Chaotic Motion in a Spinning Spacecraft with Circumferential Nutational Damper

The dynamics of a simplified model of a spinning spacecraft with a circumferential nutational damper is investigated using numerical simulations for nonlinear phenomena. A realistic spacecraft parameter configuration is investigated and is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. Such a torque, in practice, may arise in the platform of a dual-spin spacecraft under malfunction of the control system or from an unbalanced rotor or from vibrations in appendages. The equations of motion of the model are derived with Lagrange's equations using a generalisation of the kinetic energy equation and a linear stability analysis is given. Numerical simulations for satellite parameters are performed and the system is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. The motion is studied by means of time history, phase space, frequency spectrum, Poincaré map, Lyapunov characteristic exponents and Correlation Dimension. For sufficiently large values of torque amplitude, the behaviour of the system was found to have much in common with a two well potential problem such as a Duffing oscillator. Evidence is also presented, indicating that the onset of chaotic motion was characterised by period doubling as well as intermittency.     

带有姿态的绳系航天器混沌运动实验研究

利用地面物理仿真平台研究了绳系航天器的混沌动力学行为. 首先, 根据天地动力学相似原理, 通过对卫星仿真器施加喷气力和动量轮力矩来模拟空间动力学环境, 提出了两种等效方案, 给出了它们各自适用的实验工况. 数值结果表明, 在轨绳系航天器在一定的参数条件下系绳摆动为周期或概周期运动、航天器姿态发生混沌运动. 物理仿真验证了等效方案的有效性, 揭示了绳系航天器的混沌运动特征, 表明在阻尼力矩的作用下可以避免绳系航天器混沌运动的发生.      

非自旋航天器混沌姿态运动及其参数开闭环控制

研究万有引力场中受大气阻力且存在结构内阻尼的非自旋航天器在椭圆轨道上平面天平动的混沌及其参数开闭环控制问题．在建立数学模型的基础上确定出现混沌的必要条件并数值验证混沌的存在性，提出非线性振动系统混沌运动的参数开闭环控制并应用于控制航天器的混沌姿态运动．     

控制一类磁性刚体航天器的混沌姿态运动

本文研究一类磁性航天器的混沌姿态运动及其控制,建立了在近地球赤道面圆轨道上运动受万有引力矩、磁力矩作用磁性刚体航天器姿态运动的动力学方程。采用时间历程、Poincare截面、Lyapunov指数和功率谱对系统的动力学行为进行数值识别,结果表明随着磁场参数的增大系统动力学行为由准周期环面破裂而出现混沌。利用输入－输出反馈精确线性化的方法将航天器的混沌姿态控制运动控制为姿态静止和按给定的周期规律运动,数值结果表明该控制方法的有效性。     

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)     

Chaotic attitude motion of a magnetic rigid spacecraft and its control

This paper deals with chaotic attitude motion of a magnetic rigid spacecraft with internal damping in a circular orbit near the equatorial plane of the earth. The dynamical model of the problem is established. The Melnikov analysis is carried out to prove the existence of a complicated non-wandering Cantor set. The dynamical behaviors are numerically investigated by means of time history. Poincare map, power spectrum and Lyapunov exponents. Numerical simulations indicate that the onset of chaos is characterized by the intermittency as the increase of the torque of the magnetic forces and decrease of the damping. The input-output feedback linearization method is applied to control chaotic attitude motions to the given fixed point and periodic motion.     

Combined control of fast attitude maneuver and stabilization for large complex spacecraft

In remote sensing or laser communication space missions, spacecraft need fast maneuver and fast stabilization in order to accomplish agile imaging and attitude tracking tasks. However, fast attitude maneuvers can easily cause elastic deformations and vibrations in flexible appendages of the spacecraft. This paper focuses on this problem and deals with the combined control of fast attitude maneuver and sta- bilization for large complex spacecraft. The mathematical model of complex spacecraft with flexible appendages and momentum bias actuators on board is presented. Based on the plant model and combined with the feedback controller, modal parameters of the closed-loop system are calculated, and a multiple mode input shaper utilizing the modal information is designed to suppress vibrations. Aiming at reducing vibrations excited by attitude maneuver, a quintic polynomial form rotation path planning is proposed with constraints on the actuators and the angular velocity taken into account. Attitude maneuver simulation results of the control systems with input shaper or path planning in loop are sepa- rately analyzed, and based on the analysis, a combined control strategy is presented with both path planning and input shaper in loop. Simulation results show that the combined control strategy satisfies the complex spacecraft＇s require- ment of fast maneuver and stabilization with the actuators＇ torque limitation satisfied at the same time.     

Onset of chaotic motion in a gyroscopic system

We study forced vibrations of a gimbal gyro occurring if the inner ring is subjected to a perturbing torque that is the sum of the viscous friction torque and a periodic small-amplitude torque. In the absence of the perturbing torque, there exist two steady-state motions of the gimbal gyro, in which the gimbal rings are either orthogonal or coincide. These motions are respectively stable and unstable. We obtain an equation for the unperturbed system, whose separatrix passes through hyperbolic points. The distance between these points (the Melnikov distance) is calculated to find a condition for the intersection of the separatrices of the perturbed system. We find a domain in the parameter space where the distance changes sign, which indicates the onset of chaotic motion.     

Prediction of chaotic instabilities in a dragline bucket swing

The occurrence of chaotic instabilities is investigated in the swing motion of a dragline bucket during operation cycles. A dragline is a large, powerful, rotating multibody system utilised in the mining industry for removal of overburden. A simplified representative model of the dragline is developed in the form of a fundamental non-linear rotating multibody system with energy dissipation. An analytical predictive criterion for the onset of chaotic instability is then obtained in terms of critical system parameters using Melnikov's method. The model is shown to exhibit chaotic instability due to a harmonic slew torque for a range of amplitudes and frequencies. These chaotic instabilities could introduce irregularities into the motion of the dragline system, rendering the system difficult to control by the operator and/or would have undesirable effects on dragline productivity and fatigue lifetime. The sufficient analytical criterion for the onset of chaotic instability is shown to be a useful predictor of the phenomenon under steady and unsteady slewing conditions via comparisons with numerical results.     

Studies on Bifurcation and Chaos of a String-Beam Coupled System with Two Degrees-of-Freedom

In this paper, research on nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitations is presented. The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees-of-freedom. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam, and primary resonance for the string is considered. The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system. Based on the averaged equation obtained here, the techniques of phase portrait, waveform, and Poincare map are applied to analyze the periodic and chaotic motions. It is found from numerical simulations that there are obvious jumping phenomena in the resonant response–frequency curves. It is indicated from the phase portrait and Poincare map that period-4, period-2, and periodic solutions and chaotic motions occur in the transverse nonlinear vibrations of the string-beam coupled system under certain conditions. An erratum to this article is available at .     

Chaotic pitch motion of an asymmetric non-rigid spacecraft with viscous drag in circular orbit

We study the pitch motion dynamics of an asymmetric spacecraft in circular orbit under the influence of a gravity gradient torque. The spacecraft is perturbed by a small aerodynamic drag torque proportional to the angular velocity of the body about its mass center. We also suppose that one of the moments of inertia of the spacecraft is a periodic function of time. Under both perturbations, we show that the system exhibits a transient chaotic behavior by means of the Melnikov method. This method gives us an analytical criterion for heteroclinic chaos in terms of the system parameters which is numerically contrasted. We also show that some periodic orbits survive for perturbation small enough.     

飞轮控制航天器的自适应姿态跟踪

本文讨论飞轮控制航天器的姿态控制问题。由于航天器在运行过程中受各种复杂环境因素的影响，难以确定其动力学参数的准确值，因此姿态控制系统必须具有自适应能力，以适应不可预计的干扰和航天器自身参数的改变。文中基于滑模控制方法提出一种改进的自适应姿态控制规律，适用于有内扰动力短存在的航天器。此控制规律可在不须提供动力学参数的情况下，使飞轮控制航天器跟踪期望的姿态变化规律以实现姿态机动。利用Lyapunov直接方法证明了控制系统的渐近稳定性。对于受随机扰动力短作用的航天器姿态跟踪的时间历程进行了数值仿真，计算结果证实上述方法的有效性。     

亚音速气流作用下薄板结构混沌运动的时滞反馈控制

研究了亚音速气流下非线性二维薄板结构在横向周期载荷作用下的混沌运动及控制问题。基于von Karman大变形板理论和分离变量法,建立了亚音速下薄板结构的运动控制方程。对于未控系统,采用Melnikov方法判断其混沌运动阈值,并用Runge-Kutta法进行数值验证。对处于混沌运动状态的系统,采用时滞反馈控制方法对混沌运动进行控制。结果表明,Melnikov方法可以有效地预测系统的混沌运动行为,时滞反馈控制方法可以有效地将系统的混沌运动转化为周期运动。     

Pseudoelastic effect in autoparametric non-ideal vibrating system with SMA spring

In this paper a three degrees of freedom autoparametric system with limited power supply is investigated numerically. The system consists of the body, which is hung on a spring and a damper, and two pendulums connected by shape memory alloy (SMA) spring. Shape memory alloys have ability to change their material properties with temperature. A polynomial constitutive model is assumed to describe the behavior of the SMA spring. The non-ideal source of power adds one degree of freedom, so the system has four degrees of freedom. The equations of motion have been solved numerically and pseudoelastic effects associated with the martensitic phase transformation are studied. It is shown that in this type system one mode of vibrations might excite or damp another mode, and that except different kinds of periodic vibrations there may also appear chaotic vibrations. For the identification of the responses of the system's various techniques, including chaos techniques such as bifurcation diagrams and time histories, power spectral densities, Poincarè maps and exponents of Lyapunov may be used.     

Nonlinear aeroelastic analysis of a two-dimensional wing with control surface in supersonic flow

Based on the piston theory of supersonic flow and the energy method, a two dimensional wing with a control surface in supersonic flow is theoretically modeled, in which the cubic stiffness in the torsional direction of the control surface is considered. An approximate method of the cha- otic response analysis of the nonlinear aeroelastic system is studied, the main idea of which is that under the condi- tion of stable limit cycle flutter of the aeroelastic system, the vibrations in the plunging and pitching of the wing can approximately be considered to be simple harmonic excita- tion to the control surface. The motion of the control surface can approximately be modeled by a nonlinear oscillation of one-degree-of-freedom. The range of the chaotic response of the aeroelastic system is approximately determined by means of the chaotic response of the nonlinear oscillator. The rich dynamic behaviors of the control surface are represented as bifurcation diagrams, phase-plane portraits and PS diagrams. The theoretical analysis is verified by the numerical results.     

单框架控制力矩陀螺输出特性分析

广泛用于航天领域的单框架控制力矩陀螺,具有力矩放大效应的优点,其理论基础为有假设条件的力矩放大原理.本文不局限于这些假设,不限定工况,解析单框架控制力矩陀螺的输出特性.考虑安装基座的运动,得到具有两维输入三维输出的单框架控制力矩陀螺力矩输出模型,提出将输出力矩分解为可调控与不可调控两部分.为分析单框架控制力矩陀螺的输出...     

A control approach for vibrations of a nonlinear microbeam system in multi-dimensional form

Microbeams are widely seen in micro-electro-mechanical systems and their engineering applications. An active control strategy based on the fuzzy sliding mode control is developed in this research for controlling and stabilizing the nonlinear vibrations of a micro-electro-mechanical beam. An Euler-Bernoulli beam with a fixed-fixed boundary is employed to represent the microbeam, and the geometric nonlinearity of the beam and loading nonlinearity from the electrostatic force are considered. The governing equation of the microbeam is established and transformed into a multi-dimensional dynamic system with the third-order Galerkin method. A stability analysis is provided to show the necessity of the derived multi-dimensional dynamic system, and a chaotic motion is discovered. Then, a control approach is proposed, including a control strategy and a two-phase control method. For describing the application of the control approach developed, control of a chaotic motion of the microbeam is presented. The effectiveness of the active control approach is demonstrated via controlling and stabilizing the nonlinear vibration of the microbeam.     

Dynamic delayed feedback control for stabilizing the giant swing motions of an underactuated three-link gymnastic robot

This paper investigates the dynamics of the giant swing motions of an underactuated three-link gymnastic robot moving in a vertical plane by means of dynamic delayed feedback control (DDFC). DDFC, being one of useful methods to overcome the so-called odd number limitation in controlling a chaotic discrete-time system, is extended to control a continuous-time system such as a 3-link gymnastic robot with passive joint. Meanwhile, a way to calculate the error transfer matrix and the input matrix which are necessary for discretization is proposed, based on a Poincaré section which is defined to regard the target system as a discrete-time system. Moreover, the stability of the closed-loop system by the proposed control strategy is discussed. Furthermore, some numerical simulations are presented to show the effectiveness in controlling a chaotic motion of the 3-link gymnastic robot to a periodic giant swing motion.     

万有引力场中陀螺体的混沌运动

研究万有引力场中沿圆轨道运行的非对称陀螺体的姿态运动,引入Ｄｅｐｒｉｔ正则变量建立系统的Ｈａｍｉｌｔｏｎ结构,利用Ｍｅｌｎｉｋｏｖ方法证明在万有引力短作用的昆体产生混沌运动的可能性。对Ｐｏｉｎｃａｒｅ截面的数值计算表明提高陀螺体的转子转速可对混沌起抑制作用。     

Chaos in a pendulum with feedback control

We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.