Instability of solution for the fourth order linear differential equation with varied coefficient

In this paper,we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient,at least one of the characteristic roots of which has positive real part,by means of Liapunov’s second method.     

Criteria for minimization of spectral abscissa of time-delay systems

Spectral abscissa(SA) is defined as the real part of the rightmost characteristic root(s) of a dynamical system, and it can be regarded as the decaying rate of the system, the smaller the better from the viewpoint of fast stabilization. Based on the Puiseux series expansion of complex-valued functions, this paper shows that the SA can be minimized within a given delay interval at values where the characteristic equation has repeated roots with multiplicity 2 or 3. Four sufficient conditions in terms of the partial derivatives of the characteristic function are established for testing whether the SA is minimized or not, and they can be tested directly and easily.     

Eigenfunction expansion method to the solution of simultaneous equations and its application in mechanics

The solution of a homogeneous vector-matrix differential equation, when the operator is of linear second order, has been established by the method of eigenfunction approach. The uniqueness of the solution has been established for both the cases when the roots of the characteristic equations are distinct and when they are repeated. Finally, the theory has been applied on two problems of mechanics and the results are compared with the existing literature.     

Plastic dynamic stability of a column under nonconservative forces

I.Introduction'Whenadeformableobjectmovinginamedium,thedirectionsofinteractiveforcessuchasfrictionalforceshouldvarywiththedeformationoftheobject.Thisisakindofnonconservativeforces.Forexample,thedirectionofgasfrictionalforceactingontherocketsandthefossiles…     

The stability on the solution of the initial value problem

In this paper,using the differentiability of the solution with respect to the initial value and the parameter,we present a method which,different from Liapunov’s direct method.will determine the stability of the non-stationary solution of the initial value problem when the non-stationary solution remains unknown.     

A method of finding the principal modes of nonlinear vibration systems and their stabilities

This paper presents a new method of finding the principal modes of nonlinear vibration systems, by means of which the problem of finding principal modes of nonlinear systems is transferred to the problem of finding real roots of a set of algebraic equations. The method is applicable to various kinds of nonlinear vibration systems with many degrees of freedom, and is simple in calculation. The paper presents another new method of analyzing the stabilities of principal modes of nonlinear systems.     

Experimental study of delayed positive feedback control for a flexible beam

Recently, some researches indicate that positive feedback can benefit the control if appropriate time delay is intentionally introduced into control system. However, most work is theoretical one but few are experimental. This paper presents theoretical and experimental studies of delayed positive feedback control technique using a flexible beam as research object. The positive feedback weighting coefficient is designed by using the optimal control method. The available time delay is determined by analyzing the maximal real part of characteristic roots of the system. A DSP-based experiment system is introduced. Simulation and experimental results indicate that the delayed positive feedback control may effectively reduce the beam vibration if time delay is appropriately selected.     

CHARACTERISTIC EQUATIONS AND CLOSED-FORM SOLUTIONS FOR FREE VIBRATIONS OF RECTANGULAR MINDLIN PLATES

The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots,which is the particularity of Mindlin plate theory. And the closed-form solutions,which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness,are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.     

高层建筑风荷载反演研究

本文研究了结构参数未知条件下的高层建筑风荷载反演问题,通过将平均风速的实测特性作为识别计算中的辅助条件,文中提出了一类荷载归一化统计平均方法,结合工程实例,进行了高层建筑风荷载反演分析,结果表明,本文建议方法可以在幅值、时程等方面均给出良好的风荷载反演结果,同时识别得到的结构参数具有良好的精度。从而为高层建筑风荷载研究提供了一条可行途径。     

空间机械臂逆动力学的Liapunov方法

讨论了空间机械臂的逆动力学问题，指出了系统的非完整约束性质．以铰转角为变量的Ｌｉａｐｕｎｏｖ方法由于理论缺陷导致实践中出现计算死点而难以实际应用．文中提出以臂端载荷位形为变量的Ｌｉａｐｕｎｏｖ方法并给出算例，可以保证载荷向预定位形转移的渐近稳定性     

三维压电弹性体断裂分析的基本解

求解了在材料的特征方程有重根时，三维压电弹性体的单位集中不连续位移和不连续电势基本解。讨论了重根对断裂力学问题解的影响。     

充液Chaplygin球的稳定性

本文利用 Liapunov 直接方法证明带全充理想流体椭球腔的 Chaplygin 球的稳定性充分条件,并用于对考虑接触摩擦的 Kelvin 问题作出理论解释。     

A criterion of asymptotic stability of the trivial solution of an autonomous set of ordinary differential equations

This paper presents a new criterion of asymptotic stability for the trivial solution of an autonomous set of ordinary differential equations. Using the concepts of Liapunov's direct method the criterion is based on analysis of the sign of the integral from the derivative of the Liapunov function along the solutions of the sets the integral being taken for fixed, finite interval of time. The conditions of the criterion are weaker than those of Liapunov's theorem.     

Qualitative Analysis for a Class of Second-order Nonlinear System with Delay

Ｕｐｔｏｎｏｗ ,ｔｈｅｑｕａｌｉｔａｔｉｖｅｂｅｈａｖｉｏｒｏｆｔｈｅｓｅｃｏｎｄ＿ｏｒｄｅｒｎｏｎｌｉｎｅａｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓａｎｄｔｈｅｉｒａｐｐｌｉｃａｔｉｏｎｓｈａｖｅｂｅｅｎｄｉｓｃｕｓｓｅｄｅｘｔｅｎｓｉｖｅｌｙｉｎｌｉｔｅｒａｔｕｒｅ.ＷｅｒｅｆｅｒｔｏｔｈｅＲｅｆｓ .[1 -8] .Ｃｏｎｖｅｒｓｅｌｙ ,ｔｈｅｄｉｓｃｕｓｓｉｏｎｆｏｒｔｈｅｑｕａｌｉｔａｔｉｖｅｂｅｈａｖｉｏｒｏｆｔｈｅｄｅｌａｙｓｙｓｔｅｍｉｓｌｅｓｓ .Ｉｎｔｈｉｓｐａｐｅｒ,ｗｅｓｔｕｄｙｔ…     

Accuracy analysis of immersed boundary method using method of manufactured solutions

The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics.This paper analyzes the accuracy of the immersed boundary method.The procedure contains two parts,i.e.,the code verification and the accuracy analysis.The code verification provides the confidence that the code used is free of mistakes,and the accuracy analysis gives the order of accuracy of the immersed boundary method.The method of manufactured solutions is taken as a means for both parts.In the first part,the numerical code employs a second-order discretization scheme,i.e.,it has second-order accuracy in theory.It matches the calculated order of accuracy obtained in the numerical calculation for all variables.This means that the code contains no mistake,which is a premise of the subsequent work.The second part introduces a jump in the manufactured solution for the pressure and adds the corresponding singular forcing terms in the momentum equations.By analyzing the discretization errors,the accuracy of the immersed boundary method is proven to be first order even though the discretization scheme is second order.It has been found that the coarser mesh may not be sensitive enough to capture the influence of the immersed boundary,and the refinement on the Lagrangian markers barely has any effect on the numerical calculation.     

A method toward the construction of Liapunov function for discrete time systems and determination of their stability

The purpose of this paper is to present a new method for constructing Liapunov function and determining the stability of discrete time systems with a computer on the basis of the similarity transformation theory by directly applying the system matrix of the system under discussion instead of solving the discrete Liapunov's matrix equation.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov’s indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied.This study is a forward development of the study of thestability in the Liapunov sense;furthermore,it is of significance in practice sinceperturbations are often not single in the time domain.Malk in proved a general theoremabout thesubject.To apply the theorem,however,the user has to construct a Liapunovfunction which satisfies specified conditions and it is difficult to find such a function fornonlinear nonautonomous systems.In the light of the principle of Liapunov’s indirectmethod,which is an effective method to decide the stability of nonlinear systems in theLiapunov sense,the authors have achieved several important conclusions expressed in theform of theorems to determine the stability of nonlinear nonautonomous systems under thefrequently-acting perturbation.     

ON THE ASYMPTOTIC BEHAVIOR OF HOPFIELD NEURAL NETWORK WITH PERIODIC INPUTS

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An improved cross-correlation method for (digital) particle image velocimetry

An improved method that brings enhancement in accuracy for the interrogation of (digital) PIV images is described in this paper. This method is based on cross-correlation with discrete window offset, which makes use of a translation of the second interrogation window and rebuilds it considering rotation and shear. The displacement extracted from PIV images is predicted and corrected by means of an iterative procedure. In addition, the displacement vectors are validated at each intermediate of the iteration process. The present improved cross-correlation method is compared with the conventional one in accuracy by interrogation of synthetic and real (digital) PIV images and the interrogation results are discussed. The project supported by the National Natural Science Foundation of China (59936140 and 59876038)