应用数学和力学

Some Essential Features and Its Several Applications of a Degenerated Four-Leaved Rose Curve

Liu Hsien-chih

1982, 3(2): 131-143.

Abstract(2021) PDF(604)

Abstract:
This paper reports the investigation of possibly a new curve that the author has met in mechanics research, which has hitherto not yet been found in the most popular mathematical literature, nevertheless the references, which he had read, rather limited for anybody. Besides some noticeable properties of the curve, we have given only two instances which the author, by chance, has met personally in technical literature, even in these case, there still fails a detailed account from a mathematical point of view.

On the Uniformly Valid Asymptotic Solution of a Non-linear Dirichlet Problem

Jiang Fu-ru

1982, 3(2): 145-165.

Abstract(1722) PDF(416)

Abstract:
In this paper, Dirichlet problem for second order quasi-linear elliptic equation with a small parameter at highest derivatives is studied. In case the degenerate equation has no singular point, and parameter ε is sufficiently small, the existence and uniqueness of solution are proved The uniformly valid asymptotic approximation of solution is derived in the entire region.

Finite Element Control Systems of Elastic Structures

Huang Lin, Chen De-cheng

1982, 3(2): 167-180.

Abstract(2085) PDF(568)

Abstract:
The article deals with the problems of controllability,observability and stabilizability of an elastic-structural system treated by the finite element method. The results obtained here agree with that obtained in distributed parameter-system model, nevertheless, they are more convenient than those in carrying out the computation with a computer, at the same time the method appears much easier than the conventional one. In section one,the system's controllability and observability are studied and some conditions which are easier to be justified by computer are given. In section two, the problem of stabilizing an elastic object by the use of linear feedback is fully discussed. As the attained results there show that, so far as an elastic-structural system is concerned, it is possible to assign arbitrary frequencies of vibration only by the use of displacement feedback, however, it is impossible to stabilize the system while the system is completely controllable. While the velocity feedback can stabilize the system, but its ability is limited. The case of rigid body motion involved in the system equation has also been discussed. In section three, the control of a straight beam is treated by the finite element method. The whole system of a beam can be decomposed into four irrelevant subsystems of tension-compression, torsion, bending in two directions, their controllability and observability are also analyzed respectively. The controllability and observability of segment-shaped beam are discussed in the end.

A Gaseous Shock Wave Theory of the Galactic Spiral Structure

Hu Wen-rui

1982, 3(2): 181-196.

Abstract(1696) PDF(462)

Abstract:
In this paper, the galactic spiral structure is studied by the galactic shock wave of interstellar gas with self-gravitation. The perturbed gravitation of stars is not a necessary condition for the existence of such shock. It is proved first of all that there exists solution of local shock wave even if the perturbed gravitation is absent. The condition |ω_η0|>σ is required for such solution. The spiral structure can only be explained by the shock solution when the difference of density between the regions of arm and interarm is larger. The grand design of shock wave with self-gravitation is obtained by the iterative method. The features of shock wave can be analyzed qualitatively in the velocity plane for a special perturbed gravitation which is used to simulate the self-gravitation. of interstellar gas. As the mass distribution in proto-galactic disk is irregular initially, the grand design of the galactic shock wave was developed by the processes of winding,growth of instability and overlapping of waves. Hence, it gives a complete figure about the origin, evolution and persistance.A lot of observed phenomena and classificational features of the galactic spiral structure can be explained by adopting these ideas.

An Analysis of the Three-Dimensional Elastic Solid with Internal Rectangular Crack

Wang Kai

1982, 3(2): 197-202.

Abstract(1551) PDF(473)

Abstract:
In this paper, the three-dimensional elastic solid with internal rectangular crack is considered. Let the crack surfaces be subjected to the equal and opposite normal tractions p₀. This problem is reduced by means of Fourier transforms to the standard set of dual integral equations with two variables. Then the formulas of analytic solution of the displacements on the crack surfaces and of the stress-intensity factors of crack border are obtained.

On Periodic Solutions of Several Classes of Riccati’s Equation and Second-Order Differential Equations

Li Hong-xiang

1982, 3(2): 203-209.

Abstract(1853) PDF(833)

Abstract:
In this paper, the problem on periodic solutions of several classes of Riccati's equation with periodic coefficients is discussed, and the conditions, under which several classes of secondorder equations with periodic coefficients have periodic solutions, are given.

On the Circumferential Strain Factor Criterion of Mixed Mode Brittle Fracture

Fan Wei-xun

1982, 3(2): 211-224.

Abstract(1835) PDF(547)

Abstract:
In this paper, the criteria of mixed mode brittle fracture are carefully examined. It has been shown that, the circumferential strain factor criterion is rational and safe.With the exception of opening mode, mixed mode plane strain fracture of comparatively ductile materials(metals), in general, does not follow the theory of linear elastic fracture mechanics.Like the stress intensity factor that which is concerned is playing an important role in pure opening mode crack problems. We believe that, in mixed mode crack problems, the circumferential strain factor will become a parameter to determine the rate of fatigue crack propagation per cycle, and of stress corrosion cracking per unit time.

Nonlinear Modulation of the interfacial Waves of Two Superposed Fluids

Zou Qi-su

1982, 3(2): 225-234.

Abstract(1444) PDF(582)

Abstract:
The slow modulation of the interfacial capillary-gravity waves of two superposed fluids with uniform depths and solid walls is investigated by using the method of multiple sca.les.The evolution of a packet is described by the nonlinear Schrodinger equation, and then the stability of the so-called Stokes wave train is discussed.

On the Criterion for the Absolute Stability of the Control System

Liao Xiao-xin

1982, 3(2): 235-248.

Abstract(1958) PDF(724)

Abstract:
In this paper, firstly we give the criterion for the absolute stability of the second canonical form for the control system, including.the equation of the longitudinal motions of a plane as a particular example. The corresponding result in [8],[9] is a particular example given in this paper. Secondly, we give the criteria for the absolute stability of the first canonical form in the usual case and in the critical case. Finally, we give some criteria for the absolute stability of the general form for the direct control system.All the results in this paper merely depend upon the relations between the parameters of the system itself to give an explicit algebraic discriminant.

Bending of Cantilever Rectangular Plate with Concentrated Load

Lin Peng-cheng

1982, 3(2): 249-258.

Abstract(1852) PDF(746)

Abstract:
In this paper, the solutions for the bending of cantilever rectangular plates with concentrated load acting at any point of the middle line perpendicular to the clamped edge are given by means of a conception named modified simply supported edges and the method of superposition. Some numerical examples are presented. The total bending moment checks very well with the value determined statically.

On the Application of Mixed Method to Solve Solids of Revolution with Discrete Fixed Supports

He Qiong

1982, 3(2): 259-266.

Abstract(1486) PDF(423)

Abstract:
In this paper, the force method of statically indeterminate structure mechanics is used to treat the solids of revolution with discrete fixed supports. The reactionary forces of discrete fixed supports are considered as statically indeterminate unknown variables. The force-method canonical equations, in which the coefficient matrix and the right-hand vector are computed by semi-analytical finite element method, are solved. Then the finite element solution of solid of revolution with discrete fixed supports is calculated with the external loads superposed from the assigned external loads and the reactionary forces of discrete supports.

1982 Vol. 3, No. 2