Structural Mechanics Problems With Structures on a Two-Parameter Foundation Solved by DQEM

2005 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Numerical Algorithms ◽  
Structural Mechanics ◽  
Numerical Results ◽  
Structural Problems ◽  
Governing Differential Equation ◽  
Analysis Models ◽  
Element Boundary ◽  
Two Parameter

The development of DQEM solution of structural problems with structures resting on a two-parameter foundation was carried out. The DQEM uses DQ or EDQ to discretize the governing differential equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Some EDQ models can be generated by DQ. They are DQ generated EDQ. Numerical results solved by the developed numerical algorithms are presented. The convergence of the developed DQEM analysis models is efficient.

Vibration of Timoshenko Beam Structures Resting on a Two-Parameter Elastic Foundation Solved by DQEM Using Chebyshev DQ Model

2008 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Timoshenko Beam ◽  
Numerical Algorithms ◽  
Numerical Results ◽  
Eigenvalue Equation ◽  
Beam Structures ◽  
Analysis Models ◽  
Element Boundary ◽  
Two Parameter

The development of DQEM solution of vibration of Timo-shenko beam structures resting on a two-parameter elastic foundation was carried out. The DQEM uses Chebyshev DQ model to discretize the governing differential eigenvalue equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results solved by the developed numerical algorithms are presented. The convergence of the developed DQEM analysis models is efficient.

Size-Dependent Stress Analysis of Single-Wall Carbon Nanotube Based on Strain Gradient Theory

2017 ◽  
Vol 09 (06) ◽  
pp. 1750087 ◽  
Author(s):  
Mohammad Hosseini ◽  
Hamid Haghshenas Gorgani ◽  
Mohammad Shishesaz ◽  
Amin Hadi
Keyword(s):  
Differential Equation ◽  
Carbon Nanotube ◽  
Boundary Conditions ◽  
Strain Gradient ◽  
Numerical Results ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Classical Elasticity ◽  
Size Dependent ◽  
Governing Differential Equation

This paper studies stress distribution in a single-walled carbon nanotube (SWCNT) under internal pressure with various chirality. Strain gradient theory is used to capture the size-dependent behavior of the SWCNT. Minimum total potential energy principle is successfully applied to derive the governing differential equation and its associated boundary conditions. Due to complexity of the governing differential equation and boundary conditions, numerical scheme is used to solve the problem. Comparing the results based on strain gradient theory and that of classical elasticity shows a major difference between these two methods. However, a close examination of the results indicates that both theories predict the same trend for variations in the radial displacement along the SWCNT radius. Numerical results also indicate that the proposed model can lead into the classical elasticity model, provided the material length scale parameters are taken to be zero. Additionally, for plane strain condition, the radial displacements predicted by strain gradient theory are lower than those predicted by classical elasticity theory. Moreover, numerical results show that in a SWCNT, the non-dimensional radial and circumferential stresses along the wall thickness of the SWCNT increase as the radius is increased. The opposite behavior is true for non-dimensional high-order stresses.

Static Deflection and Free Vibration of Nonprismatic Beam Structures Solved by DQEM Using EDQ

2000 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Numerical Algorithm ◽  
Equilibrium Equation ◽  
Static Deflection ◽  
Beam Structures ◽  
Prismatic Beam ◽  
Vibration Problems ◽  
Analysis Models ◽  
Element Boundary

Abstract The development of (DQEM) analysis models of static deformation and free vibration problems of generic non-prismatic beam structures was carried out. The DQEM uses the extended differential quadrature (EDQ) to discretize the buckling equilibrium equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results solved by the developed numerical algorithm are presented. They prove that the DQEM efficient.

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

A 3D BOUNDARY ELEMENT METHOD FOR DETERMINATION OF ACOUSTIC EIGENFREQUENCIES CONSIDERING ADMITTANCE BOUNDARY CONDITIONS

1993 ◽  
Vol 01 (04) ◽  
pp. 455-468 ◽  
Author(s):  
Z. S. CHEN ◽  
G. HOFSTETTER ◽  
H. A. MANG
Keyword(s):  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Large Scale ◽  
Numerical Study ◽  
Numerical Algorithms ◽  
Numerical Results ◽  
Simple Problem ◽  
Element Method

A 3D boundary element method for the determination of the acoustic eigenfrequencies of car compartments, characterized by a unified treatment of Robin, Dirichlet, and Neumann boundary conditions, is presented. The drawback of frequency-dependent matrices of the eigenvalue problem is overcome by means of the Particular Integral Method. Thus, the standard numerical algorithms for the extraction of eigenvalues can be applied. The numerical study contains both a comparison of numerical results with analytical solutions of a simple problem with different types of boundary conditions and a comparison of numerical results of a large-scale problem with respective numerical results, computed on the basis of the finite element method. In addition, for the latter example, different numerical algorithms for the eigenvalue extraction are examined.

Dynamic Bending Stress in a Disk-Type Gyroscope Rotor Under Steady Precession

1970 ◽  
Vol 92 (1) ◽  
pp. 219-225
Author(s):  
R. I. Sann
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Lagrange Equation ◽  
Iterative Solution ◽  
Numerical Results ◽  
Matrix Eigenvalue Problem ◽  
Matrix Eigenvalue ◽  
Bending Stresses ◽  
The Web

This paper derives the equations which govern the cyclic bending stresses in the web of a precessing gyro rotor, and discusses methods of solution. These stresses are important because they contribute to fatigue failure. Starting from the well-known partial differential equation describing the free lateral vibration of a thin variable thickness plate in the presence of initial centrifugal stresses, an ordinary differential equation for the mode displacement as a function of radius is obtained. Boundary conditions consist of a light, flexible shaft at the inside diameter of the web and a rigid, heavy rim at the outside diameter of the web. Three methods of solving for the modal functions and resonant frequencies are described. These are 1 Reduction to a matrix-eigenvalue problem by collocation, 2 Reduction to a matrix-eigenvalue problem by finite differences, and 3 An iterative solution based on numerical integration of the differential equation. Newton-Raphson interpolation against the eigenvalue is used to satisfy the boundary conditions. The forced vibration response to steady precession rate is evaluated from the Lagrange equation governing excitation of the fundamental normal coordinate. This coordinate corresponds to the lowest “fan” vibration made of the system, i.e., a mode in which the web has one diametral nodal line and no interior nodal circles. Numerical results show the variation of fan mode frequency with rotor spin rate, using web thickness as a parameter. Maximum radial and tangential bending stresses in the web are plotted against radius, using spin rate as a parameter. The numerical results indicate existence of an optimum rotor spin-rate, at which the allowable precession torque, based on web fatigue, is maximum for a given rotor structure.

scholarly journals Analysis on free vibration and critical buckling load of a FGM porous rectangular plate

2021 ◽  
Vol 39 (2) ◽  
pp. 317-325
Author(s):  
Zhaochun Teng ◽  
Pengfei Xi
Keyword(s):  
Differential Equation ◽  
Elastic Modulus ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Natural Frequencies ◽  
Gradient Index ◽  
Rectangular Plates ◽  
Buckling Loads ◽  
Governing Differential Equation

The properties of functionally gradient materials (FGM) are closely related to porosity, which has effect on FGM's elastic modulus, Poisson's ratio, density, etc. Based on the classical theory of thin plates and Hamilton principle, the mathematical model of free vibration and buckling of FGM porous rectangular plates with compression on four sides is established. Then the dimensionless form of the governing differential equation is also obtained. The dimensionless governing differential equation and its boundary conditions are transformed by differential transformation method (DTM). After iterative convergence, the dimensionless natural frequencies and critical buckling loads of the FGM porous rectangular plate are obtained. The problem is reduced to the free vibration of FGM rectangular plate with zero porosity and compared with its exact solution. It is found that DTM gives high accuracy result. The validity of the method is verified in solving the free vibration and buckling problems of the porous FGM rectangular plates with compression on four sides. The results show that the elastic modulus of FGM porous rectangular plate decreases with the increase of gradient index and porosity. Furthermore, the effects of gradient index and porosity on dimensionless natural frequencies and critical buckling loads are further analyzed under different boundary conditions with constant aspect ratio, and the effects of aspect ratio and load on dimensionless natural frequencies under different boundary conditions.

Triangular Plates Analyzed by Point Matching

1962 ◽  
Vol 29 (4) ◽  
pp. 755-756 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Numerical Results ◽  
Point Matching ◽  
Simply Supported ◽  
Special Adaptation ◽  
Matching Technique

This brief note analyzes uniformly loaded triangular plates with either clamped or simply supported edges using a special adaptation of the point-matching technique, the functions satisfying the differential equation, also being chosen to satisfy exactly the boundary conditions on one edge. Numerical results are tabulated for three geometries.

Governing Equations for Vibrating Constrained-Layer Damping Sandwich Plates and Beams

1972 ◽  
Vol 39 (4) ◽  
pp. 1041-1046 ◽  
Author(s):  
M.-J. Yan ◽  
E. H. Dowell
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Numerical Results ◽  
Sandwich Plates ◽  
Constrained Layer Damping ◽  
Governing Equations ◽  
Natural Boundary ◽  
Comparison With Experiment ◽  
Preliminary Comparison ◽  
Natural Boundary Conditions

For constrained-layer damping a simple differential equation for nonsymmetric sandwich plates or beams made of isotropic and homogeneous layers is deduced. The natural boundary conditions associated with this equation are also derived. Typical numerical results are presented including a preliminary comparison with experiment.

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