应用数学和力学

1985 Vol. 6, No. 11

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The Global Analysis of Higher Order Non-linear Dynamical Systems and the Application of Cell-to-Cell Mapping Method

C. S. Hsu, Xu Jian-xue

1985, 6(11): 953-962.

Abstract(2109) PDF(849)

Abstract:
In this paper, the general characteristics and the topological consideration of the global behaviors of higher order nonlinear dynamical systems and the characteristics of the application of cell-to-cell mapping method in this analysis are expounded. Specifically, the global analysis of a system of two weakly coupled van der Pol oscillators using cell-to-cell mapping method is presented.The analysis shows that for this system, there exist two stable limit cycles in 4-dimensional state space, and the whole 4-dimensional state space is divided into two almost equal parts which are, respectively, the two asymototically stable domains of attraction of the two periodic motions of the two stable limit cycles. The validities of these conclusions about the global behaviors are also verified by direct long term numerical integration. Thus, it can be seen that the cell-to-cell mapping method for global analysis of fourth order nonlinear dynamical systems is quite effective.

Tsai Shu-tang, Liu Yi-xin

1985, 6(11): 963-968.

Abstract(1858) PDF(585)

Abstract:
In the preceding article^[1], the relation between thermodynamics condition and inceptive cavitation stage has been discussed. In this paper, we try to introduce the volume function of inceptive cavitation bubbles, z₀(r), instead of the energy equation used in the reference ^[1],for discussing cavitation similarity problem in similar flow systems. Under this condition, theoretical result shows that the inception cavitation number K_i increases with a rise of geometry linear scale. In other words, if two flow systems are kept with similar pattern and liquid, it is impossible to satisfy the cavitation similarity. This conclusion accords with practice.

Generalized Airy Functions with Complex Variables

Li Jia-chun, Zhao Da-gang

1985, 6(11): 969-975.

Abstract(1947) PDF(631)

Abstract:
In the present paper, we have made a detailed study of the generalized Airy functions, which have found wide applications in the field of wave propagation and hydrodynamic stability. A series of tables and graphs are provided for the above functions with complex variables. The results are proved satisfactory.

The Mixed Mode Brittle Fracture Criteria in Sliding Mode Fracture

Lin Bai-song

1985, 6(11): 977-983.

Abstract(1896) PDF(624)

Abstract:
It is well-known that the present mixed mode brittle fracture criteria are all the opening mode fracture criterion. We consider that mixed mode brittle fracture of sliding mode fracture exists too. Hence we propose three criteria of mixed mode brittle fracture of sliding mode fracture; the radial shearing stress criterion, the maximum shearing stress criterion and the distortional strain-energy-density criterion. Thus, we can overall explain the phenomena of brittle fracture in the structural elements with cracks.

Application of the Reciprocal Theorem for Calculating the Natural Frequencies of Rectangular Elastic Thin Plates

Fu Bao-lian

1985, 6(11): 985-997.

Abstract(1805) PDF(844)

Abstract:
This paper further extends the applications of the reciprocal theorem to calculating the natural frequencies of rectangular elastic thin plates on the basis of [1]. Applying the presented method, there is no need to solve governing differential equations, it is only necessary to solve a simple integral equation after using the reciprocal theorem between the basic system, and the actual system.Using the idea of the generalized edge simply supported and introducing the frequency matrix, then all frequency equations of the rectangular plates with two opposite edges simply supported and other two opposite edges variously supported are obtained together.This is a simple, convenient and general method for calculating the frequency equations of the rectangular plates.

Nonlinear Kelvin Helmholtz Instability

Chen Le-shan

1985, 6(11): 999-1012.

Abstract(2235) PDF(621)

Abstract:
A non-linear analysis is presented with derivative expansion method for the inter facial stability of a liquid film adjacent to a subsonic gas flow under the influence of body force and surface tension. The non-linear Rayleigh-Taylor instability is included as a special case. The gas and liquid are considered to be inviscid. Though Nayfeh (1971) gave consideration into this case, there is something omitted in his third-order equation (e.g. p. 213 expression (2.29)) and inconsistent with his solutions (e.g. the first-order solution (2.31) does not satisfy his initial conditions (2.20)). Besides, in this paper, our solution near the cut-off wave number is extended to include the case of travelling waves and a new conclusion is drawn.

On the Finite Displacement Problem of a Hollow Sphere under Internal and External Pressures

Huang Ze-yan

1985, 6(11): 1013-1018.

Abstract(2061) PDF(609)

Abstract:
Taking advantage of successive approximations, the present boundary-value problem is solved. We find the first-order and second-order solutions, and therefore we obtain the formulae in the second approximation for the displacement, strain, and stress fields.

Stress Intensity Factor Considering the Material Plasticity Limited in Scope

1985, 6(11): 1019-1026.

Abstract(1717) PDF(644)

Abstract:
For the practical metallic material, preceding the fracture, there always exist the domains of plasticity. For the plasticity limited in scope, the linear elastic mechanics is still applicable, but it is neccesary to correct the influence of the domain of plasticity. The traditional method of correction (Irwin's method) is to introduce the conception of the effective fracture length, namely, owing to the existence of the domain of plasticity, the practical length of the fracture increases. If we take the length of the fracture to be equal to the effective fracture length 2(a+r_y), where 2a is the original length of the fracture and 2r_y is the added length, we may not regard the existence of the domain of plasticity and still use the linear elastic fracture mechanics to deal with.In this paper, we regard that owing to the existence of the domain of plasticity, the practical fracture length and the extremal applied stress both increase, i.e., the value of the two parameters a and σ₁ (external applied stress) both change.In this paper, it is pointed out that the stress intensity factor being determined by the method in this paper eq. (3.2) is closer to Duffy's experimental eq. (3.6) coinciding with experiment than the commonly used eq. (3.4).

Proper Solutions and Limit Boundary Value Problems of Nonlinear Second-Order Systems of Differential Equations

Liang Zhong-chao, Chen Shao-zhu

1985, 6(11): 1027-1034.

Abstract(2222) PDF(626)

Abstract:
For the system of differential equations , where a(t)>0, r(t)>0 for t≥t₀, f(x) >0 and is decreasing for x>0 g(y)>0, we give necessary and sufficient condition of the existence of a proper solution, a bounded proper solution or solutions of two kinds of boundary value problems on an infinite interval [c,∞] c≥t₀. Several examples are given to illustrate the conditions of these results.

Effects of Hunting on the Stability of Ecosystems

Wang Fu-jun

1985, 6(11): 1035-1041.

Abstract(2088) PDF(612)

Abstract:
In this present paper, we examine the effects of hunting on the stability of prey-predator systems and grazing systems. The theory of stability and singular perturbation method are applied for analysing.