Structural Analysis by the Differential Quadrature Method Using Modified Weighting Matrices

1998 ◽  
Author(s):  
Siu-Tong Choi ◽  
Yu-Tuan Chou
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Sampling Point ◽  
Order Partial Differential Equation ◽  
Quadrature Method ◽  
The Finite Element Method ◽  
Computationally Efficient ◽  
Base Points ◽  
Sampling Points

Abstract The differential quadrature method has lately been more and more often used for analysis of engineering problems as an alternative for the finite element method or finite difference method. In this paper, static, dynamic and buckling analyses of structural components are performed by the differential quadrature method. To improve the accuracy of this method, an approach is proposed for selecting the sampling points which include base points and conditional points. The base points are taken as the roots of the Legendre polynomials. Accuracy of the problems analyzed will be increased by using the base points. The conditional points are determined according to boundary conditions and specified conditions of external load. A modified algorithm is proposed for applying two or more boundary conditions in a sampling point at boundary of domain, such that the higher-order partial differential equation can be solved without adding new sampling points. By applying this approach to variety problems, such as deflections of beam under nonuniformly distributed loading, vibration and buckling analyses of beam and plate, it is found that numerical results of the present approach are more accurate than those obtained by the equally-spaced differential quadrature method and is computationally efficient.

scholarly journals Differential Quadrature and Differential Transformation Methods in Buckling Analysis of Nanobeams

2019 ◽  
Vol 6 (1) ◽  
pp. 68-76 ◽  
Author(s):  
Subrat Kumar Jena ◽  
S. Chakraverty
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Transformation Method ◽  
Nonlocal Theory ◽  
Buckling Analysis ◽  
Differential Quadrature ◽  
Load Parameter ◽  
Computationally Efficient ◽  
Differential Transformation

AbstractIn this paper, two computationally efficient techniques viz. Differential Quadrature Method (DQM) and Differential Transformation Method (DTM) have been used for buckling analysis of Euler-Bernoulli nanobeam incorporation with the nonlocal theory of Eringen. Complete procedures of both the methods along with their mathematical formulations are discussed, and MATLAB codes have been developed for both the methods to handle the boundary conditions. Various classical boundary conditions such as SS, CS, and CC have been considered for investigation. A comparative study for the convergence of DQM and DTM approaches are carried out, and the obtained results are also illustrated to demonstrate the effects of the nonlocal parameter, aspect ratio (L/h) and the boundary condition on the critical buckling load parameter.

scholarly journals A Mixed Layerwise-Differential Quadrature Method for 3-D Vibration and Buckling Analyses of Orthogonally Stiffened Composite Conical Shell

2019 ◽  
Vol 24 (2) ◽  
pp. 217-227
Author(s):  
Mostafa Talebitooti
Keyword(s):  
Boundary Conditions ◽  
Conical Shell ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Algebraic Equations ◽  
Discrete Elements ◽  
Arbitrary Boundary ◽  
Stiffened Shells

A layerwise-differential quadrature method (LW-DQM) is developed for the vibration analysis of a stiffened laminated conical shell. The circumferential stiffeners (rings) and meridional stiffeners (stringers) are treated as discrete elements. The motion equations are derived by applying the Hamilton’s principle. In order to accurately account for the thickness effects and the displacement field of stiffeners, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations applying the DQM in the meridional direction. The advantage of the proposed model is its applicability to thin and thick unstiffened and stiffened shells with arbitrary boundary conditions. In addition, the axial load and external pressure is applied to the shell as a ratio of the global buckling load and pressure. This study demonstrates the accuracy, stability, and the fast rate of convergence of the present method, for the buckling and vibration analyses of stiffened conical shells. The presented results are compared with those of other shell theories and a special case where the angle of conical shell approaches zero, i.e. a cylindrical shell, and excellent agreements are achieved.

Analysis of free vibration of an isotropic plate with surface or internal long crack using generalized differential quadrature method

2019 ◽  
Vol 55 (1-2) ◽  
pp. 42-52
Author(s):  
Milad Ranjbaran ◽  
Rahman Seifi
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Isotropic Plate ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Generalized Differential Quadrature Method ◽  
Generalized Differential

This article proposes a new method for the analysis of free vibration of a cracked isotropic plate with various boundary conditions based on Kirchhoff’s theory. The isotropic plate is assumed to have a part-through surface or internal crack. The crack is considered parallel to one of the plate edges. Existence of the crack modified the governing differential equations which were formulated based on the line-spring model. Generalized differential quadrature method discretizes the obtained governing differential equations and converts them into an algebraic system of equations. Then, an eigenvalue analysis was used to determine the natural frequencies of the cracked plates. Some numerical results are given to demonstrate the accuracy and convergence of the obtained results. To demonstrate the efficiency of the method, the results were compared with finite element solutions and available literature. Also, effects of the crack depth, its location along the thickness, the length of the crack and different boundary conditions on the natural frequencies were investigated.

Vibration analysis of a nano-turbine blade based on Eringen nonlocal elasticity applying the differential quadrature method

2016 ◽  
Vol 23 (19) ◽  
pp. 3247-3265 ◽  
Author(s):  
Majid Ghadiri ◽  
Navvab Shafiei
Keyword(s):  
Boundary Conditions ◽  
Nonlocal Elasticity ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Small Scale ◽  
Quadrature Method ◽  
Bending Vibrations ◽  
Classical Plate Theory ◽  
Axial Forces

This study investigates the small-scale effect on the flapwise bending vibrations of a rotating nanoplate that can be the basis of nano-turbine design. The nanoplate is modeled as classical plate theory (CPT) with boundary conditions as the cantilever and propped cantilever. The axial forces are also included in the model as the true spatial variation due to the rotation. Hamilton’s principle is used to derive the governing equation and boundary conditions for the classic plate based on Eringen’s nonlocal elasticity theory and the differential quadrature method is employed to solve the governing equations. The effect of the small-scale parameter, nondimensional angular velocity, nondimensional hub radius, setting angle and different boundary conditions in the first four nondimensional frequencies is discussed. Due to considering rotating effects, results of this study are applicable in nanomachines such as nanomotors and nano-turbines and other nanostructures.

Analysis of Toroidal Shells Using the Differential Quadrature Method

2003 ◽  
Vol 03 (02) ◽  
pp. 215-226 ◽  
Author(s):  
D. Redekop ◽  
T. Muhammad
Keyword(s):  
Finite Element ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
The Finite Element Method ◽  
Complementary Method ◽  
Finite Element Calculations ◽  
Parametric Studies ◽  
New Applications ◽  
Toroidal Shells

In new applications of toroidal shells it is often necessary to solve problems of statics, response, vibration, and buckling. While the finite element method can serve as the main means of analysis it is desirable to have available a second, complementary, method that can be used for verification, parametric studies, and specialized analyses. In this paper the use of the new differential quadrature method in such a complimentary role is investigated. Problems involving the statics, response, vibration, and buckling of toroidal shells are analyzed. Numerical results obtained are compared with finite element calculations. Finally conclusions are drawn concerning the agreement between the methods, and the usefulness of the new method for specialized studies.

Analysis of the vibration of axially moving viscoelastic plate with free edges using differential quadrature method

2017 ◽  
Vol 24 (17) ◽  
pp. 3908-3919 ◽  
Author(s):  
Mouafo Teifouet Armand Robinson
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Viscoelastic Plate ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Simply Supported ◽  
System Parameters ◽  
Complex Eigenvalues ◽  
Axially Moving ◽  
Viscoelastic Rectangular Plate

The two-dimensional viscoelastic differential constitutive relation is employed in this paper, in order to establish the equation of motion of axially moving viscoelastic rectangular plate. Two boundary conditions are investigated, namely the clamped free and two opposite edges simply supported and two others free. The differential quadrature method is used to solve the resulting complex eigenvalues equation. The influence of boundary conditions on the instability of a moving viscoelastic plate is analyzed firstly, and secondly the effects of system parameters such as plate's viscosity and aspect ratio on the vibration frequencies are presented.

DQ Based Methods

2012 ◽  
pp. 316-346 ◽  
Author(s):  
Stefania Tomasiello
Keyword(s):  
Numerical Technique ◽  
Differential Quadrature Method ◽  
Free Vibrations ◽  
Computational Effort ◽  
Differential Quadrature ◽  
Time Dependent ◽  
Quadrature Method ◽  
The Finite Element Method ◽  
Time Domains ◽  
Typical Model

Though relatively unknown, the Differential Quadrature Method (DQM) is a promising numerical technique that produces accurate solutions with less computational effort than other numerical methods such as the Finite Element Method. There are different versions of the former method, so one can refer to a class of methods based on the differential quadrature (DQ) approach. This chapter provides systematic steps to understand DQ based methods: how they work, how to use and develop them, as well as exemplary problems. It is divided into two sections: the first provides fundamentals and theories related to the DQ approach, while the second presents the application of DQ based methods to three significant problems, i.e. free vibrations of a multi-span beam (a typical model for bridge decks and floors), time dependent heat transfer, and vibrations of a rectangular membrane. The main focus is on the application of the method to space-time domains as a whole, an issue not well covered by the current literature.

Sign in / Sign up

Export Citation Format

Share Document