应用数学和力学

1991 Vol. 12, No. 7

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On the Method of Orthogonality Conditions for Solving the Problem of Large Deflection of Circular Plate

Dai Shi-qiang

1991, 12(7): 579-586.

Abstract(1830) PDF(652)

Abstract:
In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zangfor solving the problem of large deflection of a circular plate, and find that the method could be regarded as the method of strained parameters in the singular perturbation theory. In terms of the parameter representing the ratio of the center deflection to the thickness of the plate, we make the asymptotic expansions of the deflection, membrane stress and the parameter of load as in Ref. [1], and then give the orthogonality conditions(i.e. the solvability conditions) for the resulting equations, by which the stiffness characteristics of the plate could be determined. It is pointed out that with the solutions for the small deflection problem of the circular plate and the orthogonality conditions, we can derive the third order approximate relations between the parameter of load and the center deflection and the first-term approximation of membrane stresses at the center and edge of the plate without solving the differential equations. For some special cases(i.e. under uniform load, under compound toad, with different boundary conditiors), we deduce the specific expressions and obtain the results in agreement with the previous ones given by Chien Wei-zang, Yeh kai-yuan and Hwang Chien in Refs. [1-4].

Completed and Incompleted Stochastic Webs

Liu Zeng-rong, Xu Zheng-fan, Wang Qi-gao, Hou Dian-hui

1991, 12(7): 587-592.

Abstract(1897) PDF(587)

Abstract:
In this paper, the completed stochastic web and incompleted stochastic web produced by the perturbed saddle separatrix net are given. The structural properties of two kinds of web are discussed by means of the dynamical system theory.

General Solution for Dynamical Problem of Infinite Beam on an Elastic Foundation

Zheng Xiao-ping, Wang Shang-wen, Chen Bai-ping

1991, 12(7): 593-597.

Abstract(2041) PDF(787)

Abstract:
Based on the theory of Eider-Bernoulli beam and Winkler assumption for elastic foundation, a mathematical model is presented. By using Fourier transformation for space variable, Laplace transformation for time variable and convolution theorem for their inverse transformations, a general solution for dynamical problem of infinite beam on an elastic foundation is obtained. Finally, the cases of free vibration,impulsive response and moving load are also discussed.

Quasi-Variationai Inequalities and Social Equilibrium

Ding Xie-ping

1991, 12(7): 599-606.

Abstract(2266) PDF(722)

Abstract:
A quasi-variational inequality is proved in paracompact setting which generalizes the results of Zhou Chen andAubin. As applications, two existence theorems on the solutions of optimization problems and social equilibria ofmetagames are showed which improve and extend the recent results of Kaczynski-Zeidan and Aubin.

Study on the Complicated Dynamical Behaviors of Nonlinear Ecosystems(Ⅲ)

Zan Ting-quan, Liu Zong-chao

1991, 12(7): 607-612.

Abstract(1688) PDF(555)

Abstract:
This paper is a continuous study of references [1-2]. At first, we introduce the concept of assembles of ecosystems, and then discuss macro- and micro-synergetical methods for ecology study. Combining the two methods, by use of the macroscopic data(observable) outputted from the ecosystems, we can construct their GGLE, master their information changes between, before and after the generalized phase changes(e.g. community successions), and find out the action mechanisms of microscopic factors on macroscopic results. This may be a new approach to the study of action mechanisms of complex ecosystems.

Perfectly Plastic Fields at a Rapidly Propagating Plane-Stress Crack Tip

Lin Bai-song

1991, 12(7): 613-620.

Abstract(1930) PDF(618)

Abstract:
Under the condition that all the perfectly plastic stress components at a crack tip are the functions of θ only, making use of the Mises yield condition, steady-state moving equations and elastic perfectly-plastic constitutive equations, we derive the generally analytical expressions of perfectly plastic fields at a rapidly propagating plane-stress crack tip. Applying these generally analytical expressions to the concrete crack, we obtain the analytical expressions of perfectly plastic fields at the rapidly propagating tips of modes I and Ⅱ plane-stress cracks.

The Method of the Reciprocal Theorem of Forced Vibration for the Elastic Thin Rectangular Plates(Ⅲ)—Cantilever Rectangular Plates

Fu Bao-lian, Li Nong

1991, 12(7): 621-638.

Abstract(2401) PDF(652)

Abstract:
In this paper, applying the method of the reciprocal theorem, we give the stationary solutions of the forced vibration of cantilever rectangular plates under uniformly distributed harmonic load and concentrated harmonic load acting at any point of the plates, the figures and tables of number value of bending moment and the deflection amplitudes as well.

The Sufficient and Necessary Conditions of Unconditional Stability and the Delay Bound of the Third-Order Neutral Delay Differential Equation

Chen Jun-ping, Li Zhi-yong

1991, 12(7): 639-643.

Abstract(1720) PDF(533)

Abstract:
In this paper, the sufficient and necessary conditions of the unconditional stability, and the delay bound of the third-order neutral delay differential equation with real constant coefficients are given. The conditions are brief and practical algebraic criterions Furthermore, we get the delay bound.

The Cauchy Integral of Many Complex Variables, Passive Operators and Multidimensional Dispersion Relations

Huang Lie-de

1991, 12(7): 645-652.

Abstract(1644) PDF(632)

Abstract:
This note illustrates the multidimensional dispersion relations that connect the real and imaginary parts of the matrix z(p)(it is the boundary value of the impedancez(ζ) .

On the Asymptotic Solutions for a Class of Second Order Differential Equations with Slowly Varying Coefficients

Qiao Zong-chun

1991, 12(7): 653-660.

Abstract(2091) PDF(569)

Abstract:
In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted, and the results in [1-4],[5-7] are improved and extended by means of the modified method of multiple scales^[8].

Perturbation Method for Reanalysis of the Matrix Singular Value Decomposition

Lü Zhen-hua, Feng Zhen-dong

1991, 12(7): 661-670.

Abstract(1914) PDF(686)

Abstract:
The perturbation method for the reanalysis of the singular value decomposition(SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of great worth to enhance computational efficiency of the iterative analysis problems that require matrix singular value decomposition repeatedly. The asymptotic estimate formulas for the singular values and the corresponding left and right singular vectors up to second-order perturbation components are derived. At the end of the paper the way to extend the perturbation method to the case of general complex matrices is advanced.

The Theorem of the Stability of Nonlinear Nonautonomous Systems under the Frequently-Acting Perturbation——Liapunov’s Indirect Method

Zhang Shu-shun, Shang Da-zhang

1991, 12(7): 671-678.

Abstract(1941) PDF(556)

Abstract:
In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov's indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.