Necessary and sufficient criteria for absolute stability of the direct control system

ＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＲＩＴＥＲＩＡＦＯＲＡＢＳＯＬＵＴＥＳＴＡＢＩＬＩＴＹＯＦＴＨＥＤＩＲＥＣＴＣＯＮＴＲＯＬＳＹＳＴＥＭＺｈａｎｇＪｉ－ｙｅ（张继业）;ＳｈｕＺｈｏｎｇ－ｚｈｏｕ（舒仲周）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉ...     

Nonlinear MHD Kelvin-Helmholtz instability in a pipe

Nonlinear MHD Kelvin-Helmholtz (K-H) instability in a pipe is treated with the derivative expansion method in the present paper. The linear stability problem was discussed in the past by Chandrasekhar (1961)^[1] and Xu et al. (1981).^[6]Nagano (1979)^[3] discussed the nonlinear MHD K-H instability with infinite depth. He used the singular perturbation method and extrapolated the obtained second order modifier of amplitude vs. frequency to seek the nonlinear effect on the instability growth rate γ. However, in our view, such an extrapolation is inappropriate. Because when the instability sets in, the growth rates of higher order terms on the right hand side of equations will exceed the corresponding secular producing terms, so the expansion will still become meaningless even if the secular producing terms are eliminated. Mathematically speaking, it's impossible to derive formula (39) when γ ₀ ² is negative in Nagano's paper.^[3]Moreover, even as early as γ ₀ ² → O⁺, the expansion becomes invalid because the 2nd order modifier γ₂ (in his formula (56)) tends to infinity. This weakness is removed in this paper, and the result is extended to the case of a pipe with finite depth. Theproject is supported by the National Natural Science Foundation of China.     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATE（II）

ＴＨＥＰＲＯＢＬＥＭＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＵＮＳＹＭＭＥＴＲＩＣＡＬ．ＢＥＮＤＩＮＧＦＯＲＣＹＬＩＮＤＲＩＣＡＬＬＹＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥ（ＩＩ）ＨｕａｎｇＪｉａｙｉｎ（黄家寅）;ＱｉｎＳｈｅｎｇｌｉ（秦圣立）;Ｘｉ...     

An application of Ungar's differential transform to elastodynamics

In recent years, a lot of writers have used Cagniard-de Hoop’s method^[1][2] to solve some problems of elastic wave. But it is a difficult and complicated task to change the path of integration when we use this method. A differential transform by A.Ungar^[3,6] can obviate this difficulty. In this paper, weuse Ungar’s differential transform to solve a case of Lamb’s problem^[1][2].     

The wedge subjected to general tractions: A paradox resolved

A wedge subjected to tractions in proportion tor ⁿ (n≥0), is considered. The stresses in the solutions of the classical theory of elasticity become infinite when the angle of the wedge is ρ or 2ρ. The paradox has been resolved by Dempsey^[4] and T.C.T. Ting^[5] whenn=0. The purpose of this paper is to resolve the paradox whenn>0.     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULA RPLATE（I）

ＴＨＥＰＲＯＢＬＥＭＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＵＮＳＹＭＭＥＴＲＩＣＡＬＢＥＮＤＩＮＧＦＯＲＣＹＬＩＮＤＲＩＣＡＬＬＹＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥ（Ｉ）ＱｉｎＳｈｅｎｇ－Ｉｉ（秦圣立）ＨｕａｎｇＪｉａ－ｙｉｎ（黄家寅）（Ｑｕｆ...     

The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

Large deflection problem of a clamped elliptical plate subjected to uniform pressure

In this paper, the perturbation solution of large deflection problem of clamped elliptical plate subjected to uniform pressure is given on the basis of the perturbation solution of large deflection problem of similar clamped circular plate (1948), (1954). The analytical solution of this problem was obtained in 1957. However, due to social difficulties, these results have never been published. Nash and Cooley (1959) published a brief note of similar nature, in which only the case λ=a/b=2 is given. In this paper, the analytical solution is given in detail up to the 2nd approximation. The numerical solutions are given for various Poisson ratios v=0.25, 0.30, 0.35 and for various eccentricities λ=1, 2, 3, 4, 5, which can be used in the calculation of engineering designs.     

Indentation of the semi-infinite elastic solid by a concave rigid punch

A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole. , , . , . . .     

The plane stress crack-tip field for an incompressible rubber material

ＴＨＥＰＬＡＮＥＳＴＲＥＳＳＣＲＡＣＫ－ＴＩＰＦＩＥＬＤＦＯＲＡＮＩＮＣＯＭＰＲＥＳＳＩＢＬＥＲＵＢＢＥＲＭＡＴＥＲＩＡＬＧａｏＹｕ－ｃｈｅｎ（高玉臣）,ＳｈｉＺｈｉ－ｆｅｉ（石志飞）（ＨａｒｂｉｎＳｈｉｐｂｕｉｌｄｉｎｇＥｎｇｎｅｅｒｉｎｇＩｎｓｔ...     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

Geodesic precession in the (Ω,A ab)-field theory

This paper aims to determine the geodesic precession in Yu’s (Ω, A_ab)-field theory[1], and to compare the result with that of the Schwarzschild orbit.     

The application of the finite element method to solving Biot's consolidation equation

Biot's theory of consolidation of saturated soil, regards the consolidation process as a coupling problem between stress of elastic body and flow of fluid existing in pores^[1]. It can more precisely reflect the mechanism of consolidation than Terzhigi's theory^[2]. In this article, we obtain the general Biot's finite element equations of consolidation with classical variational principles. The equations have clear physical meaning and have been applied to analysing the consolidation of Bajiazui earth dam. The computational results are in accord with engineering practice.     

Asymptotic solution to a nonlinear diffusion process by Chien-Latta's method

In this paper, by using Chien Wei-zang — Latta's composite expansion method^[5], we have obtained the first-order asymptotic solution to a system of equations for a nonlinear diffusion process, thus simplifying and improving the previous work^[4] considerably. Moreover, a kind of complete analytical solution has been given for a special case, and the periodic solution at the bifurcation point has been discussed, the related results being in agreement with the experiments.     

Billiards and Two-Dimensional Problems of Optimal Resistance

A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ì \mathbbR²{B \subset \mathbb{R}^2} we assign a measure ν _B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in \mathbbR² \B{\mathbb{R}^2 \setminus B} and characterize the set of measures {ν _B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.     

Bidirectional soliton street

The bidirectional long-wave model introduced by Wu (1994)^[1] and Yih & Wu (1995)^[2] is applied to evaluate interactions between multiple solitary waves progressing in both directions in a uniform channel of rectangular cross-section and undergoing collisions of two classes, one being head-on and the other overtaking collisions between these solitons. For a binary head-on collision, the two interacting solitary waves are shown to merge during a phase-locking period from which they reemerge separated, each asymptotically recovering its own initial identity while both being retarded in phase from their original pathlines. For a binary overtaking collision between a soliton of height α₁ overtaking a weaker one of height α₁, the two solition peaks are shown to either pass through each other or remain separated throughout the encounter according as α₁/α₂ or <3, respectively. With no phase locking during the overtaking, the two solitary waves re-emerge afterwards with their initial forms recovered and with the stronger wave being advanced whereas the weaker one retarded in phase from their original pathlines. By extension, the theory is generalized to apply to uniform channels of arbitrary cross-sectional shape. The Inaugural Pei-Yuan Chou Memorial Lecture, presented at The Sixth Asian Congress of Fluid Mechanics. Singapore, 21–26 May 1995     

Deformation localization and shear band fracture in strong anisotropy sheet tension

The tensile deformation localization and the shear band fracture behaviors of sheet metals with strong anisotropy are numerically simulated by using Updating Lagrange finite element method, Quasi-flow plastic constitutive theory^[1] and B-L planar anisotropy yield criterion^[2]. Simulated results are compared with experimental ones. Very good consistence is obtained between numerical and experimental results. The relationship between the anisotropy coefficientR and the shear band angle θ is found. The project supported by the National Natural Science Foundation of China and the Excellent Youth Teacher Foundation of the State Education Commission of China     

An accurate solution for the surface of elastic layer under normal concentrated load acting on a rigid horizontal base

On the basis of ref.^[1], this paper deduces an accurate solution for the surface of elastic layer under normal concentrated load acting on a rigid horizontal base, and gives numerical results, which suit civil engineers for reference.     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

The bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads

On the basis of the stepped reduction method suggested in [1], we investigate the problem of the bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads. The general solution of this problem is obtained so that it can be used for the calculations of strength and rigidity of practical problems such as arch, tunnel etc. In order to examine results of this paper and explain the application of this new method, an example is brought out at the end of this paper. Circular ring and arch are commonly used structures in engineering. Timoshenko, S.^[2], Barber, J. R.^[3], Tsumura Rimitsu^[4] et al. have studied these problems of bending, but, so far as we know, it has been solely restricted to the general solution of homogeneous uniform cross section ring. The only known solution for the problems with variable cross section ones has been solely restricted to the solution of special case of flexural rigidity in linear function of coordinates. On account of fundamental equations of the non-homogeneous variable cross section problem being variable coefficients, it is very difficult to solve them. In this paper, we use the stepped reduction method suggested in [1] to transform the variable coefficient differential equation into equivalent constant coefficient one. After introducing virtual internal forces, we obtain general solution of an elastic circular ring with non-homogeneity and variable cross section under the actions of arbitrary loads.