HOVERING ORBITS DESIGN FOR PERTURBED ASTEROIDS WITH PARAMETRIC EXCITATION RESONANCE 1)

本文将太阳引力摄动视为受摄不规则小行星系统的组成部分,借鉴非线性振动理论中参数激励共振的概念,创新性地设计了不规则小行星平衡点附近稳定的悬停观测轨道.为了同时考虑不规则小行星引力和太阳引力, 本文采用受摄粒杆模型描述系统.通过对未扰系统平衡点以及固有频率的分析, 给出系统存在参激共振轨道的条件.再以第二类参激主共振和1:3内共振为例,采用多尺度方法求得参数激励共振轨道的稳态解, 并对稳态解的稳定性进行判断.通过受摄小行星系统的幅频响应曲线以及力频响应曲线分析了系统的非线性特性以及参数激励效应.此外, 对内共振引起的长短周期能量转移现象进行了分析.本文的研究成果可以拓展现有小行星系统周期轨道族设计方法.     

基于参激共振的受摄小行星悬停轨道设计方法

本文将太阳引力摄动视为受摄不规则小行星系统的组成部分,借鉴非线性振动理论中参数激励共振的概念,创新性地设计了不规则小行星平衡点附近稳定的悬停观测轨道.为了同时考虑不规则小行星引力和太阳引力, 本文采用受摄粒杆模型描述系统.通过对未扰系统平衡点以及固有频率的分析, 给出系统存在参激共振轨道的条件.再以第二类参激主共振和1:3内共振为例,采用多尺度方法求得参数激励共振轨道的稳态解, 并对稳态解的稳定性进行判断.通过受摄小行星系统的幅频响应曲线以及力频响应曲线分析了系统的非线性特性以及参数激励效应.此外, 对内共振引起的长短周期能量转移现象进行了分析.本文的研究成果可以拓展现有小行星系统周期轨道族设计方法.      

Satellite rotation stability at a Mercurian type resonance

We consider the satellite plane motion about the center of mass in a central Newtonian gravitational field in an elliptic orbit. This motion is described by a second-order differential equation known as the Beletskii equation. In the framework of the plane problem (under the assumption that the body vibrates in the unperturbed orbit plane), there exists a family of periodic solutions of the Beletskii equation near the 3: 2 resonance between the orbital revolution and axial rotation periods. A nonlinear stability analysis of these periodic solutions is carried out both in the presence of third- and fourth-order resonances and in their absence as well as on the boundaries of the stability regions in the first approximation. The problem is solved numerically. For fixed parameter values (the eccentricity of the center-of-mass orbit and the inertial parameter), the construction of a symplectic mapping of the equilibrium into itself is used to calculate the coefficients of the mapping generating function, which are further used to conclude whether the equilibrium is stable or not.     

Periodic motion around libration points in the Elliptic Restricted Three-Body Problem

Periodicity of motion around the collinear libration point associated with the Elliptic Restricted Three-Body Problem is studied. A survey of periodic solutions in the Circular Restricted Three-Body Problem is presented considering both Sun–Earth and Earth–Moon systems. Halo, Lyapunov and Vertical families around L1, L2 and L3 points are investigated, and their orbital period ranges through the entire family are reported. Resonant motions within the orbit families in the circular problem are identified and selected as suitable initial guess to find periodic orbits in the elliptic problem, which are targeted using a differential correction algorithm. Periodic solutions found are cataloged depending on the number of revolutions around libration points. Geometry, dynamical behavior and stability properties of single-revolution orbits are shown, as well as double-, triple- and quadruple-revolution solutions.     

On periodic motions of an orbital dumbbell-shaped body with a cabin-elevator

The motion of a dumbbell-shaped body (a pair of massive points connected with each other by a weightless rod along which the elevator, i.e., a third point, is moving according to a given law) in an attractive Newtonian central field is considered. In particular, such a mechanical system can be considered as a simplified model of an orbital cable system equipped with an elevator. The practically most interesting case where the cabin performs periodic ??shuttle??motions is studied. Under the assumption that the elevator mass is small compared with the dumbbell mass, the Poincaré theory is used to determine the conditions for the existence of families of system periodic motions analytically depending on the arising small parameter and passing into some stable radial steady-state motion of the unperturbed problem as the small parameter tends to zero. It is also proved that, for sufficiently small parameter values, each of the radial relative equilibria generates exactly one family of such periodic motions. The stability of the obtained periodic solutions is studied in the linear approximation, and these solutions themselves are calculated up to terms of the firstorder in the small parameter. The contemporary studies of the motion of orbital dumbbell systems apparently originated in Okunev??s papers [1, 2]. These studies were continued in [3], where plane motions of an orbit tether (represented as a dumbbell-shaped satellite) in a circular orbit were considered in the satellite approximation. In [4], in the case of equal masses and in the unbounded statement, the energy-momentum method was used to perform the dynamic reduction of the problem and analyze the stability of relative equilibria. A similar technique was used in [5], where, in contrast to the above-mentioned problems, the massive points were connected by an elastic spring resisting to compression and forming a dumbbell with elastic properties. Under such assumptions, the stability of radial configurations was investigated in that paper. The bifurcations and stability of steady-state configurations of a deformable elastic dumbbell were also studied in [6]. Various obstacles arising in the construction of orbital cable systems, in particular, the strong deformability of known materials, were discussed in [7]. In [8], the problem of orbital motion of a pair of massive points connected by an inextensible weightless cable was considered in the exact statement. In other words, it was assumed that a unilateral constraint is imposed on themassive points. The conditions of stability of vertical positions of the relative equilibria of the cable system, which were obtained in [8], can be used for any ratio of the subsatellite and station masses. In turn, these results agree well with the results obtained earlier in the studies of stability of vertical configurations in the case of equal masses of the system end bodies [3, 4]. One of the basic papers in the dynamics of three-body orbital cable systems is the paper [9]. The steady-state motions and their bifurcations and stability were studied depending on the elevator cabin position in [10].     

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.     

Bifurcation analysis for T system with delayed feedback and its application to control of chaos

In this paper, a three-dimensional autonomous nonlinear system called the T system which has potential application in secure communications is considered. Regarding the delay as parameter, we investigate the effect of delay on the dynamics of T system with delayed feedback. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associated characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a periodic orbit.     

Analysis of nonlinear vibrations and stability of rotating asymmetrical nano-shafts incorporating surface energy effects

The objective of the present work is to investigate the nonlinear vibrations of the rotating asymmetrical nano-shafts by considering surface effect. In order to compute the surface stress tensor, the surface elasticity theory is used. The governing nonlinear equations of motion are obtained with the aid of variational approach. Bubnov–Galerkin is a very effective method for exploiting the reduced-order model of the equations of motion. The averaging method is employed to analyze the reduced-order model of the system. For this purpose, the well-known Van der Pol transformation in the complex form and angle-action transformation are utilized. The effect of surface stress on the forward and backward speeds, steady state responses of the system, fixed points, close orbits and stability of the solutions is examined. The preliminary results of the research show that the absolute values of forward and backward whirling speeds in the presence of surface effect with positive residual surface stress are higher than those of regarding the system without surface effect and in the presence of surface effect with negative residual surface stress. In addition, it is seen that the undamped rotating asymmetrical nano-shaft, for specified value of detuning parameter, in the absence or presence of surface effect has various number of stable and unstable periodic solutions. Besides, there is different number of separatrix (homoclinic orbit type). Furthermore, bifurcations, number of solutions and their stability for damped rotating asymmetrical nano-shaft are investigated. Also, the above results have been obtained for rotating symmetrical nano-shaft.     

Asymptotically stable periodic orbits of a coupled electromechanical system

In this paper an electromechanical system is analyzed. The existence and asymptotic stability of a periodic orbit are obtained in a mathematically rigorous way as well as an expansion of the period by using an adequate small parameter. For the analytical results the main tool used is the regular perturbation theory. Some results, such as the growing of the period according to some powers of the parameters and the relation 2:1 between the period of the cart, which is a part of the electromechanical system, and the period of the current, are compatible with earlier numerical findings.     

激励Stuart-Landau方程的研究--周期解、稳定性及流动控制

解析得出了有外部激励的Stuart-Landau（S-L）方程的频率锁定周期解,对这些解与外部激励振幅和频率的依赖关系做了详细研究,并用周期系统稳定性理论确定了解的稳定性边界.还对S-L方程所描述的流动控制效果进行了研究,发现由于外部激励的作用,稳定的锁频解可能比原来的饱和解能量减少了,外部的控制最多能使扰动能量减少为原来的一半.     

Effects of small world connection on the dynamics of a delayed ring network

This paper presents a detailed analysis on the dynamics of a ring network with small world connection. On the basis of Lyapunov stability approach, the asymptotic stability of the trivial equilibrium is first investigated and the delay-dependent criteria ensuring global stability are obtained. The existence of Hopf bifurcation and the stability of periodic solutions bifurcating from the trivial equilibrium are then analyzed. Further studies are paid to the effects of small world connection on the stability interval and the stability of periodic solution. In particular, some complex dynamical phenomena due to short-cut strength are observed numerically, such as: period-doubling bifurcation and torus breaking to chaos, the coexistence of multiple periodic solutions, multiple quasi-periodic solutions, and multiple chaotic attractors. The studies show that small world connection may be used as a simple but efficient “switch” to control the dynamics of a system.     

非线性振动系统周期运动及其稳定性的数值研究

§1引言确定型非线性振动系统的运动可分类如下: 1.非定常运动;2.定常运动:(1)周期运动,(2)各态历经运动,(3)浑沌运动。其中非定常运动是一暂态过程,会随着时间的增长逐步衰减乃至实际上消失。定常运动中的各态历经运动,指系统至少有两个互不通约(即其比值为无理数)的振动频率,因此运动虽然局      

Modelling, Dynamics and Control of Tethered Satellite Systems

Tethered satellite systems (TSS) pose quite challenging problems concerning their modelling, derivation of the equations of motion, numerical simulation of their dynamics, deciding on stability of relative equilibria provided the system moves on a circular orbit around the Earth and the occurrence of chaotic dynamics. Moreover, for the processes of deployment or retrieval of one satellite from or to another satellite certain control strategies, for example time or energy optimal control, are necessary. All these problems are considered in this paper.     

Bifurcation analysis in a delayed Lokta�CVolterra predator�Cprey model with two delays

In this paper, a class of delayed Lokta?CVolterra predator?Cprey model with two delays is considered. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also provided. Finally, main conclusions are given.     

A stability analysis of periodic solutions to the steady forced Korteweg–de Vries–Burgers equation

The stability of periodic solutions to the steady forced Korteweg–de Vries–Burgers (fKdVB) equation is investigated here. This family of periodic solutions was identified by Hattam and Clarke (2015) using a multi-scale perturbation technique. Here, Floquet theory is applied to the governing equation. Consequently, two criteria are found that determine when the periodic solutions are stable. This analysis is then confirmed by a numerical study of the steady fKdVB equation.     

Nonlinear response and nonsmooth bifurcations of an unbalanced machine-tool spindle-bearing system

In this effort, a six-degree-of-freedom (DOF) model is presented for the study of a machine-tool spindle-bearing system. The dynamics of machine-tool spindle system supported by ball bearings can be described by a set of second order nonlinear differential equations with piecewise stiffness and damping due to the bearing clearance. To investigate the effect of bearing clearance, bifurcations and routes to chaos of this nonsmooth system, numerical simulation is carried out. Numerical results show when the inner race touches the bearing ball with a low speed, grazing bifurcation occurs. The solutions of this system evolve from quasi-periodic to chaotic orbit, from period doubled orbit to periodic orbit, and from periodic orbit to quasi-periodic orbit through grazing bifurcations. In addition, the tori doubling process to chaos which usually occurs in the impact system is also observed in this spindle-bearing system.     

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。     

求解非线性动力系统周期解的改进打靶法

针对有周期解的动力系统边值问题可以转化为初值问题这一特点，改进了周期解的打靶 法数值求解. 在计算边界条件代数方程关于待定初值参数导数的过程中利用前一次 Runge-Kutta方法计算得到的节点函数值并通过再次利用Runge-Kutta方法获得了该导数值. 用此方法求解了Duffing方程及非线性转子---轴承系统的周期解,用Floquet理论判断了 周期解的稳定性,与普通打靶法作了比较,验证了方法的有效性.     

A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics

In this paper a comparative study of five different stability computational methods based on the Floquet theory is presented. These methods are compared in terms of accuracy and CPU performance. Tests are performed on a set of nonlinear problems relevant to rotating machinery with rotor-to-stator contact and a variable number of degrees of freedom, whose periodic solutions are computed with the Harmonic Balance Method (HBM).     

Dynamics of Relative Phases: Generalised Multibreathers

For small Hamiltonian perturbation of a Hamiltonian systemof arbitrary number of degrees of freedom with anormally non-degenerate submanifold of periodic orbits we construct a nearbysubmanifold and an `effective Hamiltonian' on it such that the differencebetween the two Hamiltonian vector fields is small. The effective Hamiltonianis independent of one coordinate, the `overall phase', and hence thecorresponding action is preserved. Unlike standard averaging approaches,critical points of our effective Hamiltonian subject to given actioncorrespond to exact periodic solutions. We prove there has to be at least acertain number of these critical points given by global topological principles.The linearisation of the effective Hamiltonian about critical points is provedto give the linearised dynamics for the full system to leading order in theperturbation. Hence in the case of distinct eigenvalues which move at non-zerospeed with ,the linear stability type of the periodic orbit can be read offfrom the effective Hamiltonian. Our principal application is to networks ofoscillators or rotors where many such submanifolds of periodic orbits occurat the uncoupled limit – simply excite a number N 2 of the units inrational frequency ratio and put the others on equilibria, subject to anon-resonance condition. The resulting exact periodic solutions for weakcoupling are known as multibreathers. We call the approximate solutions givenby the effective Hamiltonian dynamics, `generalised multibreathers'. Theycorrespond to solutions which look periodic on a short time scale but therelative phases of the excited units may evolve slowly. Extensions aresketched to travelling breathers and energy exchange between degrees offreedom.