应用数学和力学

1990 Vol. 11, No. 2

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A Non-Dualistic Unified Field Theory of Gravitation,Electromagnetism and Spin

Yu Xin

1990, 11(2): 95-109.

Abstract(1774) PDF(448)

Abstract:
The wisdom of classicalunified field theories in the conceptual framework of Weyl,Eddington,Einstein and Schrodinger has often been doubted and in particular there does not appear to be any empirical reason why the Einstein-Maxwell(E-MJ theory needs to be geometrized.The crux of the matter is,however not whether the E-M theory is aesthetically satisfactory but whether it answers all the modern questions within the classical context.In particular,the E-M theory does not provide a classical platform from which the Dirac equation can be derived in the way Schrodinger's equation is derived from classical mechanics via the energy equation and the Correspondence Principle.The present paper presents a non-dualistic unified field theory(UFT) in the said conceptual framework as propounded by M.A.Tonnelat.By allowing the metric form ds²=g_μγdx^γdx^γ and the non-degenerate two-form F=(1/2t)φ_μγ dx^γΛdx^γ; to enter symmetrically into the theory we obtain a UFT which contains Einstein's General Relativity and the Born-Infeld electrodynamics as special cases.Above all,it is shown that the Dirac equation describing the electron in an "external" gravito-electromagnetic field can be derived from the non-dualistic Einstein equation by a simple factorization if the Correspondence Principle is assumed.

Effects of Bottom Topography on the Standing Waves in Circular Basins

Ma Yun-li, Li Jia-chun

1990, 11(2): 111-118.

Abstract(1408) PDF(539)

Abstract:
Standing waves in the cylinder hasins with inhomogeneous bottom are considered in this paper.We assume that the inviscid,incompressible fluid is in irrotational undulatory motion.For convenience sake,cylindrical coordinates are chosen.The velocitv potentials,the wave profiles and the modified frequencies are determined(to the third order) as power series in terms of the amplitude divided by the wavelength.Axisymmetrical analytical solutions are worked out.When ω₁=0,the second order frequency are gained.As an example,we assume that cylinder bottom is an axisymmetricat paraboloid.We find out that the uneven bottom has influences on standing waves.In the end.we go into detail on geometric factors.

On the Existence of Limit Cycles of Liénard Equation

Huang An-ji, Cao Deng-qing

1990, 11(2): 119-130.

Abstract(1725) PDF(452)

Abstract:
In this paper,we have proved several theorems which guarantee that the Liénard equation has at least one or n limit cycles without using the traditional assumption G(±∞)=+∞ Thus some results in [3-5] are extended.The limit cycles can be located by our theorems.Theorems 3 and 4 give sufficient conditions for the existence of n limit cycles having no need of the conditions that the function F(x) is odd or "nth order compatible with each other" or "nth order contained in each other".

The Criterion Algorithm of Relation of Implication between Periodic Orbits(Ⅱ)

Zhang Jing-zhong, Yang Lu, Zhang Lei

1990, 11(2): 131-140.

Abstract(1713) PDF(436)

Abstract:
In recent years,there is a wide interest in Sarkovskii's theorem and related study.According to Sarkov.lkii's theorem,if the continuous self-map f of the closed interval has a 3-periodic orbit,then f must has an n-periodic orbit for any positive integer n.But f can not have all n-periodic orbits for some n.Example.Let Evidently,f has only one kind of 3-periodic orbit in the two kinds of 3-periodic orbits,which explains that it isn't far enough to uncover the relation between periodic orbits by the information which Sarkovskii's theorem has offered.In this paper,we raise the concept of type of periodic orbits,and give a feasible algorithm which decides the relation of implication between the two kinds of periodic orbits.

Optimizing Formulation of Hybrid Model for Thin and Moderately Thick Plates

Di Sheng-lin, Wu Chang-chun, Song Qi-gen

1990, 11(2): 141-148.

Abstract(1834) PDF(476)

Abstract:
A 20-DOF hybrid stress element based upon Mindlin plate theory is developed using the optimization design method for thin and moderately thick plates.Numerical tests consist of the convergency and performance to the plates with arbitrary thickness and shape and of the ultimate thin plate problems.

The Sudden Starting of a Floating Body in Deep Water

Tao Ming-de

1990, 11(2): 149-154.

Abstract(1664) PDF(403)

Abstract:
This paper applies Lagrangian method to discuss the sudden starting of a floating hotly in deep water and the analytical'solutions are obtained.It is known from the numerical results that the disturbing domain extends and the dynamic pressure also increases when the breadth of the floating body keeps constant and its depth increases.

Exact Solutions for Stationary Responses of Several Classes of Nonlinear Systems to Parametric and/or External White Noise Excitations

Zhu Wei-qiu

1990, 11(2): 155-164.

Abstract(1831) PDF(632)

Abstract:
The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach.The conditions for the existence and uniqueness and the behavior of the solutions are discussed.All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems.It is shown taht for each of the four classes of G.E.D.nonlinear stochastic systems there is a family of non-G.E.D.systems which are equivalent to the G.E.D.system in the sense of having identical stationary solution.The way to find the equivalent stochastic systems for a given G.E.D.system is indicated and.as an example,the equivalent stochastic systems for the second order G.E.D.nonlinear stochastic system are given.It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D.nonlinear stochastic systems may he found by searching the equivalent G.E.D.systems.

General Dynamic Equation and Dynamical Characteristics of Viscoelastic Timoshenko Beams

Xiao Can-zhang, Ji Yi-zhou, Chang Bao-ping

1990, 11(2): 165-172.

Abstract(1813) PDF(681)

Abstract:
In this paper,a governing differential equation of viscoelastic Timoshenko beam including both extension and shear viscosity is developed in the time domain by direct method.To measure the complex moduli and three parameters of standard linear solid,the forced vibration technique of beam is successfully used for PCL and PMMA specimens.The dynamical characteristics of viscoelastic Timoshenko beams,especially the damping properties,are derived from a considerable number of numerical computations.The analyses show that the viscosity of materials has great influence on dynamical characteristics of structures,especially on damping,and the standard linear solid model is the better one for describing the dynamic behavior of high viscous materials.

The Application of Boundary Element Method in Stress Analysis of Autofretted Tube with Notch

Song Shun-cheng, Zhang Yu-cheng, Feng De-zhen

1990, 11(2): 173-178.

Abstract(1732) PDF(418)

Abstract:
In this paper,the boundary element method is applied to investigate the internal state of stress of autofretted tube with notch and the calculated results are important in the practical design.

Calculation for Cylindrical Shell under Local Vertical Loadings

Sun Ren-bo, Wang Dai-yu, Zou Ding-qi

1990, 11(2): 179-188.

Abstract(1781) PDF(627)

Abstract:
In this paper,we use the method of mixed-type series to derive the analytical solutions of cylindrical shell,which is simply supported along the transverse edges and subjected to the local vertical loads,and give the analytical expressions of the solutions for this kind of shell under five types of local vertical loading.A numerical example for a cylindrical shell roof,which is simply supported along the trans verse edges and is free along the longitudinal edges,is given in this paper and from the calculated results it may he seen that the convergence of the solutions is considerably satisfactory.Using the solutions of this paper,we can deal with some practical problems of underground structure.