebuckling nonlinearity is shown to be significant.

Transformed differential quadrature solution to the coupled thermal–mechanical problem in transversely isotropic media

This study provides a novel solution to the coupled thermal–mechanical problem in transversely isotropic (TI) media by using the transformed differential quadrature method (TDQM). The governing equations are transformed and then discretized by the proposed TDQM. After introducing the internal and boundary conditions, we obtain the thermal–mechanical solution of a medium with various heat sources. Several examples are set up to verify the proposed solution and to discuss the convergence of the method. Also, key parameters including the transverse isotropy of skeleton, the linear expansion coefficients and the anisotropic thermal diffusivity are investigated. Compared with the traditional differential quadrature method (DQM), the TDQM introduces the integral transform theorem to simplify the derivation process and reduce the number of algebraic equations. Consequently, it avoids the exponential increase in the number of algebraic equations when solving multi-dimensional problems.

Inverse differential quadrature solutions for free vibration of arbitrary shaped laminated plate structures

An essential aspect of design of laminated plate structures in many engineering applications is the analysis of free vibration behaviour in order to model the structure for random excitations. In this regard, numerical solutions to the systems of high-order partial differential equations governing free vibration response of the structure become important. Direct approximation of such high-order systems are prone to error arising from the sensitivity of high-order numerical differentiation to noise necessitating the demand for improved solution techniques. In this work, a novel generalised inverse differential quadrature method is developed to study the dynamic behaviour of first-order shear deformable arbitrary-shaped laminated plates. The ensuing underdetermined system is operated upon by Moore-Penrose pseudo-inverse preconditioning to form a squared eigenvalue system. Free vibration solutions of square, skew, circular, and annular sector plates for different boundary conditions are obtained and validated against exact and numerical solutions in the literature and ABAQUS. It is demonstrated with numerous examples that iDQM solutions are in excellent agreement with exact solutions for square plates and the results for arbitrary shaped plates are comparable with solutions in the literature while saving up to 96% degrees of freedom required for ABAQUS solution. Finally, refined parametric studies conducted reveal that, subject to varying geometric configurations, iDQM solutions are numerically stable and potentially converge faster than DQM.

A variational differential quadrature solution to finite deformation problems of hyperelastic shell-type structures: a two-point formulation in Cartesian coordinates

A variational differential quadrature solution to finite deformation problems of hyperelastic shell-type structures: a two-point formulation in Cartesian coordinates

Curve Fitting Initial Guess for Iterative Differential Quadrature Solution of Burgers Equation

According to presented numerical studies in the literature, the solution of Burgers Equation (BE) performed for dt=0.001 and dt=0.0001 commonly. In this study, numerical solution of BE carried out by using the Iterative Differential Quadrature Method (I-DQM), as dt=0.01. Convergence speed and accuracy of iterative methods depends on the initial guess. Every Partial Differential Equation (PDE) describes one or more than one physical problems from the perspective of the engineering view. Unlike the previous iterative studies, in this work, an initial guess value is used in accordance with the physical nature of the discussed problem by using curve fitting. Absolute error analysis of obtained results performed for comparison with some previous studies. The consequence of comparisons shows that more accurate results and faster solution than other studies could be obtained by using I-DQM with curve fitting initial guess.

Differential quadrature solution for composite flat plates with delamination using higher-order layerwise models

This paper deals with the numerical modeling of delaminated composite plates by applying the differential quadrature method. A semi-layerwise modeling technique developed by the author including the first-, second- and third-order shear deformation plate theories is utilized. The proposed model captures separately the delaminated and the intact parts of the plates with straight delamination front. Firstly, the governing differential equations are presented and then it is shown how these were discretized by using a recommended grid structure and the required boundary and continuity conditions. Some plate problems with simply supported and rigidly fixed edges are solved including some variations of the size and the thicknesswise position of the material defect. One of the non-trivial aspects of the differential quadrature solution is how the continuity between the different regions can be imposed. It will be shown that at the mutual grid points of the neighboring regions it is not possible to satisfy each continuity condition against the stress resultants, and thus although some conditions are violated, the problems can be solved accurately by properly choosing the actual conditions. Once, the differential quadrature solution became available the mechanical fields were determined, too. Moreover, the J-integral can be determined and together with a simple mode separation technique even the mode-II and mode-III components became available. The comparison of results to those by spatial finite element models shows very good accuracy and the effectiveness of the proposed differential quadrature solution.

Motion limiting nonlinear dynamics and frequency characteristics of pipes reinforced with carbon nanotube agglomerations coupled with piezoelectric actuator

In the current work, for the first time, the motion limiting nonlinear dynamics and frequency analysis of a nanopipe reinforced with carbon nanotube agglomerations (CNTAs) is presented. The current composite structure is coupled with a piezoelectric actuator for electrical purposes. The first-order shear deformation theory (FSDT) is presented for displacement fields of two layers (composite and piezoelectric). Nonlocal strain gradient theory (NSGT) is used to consider the size effects. For mathematical modeling of the nanocomposite layer, the effective Poisson’s ratio, Young’s modulus, thermal expansion in addition to carbon nanotubes (CNTs) clusters/polymer density are presented. The deflection of the pipe is described through Cartesian and cylindrical coordinates. The nonlinear problem equations are extracted as two different 1-D models which are perpendicular to each other. Generalized differential quadrature method (GDQM) and Multiple scales solution procedure (MSSP) were used for solving the nonlinear equations in displacement and time domains, respectively. Finally, the outcomes of the current report demonstrate that the area ratio of piezoelectric layer, CNTs’ volume fraction, boundary conditions, and size-dependent parameter have an important role in the nonlinear dynamics of the composite nanopipes coupled with the piezoelectric layer.

Inverse differential quadrature method for structural analysis of composite plates

A novel two-dimensional inverse differential quadrature method is proposed to approximate the solution of high-order system of differential equations. A critical aspect of the proposed scheme is to circumvent the error arising from high sensitivity to noise associated with high-order numerical differentiation operations during direct approximation. A general framework for approximating arbitrary functions from high-order partial derivatives is developed and a comprehensive insight for the implementation of different orders of the analysis within the context of first-order shear deformation theory kinematical assumptions governing laminated plates bending and buckling behaviours is presented. Concerning buckling analysis, a Moore–Penrose pseudo-inverse preconditioning procedure is further proposed to formulate the eigenvalue problem, since the developed algebraic expressions constitute an underdetermined system. According to the numerical tests for bending and buckling analyses of laminated plates under different loading and boundary conditions, the accuracy of current solutions compares satisfactorily with Navier’s solution, differential quadrature method and commercial finite element analysis models. Furthermore, the developed method demonstrates the potential for improved convergence in comparison with differential quadrature solutions highlighting its computational merits.

Differential quadrature solution for vibration control of functionally graded beams with Terfenol-D layer

The governing differential equation of motion for vibration control of a functionally graded material (FGM) beam using magnetostrictive layers is solved using differential quadrature method(DQM). It is known that, when differential quadrature is implemented directly for the solution of governing differential equation for vibration control of beam, it is required to convert the generalised eigenvalue problem into standard eigenvalue problem. However in the present work, the original differential equation of vibration control of beam is be separated into two simpler second and fourth order differential equations using the separation of variables in conjunction with the characteristics equation of damped single degree of freedom system. Solution of corresponding two simpler differential equation also yields damped natural frequency and damped factor comparable to that of the former approach. It is to be noted that using either of the solutions using differential quadrature method δ point description of the physical domain at boundary is used to obtained the differential quadrature equations for the various boundary conditions of the beam. In order to assure the accuracy of formulation and solution using DQM, convergence behavior of natural frequencies is examined for five combinations of boundary conditions and comparison studies from the two solution approaches is presented. The effect of the location of the magnetostrictive layers, material properties and control parameters on the vibration suppression are investigated.

Free vibration and static deflection analysis of functionally graded and porous micro/nanoshells with clamped and simply supported edges

The static deflection and free vibration problem of functionally graded porous (FGP) cylindrical micro/nanoshells are analyzed using the concept of the modified couple stress theory. The governing equations of first order shear deformation theory (FSDT) are employed and solved by generalized differential quadrature (GDQ) solution method. Using the power low for the FG properties, and incorporating pore content effect, a modified power function is considered for modelling FGP material properties. The transverse deflection under transverse loading and free vibration are numerically presented for a cylindrical micro/nanoshell with simply and clamped edges. The influences of porosity type, porosity volume fraction, material properties, size scale parameter, and type of the boundary conditions on the static bending and free vibration of FGP micro/nanoshells are investigated.

Vibration analysis of curved composite sandwich beams with viscoelastic core by using differential quadrature method

This paper focuses on the vibration analysis of three-layered curved sandwich beams with elastic face layers and viscoelastic core. First, the equations of motion that govern the free vibrations of the curved beams together with the boundary conditions are derived by using the principle of virtual work, in the most general form. Then, these equations are solved by using the generalized differential quadrature method in the frequency domain, for the first time to the best of the authors’ knowledge. Verification of the proposed beam model and the generalized differential quadrature solution is carried out via comparison with the results that already exist in literature and the ANSYS finite element solution combined with the modal strain energy method. The effect of system parameters, i.e. layer thicknesses, the lamination angle of layers and the curvature on the vibration and damping characteristics of a curved sandwich beam with laminated composite face layers and a frequency dependent viscoelastic core is investigated in detail.

Time dependent creep analysis in thick FGM rotating disk with two-dimensional pattern of heterogeneity

Combining a novel modeling algorithm and Generalized Differential Quadrature (GDQ) solution method of differential equations, a time depended creep analysis is performed for a Functionally Graded (FG) rotating thick disk featuring a two-dimensional type of heterogeneity. The disk is made of SiC reinforcement particles distributed functionally both radially and axially in Aluminum based matrix continuously. All mechanical and thermal properties are functions of SiC volume fraction. Primary and secondary creep behaviors are based on Norton's creep law. Creep parameters are depended on temperature, particle size and volume fraction of particles. Using equilibrium, constitutive and strain-displacement equations, nonlinear displacement-based creep equations are obtained. A solution algorithm is developed to solve the nonlinear form of equations. The application of 2-D GDQ method is examined in solution of resulting system of equations. Exemplary case studies confirm that in comparison with one-dimensional models, two-dimensional methodologies are more versatile for the study of FGM structures.

Differential quadrature solutions for the nonconservative instability of a class of single-walled carbon nanotubes

PurposeThe purpose of the present paper is to investigate the nonconservative instability of a single-walled carbon nanotube (SWCNT) with an added mass through nonlocal theories. The governing equations are discretized by means of the differential quadrature (DQ) rules, as introduced by Bellman and Casti. DQ rules have been largely used in engineering and applied sciences. Recently, they were applied to enhance some numerical schemes, such as step-by-step integration schemes and Picard-like numerical schemes.Design/methodology/approachIn the present paper, the DQ rules are used to investigate the nonconservative instability of a SWCNT through nonlocal theories.FindingsTo show the sensitivity of the SWCNT to the values of added mass and the influence of nonlocal parameter on the fundamental frequencies values, some numerical examples have been performed and discussed. Yet, the effect of the different boundary conditions on the instability behaviour has been investigated. The validity of the present model has been confirmed by comparing some results against the ones available in literature.Originality/valueApplying the nonlocal elasticity theory, this paper presents a re-formulation of Hamilton’s principle for the free vibration analysis of a uniform Euler–Bernoulli nanobeam. The main purpose of this paper is to investigate the free vibration response of an SWCNT with attached mass and for various values of small scale effects.

A hybrid DQ-TLBO technique for maximizing first frequency of laminated composite skew plates

The differential quadrature (DQ) and teaching-learning based optimization (TLBO) methods are coupled to introduce a hybrid numerical method for maximizing fundamental natural frequency of laminated composite skew plates. The fiber(s) orientations are selected as design variable(s). The first-order shear deformation theory (FSDT) is used to obtain the governing equations of the plate. The equations of motion and the related boundary conditions are discretized in space domain by employing the DQ method. The discretized equations are transferred from the time domain into the frequency domain to obtain the fundamental natural frequency. Then, the DQ solution is coupled with the TLBO method to find the maximum frequency of the plate and its related optimum stacking sequences of the laminate. Convergence and applicability of the proposed method are shown and the optimum fundamental frequency parameter of the plates with different skew angle, boundary conditions, number of layers and aspect ratio are obtained. The obtained results can be used as a benchmark for further studies.

Iterative differential quadrature solutions for Bratu problem

A numerical scheme based on differential quadrature methods, is introduced for solving Bratu problem. The problem is firstly reduced to an iterative one. Then, both of differential quadrature method (DQM) and moving least squares differential quadrature method (MLSDQM) are applied to solve iteratively the nonlinear problem. The proposed scheme successfully computes multiple solutions to Bratu’s problem. The obtained results agree with the 1D and 2D closed forms. Further a parametric study is introduced to investigate the computational characteristics of the proposed scheme.

2D and 3D shell models for the free vibration investigation of functionally graded cylindrical and spherical panels

The paper shows the free vibration investigation of simply supported functionally graded material (FGM) shells. Spherical and cylindrical shell geometries are investigated for two different material configurations which are one-layered FGM structures and sandwich structures embedding an internal FGM core. A three-dimensional (3D) exact shell model and different two-dimensional (2D) computational models are compared in terms of frequencies and vibration modes. The proposed numerical solutions are typical 2D finite elements (FEs), and classical and advanced generalized 2D differential quadrature (GDQ) solutions. High and low frequency orders are investigated for thin and thick simply supported shells. Vibration modes are fundamental to compare the 3D exact shell model and 2D numerical solutions. The 2D finite element results based on the classical Reissner–Mindlin theory are calculated using a typical commercial FE software. Classical and advanced GDQ 2D models use the generalized unified formulation. The 3D exact shell model uses the differential equations of equilibrium written in general orthogonal curvilinear coordinates and developed in layer-wise (LW) form. The differences between 2D numerical results and 3D exact results depend on the thickness ratio and geometry of the structure, the considered mode and the frequency order, the lamination sequence and materials.

Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory

In this paper, analysis of shear deformable functionally graded (FG) nanobeams in postbuckling based on modified couple stress theory is presented. The nonlinear behavior of postbuckling is defined based on von-Karman geometric nonlinearity. A material-length scale parameter is used to capture the size-dependent behavior of small-scale beams. It is assumed that the material properties (Young׳s modulus and material-length scale parameter) of FG beam vary across the thickness according to a power law; however the Poisson׳s ratio is held constant. General beam theory is applied to describe the shear deformation effect. Governing differential equations and boundary conditions are derived using the principle of minimum potential energy. This study presents exact and generalized differential quadrature (GDQ) solutions for the static postbuckling response of FGM nanobeams under different boundary conditions. Results of both methods are compared. Difference between postbuckling response obtained by first-order and higher-order beam theories reveals the significant effect of the shear deformation. Moreover, the obtained results indicate that the first-order beam theory has some errors in estimating the amplitude of buckling. Also, effects of length-scale parameter, material gradient, length-to-thickness ratio and Poisson׳s ratio are presented.

Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates

Most of the early studies on plates vibration are focused on two-dimensional theories, these theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling leading to simpler expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick plates. The main objective of this research paper is to present 3-D elasticity solution for free vibration analysis of continuously graded carbon nanotube-reinforced (CGCNTR) rectangular plates resting on two-parameter elastic foundations. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this study, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The formulations are based on the three-dimensional elasticity theory. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. The novelty of the present work is to exploit Eshelby-Mori-Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various coefficients of foundation and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR rectangular plates. The new results can be used as benchmark solutions for future researches.