Some approximations for the dynamics of spacecraft tethers

The formulation of approximate solutions to equations that embody the dominant characteristics of the orbital motion of a two-satellite tethered system are studied. The orbital motion of the system is viewed as perturbed two-body motion, and a restricted tether problem is obtained by neglecting librational motion. An exact analytical solution to this restricted problem in terms of elliptic functions is presented. An approximate solution to the restricted tether problem obtained by applying the method of averaging is also provided. An approximation for small-amplitude librational motion is formulated, whose solution is based on methods for solving equations with variable coef cients. The analytical solutions are good approximations to the orbital motion of the tetherperturbed satellite and the librational motion of the system when the libration is small. The restricted tether motion approximation is then utilized to solve the identi cation problem of a tethered satellite system. Nomenclature a ¤ = apparent semimajor axis e ¤ = apparent eccentricity F = incomplete elliptic integral of the rst kind h = nondimensionalangular momentum K = complete elliptic integral of the rst kind K 0 = associate complete elliptic integral of the rst kind k = modulus of Jacobian elliptic functions and integral k 0 = complementary modulus of Jacobian elliptic functions and integral m; m p = masses of satellites q = Jacobi's nome r = nondimensionalradial distance, radial distance/r E r E = radius of the Earth, 6,378,000 m sn = Jacobian elliptic function t = time t ¤ = nondimensionaltime, t p .¹=r 3 E / D t £ 0:0012394 u = 1=r X = state vector ®;¯;°; » = variable coef cients "= tether parameter, m p ½=.m C m p /r E µ = orbital angle (true anomaly) µ 2 = out-of-plane libration angle µ 3 = in-plane libration angle ¹ = gravitational constant of the Earth ½ = tether length '; ' ¤ = amplitudes ! = orbital frequency

show abstract

“…(24)] to the restricted tether problem and by numerically integrating Eqs. (4)(5)(6). These results are shown in Fig.…”

Section: Approximate Solution To the Restricted Tether Problemsupporting

confidence: 67%

“…Equations for the restricted tether motion are obtained by neglecting µ 3 and its derivatives in Eqs. (4) and (5):…”

Section: Restricted Tether Orbital Motionmentioning

confidence: 99%

See 1 more Smart Citation

Approximate Solutions for Tethered Satellite Motion

Cho

Lovell

Cochran

et al. 2001

Journal of Guidance, Control, and Dynamics

View full text Add to dashboard Cite

show abstract

Nonstationary nonplanar free motions of an orbiting string with multiple internal resonances

1996

View full text Add to dashboard Cite

The paper discusses the nonlinear free dynamics of an orbiting string satellite system. The focus is on the transversal oscillations, which are governed by two partial integro-differential equations in two transversal displacement components with quadratic nonlinearities. The system is weakly nonlinear but in practice works in conditions of simultaneous internal resonance. The investigation focuses on nonstationary motions arising from perturbed steady-state nonplanar oscillations. A four-mode model is used to study the problem: two modes are necessary to describe the basic oscillation and at least two other modes are involved in the resonance phenomena when the motion is perturbed. The multiple time scales method is used to obtain the equations that govern the amplitude and phase modulations. For increasing levels of system energy, fundamental and bifurcated paths of fixed points of the seven first-order differential equations are determined and their stability is investigated. The trajectories of motion of periodically modulated amplitude solutions and their stability are also studied. A model with a higher number of modes is used to evaluate the accuracy of the stability analysis of two-mode nonplanar oscillations perturbed by a two-mode disturbance

show abstract