含外激励van der Pol-Mathieu方程的非线性动力学特性分析

本文针对含有自激励,参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析,发现其准周期运动的频谱中含有均匀边频带这一新的特性.首先,采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mathieu方程的周期响应,得到了其非线性频率响应曲线;再利用F...     

Solution of the optimal turn problem for a spherically symmetric rigid body with arbitrary boundary conditions in the class of generalized conical motions

We consider the problem of time- and energy consumption-optimal turn of a rigid body with spherical mass distribution under arbitrary boundary conditions on the angular position and angular velocity of the rigid body. The optimal turn problem is modified in the class of generalized conical motions, which allows one to obtain closed-form solutions for equations of motion with arbitrary constants. Thus, solving the optimal control boundary value problem is reduced to solving a system of nonlinear algebraic equations for the constants. Numerical examples are considered to illustrate the proximity between the solutions of the traditional and modified problems of optimal turn of a rigid body.     

Periodic Response and Chaos in Nonlinear Systems with Parametric Excitation and Time Delay

In this paper, the periodic motions of a nonlinear system with quadratic,cubic, and parametrically excited stiffness terms and with time-delayterms are obtained by the incremental harmonic balance (IHB) method. Theelements of the Jacobian matrix and residue vector arising in the IHBformulation are derived in closed form. A mechanism model representingthe one-mode oscillation of beams and plates is considered as anexample. A path-following algorithm with an arc-length parametriccontinuation procedure is used to obtain the response diagrams. Thesystem also exhibits chaotic motion through a cascade of period-doublingbifurcations, which is characterized by phase planes, Poincaré sectionsand Lyapunov exponents. The interpolated cell mapping (ICM) procedure isused to obtain the initial condition map corresponding to multiplesteady-state solutions.     

PMUCR方法在高维非线性动力学系统中的应用

对现有的若干种胞映射方法应用于高维非线性动力学系统全局分析时存在的局限性进行了分析,在此基础上,提出了胞映射方法应用于高维系统时需遵循的几个原则:选择合适的分析平面;减少数据量;确定吸引子在相空间中的位置。根据这些原则,对胞参考点映射法(PMUCR)进行了改进,并对该改进方法应用于2维系统和4维系统时消耗的相对CPU时间进行了比较,结果表明,该改进方法能有效地应用于高维非线性动力学系统中。最后以船舶机械非线性隔振系统为例,分别分析了该改进方法在系统呈现周期运动和混沌运动时的应用。     

A note on longitudinal oscillations of a generalized Burgers fluid in cylindrical domains

The starting solutions for the oscillating motion of a generalized Burgers fluid due to longitudinal oscillations of an infinite circular cylinder, as well as those corresponding to an oscillating pressure gradient, are established as Fourier–Bessel series in terms of some suitable eigenfunctions. These solutions, presented as sum of steady-state and transient solutions, describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the motion of the fluid is described by the steady-state solutions which are periodic in time and independent of the initial conditions. These solutions are also presented in simpler but equivalent forms in terms of modified Bessel functions of first and second kind. In both forms, the steady-state solutions can be specialized to give the similar solutions for Burgers, Oldroyd-B, Maxwell, second grade and Newtonian fluids performing the same motions. Finally, the required time to reach the steady-state for cosine and sine oscillations of the boundary is obtained by graphical illustrations.     

Effect of External Excitations on a Nonlinear System with Time Delay

The trivial equilibrium of a two-degree-of-freedom autonomous system may become unstable via a Hopf bifurcation of multiplicity two and give rise to oscillatory bifurcating solutions, due to presence of a time delay in the linear and nonlinear terms. The effect of external excitations on the dynamic behaviour of the corresponding non-autonomous system, after the Hopf bifurcation, is investigated based on the behaviour of solutions to the four-dimensional system of ordinary differential equations. The interaction between the Hopf bifurcating solutions and the high level excitations may induce a non-resonant or secondary resonance response, depending on the ratio of the frequency of bifurcating periodic motion to the frequency of external excitation. The first-order approximate periodic solutions for the non-resonant and super-harmonic resonance response are found to be in good agreement with those obtained by direct numerical integration of the delay differential equation. It is found that the non-resonant response may be either periodic or quasi-periodic. It is shown that the super-harmonic resonance response may exhibit periodic and quasi-periodic motions as well as a co-existence of two or three stable motions.     

Influence of micro-structural defects on post-buckling and large-amplitude vibration of geometrically imperfect gradient plate

We propose a polynomial expansion method to investigate periodic motions around both collinear and triangular libration points in the planar circular restricted three-body problem. The polynomial expansion method focuses on the approximate nonlinear polynomial relations between the two directions in the motion plane, which provide an alternative way to inspect the periodic motions. Based on the nonlinear polynomial relations, the planar two-dimensional system is decoupled. As an example, the \(3\mathrm{rd}\) order analytical solutions for the periodic motion are determined by solving the decoupled system. The efficiency of the polynomial expansion method has been validated by numerical method.     

Bifurcation and chaos analysis of multistage planetary gear train

A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.     

Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders

The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and at t = 0＋ both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies Ω1 and Ω2. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)     

完全笛卡尔坐标描述的多体系统动力学

用完全笛卡尔坐标描述多体系统的运动学和动力学在提高计算效率方面有突出优点．导出用完全笛卡尔坐标表示的刚体及多体系统的动量和动量矩的解析式，给出与之对应的广义惯量矩阵概念，建立无力矩状态下用完全笛卡尔坐标描述的多体系统动力学的一阶微分方程组，用于多体航天器的姿态运动分析     

Periodic motions of coupled simple pendulums with periodic disturbances

Several theorems on the existence of oscillatory, rotary, and mixed periodic motions of n, coupled simple pendulums are proved. These theorems are very general and include subharmonic and ultraharmonic type solutions as well as harmonic type solutions and cover cases of large coupling and disturbances as well as small. The results are also extended to include more general systems of nonlinear oscillators.     

非线性转子-轴承系统的周期解及近似解析表达式

通过对普通打靶方法进行改造提出一种确定非线性系统周期轨道及周期的新型打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参与打靶法的迭代过程，迭代过程包含对周期轨道和周期的求解，迭代过程中的增量通过优化方法选择，从而能迅速确定出系统的周期轨道及其周期。应用所求的结果结合谐波平衡方法求得了非线性系统的周期轨道的近似解析表达式，理论上通过增加谐波的阶数任何精度的周期解都可以得到。最后将该方法应用于非线性转子轴承系统，求出了在某些参数下转子的周期解及其近似解析表达式，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性，计算结果对于转子系统运动的定量控制有重要理论指导意义。     

Periodic motions of spinning rigid spacecraft under influence of gravitational and magnetic fields

The motion of a magnetized axisymmetric spacecraft about its center of mass in a circular orbit is considered, taking the gravitational and magnetic effects of the central body into account. Equations of motion of the reduced system are transformed to equations of plane motion of a charged particle under the action of electric and magnetic fields. Stationary motions of the system are determined and periodic motions near to them are constructed using the Lyapounoff theorem of the holomorphic integral.     

Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part I: Theoretical Analysis

In a previous paper, the dynamic behaviour of a Jeffcott rotor was studied in the presence of pure static unbalance and nonlinear elastic restoring forces. The present paper extends the analysis to a rigid rotor with an axial length such as to make the transverse moment of inertia greater than the axial one. As in the previous investigation, the elastic restoring forces are assumed to be nonlinear and the effects of couple unbalance are also included but, unlike the Jeffcott rotor, the system exhibits six degrees-of-freedom. The Lagrangian coordinates were fixed so as to coincide with the three coordinates of the centre of mass of the rotor and the three angular coordinates needed in order to express the rotor's rotations with respect to a reference frame having its origin in the centre of mass. The precession motions of such a rotor turn out to be cylindrical at low angular speeds and exhibit a conical aspect when operating at higher speeds. The motion equations of the rotor were written with reference to a system that was subsequently adopted for the experimental analysis. The particular feature of this system was the use of a steel wire (piano wire) for the rotor shaft, suitably constrained and with the possibility of regulating the tension of the wire itself, in order to increase or reduce the nonlinear character of the system. The numerical analysis performed with integration of the motion equations made it possible to point out that chaotic solutions were manifested only when the tension in the wire was given the lowest values – i.e. when the system was strongly nonlinear – in the presence of considerable damping and rotor unbalance values that were so high as to lose any practical significance. Under conditions commonly shared by analogous real systems characterised by poor damping, where the contribution to nonlinearity is almost entirely due to elastic restoring forces, the analysis pointed out that precession motions may be manifested with a periodic character, whether synchronous or not, or a quasi-periodic character, but in no case is the solution chaotic.     

A field method and its application to the theory of vibrations

The problem of integration of the differential equations of motion of a nonconservative dynamical system is replaced by an equivalent problem of finding a complete integral of a quasi-linear partial differential equation of the first order. In the second part, these complete integrals are combined with the two time scales perturbation method in the study of non-linear oscillatory motions.     

Beck's rod with shear,compressibility and pulsating force

In-plane motion of an elastic rod, subjected to a compressive follower force is analyzed. Equations of motions are derived for the case when deformations are not small. It is assumed that the compressive force has a periodic component, so that a parametric instability is possible. The stability bound is estimated analytically and determined numerically. In deriving the differential equations of motion a generalized constitutive equation taking into account compressibility of the rod axis and the influence of shear stresses is used.     

一种确定非线性裂纹转子解的形式的新方法

将小波变换与Ｐｏｉｎｃａｒｅ映射相结合，即用Ｐｏｉｎｃａｒｅ映射确定周期解，用谐波小波变换区分拟周期响应和混沌运动，提出了一种分析非线性裂纹转子系统解的形式随参数变化的新方法．结果表明这种方法是非常有效的，它比以前所用的计算Ｌｉａｐｕｎｏｖ指数的方法节约了计算时间，并且较易实施．     

Motion types and chaos of multi-body systems vibrating with impacts

In this paper, the limit sets theory for an autonomous dynamical system is generalized to a multi-body system vibrating with impacts. We discover that if every motion of the system is bounded, it has only four different types: periodic motion γ1, non·periodic recurrent motion γ2, and non-Poisson stable motions γ3 and γ4 approaching γ1 and γ2, respectively. γ2 is the source of chaos. It is very interesting that chaotic motions seem stochastic but possess the character of recurrence. By way of example, we discuss chaotic motions of a small ball bouncing vertically on a massive vibrating table. The result obtained by us is different from that obtained by Holmes. The project supported by National Natural Science Foundation of China