Vibration analysis of spherical structural elements using the GDQ method

This work deals with the dynamical behaviour 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 the congruence relationships. The boundary conditions considered are clamped (C) and free (F) edge. Numerical solutions have been computed by means of the technique known as the Generalized Differential Quadrature (GDQ) Method. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. At the moment it can only be pointed out that by using the GDQ technique the numerical statement of the problem does not pass through any variational formulation, but deals directly with the governing equations of motion. Referring to the formulation of the dynamic equilibrium in terms of harmonic amplitudes of mid-surface displacements and rotations, in this paper the system of second-order linear partial differential equations is solved, without resorting to the onedimensional formulation of the dynamic equilibrium of the shell. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved. Several examples of parabolic shell elements are presented to illustrate the validity and the accuracy of GDQ method. The convergence rate of the 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. The effect of the distribution choice of sampling points on the accuracy of GDQ solution is investigated. GDQ results, which are based upon the FSDT, are compared with the ones obtained using commercial programs such as Ansys, Femap/Nastran, Abaqus, Straus, Pro/Engineer.

Static analysis of thick laminated shells with different boundary conditions using GDQ

Orthotropic laminated structures have attracted much interest in many applications because of their desirable properties such as improved thermal and mechanical properties, high strength to weight ratio, and high stiffness to weight ratio. In this paper, an improved version of the differential quadrature method, called the generalized differential quadrature (GDQ) method, is used for the first time to study the vibration of antisymmetric angle-ply laminated cylindrical panels with boundary conditions other than the simply supported type on its four edges. The GDQ method has shown computational advantages over the existing differential quadrature methods in that there is no restriction on the number of grid points in the approximation and the weighting coefficients are determined using a recurrence relation. The GDQ method has shown superb accuracy, efficiency and convenience, and great potential in solving the differential equations of structural problems. The present analysis is formulated using Donnell's shell theory and is validated by comparing results with those in the literature.

Large displacement transient analysis of FGM super-elliptic shells using GDQ method

In this paper, static analysis of a nano rectangular plate subjected to uniform distributed load is studied based on modified couple stress theory (MSCT) by using Generalized Differential Quadrature (GDQ) Method. The inclusion of an additional material parameter enables the new plate model to capture the size effect. The new non-classical plate model reduces to the classical plate model when the length scale parameter is set to zero. In deriving of the governing equations, the minimum total potential energy principle is used. In the solution of the governing equations, the DQM method is used for simply-supported nano plate within the Kirchhoff-Love plate theory. In the numerical results, the influences the material length scale parameter and the dimension parameters of the plate on the static deflection of the nano plate is presented. Also, the difference between the classical theory (CT) and MSCT is investigated for static responses of nano plate.

Numerical analyses of piezoelectric materials with the GDQ method

Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels

Free vibration analysis of Mindlin plates partially resting on Pasternak foundation

Modal analysis of shear-induced flexural vibration of FGPM beam using Generalized Differential Quadrature method

The vibrations of rotating joined conical–conical shells with classical supported conditions have been studied extensively. As a matter of fact, in some cases, these classical boundary conditions cannot exactly model actual situations. Moreover, theoretical frameworks on them are still limited. This research aims to investigate the fundamental frequencies and dynamic mode shapes of the traveling wave of the rotating porous metal material joined conical–conical thin shells (PJCS) with elastic supports. By utilizing artificial spring technology, arbitrary elastic supported boundary conditions and classical boundary conditions are achieved efficiently. A new dynamic model has been formulated with the help of the first-order shear deformation theory (FSDT) and Hamilton’s principle. By employing the generalized differential quadrature (GDQ) method along with stress boundary conditions and generalized eigenvalues, various factors such as porosity, semi-vertex angles and stiffness are analyzed for their impact on the fundamental frequencies of forward wave (FW), backward wave (BW) and mode shapes. The presented results are validated through the convergence and comparison studies from literatures. The interesting and novel results indicate that the in-plane displacement constraints have the most significant impact on the critical speed, while the lateral displacement constraint has the least effect. The vibrations are more easily excited for the part with a larger half vertex angle. Rotating PJCS with Type 1 has the biggest critical rotating speed.

This article describes a compact numerical algorithm based on the generalized differential quadrature (GDQ) method for the numerical analysis of natural convection in a differentially heated square cavity. The velocity–vorticity form of the Navier–Stokes equations and energy equation are used to represent the mass, momentum, and energy conservations of the fluid medium in the cavity. The GDQ form of the governing equations and the vorticity definition at the boundaries are solved by a coupled solution algorithm using a global matrix scheme for all the field variables. The vorticity values at the boundary are correctly imposed using the GDQ method, which approximates a given space derivative with higher-order accuracy compared to the existing schemes based on Taylor's series expansion. This has assured a divergence-free solution for the flow field by satisfying the continuity constraint, though the pressure term is not used directly in the present formulation. The proposed algorithm is validated for a lid-driven cavity flow for Reynolds number Re = 100, 400, and 1,000, and the predicted velocity profiles are in excellent agreement with the benchmark solutions. The algorithm is then used to compute the average Nusselt number and flow parameters for natural convection in a square cavity for Rayleigh number Ra = 103, 104, 105, and 106. These results are in better agreement with the benchmark solutions than the results obtained by other numerical schemes, which used much finer grids compared to the present scheme.

Higher order theories for the modal analysis of anisotropic doubly-curved shells with a three-dimensional variation of the material properties

Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method

Buckling analysis of axially-loaded functionally graded carbon nanotube-reinforced composite conical panels using a novel numerical variational method

In this article, the vibrational analysis of temperature-dependent cylindrical functionally graded (FG) microshells surrounded by viscoelastic a foundation is investigated by means of the modified couple stress theory (MCST). MCST is applied to this model to be productive in design and analysis of micro actuators and micro sensors. The modeled cylindrical FG microshell, its equations of motion and boundary conditions are derived by Hamilton’s principle and the first-order shear deformation theory (FSDT). For the first time, in the present study, functionally graded length scale parameter which changes along the thickness has been considered in the temperature-dependent cylindrical FG microshell. The accuracy of the present model is verified with previous studies and also with those obtained by analytical Navier method. The novelty of the current study is consideration of viscoelastic foundation, various thermal loadings and size effect as well as satisfying various boundary conditions implemented on the temperature-dependent cylindrical FG microshell using MCST. Generalized differential quadrature method (GDQM) is applied to discretize the equations of motion. Then, some factors are investigated such as the influence of length to radius ratio, damping, Winkler and Pasternak foundations, different temperature changes, circumferential wave numbers, and boundary conditions on natural frequency of the cylindrical FG microshell. The results have many applications such as modeling of microrobots and biomedical microsystems.