On the asymptotical behaviour of solutions of a class of nonlinear control systems

Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｃｏｎｔｒｏｌｓｙｓｔｅｍ：·ｘ＝Ａｘ＋Ｂ（σ）,σ＝ＣＴｘ,（１）ｗｈｅｒｅｘ＝（ｘ１,ｘ２,…,ｘｎ）Ｔ∈Ｒｎ,ｔｈｅｎｏｔｅＴｅｘｐｒｅｓｓｅｓｔｈｅｔｒａｎｓｐｏｓｅｏｆｍａｔｒｉｘａｎｄｖｅｃｔｏｒ,Ａ,ＢａｎｄＣａｒｅｎ×ｎ,ｎ×ｍａｎｄｎ×ｍｃｏｎｓｔａｎｔｍａｔｒｉｃｅｓｒｅｓｐｅｃｔｉｖｅｌｙ,σ＝（σ１,σ２,…,σｍ）Ｔ∈Ｒｍａｎｄ（σ）＝（１（σ１）,２（σ２）,…,ｍ（σｍ））Ｔ：Ｒｍ→Ｒｍｉｓｃｏｎｔ…     

集中载荷作用下不同边界条件的弯曲厚矩形板的受迫振动

运用边界积分法研究了四边简支、两对边固定另两对边简支、四边固定三种复杂边界条件下厚矩形板的受迫振动问题,求解过程清晰,从而给出了受迫振动控制方程和挠曲面方程。通过在Matlab平台上进行数值计算,得出了图表形式的计算结果,并与有限元模拟值进行对照。研究表明,边界积分法用于求解厚矩形板的受迫振动问题的准确性,本文推导的控制方程和挠曲面方程的正确性,进而对工程实际中的各种相关问题具有一定的现实意义,也为求解此类问题提供了一种新途径,可以直接运用到工程实际中。     

Bending of cantilever rectangular plate with concentrated load

In this paper the solutions for the bending of cantilever rectangular plates with concentrated load acting at any point of the middle line perpendicular to the clamped edge are given by means of a conception named modified simply supported edges and the method of superposition. Some numerical examples are presented. The total bending moment checks very well with the value determined statically.     

Bending of rectangular plate with each edges arbitrary a point supported under a concentrated load

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｅｎｇｉｎｅｅｒｉｎｇｔｅｃｈｎｏｌｏｇｙ,ｔｈｅｂｅｎｄｉｎｇｐｒｏｂｌｅｍｏｆｅｌａｓｔｉｃｔｈｉｎｐｌａｔｅｈａｓｉｍｐｏｒｔａｎｔｐｒａｃｔｉｃａｌｓｉｇｎｉｆｉｃａｎｃｅ,ａｎｄａｌｏｔｏｆｒｅｓｕｌｔｓｈａｖｅｂｅｅｎｏｂｔａｉｎｅｄ.Ｂｅｃａｕｓｅｔｈｅｐｒｏｂｌｅｍｓｏｆｅａｃｈｅｄｇｅｓｐｏｉｎｔｓｕｐｐｏｒｔｅｄａｒｅｃｏｍｐｌｅｘｅｒ,ｔｈｅｒｅａｒｅｆｅｗｒｅｓｕｌｔｓｏｆｔｈｅｐｒｏｂｌｅｍｓ.Ｉｎｔｈｅｐａｐｅｒ[1],ｔｈｅｍｅｔｈｏｄｏｆｓｕｐ…     

Nonlinear problems of circular plates with variable thickness under central concentrated load

Nonlinearly bending problems of circular thin plates with linearly varying thickness under central concentrated load are discussed.     

On the nonlinear stability of a truncated shallow spherical shell under a concentrated load

In this paper, the axisymmetric nonlinear stability of a clamped truncated shallow spherical shell with a nondeformable rigid body at the center under a concentrated load is investigated by use of the modified iteration method. The analytic formulas of second approximation for determining the upper and lower critical buckling loads are obtained. This paper was read at The Third East China Symposium on Solid Mechanics, Jiuhuashan, October, 1986.     

A perturbation-variational solution of the large deflection of rectangular plates under uniform load

In this paper, von Karman’s set of nonlinear equation for large deflection of rectangular plates is at first converted into several sets of linear equations by taking central dimensionless deflection as perturbation parameter, and then, the sets of linear equations for plates with various ratio λ of length to width are solved with application of variational method. The analytical expressions for displacements and stresses as well as formulas for numerical calculation are worked out. The figures of maximum deflection-load end maximum stress with ratio H of length to width as a parameter are given in this paper. Through comparison, it is found that the results of this paper are quite in accord with experiments.     

Bending of elastic plates with a physically nonlinear inclusion

An infinite elastic isotropic plate with an elliptical, physically nonlinear inclusion loaded at infinity by uniformly distributed moments is considered. Surface loads are absent. The problem of the stress-strain state of the plate is solved in a closed form. It is shown that, for reasonably general stress-strain relations for the inclusion, the bending-moment field (and the corresponding curvatures) in the inclusion is homogeneous. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 152–157, November–December, 2006.     

Buckling and postbuckling of moderately thick plates

This paper gives the basic differential equations for finite deflections of elastic plates according to Reissner’s approximate stress distributions. The buckling and postbuckling problems of elastic rectangular plates, including the effect of transverse shear deformation, are solved and discussed, by using perturbation method suggested in ref. [8]. The postbuckling equilibrium paths of perfect and imperfect moderately thick rectangular plates are presented and compared with the results based on thin plate theory.     

Further research on the bending of the cantilever rectangular plates under a concentrated load

In this paper we apply the reciprocal theorem^[1] to further research on the bending problem of the cantilever rectangular plate under a concentrated load acting at any of its points. This method is even simpler and more general.     

ASYMPTOTIC SOLUTIONS OF THE NONLINEAR STABILITY OF A TRUNCATED SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

In this paper,the nonlinear stability problem of a clamped truncated shallow sphericalshell with a nondeformable rigid body at the center under a concentrated load is studied bymeans of the singular perturbation method.When the geometrical parameter k is large,theuniformly valid asymptotic solutions are obtained.     

Bending of cantilever rectangular plates with the effect of transeverse shear deformation

On the basis of Reissner's theory, the exact solutions of the bending of cantilever rectangular plates are obtained by means of the concept of generalized simply-supported boundary. From the results obtained, it can be found that the method is valid.     

Global analysis of some epidemic models with general contact rate and constant immigration

An epidemic models of SIR type and SIRS type with general contact rate and constant immigration of each class were discussed by means of theory of limit system and suitable Liapunov functions. In the absence of input of infectious individuals, the threshold of existence of endemic equilibrium is found. For the disease-free equilibrium and the endemic equilibrium of corresponding SIR model, the sufficient and necessary conditions of global asymptotical stabilities are all obtained. For corresponding SIRS model, the sufficient conditions of global asymptotical stabilities of the disease-free equilibrium and the endemic equilibrium are obtained. In the existence of input of infectious individuals, the models have no disease-free equilibrium. For corresponding SIR model, the endemic equilibrium is globally asymptotically stable ; for corresponding SIRS model, the sufficient conditions of global asymptotical stability of the endemic equilibrium are obtained.     

Control of chaotic oscillations of a satellite

Analytical conditions and practical methods of their realization are proposed to solve a problem of a command signal tracking for a nonlinear disturbed system.Non- linear disturbed plants consisting of linear dynamic block and nonlinear block in feedback are considered.Nonlinear part of the plant and disturbance are unknown and bounded. The paper illustrates a possibility of applications of proposed algorithms to control libra- tion angle of satellite.     

四边简支矩形中厚板的弯曲

本文采用Reissner中厚板理论求解了四边简支矩形中厚板的弯曲问题。文中首先对Reissner中厚板理论的控制方程进行了适当的变更,使之成为非耦联的二阶偏微分方程组,然后利用有限积分变换法求解所得新的控制方程,得到了四边简支矩形中厚板受均布载荷作用下的解析解。文中所述方法可用以求解具有其它边界条件和载荷的矩形中厚板的弯曲问题,同时还可移植应用于其它中厚板理论。     

Initial value problem for a class of fourth-order nonlinear wave equations

In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.     

Nonlinear vibrations of a viscoelastic plate with concentrated masses

The problem of vibrations of a viscoelastic plate with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate, the action of the concentrated masses is taken into account using Dirac δ-functions. The problem is reduced to solving a system of Volterra type ordinary nonlinear integrodifferential equations using the Bubnov-Galerkin method. The resulting system with a singular Koltunov-Rzhanitsyn kernel is solved using a numerical method based on quadrature formulas. The effect of the viscoelastic properties of the plate material and the location and amount of concentrated masses on the vibration amplitude and frequency characteristics is studied. A comparison is made of numerical calculation results obtained using various theories. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 158–169, November–December, 2007.     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

The application of the asymptotic method to a clas of strongly nonlinear systems

In this paper, according to the form of the asymptotic solution of papers [1, 2], the asymptotic method is extended to the following a class of more general strong nonlinear vibration systems where g and f are the nonlinear analytical-functions of x and x, and ε>0 is a small parameter. We assume that the derivative system corresponding to ε=0 has periodic solution. The recurrence equations of the asymptotic solution for the system(0.1)are deduced in this paper, and they are applied to practical examples.     

Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｅｌｌｉｐｔｉｃｓｙｓｔｅｍ(1λ)　-Δｕ=ｆ(λ,ｘ,ｕ)-ｖ　　(ｉｎΩ),-Δｖ=δｕ-γｖ(ｉｎΩ),ｕ=ｖ=0(ｏｎΩ),ｗｈｅｒｅΩｉｓａｓｍｏｏｔｈｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮ(Ｎ≥2)ａｎｄλｉｓａｒｅａｌｐａｒａｍｅｔｅｒ.Ｔｈｅｓｏｌｕｔｉｏｎｓ(ｕ,ｖ)ｏｆｔｈｉｓｓｙｓｔｅｍｒｅｐｒｅｓｅｎｔｓｔｅａｄｙｓｔａｔｅｓｏｌｕｔｉｏｎｓｏｆｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｓｙｓｔｅｍｓｄｅｒｉｖｅｄｆｒｏｍｓｅｖｅｒａｌａｐ…