Applied Mathematics and Mechanics (English Edition)

ON THE SENSITIVE DYNAMICAL SYSTEM AND THE TRANSITION FROM THE APPARENTLY DETERMINISTIC PROCESS TO THE COMPLETELY RANDOM PROCESS

岳曾先;张彬

1985, 6(3): 193-211.

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The detailed analysis of the dynamical process of coin tossing is made. Through calculalions, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final conflguration, i.e. "head" or "tail", becomes more and more sensitive to the initial parameters (the initial velocity angular velocity,and the initial orientation). the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the "head" upward initially hut allow a small range for the change of somt other initial parameters, the frequency that thefmal conflguration is "head", would be I if the initial height h of the mass center is sufficiently small, and would be close to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a "transition" is very similar to the transition from laminar t o turbulent flows. A basic difference between the "transition stage" and the"completely rondom stage" is indicated: In the "completely rondom stage", the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters. but the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the "transition stage", both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the "transition stage" is the existence of the "long-train structure" in the parameter space. The illuminalions of this analysis on some other rondom phenomena are discussed.

THE SPECTRAL PROBLEM OF POISEUILLE FLOW

董明德

1985, 6(3): 213-218.

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For Poiseuille flow the Orr-Sommerfeld equation is solved exactly. Regular solutions are obtained, thereby eigenvalue equation can be analyzed analytically and explicitly. The bifurcation solutions will be discussed subsequently.

THEOREM OF THE UNIQUENESS OF DISPLACEMENT AND STRESS FIELDS OF LINE-LOADED INTEGRAL EQUATION METHOD

云天铨

1985, 6(3): 219-222.

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According to Fredholm’s theorem, this paper proves that due to the virtual fundaemental loods which satisfy the boundary conditions and being distributed outside the elastic body occupied region the displacement and stress fields in the elastic body occupied region are unique. This theorem forms a theoretical basis of the applications of the line-loaded integral equation method.

NEW FORMULATIONS OF THE SECOND FUNDAMENTAL PROBLEM IN PLANE ELASTICITY

路见可

1985, 6(3): 223-231.

摘要 ( 411 )

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Some general formulations of the second fundamental problem in plane elasticity are proposed here when the displacements given on the closed boundary contours of a multi-connected elastic region are relative to certain rigid motions which are different to each other for different boundary contours. In such case, it is proved that, for the unique existence ofsolution, there must be given in addition to the principal vector and the principal moment of the extemal forces on each boundary contour. A method of solution is given also together with some illustrative examples.

FORCED VIBRATIONS OF ELASTIC SHALLOW SHELL DUE TO THE MOVING MASS

成祥生

1985, 6(3): 233-240.

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This paper discusses the forced vibrations of the elastic shailow shell due to the moving moss by means of the variational method. In the text mentioned above a series of problems such as the forced vibrations, resonance conditions and critical speed and so forth.

ELASTOPLASTIC ANALYSIS OF CYLINDRICALLY ORTHOTROPIC COMPOSITE THICK-WALLED TUBE

周次青;丘益元

1985, 6(3): 241-248.

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In the present paper, Tsai-Hill Yield Criterion in a state of plane strain is derived for incompressible and compressible materials respectively The elastoplastic stress and displacement fields of cylindrically orthotropic composite thick-walled tube under uniform radial pressure are studied. The formulas of the yield pressure, Iimit pressure and region of shakedown are obtained.

APPLICATION OF SHOOTING METHODS FOR TWO-POINT BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS WITH PARAMETERS IN HYDRODYNAMICS

郭本铁;应玉燕;陈帮国;徐宝智

1985, 6(3): 249-257.

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In this paper, first we introduce the shooting method (including the method of adjoints and the method of quasilinearization) for two-point boundary value problems of ordinary differential equations with parameters, and give out an example of solving hydrodynamic lubrication equation by using the shooting method. The calculated results are satisfactory.

ONE-DIMENSIONAL NONUNIFORM TEMPERATURE FLOW WITH HEAT TRANSFER BY CONVECTION

袁镒吾

1985, 6(3): 259-268.

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In ref. [1], V. E. Najenov studied the conditions that when the viscosity of the liquid is on exponential function of temperature, the pipe flow, hoving sleady heat transfer, is one-dimensional and with nonuniform temperature. For plane canal and circular pipe he still studied the velocity and the temper turefields.In thispaper, the author presents two new methodsfor solving the same problem. The method as in ref. [1] may be regarded as the natural branch of the methods of this paper. One of our new methods only can solve the same problem as in ref.[1] and the complex degree of its computing process is nearly the same as that in ref. [1]. But the other can go beyond the studying scope of ref. [1], namely, for the case ihal the curvatures of circumference of the cross section of the pipe are not equivalent everywhere, the problem may also be solved.

CRITERIA FOR THE EXISTENCE OF SOLUTIONS TO RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS

丁协平

1985, 6(3): 269-275.

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In [1], we proved a general rondom fixed point theorem and gave some applications. In this paper, we shall give further applications of the theorem. We first obtain a rondom Darbo’s fixed point theorem, using the theorem, we give the criteria for the existence of solutions under compactness hypotheses to nonlinear rondom volterra integral equations and the Cauchy problem of nonlinear rondom differential equations. Our theorems improve andgeneralize some main results of Lakshmikanthem^[2], Vaughn^[3,4] as well as De Blasi and Myjak^[5].

RECURRENCE FORMULA FOR B-SPLINES WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS

许跃生

1985, 6(3): 277-281.

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In this paper, we give recurrence formula for normalized B-Splines with respect to Standard system {φ^i-1(x)}_i-1^m which is the basic set of solutions to a class of differential operators in view of generalized divided difference——Green function——B-Splines.

A NEW METHOD OF INTEGER LINEAR PROGRAMMING-BRANCH DIRECTION SEARCH METHOD

夏德麟

1985, 6(3): 283-289.

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