动量矩变量描述的陀螺体永久转动的分岔特性

本文建立以动量矩分量和Ｄｅｐｒｉｔ正则变量为独立变量的陀螺体动力学方程，用以描述无力矩非对称陀螺体的姿态运动。在动量矩空间内讨论了陀螺体永久转动轴的数目和稳定性随转子转速变化的分岔特性。     

基于Rodrigues参数的陀螺体受控运动

经典刚体动力学中表示刚体姿态的参数中，Euler角、Cardan角和Euler参数在工程技术中使用最为普遍．近期在航天器姿态控制问题中使用Rodrigues参数的报道也引起注意，Rodrigues参数以其表达形式简明和代数运算特点而具有独特优点．航天器姿态控制系统必须具有自适应性以适应参数的变化，建立用Rodrigues参数表达的无力矩陀螺体受控运动方程，提出基于Rodrigues参数的自适应姿态控制方案，并应用Lyapunov定理证明受控运动的渐近稳定性。     

Global robust synchronization for time-varying phase mismatching electro-mechanical gyrostat systems under variable substitution control

A master–slave scheme for global robust synchronization of two electro-mechanical gyrostat systems with time-varying phase mismatches under variable substitution control is presented in this paper. Under this scheme, a sufficient criterion for the global robust synchronization with bounded error is rigorously proved in the form of matrix inequality and the corresponding estimated error bound is mathematically given. On the basis of the criterion, further derivation brings some simple and optimized algebraic criteria for various single-variable coupling, whose performance is then verified through some numerical examples.     

Adaptive finite-time stabilization of uncertain non-autonomous chaotic electromechanical gyrostat systems with unknown parameters

This paper deals with the problem of robust finite-time stabilization of non-autonomous chaotic gyrostat systems. It is assumed that the parameters of the gyrostat system are completely unknown in advance and the system is perturbed by unknown uncertainties and disturbances. Some update laws are proposed to estimate the unknown parameters. Based on the finite-time control idea and the update laws, appropriate control laws are designed to ensure the stabilization of the closed-loop system in a finite time. The finite-time stability and convergence of the closed-loop system are analytically proved. A numerical simulation is given to demonstrate the applicability and robustness of the proposed finite-time controller and to verify the theoretical results.     

非轴对称陀螺体自激运动的定性分析

研究非轴对称陀螺体在自激控制力矩作用下的姿态运动.将无力矩状态的两个初积分的积分常数T和G2作为描述自激运动的状态变量建立其微分方程.利用受扰运动在(T,G2)空间中的相轨迹判断稳态运动的稳定性,并进行了数值验证.     

Qualitative analysis of the phase flow of an integrable approximation of a generalized roto-translatory problem

In this paper, we consider an integrable approximation of the planar motion of a gyrostat in Newtonian interaction with a spherical rigid body. We then describe the Hamiltonian dynamics, in the fibers of constant total angular momentum vector of an invariant manifold of motion. Finally, using the Liouville-Arnold theorem and a particular analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. The results can be applied to study two-body roto-translatory problems where the rotation of one of them has a strong influence on the orbital motion of the system.      

Stability of equilibrium states in the Zhukovski case of heavy gyrostat using algebraic methods

We study the stability of the equilibrium points of a skew product system and analyze the possibility to construct a Lyapunov function by using a set of conserved quantities and solving an algebraic system. We apply the theoretical results to study the stability of an equilibrium state of a heavy gyrostat in the Zhukovski case. Copyright © 2012 John Wiley & Sons, Ltd.     

Lagrangian relative equilibria for a gyrostat in the three-body problem

In this paper we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric mechanics methods, we study the approximate Poisson dynamics that arise when we develop the potential of the system in Legendre series and truncate this to an arbitrary order k. After reduction of the dynamics by means of the two symmetries of the system, we consider the existence and number of equilibria which we denominate of Lagrangian type, in analogy with classic results on the topic. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order k, and explicit expressions for these equilibria are given, this being useful for the subsequent study of their stability. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order k for k ≥1 is independent of the order of truncation of the potential, if the gyrostat S ₀ is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies.      

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。