Solution set on the natural satellite formation orbits under first-order earth＇s non-spherical perturbation

Using the reference orbital element approach, the precise governing equations for the relative motion of formation flight are formulated. A number of ideal formations with respect to an elliptic orbit can be designed based on the relative motion analysis from the equations. The features of the oscillating reference orbital elements are studied by using the perturbation theory. The changes in the relative orbit under perturbation are divided into three categories, termed scale enlargement, drift and distortion respectively. By properly choosing the initial mean orbital elements for the leader and follower satellites, the deviations from originally regular formation orbit caused by the perturbation can be suppressed. Thereby the natural formation is set up. It behaves either like non-disturbed or need little control to maintain. The presented reference orbital element approach highlights the kinematics properties of the relative motion and is convenient to incorporate the results of perturbation analysis on orbital elements. This method of formation design has advantages over other methods in seeking natural formation and in initializing formation.     

USEFUL RELATIVE MOTION DESCRIPTION METHOD FOR PERTURBATIONS ANALYSIS IN SATELLITE FORMATION FLYING

Nomenclature OXYZEarth’sequatorialinertialreferenceframeωArgumentofperigee SlxyzLeadingsatelliteorbitframeMMeananomaly SfxyzFollowingsatelliteorbitframefTrueanomalyaSemi majoraxisθ=ω fArgumentoflatitude eEccentricitynMeanmotion iOrbitinclinationrSatel…     

一类刚-梁耦合系统平面稳态运动的稳定性

研究了中心力场中的一类刚-弹耦合系统的平面运动动力学，模型是带有一悬臂 梁的刚体. 综合考虑了系统轨道运动与姿态运动，在Lagrange力学体系下给出了系统的运 动方程，在保守系统和考虑梁的材料黏滞阻尼两种情况下，利用能量-动量方法给出了一类 相对平衡点稳定性的充分条件.     

Review of station keeping strategies for elliptically orbiting constellations in along-track formation

Three techniques for station keeping an orbiting constellation of satellites in an elliptical orbit are developed: (1) based on an application of the linearized Tschauner–Hempel (TH) equations for the motion of a daughter satellite relative to a reference (mother) satellite together with the linear quadratic regulator (LQR) control strategy which can be used in a piecewise adaptive manner; (2) since the mathematical model is inherently nonlinear and time varying, a control law based on a non-linear Lyapunov function is applied to daughter satellites’ osculating orbital elements; (3) by carefully selecting relative orbital design parameters so that the relative secular drifts due to the non-spherical Earth’s perturbation in the longitude of the ascending node, the argument of perigee and mean anomaly could vanish or be constrained to a desired value.     

A boundary element method and spectral analysis model for small‐amplitude viscous fluid sloshing in couple with structural vibrations

A boundary element method (BEM) is presented for the coupled motion analysis of structural vibrations with small‐amplitude fluid sloshing in two‐dimensional space. The linearized Navier–Stokes equations are considered in the frequency domain and transformed into a Laplace equation and a Helmholtz equation with pure imaginary constant. An appropriate fundamental solution for the Helmholtz equation is provided. The conditions of zero stress are imposed on the free surface, and non‐slip conditions of fluid particles are imposed on the walls of the container. For rigid motion models, the expressions for added mass and added damping to the structural motion equations are obtained. Numerical examples are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Dynamics of a satellite carrying a point mass moving about it

We study the dynamics of a complex system consisting of a solid and a mass point moving according to a prescribed law along a curve rigidly fixed to the body. The motion occurs in a central Newtonian gravitational field. It is assumed that the orbit of the system center of mass is an ellipse of arbitrary eccentricity.We obtain equations that describe the motion of the carrier (satellite) about its center of mass. In the case of a circular orbit, we present conditions that should be imposed on the law of the relative motion of the mass point carried by the satellite so that the latter preserves a constant attitude with respect to the orbital coordinate system. In the case of a dynamically symmetric satellite, we consider the problem of existence of stationary and nearly stationary rotations for the case in which the carried point moves along the satellite symmetry axis.We consider several problems of dynamics of the satellite plane motion about its center of mass in an elliptic orbit of arbitrary eccentricity. In particular, we present the law of motion of the carried point in the case without eccentricity oscillations and study the stability of the satellite permanent attitude with respect to the orbital coordinate system.     

Stability and control of tethered satellite with chemical propulsion in orbital plane

The tethered satellite with chemical propulsion has broad application prospects in the disposal of abandoned satellites, the orbital rescue of spacecrafts, and the transportation of space supplies, which is completely different from the traditional applications of tethered satellites. Therefore, new research on its dynamics, stability, and control becomes useful and interesting. In this article, based on a dumbbell model of tethered satellite, the dynamics equations of tethered system in orbital maneuvering are established. Furthermore, according to the definitions of transversal and radial propulsive coefficients, analytical solutions of the equilibrium position for librational angle are derived during maneuvering in orbital plane; meanwhile, the effects of propulsive coefficients on librational stability are analyzed, which provides a basis for a selection of expected attitude trajectory. Then, a method of hierarchical sliding-mode tension control is presented to track the expected in-plane angle. This method can address the underactuated problem of tethered systems without either complex coordinate transformation for the system state model or constraint equation restrictions. During orbital flight, in-plane and out-of-plane angles are decoupled, so the tether tension control cannot be conducted to inhibit the out-of-plane angle. To solve this problem, the binormal component of thrust acceleration normal to the orbital plane is adopted as a control variable, and a feedback linearization-based thrust controller is designed to damp out the out-of-plane angle. Afterwards, orbital transfer cases between two circular orbits are studied to demonstrate the effectiveness of the tethered satellite with chemical propulsion. Numerical simulation results indicate that the stability of librational angles has a close relation to propulsive coefficients, and distributions of stable centers and unstable saddle points are totally different on both sides of bifurcation point. In addition, tracking control requirements for tethered satellite are guaranteed by designed controllers, which ensure flight safety in orbital maneuvering.     

卫星编队飞行中相对轨道的J2摄动分析

详细分析了$J_{2}$摄动对编队卫星相对轨道构形的影响. $J_{2}$ 摄动对相对轨道的影响分为相对轨道构形的漂移、相对轨道平面的章动和进动. 首先, 分析了相对轨道构形漂移速度、章动角速度和进动角速度的一阶近似表达式的数量级及其影 响因素. 其次,给出一个准则,来判断同一相对轨道的漂移和转动之间的关系. 最后, 利用该准则,分析了主、从星的轨道根数差对相对轨道的漂移和转动的影响.     

Hohmann转移释放绳系卫星的方法研究

借助Ｈｏｈｍａｎｎ轨道转移的思想，研究一种绳系卫星释放方法并进行动力学分析．从轨道方程和Ｋｅ ｐｌｅｒ方程导出轨道转移条件．由数值计算对此方法加以说明．     

航天器集群编队最优单脉冲机动

对航天器集群编队最优单脉冲机动问题进行了研究. 针对不同的任务约束,基于非线性相对运动的周期性条件,以解析的思路分别研究了机动时刻给定和机动时刻未定情况下集群编队的最优单脉冲机动问题. 对于机动时刻给定的情况,从高斯变分方程和基于能量匹配条件的拉格朗日乘子法两个角度分别进行了探讨,将问题转化为对一元二次方程求极值或对一个单零点非线性方程求根;对于机动时刻未定的情况,将问题转化为对一个多零点非线性方程求根,通过傅里叶-贝塞尔级数展开可以得到任意高阶近似解. 对于每种情况,推导得到二范数意义下能量最省对应的最优参考长半轴,以及所施加的最优速度脉冲. 数值仿真验证了本文方法的正确性,并对仿真结果进行了解释和分析.      

Long-term evolution of mid-altitude quasi-satellite orbits

Quasi-satellite orbits are of great interest for the exploration of planetary moons because of their dynamical features and close proximity with respect to the surface of scientifically relevant objects like Phobos and Deimos. This paper explores the equations of the elliptical Hill problem, offering a new analytical insight into the long-term evolution of mid-altitude quasi-satellite orbits. Our developments are based on the Yamanaka–Ankersen solution of the Tschauner–Hempel equations and capture the effects of the secondary’s gravity and orbital eccentricity on the shape and orientation of near-equatorial retrograde relative trajectories. The analytical solution of the in-plane and out-of-plane components of the secular motion is achieved by averaging over the relative longitude of a spacecraft as seen from the co-rotating frame of the two primaries. Developments are validated against the numerical integration of quasi-periodic trajectories that densely cover the surface of three-dimensional invariant tori. This analysis confirms the stable nature of quasi-satellite orbits and provides new tools for future spacecraft missions such as the Martian Moons eXploration envisaged by JAXA.

     

一种基于直接计算高阶奇异积分的断裂力学双边界积分方程分析法

相对于有限元法,边界单元法在求解断裂问题上有着独特的优势,现有的边界单元法中主要有子区域法和双边界积分方程法.采用一种改进的双边界积分方程法求解二维、三维断裂问题的应力强度因子,对非裂纹边界采用传统的位移边界积分方程,只需对裂纹面中的一面采用面力边界积分方程,并以裂纹间断位移为未知量直接用于计算应力强度因子.采用一种高阶奇异积分的直接法计算面力边界积分方程中的超强奇异积分;对于裂纹尖端单元,提供了三种不同形式的间断位移插值函数,采用两点公式计算应力强度因子.给出了多个具体的算例,与现存的精确解或参考解对比,可得到高精度的计算结果.      

挠性联结双体航天器的稳定性与分岔

研究圆轨道内受万有引力矩作用的挠性联结双体航天器在轨道平面内的姿态运动,讨论其相对轨道坐标系统平衡状态的稳定性与分岔。提出判平衡方程非平凡解存在性的几何方法,并应用Ｌｉａｐｕｎｏｖ直接法、Ｌｉａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ约化方法和奇异性理论导出解析形式的稳定性与分岔的充要条件,从而对系统的全局运动性态作出定性的描述。     

Dynamic analysis of a tethered satellite system with a moving mass

This paper presents a dynamic analysis of a tethered satellite system with a moving mass. A dynamic model with four degrees of freedom, i.e., a two-piece dumbbell model, is established for tethered satellites conveying a mass between them along the tether length. This model includes two satellites and a moving mass, treated as particles in a single orbital plane, which are connected by massless, straight tethers. The equations of motion are derived by using Lagrange’s equations. From the equations of motion, the dynamic response of the system when the moving mass travels along the tether connecting the two satellites is computed and analyzed. We investigate the global tendencies of the libration angle difference (between the two sections of tether) with respect to the changes in the system parameters, such as the initial libration angle, size (i.e. mass) of the moving mass, velocity of the moving mass, and tether length. We also present an elliptic orbit case and show that the libration angles and their difference increase as orbital eccentricity increases. Finally, our results show that a one-piece dumbbell model is qualitatively valid for studying the system under certain conditions, such as when the initial libration angles, moving mass velocity, and moving mass size are small, the tether length is large, and the mass ratio of the two satellites is large.     

Segregated finite element algorithms for the numerical solution of large-scale incompressible flow problems

This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solution algorithms for the simulation of large-scale flow problems. Two main steps were taken towards achieving this goal. The first step was to employ segregated solution schemes as opposed to the fully coupled solution approach traditionally used in many finite element solution algorithms. The second step was to replace the direct Gaussian elimination linear equation solvers used in the first step with iterative solvers of the conjugate gradient and conjugate residual type. The three segregated solution algorithms developed in step one are first presented and their integrity and relative performance demonstrated by way of a few examples. Next, the four types of iterative solvers (i.e. two options for solving the symmetric pressure type equations and two options for solving the non-symmetric advection–diffusion type equations resulting from the segregated algorithms) together with the two preconditioning strategies employed in our study are presented. Finally, using examples of practical relevance the paper documents the large gains which result in computational efficiency, over fully coupled solution algorithms, as each of the above two main steps are introduced. It is shown that these gains become increasingly more dramatic as the complexity and size of the problem is increased.     

Coupled Groundwater Flow and Transport in Porous Media. A Conservative or Non-conservative Form?

A new formulation for the modeling of density coupled flow and transport in porous media is presented. This formulation is based on the development of the mass balance equation by using the conservative form. The system of equations obtained by coupling the flow and transport equations using a state equation is solved by a combination of the mixed hybrid finite element method (MHFEM) and the discontinuous finite element method (DFEM). The former is applied in order to solve the flow equation and the dispersive part of the transport equation, whilst the latter is used to solve the advective part of the transport equation. Although the advantages of the MHFEM are known (efficiency calculation of velocity field and continuity of fluxes from one element to an adjacent one), its application in a classical development form (volumetric fluxes as unknowns) leads to the non-conservative version of the mass balance equation. The associated matrix of the system of equations obtained by hybridization is positive definite but non-symmetrical. By using a new approach (mass fluxes as unknowns) the conservative form of the continuity equation is preserved and the associated matrix of the system of equations obtained by hybridization becomes symmetrical. When applied to Elder's problem involving a strong density contrast, this new approach, with a lower calculation cost, leads to similar or identical results to those found in the specialized literature. The comparison between the conservative and non-conservative formulations solved with the same MHFEM and DFEM combination emphasizes the rigor and the pertinence of this new approach. Furthermore, we show the existence of a limit refinement defining the stability of the numerical solution for Elder's problem.     

Direct numerical simulation of fluid flow laden with many particles

Two-dimensional channel flow of fluid laden with many particles is studied by direct numerical simulation using the Navier–Stokes equation coupled with the equation of motion for respective particles. Fractional four-step method with Crank–Nicolson scheme and ALE technique is adopted for P2P1 mixed finite element formulation of the governing equations for fluid motion. The motion and distribution of particles in the fluid is virtually described and the calculated relative viscosity is compared with previous results within the limits of possibility. The effect of the ratio of channel gap to particle diameter on the relative viscosity and the tubular pinch effect are also delineated.     

A boundary element method for small-amplitude viscous fluid sloshing in couple with structural vibration

A boundary element method is presented for the coupled motion analysis of structural vibration with small-amplitude fluid sloshing in two-dimensional space. The linearized Navier-Stokes equations are considered in frequency domain and transformed into boundary integral equations. An appropriate fundamental solution for the Helmholtz equation with pure imaginary constant is found. The condition of zero-stress is imposed on the free surface, and non-slip condition of fluid particles is imposed on the walls of the container. For rigid motion models, the expressions for added mass and added damping to the structural motion equations are obtained. Some typical numerical examples are presented.     

Foucault Pendulum-like problems: A tensorial approach

The paper offers a comprehensive study of the motion in a central force field with respect to a rotating non-inertial reference frame. It is called Foucault Pendulum-like motion and it is a generalization of a classic Theoretical Mechanics problem. A closed form vectorial solution to this famous problem is presented. The vectorial time-explicit solution for the classic Foucault Pendulum problem is obtained as a particular case of the considerations made in the present approach. New interesting conservation laws for the Foucault Pendulum-like motion are deduced by using simple differential and vectorial computations. They help to visualize the shape of the trajectories. Exact vectorial expressions for the law of motion and the velocity are also offered. The case of the driven Foucault Pendulum is also analyzed, and a closed form solution is deduced based on the general considerations. In the end, an new tensorial prime integral for the Foucault Pendulum problem is offered. It helps to reveal in a concise form, within a single entity, all the scalar and vectorial conservation laws for the Foucault Pendulum motion.Two important engineering applications to this approach are presented: the motion of a satellite with respect to a rotating reference frame and the Keplerian relative orbital motion. The latter has a great importance in modeling the problems concerning satellite formation flying, satellite constellations and space terminal rendezvous. The classic problem of the harmonic oscillator in an electromagnetic field is also solved by using the instruments presented in this paper.     

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.