应用数学和力学

1993 Vol. 14, No. 2

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Singular Perturbation of Initial Value Problem for a Nonlinear Second Order Systems

Lin Zong-chi, Lin Su-rong

1993, 14(2): 95-100.

Abstract(1795) PDF(581)

Abstract:
In this paper,the singular perturbation of initial value problem for nonlinear second order vector differential equations ε'x"=f(i,x,x',ε) x(0,ε)=α,x'(0,ε)=β is discussed,where r>0 is an arbitrary constant,ε>0 is a small parameter,x,f,a and Under suitable assumptions,by using the method of many-parameter expansion and the technique of diagonalization,the existence oj the solution of perturbation problem is proved and its uniformly valid asymptotic expansion of higher order is derived.

Incompatible Curved Quadrilateral Plate Bending Element with 12 Degrees of Freedom

Ji Zhen-yi, Yeh Kai-yuan

1993, 14(2): 101-107.

Abstract(1707) PDF(592)

Abstract:
This paper presents a new curved quadrilateral plate element with 12-degree freedom by the exact element method^[1].The method can be used to arbitrary non-positive and positive definite partial differential equations without variation principle.Using this method,the compatibility conditions between element can be treated very easily,if displacements and stress resultants are continuous at nodes between elements.The displacements and stress resultants obtained by the present method can converge to exact solution and have the second order convergence speed.Numerical examples are given at the end of this paper,which show the excellent precision and efficiency of the new element.

Pansystems Philosophy,Pansystems Mathematics(PPPM) and Applications to APTMS:Affairology,Physics,Technology,Medicine and Strategics(Ⅱ)

Wu Xue-mou

1993, 14(2): 109-117.

Abstract(1645) PDF(558)

Abstract:
A new type of philosophy and mathematics from the pansystems view is introduced here,including the 7 philosophy theories(7PT) and related mathematizing researches.Many second/third philosophies are developed within pansystems framework and related applications to APTMS.

The Numerical Stabilities of Multiderivative Block Method

Kuang Jiao-xun, Lin Yu-hua

1993, 14(2): 119-126.

Abstract(1711) PDF(555)

Abstract:
In [1],a class of multiderivative block methods(MDBM) was studied for the numerical solutions of stiff ordinary differential equations.This paper is aimed at solving the problem proposed in [1] that what conditions should be fulfilled for MDBMs in order to guarantee the A-stabilities.The explicit expressions of the polynomials P(h) and Q(h) in the stability functions ξ^k(h)=P(h)/Q(h)are given.Furthermore,we prove P(-h)=Q(h).With the aid of symbolic computations and the expressions of diagonal Fade approximations,we obtained the biggest block size k of the A-stable MDBM for any given l(the order of the highest derivatives used in MDBM,l≥1)

Consolidation Theory of Unsaturated Soil Based on the Theory of Mixture(Ⅰ)

Chen Zheng-han, Xie Ding-yi, Liu Zu-dian

1993, 14(2): 127-137.

Abstract(2741) PDF(928)

Abstract:
Unsaturated soil is a three-phase media and is composed of soil grain,water and gas.In this paper,the consolidation problem of unsaturated soil is investigated based on the theory of mixture.A theoretical formula of effective stress on anisotropic porous media and unsaturated soil is derived.The principle of effective stress and the principle of Curie symmetry are taken as two fundamental constitutive principles of unsaturated soil.A mathematical model of consolidation of unsaturated soil is proposed,which consists of 25 partial differenfial equations with 25 unknowns.With the help of increament linearizing method,the model is reduced to 5 governing equations with 5 unknowns,i.e.,the three displacement components of solid phase,the pore water pressure and the pore gas pressure.7 material parameters are involved in the model and all of them can he measured using soil tests.It is convenient to use the model to engineering practice.The well known Biot's theory is a special case of the model.

Problem of Hydrodynamic Pressure on Suddenly Starting Vessel

Tao Ming-de, Shi Xiao-ming

1993, 14(2): 139-145.

Abstract(1646) PDF(602)

Abstract:
In this paper,Lagrangian method is applied to discuss the problem of the hydrodynamic pressure on a suddenly starting vessel.The free surface profile and the coefficients of the hydrodynamic pressure im the vessel wall are obtained.And it is verified that the singularity of the pressure near the free surface is only logarithmic.

Correction for Housne’s Equation of Bending Vibration of a Pipe Line Containing Flowing Fluid

Zhang Xi-de, Du Tao, Zhang Wen, Shen Wen-jun

1993, 14(2): 147-149.

Abstract(1999) PDF(769)

Abstract:
This paper points out that Housner's equation of bending vibration of a pipe line containing flowing fluid is approximate and makes correction to it.An exact form of the vibration equation is given.

A Polynomial Method for Solving the Problems for Lateral Instability of Cantilever Plates

Yuan Yi-wu

1993, 14(2): 151-155.

Abstract(1525) PDF(647)

Abstract:
The present article researches the problems of the lateral instability of cantilever rectangular plates under a concentrated force or a uniformly distributed load respectively.We select the polynomial(2.1) instead of the cosecant function in Ref.[1] as the flexural functions.The minimum critical load obtained here is more exact than the results obtained in Ref.[1].

Anisotropic Plastic Fields at a Rapidly Propagating Crack-Tip

Lin Bai-song

1993, 14(2): 157-163.

Abstract(1672) PDF(517)

Abstract:
Under the condition that all the stress components at a crack-tip are the functions of θ only,making use of the equations of steady-slate motion,stress-strain relations and Hill anisotropic yield conditions,we obtain the general solutions at a crack-tip in both the cases of anti-plane and in-plane strains.Applying these general solutions to the concrete cracks,the anisotropic plastic fields at the rapidly propagating tips of mode Ⅲ and mode Ⅰ cracks are derived.

Vibration Problems of Flexible Circular Plates with Initial Deflection

Wang Jin-ying, Chen Ke-jin

1993, 14(2): 165-171.

Abstract(1892) PDF(606)

Abstract:
In this paper,the differential equations of flexible circular plates with initial deflection are derived.The stability of motion is investigated in phase plane.The periodical solutions of nonlinear vibration for circular plates with initial deflection are obtained by use of Galerkin method and Lindstedt-Poincaré perturbation method.The effect of initial deflection on the dynamic behavior of the flexible plates are also discussed.

Numerical Residual Elimination Method of Finite Differential Equations

Huang Ping, Dong Zheng-zhu

1993, 14(2): 173-179.

Abstract(2258) PDF(549)

Abstract:
In this paper,a new elimination of finite differential equations has been discussed.It applies the numerical direct iteration to obtain the residual equations,in which the number of unknowns has been reduced greatly.The solution process is simple and efficient,and the solution is exact.

Some Multiplicity Results for an Elastic Beam Equation at Resonance

Ma Ru-yun

1993, 14(2): 181-188.

Abstract(1871) PDF(586)

Abstract:
This paper deals with multiplicity results for nonlinear elastic equations of the type -d⁴u/dx⁴+π⁴u+g(x,u)=e(x)(0R→R satisfies Carathéodory condition e∈L²[0,1].The solvability of this problem has been studied by several authors,but there isn't any multiplicity result until now to the author's knowledge.By combining the Lyapunov-Schmidt procedure with the technique of connected set,we establish several multiplicity results under suitable condition.