应用数学和力学

1990 Vol. 11, No. 8

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The Double and Triple Velocity Correlations of Small Vortexes in Three-Vortex Model Theory and the Decay of Grid Produced Turbulence

Lin Duo-min, Cai Shu-tang

1990, 11(8): 659-664.

Abstract(1683) PDF(493)

Abstract:
In the three-vortex model theory of turbulence^[1,2] and double and the triple velocity correlation functions of small vortexes were employed. In this paper, the double and the triple velocity correlation functions of small vortexes are further discussed, and the expressions of some coefficients in the expansions in terms of relative displacement of two points are given. Finally, by using these coefficients, the decay of grid-produced trubulence is calculated. The result of calculations gives good agreement with the experimental data of G.K. Batchelor and A.A.Townsend^[3].

Numerical Solution of the Singularly Perturbed Problem for the Hyperbolic Equation with Initial Jump

Su Yu-cheng, Lin Ping

1990, 11(8): 665-670.

Abstract(1669) PDF(493)

Abstract:
In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.

Stress Intensity Factors of a Plate with Two Cracks Emanating from an Arbitrary Hole

Wang Yuan-han

1990, 11(8): 677-686.

Abstract(1864) PDF(552)

Abstract:
In this paper, Muskhelishvili complex function theory and boundary collocation method are used to calculate the stress intensity factors (SIF) of a plate with two cracks emanating from an arbitrary hole. The calculated examples include a circular, elliptical, rectangular, or rhombic hole in a plate. The principle and procedure by the method is not only rather simple, but also has good accuracy. The SIF values calculated compare very favorably with the existing solutions. At the same time,the method can be used far different finite plate with two cracks emanating from a hole with more complex geometrical and loading conditions. It is an effective unified approach for this kind of fracture problems.

Analysis of Surface Crack on Forward Extruding Bar during Axisymmetric Cup-Bar Combined Extrusion Process

Li Miao-quan, Wu Shi-chun, Tang Cai-rong

1990, 11(8): 687-695.

Abstract(1759) PDF(454)

Abstract:
In this paper, the kinematically admissible velocity field with surface crack on forward extruding bar is put forward during the axisymmetric cup-bar combined extrusion process, in accordance with the results of model experiments.On the basis of velocity field, the necessary condition for surface crack formation on the forward extruding bar is derived, with the help of upper bound theorem and the minimum energy principle. Meanwhile, the relationships between surface crack formation and combination of reduction in area for the part of forward and backward extursions (ε_b,ε_f) relative residual thickness of billet (T/R₀),frictional factor (m) or relative land length of ram and chamber (l_b|R₀,l_f|R₀) are calculated during the extrusion process. Therefore, whether the surface crack on forward exturding bar occurs can be predicted before extruding the lower-plasticity metals for axisymmetric cup-bar combined extrusion process.The analytical results agree very well with experimental results of aluminium alloy LY12 (ASTM 2024) and LC4 (ASTM 7075).

Extraction of Characteristic Roots on BesseLl-Neumann’s Mixed Equations

Yang Wen-xiong, Gu Er-zuo, Zhu Min

1990, 11(8): 705-712.

Abstract(1796) PDF(419)

Abstract:
This paper discusses the problem of the extraction of characteristic roots {λ_t}(λ₀<λ₁<λ₂…) on Bessel-Neumann's mixed equations. It gives the expressions and the evaluation of the minimum root. The advantage of the method has no use for the table of the multi-figure number Bessel function and it does not need computer but can calculate all the characteristic roots {λ_t}. The precision of these roots is still high.

The Integral as a Function of the Upper Limit and an Analytical Solution to Plane Strain Drawing and Extrusion

Zhao De-wen, Zhao Zhi-ye, Zhang Qiang

1990, 11(8): 713-718.

Abstract(1896) PDF(545)

Abstract:
A kinematically admissible velocity field which is different from Avitzur's is established in Cartesian Coordinates. An upper-bound analytical solution to strip drawing andextrusion is obtained by using the integral as a function of the upper limit in this paper.

Fuzzy Cluster Analysis of Turbulent Scales

Liang Zai-chao Liu Shi-he

1990, 11(8): 719-724.

Abstract(1742) PDF(450)

Abstract:
Turbulent motion could be regarded as the superposition of fluctuations with different scales. It's of great theoretical and practical importance to determine the classification of turbulent scales quantitatively to the better description of vortex motions with different scales, and to the research of the interaction among different sclaes of vortex and the construction of better turbulent models. The mathematical method, which carries out the classification on a certain requirement, is called cluster analysis. In this paper, fuzzy cluster analysis method is used to study the classification of turbulent scales quantitatively in smooth and rough wall boundary conditions. Furthermore, the properties and interactions among all kinds of flow structures are also studied. The results are helpful to gain some insight into the properties and interactions of all kinds of turbulent scales in wall turbulent shear flow.

Noise at Inception and Collapse of a Cavity

Huang Jing-chuan

1990, 11(8): 725-730.

Abstract(1776) PDF(640)

Abstract:
The paper analyzes the noise at inception and collapse of an isolated bubble cavity filled with gas and vapour. The expressions and their numerical solutions of the sound pressure and the vibration velocity are presented.The results indicate that the noise occurs at every stage of a cavity. The noise has comparatively big value only at the late period of collapse. The sound pressure is of magnitude 100db.

Stability of Nonlinear Comparison Equations for Discrete Large-Scale Systems

Shu Huang

1990, 11(8): 731-737.

Abstract(1719) PDF(449)

Abstract:
On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparison equations was studied in the past. In this paper, various criteria of stability for discrete nonlinear autonomous comparison equations are completely established. Among them, a criterion for asymptotic stability is not only sufficient, but also necessary, from which a criterion on the function class C₁, is derived. Both of them can be used to determine the unexponential stability, even in the large, for discrete nonlinear (autonomous or nonautonomous) systems. All the criteria are of simple algebraic forms and can be readily used.

Some Properties of Far Field Patterns of Acoustic Waves in an lnhomogeneous Medium

Cheng Jin

1990, 11(8): 739-746.

Abstract(1683) PDF(428)

Abstract:
In this paper, by using functional analysis and integral equation method, we obtain some results about the properties of far field of acoustic waves in an inhomogeneous medium. And we also discuss some ill-posed inverse scattering problems by Tikhonov regularization method.

On the Limit Case of the Step-Reduction Method for Calculating Non Uniform Beam with Various Sections

Wu Jiong-yu

1990, 11(8): 747-752.

Abstract(1817) PDF(541)

Abstract:
In this paper, the step reduction method is discussed, which was advanced by Prof, Yeh Kai-yuan for calculating a non-uniform beam with various sections. The following result is proved. The approximate solution by this method approaches the true solution if the number of the steps approaches the infinity. However, the measure of the error between the limit solution and the ture solution is not in the pure mathematics sense but in the mechanics sense.

A Series Boundary Integration Method for the Bending Analysis of Anisotropic Plates

Liang Li-ping

1990, 11(8): 897-704.

Abstract(1768) PDF(611)

Abstract:
A series boundary integration method is given which exactly satisfies the fundamental equation of bending analysis of general anisotropic plates or laminated plates in Kirchhoffs sense. With a unified deflection series, the method may be applied to the plates having different planforms and support conditions. Several groups of representative examples are calculated. The examples include circular, square and triangular plates, and their boundaries include clamped edges, simply supported edges, free edges and free corner. Numerical results indicate rapid convergency for both deflection and stress resultants and demonstrate wide applicability of the method.