Global dynamical behavior of a three-body system with flexible connection in the gravitational field

Summary In this paper, the global behavior of relative equilibrium states of a three-body satellite with flexible connection under the action of the gravitational torque is studied. With geometric method, the conditions of existence of nontrivial solutions to the relative equilibrium equations are determined. By using reduction method and singularity theory, the conditions of occurrence of bifurcation from trivial solutions are derived, which agree with the existence conditions of nontrivial solutions, and the bifurcation is proved to be pitchfork-bifurcation. The Liapunov stability of each equilibrium state is considered, and a stability diagram in terms of system parameters is presented. Received 10 March 1998; accepted for publication 21 July 1998     

On a class of quasilinear Schrodinger equations

A type of quasilinear Schrodinger equations in two space dimensions which describe attractive Bose-Einstein condensates in physics is discussed. By establishing the property of the equation and applying the energy method, the blowup of solutions to the equation are proved under certain conditions. At the same time, by the variational method, a sutficient condition of global existence which is related to the ground state of a classical elliptic equation is obtained.     

挠性联结双体航天器的稳定性与分岔

研究圆轨道内受万有引力矩作用的挠性联结双体航天器在轨道平面内的姿态运动,讨论其相对轨道坐标系统平衡状态的稳定性与分岔。提出判平衡方程非平凡解存在性的几何方法,并应用Ｌｉａｐｕｎｏｖ直接法、Ｌｉａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ约化方法和奇异性理论导出解析形式的稳定性与分岔的充要条件,从而对系统的全局运动性态作出定性的描述。     

Finite-dimensional approximation to global minimizers in functional spaces with R-convergence

A new concept of convergence (R-convergence) of a sequence of measures is applied to characterize global minimizers in a functional space as a sequence of approximate solutions in finite-dimensional spaces. A deviation integral approach is used to find such solutions. For a constrained problem, a penalized deviation integral algorithm is proposed to convert it to unconstrained ones. A numerical example on an optimal control problem with non-convex state constraints is given to show the effectiveness of the algorithm.     

A criterion of robustness intelligent nonlinear control for multiple time-delay systems based on fuzzy Lyapunov methods

A control system with state feedback controllers, in which the fuzzy Lyapunov approach is developed for the stability criterion, is studied. The proposed intelligent design provides a systematic and effective framework for the control systems. The global nonlinear controller is constructed based on T–S (Takagi–Sugeno) fuzzy controller design techniques, blending all such local state feedback controllers. Based on this design, the stability conditions of a multiple time-delay system are derived in terms of the fuzzy Lyapunov theory. The effectiveness and the feasibility of the proposed controller design method are demonstrated through numerical simulations.     

Stability of Semiconductor States with Insulating and Contact Boundary Conditions

We prove the existence of global smooth solutions near a given steady state of the hydrodynamic model of the semiconductors in a bounded domain with physical boundary conditions. The steady state and the doping profile are permitted to be of large variation but the initial velocity must be small. Two cases are considered. In the first one the problem is three-dimensional, the boundary conditions are insulating and the steady state velocity vanishes. In the second one, the problem is one-dimensional, the boundary is of contact type and the steady state velocity does not vanish.     

基于滑模理论的建筑结构分散稳定控制

基于大系统分散控制思想,将大尺度高阶建筑结构系统分解为多个子结构系统;子结构之间的相互耦合作用视为有界广义力,得到以状态方程形式的子结构模型。利用滑模理论的抗摄动条件,设计具有全局稳定的子结构滑动模态轨迹,利用子结构系统局部状态实现全局稳定的控制力条件,并以参数ρi实现各子结构间的调节,建立稳定的分散控制格式。在控制算法中采用了准滑模控制方法,克服变结构滑动模态中的抖振影响。利用本文方法,对20层钢结构基准模型在地震激励下的控制进行设计并数值仿真,验证了该方法的有效性。     

The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.     

Periodic solutions of an epidemic model with saturated treatment

Based on the fact that many infectious diseases exhibit periodic fluctuations and there is a saturated phenomenon during disease treatment, we study an SIR model with periodic incidence rate and saturated treatment function. Firstly, we find that the basic reproduction number less than 1 cannot insure the global stability of disease-free equilibrium and it needs to add other conditions. Moreover, we establish sufficient conditions for the multiplicity of positive periodic solutions. We also apply the numerical method to confirm theoretical results and show the stability of the periodic solutions. We observe that there are two periodic solutions in the system where one is stable and the other one is unstable. These results will provide some guidance for control measures of disease.     

非线性系统全局动态特性分析的PCM法及其在转子轴承系统中的应用

本文将Ｐｏｉｎｃａｒｅ映射的思想与胞映射法相结合，提出了可用于高维非线性动力系统全局稳定性分析的新型数值方法：ＰＣＭ（Ｐｏｉｎｃａｒｅ－Ｃｅｌｌ－Ｍａｐｐｉｎｇ）法，和胞映射法相比，新方法在实用上具有明显的优点。为说明ＰＣＭ法的有效性，本文应用此方法对平衡转子轴承非线性动力系统进行了全局稳定性分析，同时给出了一确定状态空间中存在的所有周期解及其吸引域。     

Representation of the solution of refined bending theory for isotropic plates

A solution of the bending problem for isotropic plates in a refined statement based on the system of six-order differential equations is proposed. A procedure for determining the general solutions of the corresponding biharmonic and metaharmonic equations is suggested. A method for satisfying the boundary conditions is given. The results of numerical studies of the stress state of an infinite plate with an elliptic cavity are given.     

Semi-analytical three-dimensional elasticity solutions for generally laminated composite plates

Semi-analytical solutions for bending and free vibration of composite laminated plates have been derived based on three-dimensional elasticity theory using a newly developed hybrid analysis, which perfectly combines the state space approach (SSA) and the technique of differential quadrature (DQ). The thickness direction of laminates is selected as the transfer direction in SSA, and the DQ technique is employed to discretize the in-plane domains. This actualizes the transformation of the original partial differential equations into a state equation consisting of first-order ordinary differential equations. In particular, the use of DQ technique makes ease of the treatment of various boundary conditions, which cannot be considered in the conventional exact SSA. To avoid numerical instabilities in the conventional transfer matrix method, artificial interfaces are introduced to divide each layer into several sub-layers to reduce the transfer distance and the joint coupling matrices are established according to the continuity conditions at actual and artificial interfaces to implement the global analysis. Comprehensive numerical examples are preformed to validate the present hybrid method. Effects of some parameters on mechanical properties of the laminates are discussed.     

Acoustic wave interaction with a laminated transversely isotropic spherical shell with imperfect bonding

An exact analysis is carried out to study interaction of a time-harmonic plane-progressive sound field with a multi-layered elastic hollow sphere made of spherically isotropic materials with interlaminar bonding imperfections. A modal state equation with variable coefficients is set up in terms of appropriate displacement and stress functions and their spherical harmonics, ultimately leading to calculation of a global transfer matrix. A linear spring model is adopted to describe the interlaminar adhesive bonding whose effects are incorporated into the global transfer matrix by introduction of proper interfacial transfer matrices. The solution is first used to correlate the perturbation in the material elastic constants of an evacuated and water submerged steel (isotropic) spherical shell to the sensitivity of resonances appearing in the backscattered amplitude spectrum. The backscattering form function, in addition to the acoustic radiation force acting on selected transversely isotropic spherical shells with distinct degrees of material anisotropy, is subsequently calculated and discussed. An illustrative numerical example is given for a multi-layered hollow sphere with two distinct interlaminar interface conditions (i.e., perfectly and imperfectly bonded layers). Limiting cases are considered and fair agreements with solutions available in the literature are established.     

Global sliding mode control and application in chaotic systems

This paper is concerned with the stabilization problem for a class of nonlinear systems. Using the global sliding mode control approach, a novel robust control law is established to make the state of system stable and to improve the robustness and the stability of system. A new reaching law is introduced to reduce the chattering. Finally, the method is applied to chaotic systems and an example of the chaotic system is given to illustrate the advantage of the proposed method.     

Hamiltonian principle based stress singularity analysis near crack corners of multi-material junctions

This paper presents a new method for the stress singularity analysis near the crack corners of a multi-material junctions. The stress singularities near the crack corners of multi-dissimilar isotropic elastic material junctions are studied analytically in terms of the methods developed in Hamiltonian system. The governing equations of plane elasticity in a sectorial domain are derived in Hamiltonian form via variable substitution and variational principle respectively. Both of the methods of global state variable separation and symplectic eigenfunction expansion are used to find the analytical solution of the problem. The relationships among the state vectors in different material spaces are obtained by means of coordinate transformation and consistent conditions between the two adjacent domains. The expression of the original problem is thus changed into a new form where the solutions of symplectic generalized eigenvalues and eigenvectors are needed. The closed form of expressions is established for the stress singularity analysis near the corner with arbitrary vertex angles. Numerical results are presented with several chosen angles and multi-material constants. To show the potential of the new method proposed, a semi-analytical finite element is furthermore developed for the numerical analysis of crack problems.     

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method.The invariance of some sets under the flow of these problems and the vac- uum isolation of solutions are obtained by introducing a family of potential wells.Then the threshold result of global existence and nonexistence of solutions are given.Finally, the problem with critical initial conditions are discussed.     

Periodic solution of a turbidostat model with impulsive state feedback control

In this paper, a turbidostat model with impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control may have order one periodic solution, and the sufficient condition for existence and stability of order one periodic solution is gotten as well. For some special cases, it is shown that in the system an order two periodic solution may exist. Our results show that the control measure is effective and reliable.     

Pinning synchronization of complex directed dynamical networks under decentralized adaptive strategy for aperiodically intermittent control

In this paper, the problem of global synchronization for complex directed dynamical networks via adaptive aperiodically intermittent pinning control is studied. By constructing a piecewise Lyapunov function, some sufficient conditions to guarantee global synchronization are derived based on the analytical technique and theory of series with nonnegative terms. Different from previous works, the adaptive intermittent pinning control is aperiodic with non-fixed both control period and control width, and moreover, the adaptive approach is decentralized relying only on the state information of the controlled node. Hence, the adaptive intermittent pinning control strategy proposed in this paper is more practically applicable than those in previous works. Additionally, it is noted that the derived synchronization criteria are dependent on the control rates, but not the control widths or the control periods, which makes the theoretical results are less conservative than the corresponding results given in the existing works. A numerical example is finally provided to illustrate the validity of our theoretical results.     

A NEW STATE SPACE SOLUTION FOR RECTANGULAR THICK LAMINATES WITH CLAMPED EDGES Hu Wenfeng Liu Yihua

通过把固支边上的边界位移函数作为状态变量引入状态方程,得到了含固支边矩形单层与叠层厚板的精确解. 在求解过程中,将非齐次状态方程的求解变为齐次状态方程的求解,省略了求解待定常数的中间过程,使求解过程变得简单. 所得到的解能够严格满足固支边界条件,在同一材料层内不需作分层处理,因而更加精确. 此外,对固支边的应力提出了新的计算方法,能够得到更精确的边界应力. 算例表明,本文解比现有精确解收敛快,与有限元解吻合的更好,尤其是在固支边处体现得更加明显.     

Strength calculations and optimal weight design of multilayer shell-shaped composite products under a set of loads

The stress-strain state of axisymmetric multilayer shells is analyzed using kinematic and static hypotheses that allow for the transverse shear stresses satisfying the necessary equations of state, continuity conditions at the boundaries between the layers and given boundary conditions. A numerical solution of the problem of the stress-strain state for a multilayer bar is compared with the Lekhnitskii solution (for a cantilever beam loaded by a concentrated force and moment) to asses the applicability of the employed bending equations of multilayer shells. It is shown that these solutions are in good agreement. The problem of the initial fracture of the shells considered is formulated using phenomenological strength criteria for each layer. A coordinate-wise descent method in the unit interval is proposed to solve weight optimization problems for multilayer shells of composite materials under combined loading. Regions of safe operating loads and the optimal weight distribution of layer thicknesses are determined for a multilayer bar acted upon by a uniformly distributed load and concentrated force.