Chaos in the Reorientation Process of a Dual-Spin Spacecraft With Time-Dependent Moments of Inertia

Abstract

We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it is shown that in the absence of external torques and spinning rotors the system possesses chaotic behavior in the sense that it exhibits Smale's horseshoes. We prove this statement by means of the Melnikov method.The presence of chaotic behavior results in a random spin-up operation. This randomness is visualized by means of maps of the initial conditions with final nutation angle close to zero. This phenomenon is well described by a suitable parameter that measures the amount of randomness of the process. Finally, we relate this parameter with the Melnikov function in the absence of the spinning rotor and with the presence of subharmonic resonances.

Exact solution of a triaxial gyrostat with one rotor

“…(see, for instance, Cochran et al 1982;Hall 1995a;Hall and Rand 1994;Elipe and Lanchares 1997b;Lanchares et al 1998;Iñarrea and Lanchares 2000;Vera and Vigueras 2006 and also Hughes 1986 for further references).…”

Chaotic dynamics of an unbalanced gyrostat

“…We first consider qualitative forms 1 in zone 1 and qualitative form 5 in zone 3 (Table 2), i.e., the "traditional" forms of the phase space of a free rigid body and a balanced gyrostat. 2,7,17 For form 1, the quadratic polynomial f + (L) has the multiple root L = L s ; here and below, L s refers to the value of L in the corresponding case of the saddle point and the constant h s (see Table 2). The polynomial f -(L) has two different roots L 1 and L 2 (L 2 < L 1 ), which are easily found: (3.5) For form 5, the polynomial f -(L) has the multiple root L = L s , and the polynomial f + (L) has two different roots: (3.6) Here and in relation (3.5) we have Note that the roots L 1 and L 2 (L 2 < L 1 ) correspond to the "extremum" points of the separatrices (Fig.…”

Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field

“…In Astrodynamics, gyrostats play an essential role, since they are used for controlling the attitude dynamics of a spacecraft and for stabilizing their rotations. See, for instance, Cochran [8], Hall [15,16,17], Elipe and coworkers [10,11,12,13,20,26], Vera [36], Aslanov [4,5] and also Hughes [19] for further references.…”