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.

The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

Solution of axisymmetric torsion problems by point matching

1971 ◽  
Vol 6 (2) ◽  
pp. 124-133 ◽  
Author(s):  
G J Matthews ◽  
C J Hooke
Keyword(s):  
Boundary Conditions ◽  
Numerical Technique ◽  
Point Matching ◽  
Analytical Results ◽  
Elasticity Equations ◽  
Matching Technique ◽  
Basic Solutions ◽  
Axisymmetric Bodies

A general numerical technique is presented for the solution of the problem of torsion of axisymmetric bodies. The method superimposes a number of basic solutions of the elasticity equations using the point-matching technique so as to satisfy approximately the prescribed boundary conditions of a body. Results obtained by this technique are compared with those obtained by alternative experimental and theoretical techniques for various body geometries to assess the accuracy of the method. The technique is then applied to the problem of the torsion of shouldered shafts since large discrepancies exist between the experimental and analytical results available for this type of structure.

Analytical solution for the buckling of rectangular plates under uni-axial compression with variable thickness and elasticity modulus in the y-direction

2009 ◽  
Vol 224 (1) ◽  
pp. 33-41 ◽  
Author(s):  
M Saeidifar ◽  
S N Sadeghi ◽  
M R Saviz
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Power Series ◽  
Boundary Conditions ◽  
Variable Thickness ◽  
Elasticity Modulus ◽  
Plate Motion ◽  
Transverse Displacement ◽  
Buckling Loads ◽  
Simply Supported

The present study introduces a highly accurate numerical calculation of buckling loads for an elastic rectangular plate with variable thickness, elasticity modulus, and density in one direction. The plate has two opposite edges ( x = 0 and a) simply supported and other edges ( y = 0 and b) with various boundary conditions including simply supported, clamped, free, and beam (elastically supported). In-plane normal stresses on two opposite simply supported edges ( x = 0 and a) are not limited to any predefined mathematical equation. By assuming the transverse displacement to vary as sin( mπ x/ a), the governing partial differential equation of plate motion will reduce to an ordinary differential equation in terms of y with variable coefficients, for which an analytical solution is obtained in the form of power series (Frobenius method). Applying the boundary conditions on ( y = 0 and b) yields the problem of finding eigenvalues of a fourth-order characteristic determinant. By retaining sufficient terms in power series, accurate buckling loads for different boundary conditions will be calculated. Finally, the numerical examples have been presented and, in some cases, compared to the relevant numerical results.

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.

Vibration of a Fluid Conveying Pipe Carrying a Discrete Mass

1974 ◽  
Vol 96 (4) ◽  
pp. 268-272 ◽  
Author(s):  
T. T. Wu ◽  
P. P. Raju
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Flow Velocity ◽  
General Expression ◽  
Normal Modes ◽  
Critical Value ◽  
Numerical Results ◽  
Simply Supported ◽  
Various Boundary Conditions ◽  
Discrete Mass

This paper presents a method to predict the dynamic response of a fluid conveying pipe carrying a discrete mass when the flow velocity is less than its critical value. A general expression for the normal modes of a vibrating pipe with various boundary conditions is newly derived herein. Also presented for a particular case are the numerical results of eigenfunctions and eigenvalues which can be used to calculate the dynamic response of a simply-supported pipe with an attached discrete mass at its mid-span.

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.

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.

Free-Vibration Analysis of Rectangular Plates With Clamped-Simply Supported Edge Conditions by the Method of Superposition

1977 ◽  
Vol 44 (4) ◽  
pp. 743-749 ◽  
Author(s):  
D. J. Gorman
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Formidable Challenge ◽  
Edge Conditions ◽  
Governing Differential Equation

In this paper attention is focused on the free-vibration analysis of rectangular plates with combinations of clamped and simply supported edge conditions. Plates with at least two opposite edges simply supported are not considered as they have been analyzed in a separate paper. It is well known that the family of problems considered here have presented researchers with a formidable challenge over the years. This is because they are not directly amenable to Le´vy-type solutions. It has been pointed out in the literature that most of the existing solutions are approximate in that they either do not satisfy exactly the governing differential equation or the boundary conditions, or both. In a new approach taken by the author the method of superposition is exploited for handling these dynamic problems. It is found that solutions of any degree of exactitude are easily obtained. The governing differential equation is completely satisfied and the boundary conditions are satisfied to any degree of exactitude by merely increasing the number of terms in the series. Convergence is shown to be remarkably rapid and tabulated results are provided for a large range of parameters. The immediate applicability of the method to problems involving elastic restraint or inertia forces along the plate edges has been discussed in an earlier publication.

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.

Large Time Behavior of Solutions to One Nonlinear Integro-Differential Equation

2008 ◽  
Vol 15 (3) ◽  
pp. 531-539
Author(s):  
Temur Jangveladze ◽  
Zurab Kiguradze
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Boundary Conditions ◽  
Large Time ◽  
Dirichlet Boundary ◽  
Large Time Behavior ◽  
Dirichlet Boundary Conditions ◽  
Time Behavior ◽  
Integro Differential Equation ◽  
Homogeneous Data

Abstract Large time behavior of solutions to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. The rate of convergence is given, too. Dirichlet boundary conditions with homogeneous data are considered.

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