应用数学和力学

1987 Vol. 8, No. 6

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Guo You-zhong

1987, 8(6): 473-478.

Abstract(1395) PDF(598)

Abstract:
In this paper we present the theoretical foundation concerning a famous method in structural analysis,the distribution of moment(or/and dispacement): the mathematical description,new criterion,convergence and error estimation of approximate solution and some possible generalizations to that of other constructions.

Second-Order Cnoidal Waves at the Free Surface and Interface of a Two-Fluid System

Liu Yu-lu, Dai Shi-qiang

1987, 8(6): 479-484.

Abstract(1791) PDF(521)

Abstract:
In this paper,using the PLK method and reductive perturbation method,we obtained the second approximation to cnoidal waves at the free surface and interface for the two-fluid system considered in [1].The corresponding results in [3] and [4] may be obtained as special cases in this paper.

C¹ Rational Interpolating Surface under Local Coordinate Systems

Jiang Shou-shan, Yang Peng-ji

1987, 8(6): 485-490.

Abstract(1490) PDF(461)

Abstract:
A method of constructing rational interpolating surface under local coordinate systems is presented,which can be used to solve the "large torsion problem" of surfaces.This kind of surface has better approximating effect and its properties can be easily discussed,so it has practical applications not only in CAD but also infinite element analysis and other fields.An example is given in the paper.

First-Order Perturbation Solution to the Complex Eigenvalues

Li Ji-ming, Wang Wei

1987, 8(6): 491-496.

Abstract(1682) PDF(665)

Abstract:
The matrix perturbation method is extended to discrete linear nonconservative system with unsymmetrical matrices in this article.By introducing the concept of the adjoint complex eigenvector and by making use of the orthogonality relationship in the complex mode theory,the first-order perturbation solution to the complex eigenvalues is derived.Numerical example shows that this method is efficient and practicable.

A Finite Element Explicit Algorithm for Solving the Temporal Temperature Fields

Huang Zhen-zhong

1987, 8(6): 497-504.

Abstract(1685) PDF(641)

Abstract:
Practical calculations and numerical experiments in this paper have shown that in elements relating to a common node it is acceptable and reasonable for derivaties of temperature with respect to time on nodes of those elements to be presented with one on common node,if linear interpolation shape function is taken.The relation between the derivative of temperature to time on a certain node and the temperature on other nodes around that node may therefore be established after discretization of the differential equation is made in space by the finite element method.Then an explicit scheme for calculating the temperature fields may be constructed.The obtained algebraic equations,being simple and the procedure being straight will be its two tangible advantages and its calculating will,therefore,be fast.The stability analysis by the maximum principle,as in the example quoted,proves that the stability condition is similar to that in implicit algorithms.

A Method of Finding the Principal Modes of Nonlinear Vibration Systems and Their Stabilities

Liu Lian-sheng, Huo Quan-zhong, Huang Ke-lei

1987, 8(6): 505-512.

Abstract(1664) PDF(738)

Abstract:
This paper presents a new method of finding the principal modes of nonlinear vibration systems,by means of which the problem of finding principal modes of nonlinear systems is transferred to the problem of finding real roots of a set of algebraic equations.The method is applicable to various kinds of nonlinear vibration systems with many degrees of freedom,and is simple in calculation.The paper presents another new method of analyzing the stabilities of principal modes of nonlinear systems.

Singular Perturbation of Linear Algebraic Equations with Application to Stiff Equations

Lin Wu-zhong

1987, 8(6): 513-522.

Abstract(1742) PDF(624)

Abstract:
In this paper the singular perturbation problem of linear algebraic equations with a small parameter is presented by an example in practice.The existence and uniqueness theorem of its solution is proved by the perturbation method and the estimation of error for its approximate solution is given.Finally,the example mentioned above explaining how to apply the theory to solve the stiff equations is shown.

On the Positive Definiteness of a Class of Operators

Wu Ji-ke, Yuan Yong

1987, 8(6): 523-526.

Abstract(1736) PDF(652)

Abstract:
In this paper,a proof of the positive definiteness for a class of operators is given.The operators considered are general enough to include those in two-and three-dimensional elasticity,thin plates and shells as their special cases.

Singularly Perturbed Nonlinear Second Order Elliptic Equation

Jin Shan

1987, 8(6): 527-538.

Abstract(1464) PDF(679)

Abstract:
In this paper,we study singular perturbation problems of some semi-linear second order elliptic equations with nonlinear boundary value conditions: where ε is a small positive parameter and (∂)u/(∂)l is a directional derivative,which lies on an oblique vector (x,ε).We have given a construction of the asymptotic solutions and proof of their asymptotic correctness,which is based on the principle of contraction mapping.

Further Study of the Relation of von Karman Equation for Elastic Large Deflection Problem and Schrodinger Equation for Quantum Eigenvalues Problem

Shen Hui-chuan

1987, 8(6): 539-546.

Abstract(1420) PDF(524)

Abstract:
This work is the continuation and improvement of the discussion of Ref.[1].We also improve the discussion of Refs.[2-3] on the elastic large deflection problem by results of thispaper.We again simplify the von Karman equation for elastic large deflection problem,and finally turn it into the nonlinear Schrodinger equation in this paper.Secondly,we expand the AKNS equation to still more symmetrical degree under many dimensional conditions in thispaper.Owing to connection between the nonlinear Schrodinger equation and the integrability condition for the AKNS equation or the Dirac equation,we can obtain the exact solution for elastic large deflection problem by inverse scattering method.In other words,the elastic large deflection problem wholly becomes a quantum eigenvalues problem.The large deflection problem with orthorhombic anisotropy is also deduced in this paper.

Stress Analysts of Plates with a Circular Hole Reinforced by Flange Reinforcing Member

Wang Gui-fang

1987, 8(6): 547-565.

Abstract(1390) PDF(561)

Abstract:
This paper deals with the problem of stress analysis of plates with a circular hole reinforced by flange reinforcing member.The so called flange reinforcing member here means that the reinforcing member is built up by setting shapes or bars with any section shape on both sides of the plates along the edge of the hole.Two cases of external loads are considered.In one case the external loads are stressesσ_X^(∞),σ_Y^(∞),and τ_XY^(∞) acting at infinite point of the plate,and in the other the external loads are linear distributed normal stresses.The procedure of solving the problems mentioned above consists of three steps.Firstly,the reinforcing member is taken out from the plates and considered to be a circular bar being solved to determine its deformation under the action of radial force q₀(θ) and tangential force t₀(θ) which are forces acting upon each other between reinforcing member and plate.Secondly,the displacements of plate with a circular hole under the action of q₀(θ) and t₀(θ) and external loads are determined.Finally,forces q₀(θ) and t₀(θ) are obtained by the compatibility of deformations between reinforcing member and plate.Then the internal forces and displacements of reinforcing member and plate are deduced from q₀(θ) and t₀(θ) obtained.