Applied Mathematics and Mechanics (English Edition)

Longitudinal Free Vibration of Inhomogeneous Elastic Straight Strut with a Variable Cross Section

刘先志

1980, 1(2): 149-157.

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The scope of the present article, motivated by the case of the composite wooden propeller of an airplane, is to deal tentatively with the longitudinal free vibrationproblem of an elastic straight bar with a more general mathematical treatment.In this analysis, we have assigned to the modulus of elasticity, the bar cross section as well as the mass per unit length of the bar an exponential function variation, and then found a general solution, wherein three parameters were considered as the main factors to affect the longitudinal free vibration of the inhomogeneous elastic straight bar with a variable cross section.

The Explicit Forms of Field Functions in Totrahedron Element with 16 and 20 Degrees of Freedom

钱伟长

1980, 1(2): 159-164.

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In this paper, the explicit forms of field functions in tetrehedron elements with 16 and 20 degrees of freedom are given in terms of volume coordinates L₁, L₂, L₃, L₄ of tetrahedron.

On a class of method for solving problems with random boundary notches and/or cracks

欧阳鬯

1980, 1(2): 165-173.

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As we know, problems with boundary imperfections (notches or cracks) are the more important one in practical fracture analysis. Frequently, these imperfections would appear in the boundaries of the bodies as a randomly distributed group, and under the loading circumstances would grow up to be unstable cracks which induces catastrophic fracture of the bodies. For right evaluation the fracture behavior of the bodies with such boundary imperfections, it demands mathematical solutions for problems with random boundary notches and/or cracks.

The Difference Methods for the Solution of Singular-Perturbation for the Elliptic-Parabolic Differential Equation

苏煜城;吴启光

1980, 1(2): 175-185.

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In this paper it is discussed the difference method for the solution of singular perturbation problems for the elliptic equations, involving small parameter in the higher derivatives. As ε= 0 the original equations are degenerated into the parabolic equations.Authors constructed special difference schemes by means of the boundary layer properties of the solutions of these problems as well as investigated the convergence of this scheme and asymptotic behaviour of the solutions. Finally, a numerical example is given.

Circular Shallow Spherical Shells with Central Circular Hole Under Simultaneous Actions of Arbitrary Unsteady Temperature Field and Arbitrary Dynamic Normal Load

葉开源;许剑雲

1980, 1(2): 187-209.

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In this paper, under assumption that tempeature is linearly distributed along the thickness of the shell, we deal with problems as indicated in the title and obtain general solutions of them which are expressed in analytic form.In the first part, we investigate free vibration of circular shallow spherical shells with circular holes at the center under usual arbitrary boundary conditions. As an example, we calculate fundamental natural frequency of a circular shallow spherical shell whose edge is fixed (m=0). Results we get are expressed in analytic form and check well with E. Reissner’s [1]. Method for calculating frequency equation is recently suggested by Chien Wei-zang and is to be introduced in appendix 3.In the second part, we investigate forced vibration of shells as indicated in the title under arbitrary harmonic temperature field and arbitrary harmonic dynamic normal load.In the third part, we investigate forced vibration of the above mentioned shells with initial conditions under arbitrary unsteady temperature field and arbitrary normal load.In appendix 1 and 2, we discuss how to express displacement boundary conditions with stress function and boundary conditions in the case m=1.

On the Boundary Value Problems for a Class of Ordinary Differential Equations with Turning Points

江福汝

1980, 1(2): 211-223.

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In this paper we study the boundary value problems for a class of ordinary differential equations with turning points by the method of multiple scales. The paradox in [1] and the variational approach in [2] are avoided. The uniformly valid asymptotic approximations of solutions have been constructed. We also study the case which does not exhibit resonance.

A Brief Account of G.D. Birkhoff’s Problem in the Problem of Three Bodies

董金柱

1980, 1(2): 225-230.

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The present article gives a historical survery of G.D. Birkhoffs seventh problem which is an inquiry about the topological structure of the set of definition of the reduced differential equations of motion. Recent advances in the problem and their meaning have been briefly indicated.The classical 3-body problem concerns how the three particles should move under their mutual Newtonian attraction. By a particle we mean a goometrical point endorsed with a constant positive number m which is called mass.

On The Reissner Theory of Bending of Flastic Plates

苗天德;程昌钧

1980, 1(2): 231-246.

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The Reissner equations of elastic plates are rederived on the bases of the incomplete generalized variational principle of Complementary energy. The Stress function is naturally obtained from the variational Calculation in the form of Lagrange multiplier. The stucture of solutions of the Reissner equations is thus defined. On the bases of these discussions, a simplified theory has been put forward, in which the equations of equilibrium involving the shearing influence can be reduced into a fourth order differential equation similar to those of the Classical theory of plates.

Direct Derivation of the Two-shaft System Balance Conditions for the Second Order Reciprocal Inertia Forces on the V-Type Eight Cylinders Internal Combustion Engines with a Plane Crankshaft

刘先志

1980, 1(2): 247-263.

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By employing the "four shafts balance concept" paper [1] has reported a balance regime for the second order reciprocal inertia forces on the V-type eight cylinder internal combustion engines with a plane crankshaft. Thereafter, paper [2] has acquired a "two-shafts balance" regime, but through a rather tedious roudabout degenerating manipulation. The present article has, but starting out directly from the "two-shafts balance concept", successfully acquired the same results as those in paper [2]. In addition, we propose, herein, a third balance system which might be, in general, called the slipper balance" regime.

The Fundamental Equations in Finite Element Method of Coupled Thermo-elastic Plane Problem

王洪纲

1980, 1(2): 265-277.

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The fundamental equations in finite element method for unsteady temperature field elastic plane problem are derived on the bases of variational principle of coupled thermoelastic problems. In these derivations, elastic plane is divided into three nodes triangular elements, and time interval is divided into linear time elements, in which all the variables, including displacements and temperatures at various nodal points, are varied linearly with time. Two coupled sets of linear algebraic equations of all the unknown displacements and temperatures at every nodal point in every instant (i.e. the terminal values of time elements) are obtained. They are the fundamental equations of the said problem.

Elastic Behavior of Uniformly Loaded Circular Corrugated Plate with Sine-Shaped Shallow Waves in Large Deflection

陈山林

1980, 1(2): 279-291.

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By means of modified iteration method, this paper gives approximate solution of the large deflection equations of circular corrugated plate with sine-shaped shallow waves having a central platform under uniform lateral load. A formula of initial modification coefficient is given, and an integral is obtained for the simplification of modified iteration calculations. The results of present paper show better agreement with experimental data and larger applicable range than all other existing solutions of corrugated plates.

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