应用数学和力学

1982 Vol. 3, No. 5

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Compatible Dynamic Finite Elements with Diagonalized Consistent Mass Matrix

Chien Wei-zang

1982, 3(5): 565-576.

Abstract(1964) PDF(580)

Abstract:
In this paper, the compatible dynamic finite elements with diagonalized consistent mass matrix are studied. In previous papers^[1,2], the author studied the dynamic finite elements with diagonalized consistent mass matrix, but all of them are incompatible elements. In this paper,the compatible form functions are obtained not only for the tetrahedron elements, but also for the triangular ring elements, with diagonalized consistent mass matrices. This kind of finite elements can be used for the treatment of impact problems, vibration problems, and problems involving time coordinates, including the linear and nonlinear equations.

On Some Problems in the Assembling of Multilevel Substructuring

Zhong Wan-xie, Li Xi-kui

1982, 3(5): 577-584.

Abstract(1814) PDF(496)

Abstract:
In this paper, some essential problems in the assembling of multi-level substructuring are discussed, which include the referencing of superelements in the substructural module and respective expression; the automatic formation of the node sequence as well as the qualification number of these nodes in the substructural modules.All the ideas mentioned above have been implemented in the general purpose structural analysis program JIGFEX. The assembling consideration is one of the most important practical problems in the structural analysis computer program design by the multi-level substructuring approach. In this paper those problems are discussed only from the theoretical aspect and all the practical implementation and applications are neglected however.

On Various Variationai Principles in Nonlinear Theory of Elasticity

Tai Tien-min

1982, 3(5): 585-596.

Abstract(1667) PDF(612)

Abstract:
In the present paper functionals for the various possible main variational principles in the nonlinear theory of e-lasticity are derived from the "full energy principle" and several of them are not found yet in the literatures available. Through the derivation of this paper we suggest a conjecture on the nonexistence of the eleventh and the sixth classes for the variational principles in Table 6.1 of H.C. Hu's monograph [1].

Green Function and Integral Equation Method for the Orr-Sommerfeld Equation

Li Jia-chun

1982, 3(5): 597-604.

Abstract(2557) PDF(566)

Abstract:
In this paper, an alternative method has been presented to solve the Orr-Sommerfeld equation in the linear theory of stability. To begin with, we define a Green function which is expressed in terms of matrix. Subsequently, its reciprocity has been shown. Finally, a linear integral equation equivalent to the original Orr-Sommerfeld equation is derived. The method is applied to the cases with two solid walls and any velocity distribution of the main flow at any Reynolds number.

The Theory of Similarity and the Profile of Vertical Distribution of Concentration of Sand Particles

Tsai Shu-tang

1982, 3(5): 605-612.

Abstract(1593) PDF(684)

Abstract:
In this article, we treat the problem of two-dimensional uniform steady channel flow of turbid water with theory of similarity. Under the condition of similarity of turbulent fluctuation velocity and fluctuation of concentration of sand particles, we obtain the profile of the vertical distribution of concentration of sand particles. This profile of vertical distribution of concentration of sand particles is slightly different from that obtained by diffusion theory, and departs from that obtained by gravitational theory.

Automatically Meshing Technique Used in the Mixed FEM for Stress Analysis of Reinforced Conical Branch Pipe Junctions

Xu Ci-da, Liu Yong-ren, Li Xue-chun

1982, 3(5): 613-619.

Abstract(1893) PDF(592)

Abstract:
In this paper the use of the mixed finite element method for stress analysis of reinforced conical branch pipe junctions subjected to internal water pressure is presented, branch pipe junction being considered as an intersection body of two thin conical shells. For the purpose of computing a large number of branch pipe junctions with different geometrical varieties, an automatical meshing routine has been put up in the mixed FEM program with 3 geometrical parameters of the junctions to be varied, i.e., the angle included between axes of the main pipe and the branch pipe,the thickness of the wall of the shell and that of the reinforcing pad and the ratio of diameters of the branch pipe to that of the main pipe. The computer program has been provided functions to distinguish 8 kinds of different meshes and 12 sorts of elements, and to lay automatically coordinates of nodes as well as different boundary conditions of elements. In this way, stress analyses of 101 junctions have been carried out and results of computations are excellent.

On the Free Energy and Variational Theorem of Elastic Problem in Thermal Shock

Wang Hong-gang

1982, 3(5): 621-627.

Abstract(2090) PDF(573)

Abstract:
In some investigations on variational principle for coupled thermoelastic problems, the free energy Φ(e_ij,θ),where the state variables are elastic strain e_ij and temperature increment θ, is expressed by This expression is employed only under the condition of |θ|<0(absolute temperature of reference) But the value of temperature increment is great, even greater than T₀ in thermal shock. And the material properties (λ,μ,ν,c, etc.) will not remain constant, they vary with θ. The expression of free energy for this condition is derived in this paper. Equation (0.1) is its special case.Euler's equations will be nonlinear while this expression of free energy has been introduced into variational theorem. In order to linearise, the time interval of thermal shock is divided into a number of time elements Δt_k, (Δt_k=t_k-t_k-1,k=1,2…,n), which are so small that the temperature increment θ_k within it is very small, too. Thus, the material properties may be defined by temperature field T_k=T(x₁,x₂,x₃,t_k-1) at instant t_k-1, and the free energy Φ_k expressed by eg. (0.1) may be employed in element Δt_k.Hence the variational theorem will be expressed partly and approximately.

Unsymmetrical Bending Problems for the Annular and Circular Thin Plates under Various Supports (I)

Jiang Fu-ru

1982, 3(5): 629-640.

Abstract(1683) PDF(762)

Abstract:
In this paper we consider the nonlinear and unsymmetrical bendings of annular and circular thin plates under various supports. The uniformly valid asymptotic solutions have been derived by means of the perturbation method presented in [7].

Singular Perturbation of General Boundary Va!ue Problem for Higher-Order Elliptic Equation Containing Two-Parameter

Lin Zong-chi

1982, 3(5): 641-652.

Abstract(1603) PDF(702)

Abstract:
In this paper we consider the construction of asymptotic expression of solution for general boundary value problem for higher order elliptic equation containing two parameters. By using the method of two-parameter expression, asymptotic expression of solution and estimation corresponding to the remainder term are given.These results are the extensions of [1] and [7].

The Small Disturbance Problems Associated with the Algebraic Riccati Equation of Continuous Linear Time-Invariant Systems

Hwang Ling, Zhu Wei-ling

1982, 3(5): 653-660.

Abstract(1815) PDF(545)

Abstract:
The stability problem of the disturbed algebraic Riccati equation of continuous linear time-invariant systems is discussed Ln this paper. Through matrix norm analysis the estimation (expressed in terms of the disturbance range of the system parameters) of the disturbance range in the solution of the disturbed algebraic Riccati equation is established. Apparently this method is quite convenient for the practical computational purposes.

Application of the Theory of Branching of Solutions of Nonlinear Equations to the Estimation of Pole Perturbance of Linear Closed-Loop Systems

Zhang Rong-xiang, Chen Zhao-kuan

1982, 3(5): 661-670.

Abstract(1581) PDF(482)

Abstract:
When the state and input matrices of a multivariable linear time-invariant system are perturbed, the problem of the estimation of pole perturbance of closed-loop system is considered by making use of the theory of branching of solutions for nonlinear equations.

Flexible Shaft Whirl Induced by Dry Friction and Its Stability

Zhang Wen

1982, 3(5): 671-677.

Abstract(1834) PDF(520)

Abstract:
This paper is devoted to the study of the whirling phenomena of flexible rotors due to dry friction. The mechanical model used here is a two-degree-of-freedom system in which the rubbing plane is not coincident with the rotating plane of the lumped mass. The characteristic equation of whirl speeds is derived and the whirling modes are obtained. The dynamic stability of each admissible whirling motion is also discussed. The results show that the whirl speeds are always higher than the critical speed of the shaft.

The General Derivation of Ritz Method and Trefftz Method in Elastomechanics

Shung Chu-hua

1982, 3(5): 679-687.

Abstract(1786) PDF(489)

Abstract:
This paper derives the Ritz method and Trefftz method in linear elastomechanis with the help of general mathematical expressions. Thus it is proved that Ritz method gives the upper bound of the corresponding functional extremurn, while Trefftz method gives its lower bound. At the same time it has been found that the eigenvalue problem (e.g. thenatural frequency problem) concerning the functional variational method in Trefftz method is in concord with the lower bound method of the loosened boundary condition which seeks for the eigenvalue. Of course, the results of this derivation are also applicable to the sort of functional variational method of which Euler's equation is linear positive definite.

Analytic Solutions for a Finite Crack Rupturing with Type-Ⅱ at One Tip with Exciting and Decaying Processes

Fan Ja-shen, Xu Ping

1982, 3(5): 689-706.

Abstract(1886) PDF(538)

Abstract:
Under the action of Kayleigh damping, when the shear stress exerts at the boundary of the crack and causes one tip of the crack rupturing with varying velocity. By singular perturbation method, we reduce the governing non-linear partial differential equations into a system of linear ones and solve them by using generalized Fourier series.

Generalized Expressions of Boundary Conditions of Shells of Revolution

Mei Zhan-xin

1982, 3(5): 707-713.

Abstract(1747) PDF(527)

Abstract:
As for the boundary conditions of shells of revolution, traditionally, four out of the eight quantities which are the four displacements on the middle surface u, v, w and ψ together with the four corresponding forces, are given. when the generalized displacements on the nodal circles are used as basic unknowns, the number of unknowns on a nodal circle is more than four^[1][2][3][4]. In this case, how to deal with the boundary conditions is still a problem that has not been solved satisfactorily yet. In this paper,the relations between the generalized and nongeneralized quantities of a shell's edge are derived according to the principle of virtual work. Seven types of common edges are studied and their expressions of boundary conditions in the form of generalized displacements or forces are qiven. The number of expressions for each type of edge may correspond with the number of unknowns used on a nodal circle. Kith these expressions, boundary conditions can be put directly into equations of motion of generalized displacement method so as to solve the generalized displacements. By so doing, the process of transformation and inverse transformation about unknowns in [2] is avoided. Not only is the argument simple and clear, but the calculation work is reduced.Having the set of generalized expressions of boundary conditions, the generalized displacement method of the shell of revolution may be more perfect in theory.