Heteroclinic bifurcations in completely liquid-filled spacecraft with flexible appendage

In this paper, the chaotic dynamics in an attitude transition maneuver of a rigid body with a completely liquid-filled cavity in going from minor axis to major axis spin under the influence of viscous damping and a small flexible appendage constrained to undergo only torsional vibration is investigated. The focus in this paper is on the way in which the dynamics of the liquid and flexible appendage vibration are coupled. The equations of motion are derived and then transformed into a form suitable for the application of Melnikov's method. Melnikov's integral is used to predict the transversal intersections of the stable and unstable manifolds for the perturbed system. An analytical criterion for chaotic motion is derived in terms of the system parameters. This criterion is evaluated for its significance to the design of spacecraft. The dependence of the onset of chaos on quantities such as body shape and magnitude of damping values, fuel fraction and frequency of flexible appendage vibration are investigated.     

柔性全充液航天器大角度姿态机动混沌动力学

研究了受液体燃料黏性阻尼及柔性附件扭振影响的全充液航天器由最小惯量轴向最大惯量作大角度姿态机动过程中的混沌姿态动力学, 尤其是液体燃料和柔性附件振动的耦合效应对航天器姿态动力学的影响. 推导了耦合系统的动力学方程并利用尺度化方法将其转化为扰动系统的标准形式以便应用Melnikov方法对系统进行混沌姿态预测.推导了以系统参数形式表达的混沌姿态预测的解析准则. 将利用数值方法所得到的对系统的数值仿真结果与Melnikov解析准则进行了比较和评述. 研究了诸如航天器构型、液体燃料惯量及阻尼、柔性附件固有频率等系统特征量对混沌姿态的影响.      

Analytical method for the attitude stability of partially liquid filled spacecraft with flexible appendage

In this paper, the attitude stability of liquid-filled spacecraft with flexible appendage is investigated. The motion of liquid sloshing is modeled as the spherical pen-dulum, and the flexible appendage is approached by a linear shearing beam. Nonlinear dynamic equations of the coupled system are derived from the Hamiltonian. The stability of the coupled system was analyzed by using the energy-Casimir method, and the nonlinear stability theorem of the coupled spacecraft system was also obtained. Through numerical computation, the correctness of the proposed theorem is verified and the boundary curves of the stable region are presented. The increase of the angular velocity and flexible attachment length will weaken the attitude stability, and the change of the filled ratio of liquid fuel tank has a different influence on the stability of the coupled spacecraft, depend-ing on the different conditions. The attitude stability analysis of the coupled spacecraft system in this context is useful for selecting appropriate parameters in the complex spacecraft design.     

Analytical study of nonlinear synchronous full annular rub motion of flexible rotor–stator system and its dynamic stability

The nonlinear synchronous full annular rub motion of a flexible rotor induced by the mass unbalance and the contact-rub force with rigid and flexible stator is studied analytically. The nonlinear property is due to the dry friction force between stator and rotor. The exact solutions of the synchronous full annular rub motion and its run speed regions are obtained. The stability of the synchronous full annular rub motion is discussed analytically. The stability criterion and the stability regions of the synchronous full annular rub motion are obtained. A simplified approximate criterion formula for dynamic stability is also derived under the conditions of large impact stiffness, small damping and small friction. The simplified criterion formula can be used conveniently in engineering and matches the real situations of industry.     

Chaotic attitude and reorientation maneuver for completely liquid-filled spacecraft with flexible appendage

The present paper investigates the chaotic attitude dynamics and reorientation maneuver for completely viscous liquid-filled spacecraft with flexible appendage. All of the equations of motion are derived by using Lagrangian mechanics and then transformed into a form consisting of an unperturbed part plus perturbed terms so that the system＇s nonlinear characteristics can be exploited in phase space. Emphases are laid on the chaotic attitude dynamics produced from certain sets of physical parameter values of the spacecraft when energy dissipation acts to derive the body from minor to major axis spin. Numerical solutions of these equations show that the attitude dynamics of liquid-filled flexible spacecraft possesses characteristics common to random, non- periodic solutions and chaos, and it is demonstrated that the desired reorientation maneuver is guaranteed by using a pair of thruster impulses. The control strategy for reorientation maneuver is designed and the numerical simulation results are presented for both the uncontrolled and controlled spins transition.     

一类带柔性附件充液航天器姿态机动控制

基于Lyapunov稳定性理论研究了用动量轮控制一类带轻质悬臂梁附件的充液航天器的姿态机动控制问题, 其中晃动液体用黏性力矩球摆模型代替, 悬臂梁附件用若干集中质量代替. 用动量矩定理和Lagrange方程分别推导得到航天器主刚体、等效球摆、等效集中质量的动力学方程, 所用反馈控制律包含了与动量轮角加速度密切相关的权重因子, 利用系统初、终状态和到达最终姿态所需时间解析确定此权重因子. 同时利用Lyapunov稳定性理论得到了实现最终姿态机动的稳定性判据. 数值仿真表明所用控制律的有效性, 分析附件的相对主刚体平面的转角、相对系统质心的高度、长度、刚度、质量、阻尼系数和到达最终姿态所需时间等因素对控制过程中航天器剩余章动角的影响大小.      

Exact solution of free vibration of adjacent buildings interconnected by visco-elastic dampers

With consideration of a high-rise coupled building system, a flexible beams-based analytical model is setup to characterize the dynamic behavior of the system. The general motion equation for the two beams interconnected by multiple viscous/visco-elastic dampers is rewritten into a non-dimensional form to identify the minimal set of parameters governing the dynamic characteristics. The corresponding exact solution suitable for arbitrary boundary conditions is presented. Furthermore, the methodology for computing the coefficients of the modal shape function is proposed. As an example, the explicit expression of the modal shape function is derived, provided only one damper is adopted to connect the adjacent buildings. Finally, to validate the proposed methodologies, three case studies are performed, in which the existence of the overdamping and the optimal damping coefficient are revealed. In the case of using one damper in connecting two similar buildings, the estimating equations for the first modal damping ratio are formulated.     

A novel eight-legged vibration isolation platform with dual-pyramid-shape struts

An eight-legged vibration isolation platform (ELVIP) with dual-pyramid-shape (DPS) struts is proposed in this paper to achieve six-degree-of-freedom vibration isolation. The DPS strut has higher static stiffness and lower dynamic stiffness than the equivalent linear strut due to geometric nonlinearity, and therefore the ELVIP with DPS struts possesses high-static–low-dynamic-stiffness characteristic which is desirable for widening the frequency range of isolation. The layout of the ELVIP legs is conducive to vibration decoupling and has higher reliability than six-legged platforms. Firstly, the stiffness characteristics of the DPS strut are derived; then the dynamic model of the ELVIP with DPS struts is established and the steady-state responses are obtained analytically; finally, the isolation performance is studied and the effects of damping and excitation amplitudes are investigated. It is shown that the ELVIP with DPS struts, as a passive approach, can achieve good isolation performance in all six directions with a small static deflection.

     

Dynamic analysis of a system of hinge-connected rigid bodies with nonrigid appendages

Equations of motion are derived for use in simulating a spacecraft or other complex electromechanical system amenable to idealization as a set of hinge-connected rigid bodies of tree topology, with rigid axisymmetric rotors and nonrigid appendages attached to each rigid body in the set. In conjunction with a previously published companion paper on finite-element appendage vibration equations, this paper provides a complete minimum-dimension formulation suitable for generic programming for digital computer numerical integration.     

Analytical solution of attitude motion for spacecraft with a slewing appendage

This paper investigates pitch motion and in orbital plane elastic vibration of a spacecraft with a flexible beam type appendage undergoing prescribed slew maneuver. The governing equations are transformed into a standard quasi-linear form, and then solved by Butenin's variation of parameters approach. Validity of the analytical solutions is assessed over a range of system parameters and initial conditions by comparing them with the results of numerical integration. The results show that they are very good approximations and provide extensive insight into the dynamical response of the system.     

万有引力场中带挠性板非轴对称航天器的姿态稳定性

本文讨论由非轴对称主刚体和矩形挠性板组成的航天器在万有引力场中的姿态运动。利用Ｇａｌｅｒｋｉｎ方法对动力学方程离散化，并利用Ｋｅｌｖｉｎ－Ｔａｉｔ－Ｃｈｅｔａｙｅｖ定理判断航天器在轨道坐标系内相对平衡的稳定性。导出适用于任意阶模态的解析形式稳定性充分条件。     

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.     

中心舱体与薄壁梁刚-柔耦合系统的热弹性-结构动力学分析

空间柔性结构受太阳热流冲击而诱发的振动是导致航天器失效的典型模式之一,准确预测结构热致振动的响应及稳定性是卫星设计的基础。针对常见的中心舱体与附属薄壁杆件组成的空间结构,提出了考虑刚-柔耦合、耦合热弹性和耦合热-结构三重耦合效应的热致振动分析理论模型。其中,刚-柔耦合是指舱体姿态角、顶端集中质量转动与柔性附件运动的耦合;耦合热弹性是指应变率与温度场的耦合;耦合热-结构是指舱体转动及结构变形与薄壁杆件吸收太阳热流的耦合。基于热弹性理论和Lagrange方程,推导了传热和运动的耦合方程;采用Laplace变换方法并使用Routh-Hurwitz稳定判据推导了稳定性边界方程。结果表明,该模型能够更为准确的给出热致振动响应及稳定性预测。     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

Attitude maneuver of liquid-filled spacecraft with a flexible appendage by momentum wheel

Attitude maneuver of liquid-filled spacecraft with an appendage as a cantilever beam by momentum wheel is studied.The dynamic equations are derived by conservation of angular momentum and force equilibrium principle.A feedback control strategy of the momentum wheel is applied for the attitude maneuver.The residual nutation of the spacecraft in maneuver process changes with some chosen parameters,such as steady state time,locations of the liquid container and the appendage,and appendage parameters.The results indicate that locations in the second and fourth quadrants of the body-fixed coordinate system and the second quadrant of the wall of the main body are better choices for placing the liquid containers and the appendage than other locations if they can be placed randomly.Higher density and thicker cross section are better for lowering the residual nutation if they can be changed.Light appendage can be modeled as a rigid body,which results in a larger residual nutation than a flexible model though.The residual nutation decreases with increasing absolute value of the initial sloshing angular height.     

多种非线性力作用下不平衡弹性转子的分岔特性

研究了4自由度不平衡弹性转子在非线性油膜力、非线性内阻力和非线性弹性力联合作用下的动力学特性。结果表明，当只有非线性油膜力作用时，转子只存在由于油膜失稳而导致的倍周期分岔。而当非线性油膜力与非线性内阻力共同作用时，在油膜失稳后，转子产生低频振动。转速继续增加，还会诱发内阻失稳，产生概周期运动。在倍周期分岔中，存在分岔激变现象。本文发现的由于油膜涡动而导致的内阻失稳(概周期运动)是一种未见报道的转子失稳模式(组合失稳)，它与油膜失稳(倍周期运动)一起可作为转子故障诊断的典型失稳模式。     

Singularly perturbed feedback linearization with linear attitude deviation dynamics realization

A new approach for feedback linearization of attitude dynamics for rigid gas jet-actuated spacecraft control is introduced. The approach is aimed at providing global feedback linearization of the spacecraft dynamics while realizing a prescribed linear attitude deviation dynamics. The methodology is based on nonuniqueness representation of underdetermined linear algebraic equations solution via nullspace parametrization using generalized inversion. The procedure is to prespecify a stable second-order linear time-invariant differential equation in a norm measure of the spacecraft attitude variables deviations from their desired values. The evaluation of this equation along the trajectories defined by the spacecraft equations of motion yields a linear relation in the control variables. These control variables can be solved by utilizing the Moore–Penrose generalized inverse of the involved controls coefficient row vector. The resulting control law consists of auxiliary and particular parts, residing in the nullspace of the controls coefficient and the range space of its generalized inverse, respectively. The free null-control vector in the auxiliary part is projected onto the controls coefficient nullspace by a nullprojection matrix, and is designed to yield exponentially stable spacecraft internal dynamics, and singularly perturbed feedback linearization of the spacecraft attitude dynamics. The feedback control design utilizes the concept of damped generalized inverse to limit the growth of the Moore–Penrose generalized inverse, in addition to the concepts of singularly perturbed controls coefficient nullprojection and damped controls coefficient nullprojection to disencumber the nullprojection matrix from its rank deficiency, and to enhance the closed loop control system performance. The methodology yields desired linear attitude deviation dynamics realization with globally uniformly ultimately bounded trajectory tracking errors, and reveals a tradeoff between trajectory tracking accuracy and damped generalized inverse stability. The paper bridges a gap between the nonlinear control problem applied to spacecraft dynamics and some of the basic generalized inversion-related analytical dynamics principles.     

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．