A group of chaotic motion of soft spring quadratic duffing equations

In this paper,we use the Melnikov function method to study a kind of soft Duffing equations(?) Af((?),x) x-x~(2k 1)=r[M(x,(?))cosωt N(x,(?))sinωt](k=1,2,3…)and give the condition that the equations have chaotic motion and bifurcation.The method used in this paper is effective for dealing with the Melnikov function integral of the system whose explict expression of the homoclinic or heteroclinic orbit cannot be given.     

Chaotic behavior in the nonlinear elastic beam

The dynamic response of the non-linear elastic simply supported beam subjected to axial forces and transverse periodic load is studied. Melnikov method is used to consider the dynamic behavior of the system whose post-buckling path is steady. The effect of the higher order terms in the controlling equation is taken into account. It is found that the fifth-order terms have a great influence on the dynamic behavior of the system. The result shows that there exist either homoclinic orbits or heteroclinic orbits in the system. In this paper, the critical values of the system entering chaotic states are given. The diagram of an example is shown. The project is supported by the National Natural Sciences Foundation of China.     

Anomalous dynamics response of nonlinear elastic bar

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Existence of closed orbits for a differential equation model concerning multi-molecule reactions

In this paper,the existence of closed orbits for the biochemical reaction modeldx/dt=1-x~ny~2,dy/dt=a(x~ny~2-y)is discussed,where n is a positive integer and x≥0,y≥0,a>0.We also point outthat the equation has no closed orbits or has stable limit cycles arising from Hopfbifurcations under a certain condition of a.     

Limit load in bending of welded samples with a soft welded joint

The limit load is one of the basic characteristics in estimating the workability of various structures, in particular, welded structures with a soft welded joint. In some cases, the difference in the yield stresses of the base material and the welded joint material is so pronounced that plastic strains are localized in a thin welded joint. The upper bound of the limit load of a welded sample subjected to bending under conditions of plane strains is obtained with allowance for certain specific features of such a distribution of strains. A comparison with the known solution is performed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 216–222, March–April, 2008.     

Chaotic frictional vibrations excited by a quasiperiodic load

A Duffing oscillator frictionally interacting with a moving belt under a quasiperiodic load is studied. The multiple-scales method is used to derive a system of two nonautonomous equations with small parameters, which describes the modulation of vibrations. It is shown that the system of modulation equations has a heteroclinic structure. Melnikov functions are used to analyze the domain of heteroclinic chaos __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 127–133, September 2006.     

Singular perturbation of boundary value problem for a vector fourth order nonlinear differential equation

We study the vector boundary value problem with boundary perturbations: ε~2y~((4))=f(x,y,y″,ε, μ) ( μ<χ<1-μ) y(χ,ε,μ)l_(χ-μ)= A_1(ε,μ), y(χ,ε,μ)l_(χ-1-μ)=B_1(ε,μ) y″(χ,ε,μ)l_(χ-μ)=A_2(ε,μ),y″(χ,ε,μ)l_(χ-1-μ)=B_2(ε,μ)where yf, A_j and B_j (j=1,2) are n-dimensional vector functions and ε,μ are two small positive parameters. This vector boundary value problem does not appear to have been studied, although the scalar boundary value problem has been treated. Under appropriate assumptions, using the method of differential inequalities we find a solution of the vector boundary value problem and obtain the uniformly valid asymptotic expansions.     

环板在局部线性荷载作用下的塑性极限分析的计算方法探讨

本文结合环板在线性荷载作用下的塑性极限分析问题,给出一种简便的计算方法,即奇异函数法。文中用此法计算了具有外边界或内边界支承的环板的极限荷载,并画出几何参数对极限荷载的影响曲线。所得的结果具有良好的规律性而且是比较合理的。     

具有内悬臂的环板在两种不同荷载共同作用下的塑性极限分析

针对承受两种不同荷载的具有内悬臂环板的塑性极限分析问题，应用奇异函数给出一种分析方法。文中给出一些算例，由此显示出本文方法的简单方便性。     

Road to chaos for a soft spring system under weak periodic disturbance

The Mel’nikov-Holmes method is used to discuss the road to chaos for an elastic system with soft spring constant and under weak periodic disturbance. It is found that according to how the disturbance is applied, forced vibration or parametric excitation, the response may pass to chaos through different sequences of subharmonic bifurcations, and that the orders in the sequences may be different for different disturbance frequencies.     

Limit Cycles for a Mechanical System Coming from the Perturbation of a Four–Dimensional Linear Center

We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of differential systems coming from the Mechanics. The tool for proving these results is the averaging theory.     

Evolution of Limit Cycles in the Stability Domain of a Double Pendulum under a Variable Follower Force

It is shown that stable and unstable limit cycles are born at a certain value of the follower force. As the absolute value of the force increases, the cycles move with different velocities in opposite directions. The unstable limit cycle moves toward the origin in the four-dimensional phase space. At a certain value of the follower force, this cycle gets at the origin, disturbing its stability as a singular point of the dynamic system. If the absolute value of the follower force decreases, then death of both cycles is possible. In this case, the attraction domain of the origin becomes infinite