应用数学和力学

1983 Vol. 4, No. 4

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Theorems on the Limit Analysis Dealing with the Yield Condition Expressed by the Sum of the Homogeneous Linear Form of Stress Tensor and the Homogeneous Quadratic Form of Stress Tensor

Zhao Jia-yi, Hsueh Dah-wei

1983, 4(4): 441-452.

Abstract(1874) PDF(521)

Abstract:
In this paper a generalized variational principle on the limit analysis dealing with the yield condition expressed by the sum of the homogeneous linear form of stress tensor and the homogeneous quadratic form of stress tensor is suggested.This variational principle can be applied to the limit analysis in rock mechanics and it takes the situation, in which the yield condition is expressed by the homogeneous linear form of stress tensor or the homogeneous quadratic form of stress tensor, as its special case.

On Liapounoff s Method in the Theory of Stability of Laminar Fluid Flows

Zhou Heng, Li Li

1983, 4(4): 453-462.

Abstract(2067) PDF(442)

Abstract:
In [1] Zhou extended some Liapounoff's theorems of the theory of stability in the case of plane laminar fluid flows. In [2] Zhou and Li investigated the eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation, and obtained some new results. In this paper, based on the results of [1] and [2] we shall prove: (1) For the linearized problem the definition of stability according to the eigenvalues of Orr-Sommerfeld equation and that according to the perturbation energy are equivalent; (2) The method of linearization is admissible for the stability problem of plane laminar fluid flows for sufficiently small initial disturbance.

Vector Analysis of Spatial Mechanisms——(Ⅱ) Configuration Analysis of Spatial Mechanisms by Means of the Method of Vector Equations

Yu Xin

1983, 4(4): 463-467.

Abstract(1568) PDF(479)

Abstract:
In this paper, the configurations of the P-P-G-C and the R-G-C-R Mechanisms are established by the method of vector equations. As in Part (Ⅰ), explicit solutions in terms of given vectors are obtained, independent of the polynomial equations of the kind obtained by Chace.

The Extremity Laws of Hydro-Thermodynamics

Huang Wan-li

1983, 4(4): 469-476.

Abstract(1845) PDF(766)

Abstract:
This paper presents the law of maximum rate of energy dissipation in hydrodynamics and also in general continuum dynamics as an addition to the classical conservation laws expressed in the equation of continuity and the equations of motion. The corollary of the law is Belanger-Boss theorem of minimum reserved specific energy in applied hydraulics.The mechanical energy dissipated is transformed into heat reserved in the substance. The rate of energy dissipation at a time at a given temperature gives rise to the increase in entropy production. Hence the maximum rate of energy dissipation suggests itself the idea of reformulation of the second law of thermodynamics that the rate of entropy production in mechanical motion is always the maximum possible.The proposed extremity law in continuum dynamics has been derived from the variational principle and the reformulated second law of thermodynamics analyzed microscopically in the paper. The two laws together form the extremity laws of hydro-thermodynamics.

The Calculation of Bending and Stability of a Thin Rectangular Plate for Variable Rigidity

Huang Dao-an

1983, 4(4): 477-487.

Abstract(1887) PDF(691)

Abstract:
The subject discussed in this paper is the rectangular plate. Its two opposite edges are simply supported, while the other two are arbitrary, and the rigidity of the plate is. variable along the direction parallel to the simply supported edges. In order to solve the problem, the author adopts the finite plate-strip element method, which is different from the usual finite element method or the finite strip method. The steps of the above method is no longer to establish a rigidity matrix for elements or strips and gather them into a total matrix for solution. Now the relation of transfer between the strain and inner force of every plate-strip is shown.Finally a practical example is given and this method is found to be easier and more effective.

A Numerical Treatment of the Periodic Solutions of Non-Linear Vibration Systems

Ling

1983, 4(4): 489-506.

Abstract(2415) PDF(1246)

Abstract:
Direct numerical integration can be used to find the periodic solutions for the equations of motion of nonlinear vibration systems. The initial conditions are iterated so that they coincide with the terminal conditions. The time interval of the integration (i.e.,the period) and certain parameters of the equations of motion can be included in the iterations.The integration method has a variable steplength.This shooting method can produce periodic solutions with a shorter computer time. The only error occurs in the numerical integration and it can therefore be estimated and made small enough. Using this method one can treat a variety of vibration problems, such as free conservative, forced, parameter-excited and self-sustained vibrations with one or several degrees-of-freedom. Unstable solutions and those which are sensitive to parameter changes can also be calculated.The stability of the solutions is investigated based on the theory of differential equations with periodic coefficients. The extrapolation method and the procedure of automatic steplength control are used to estimate the initial values of iterations by determining the resonance curve and other vibration characteristics.Some examples have been calculated to illustrate the applicability of the method. The non-linearity way be expressed by an analytical function or any other functions, such as a piecewise linear function. Several remarkable features of nonlinear vibrations are presented through the periodic solutions obtained. Finally, some results are compared with those obtained by other approximation methods and experiments.

Two-Phase Displacement of Non-Newtonian Fluid

Chen Li-Han, Wu Wang-yi

1983, 4(4): 507-516.

Abstract(2051) PDF(449)

Abstract:
This paper considers the problem of non-Newtonian oil displacement by water in porous media, adopting the linear permeation law with initial pressure gradient. For one-dimensional flow, the basic equation of non-Newtonian oil displacement by water in sandstone reservoirs and fractured reservoirs is derived and numerical solutions are obtained. The results are compared with the corresponding ones for Newtonian oil displacement to show the essential characteristics of non-Newtonian oil displacement by water.

Effect of Initial Imperfections in Geometry on the Elastic- Plastic Stability of a Thin Annular Plate

Jiang Zhi-qing

1983, 4(4): 517-526.

Abstract(1772) PDF(465)

Abstract:
In this paper, Neale's generalized variational principle about incremental boundary-value problems is utilized to study the effect of initial imperfections in geometry on the critical loads of elastic-plastic buckling of thin annular plates. The calculations show that, if the effect of initial imperfections in geometry is taken into account in the solutions by J₂ incremental theory, the results are very close to the bifurca-tional buckling loads of the perfect annular plates according to the plastic deformation theory.

Extended Bounding Theorems of Limit Analysis

Gao Yang

1983, 4(4): 527-538.

Abstract(1794) PDF(454)

Abstract:
This paper studies the bounding problems of the complete solution of limit analysis for a rigid-perfectly plastic medium, allowing for the discontinuity of plastic flow. A generalized variational principle involving conditions of the rigid-plastic interface and the discontinuous surface of a velocity field has been advanced for the mixed-boundary value problem. Based on this principle, a set of variational formulae of limit analysis is established. The safety factors obtained by these formulae lie between the upper and lower bounds obtained by the classical bounding theorems with the same kinematically and statically admissible field.Moreover, extended bounding theorems have been derived and proved, which hold a broader stress and velocity field than the statically and kinematically admissible field. The corollaries of these theorems indicate the relationship between the variational solution and the complete solution of limit analysis. Applications of these theorems show that a close approximation can be obtained by the proposed method with different admissible fields.

On a New Kind of Methods to Solve the Plane Problems of Two-Phase Flow Through Porous Media

Chen Zhong-xiang, Yuan Yi-rang, Wang Wen-qia

1983, 4(4): 539-550.

Abstract(1645) PDF(522)

Abstract:
This paper presents a new kind of method for solving the plane problems of two-phase flow in porous media. The elliptical partial differential equation for pressure distribution is solved by the finite element method, and then the semi-analytical solution for pressure gradient is used to determine the new saturation field according to the existing exact formula describing the saturation propagation along the streamlines. The main distinguishing feature and advantage of this kind of method are the ability to overcome the numerical dispersion which is inherent in the ordinary numerical simulation methods, and thereby, to give a precise and clear-cut position of the saturation discontinuity in the water-oil displacement front. Moreover, the saturation equation, which should commonly be solved simultaneously or alternatively with the pressure equation, is completely avoided, so that the computing time is greatly reduced.

A Boundary Perturbation Solution for the Hydrodynamic Stability of Blood Flow in a Cone-Type Blood Processor

Zhu Yueh-rui, Zhang Xiao-ci

1983, 4(4): 551-561.

Abstract(1609) PDF(498)

Abstract:
This paper may be the first attempt to solve the flow-field of homogeneous fluid between concentric cones (narrow gap) with equal angular velocities of rotation by using a boundary perturbation method, and the hydrodynamic stability of blood flow in the cone-type blood processor is verified, under conditions of the narrow gap between the cones and small axial Re numbers. This paper also verifies the hydrodynamic stability between concentric cylinders (narrow gap) with equal angular velocities of rotation by using a new mathematical technic.These theoretical analyses are in agreement with the experimental observations, which have been made by Shanghai Medical Instruments Institute.

Dynamic Analysis of Nonlinear Systems by Modal Synthesis Techniques

Zheng Zhao-chang

1983, 4(4): 563-572.

Abstract(2039) PDF(1015)

Abstract:
Different kinds of modal synthesis method have been used widely in dynamic analysis of linear structure systems, but, in general, they are not suitable for nonlinear systems.In this paper, a kind of modal synthesis techniques is extended to dynamic analysis of nonlinear systems. The procedure is based upon the method suggested in [20],[21], which is applicable to vibration analysis for complex structure systems with coupling attachments but with simplified forms of linear springs and dampers. In fact, these attachments have nonlinear characteristics as those generally known to the cases of nonlinear elasticity and nonlinear damping, e.g., piecewise-linear springs, softening or hardening springs. Coulomb damping,elas-ioplastic hysteresis damping, etc. So long as the components of structure are still linear systems, we can get a set of independent free-interface normal mode information hut only keep the lower-order for each component. This can be done by computations or experiments or both. The global equations of linear vibration are set up by assembling of the component equations of motion with nonlinear coupling forces of attachments. Then the problem is reduced to less degrees of freedom for solving nonlinear equations. Thus considerable saving in computer storage and execution time can be expected. In the case of a very high-order system, if sufficient degrees of freedom are reduced, then it may be possible for the problem to be solved by the aid of a computer of ordinary grade.As the general nonlinear vibration of multiple degrees of freedom systems is quite involved, in general, the exact solution of a nonlinear system equations is not easy to find, so the numerical method can be adopted for solving the reduced nonlinear equations to obtain the transient response of system for arbitrary excitations.

Fully Stressed Design of Superstatic Truss with a Multiple Loads (Ⅰ)

Hu Ding-zhong

1983, 4(4): 573-585.

Abstract(1982) PDF(527)

Abstract:
In this paper, we study the relationship between the number of redundant forces and the number of load conditions in a super-static truss without sick members in order to accomplish fully stressed design.This paper will not only improve the conclusion given by R. H. Gallagher et al., but also clarify the choice of load conditions (number, magnitude and form). Moreover,it will explain that some counter-examples in literatures are invalid.