Chaotic Attitude Motion of Gyrostat Satellite via Melnikov Method

Abstract

In this paper Deprit's variables are used to describe the Hamiltonian equations for attitude motions of a gyrostat satellite spinning about arbitrarily body-fixed axes. The Hamiltonian equations for the attitude motions of the gyrostat satellite in terms of the Deprit's variables and under small viscous damping and nonautonomous perturbations are suitable for the employment of the Melnikov's integral. The torque-free homoclinic orbits to the symmetric Kelvin gyrostat are derived by means of the elliptic function integral theory. With the help of residue theory of complex functions, the Melnikov's integral is utilized to analytically study the criterion for chaotic oscillations of the attitude motions of the symmetric Kelvin gyrostat under small, damping and periodic external disturbing torques. The Melnikov's integral yields an analytical criterion for the chaotic oscillations of the attitude motions in the form of an inequality that gives a necessary condition for chaotic dynamics in terms of the physical parameters. The dependence of the onset of homoclinic orbits on quantities such as body shapes, the initial conditions of the angular velocities or the two constants of motions of the torque-free gyrostat satellite is investigated in details. The dependence of the onset of chaos on quantities such as the amplitudes of the external excitation and the damping coefficients' matrix is discussed. The bifurcation curves based upon the Melnikov's integral are computed by using the combined parameters versus the frequency of the external excitation. The theoretical criterion agrees with the result of the numerical simulation of the gyrostat satellite by using the fourth-order Runge-Kutta integration algorithm. The numerical solutions show that the motions of the perturbed symmetric gyrostat satellite possess a lot of "random" characteristic associated with a nonperiodic solution.

Nonlinear Oscillations of a Suspended Gyrostat

“…The hyperbolic fixed points of the torque-free symmetric gyrostat may be computed according to the formulas established in Appendix C of Kuang et al (2001b). The rigorous algorithms for the construction of homoclinic orbits for the symmetric gyrostat under torque-free motions can be obtained from Appendix E of Kuang et al (2001b). The detailed procedures for acquiring the homoclinic orbits for the motion of the torque-free symmetric gyrostat are omitted here for brevity.…”

Spatial chaos of 3‐D elastica with the Kirchhoff gyrostat analogy using Melnikov integrals

“…The utilization of the Melnikov integrals of Wiggins and Shaw [2] for detecting the transversal intersections between the stable and unstable manifolds of a saddle point of the Poincare map of the disturbed gyrostat under the action of small periodic torques is found in Tong and Tabarrok [24], Tong et al [25], and Kuang et al [26][27][28][29]. Kuang et al [26][27][28][29] established a special version of the Melnikov integrals for the dissipative gyrostat subject to small applied torques. Based on that version of the Melnikov integrals for the forced rotations of the gyrostat, they studied the chaotic attitude dynamics for both asymmetrical and symmetrical dissipative satellites.…”

“…Based on that version of the Melnikov integrals for the forced rotations of the gyrostat, they studied the chaotic attitude dynamics for both asymmetrical and symmetrical dissipative satellites. The other new contributions in Kuang et al [27][28][29] were the investigation of the homoclinic solutions of the rotational motions of the torque-free gyrostat with different configurations. According to Rumyantsev [30], when the motions of the contained liquid were irrotational, the attitude dynamics equations of a rotational liquid-filled body are identical to those of a solid body joining a rotating gyroscope.…”

Ricci-based chaos analysis for roto-translatory motion of a Kelvin-type gyrostat satellite

“…Study of chaos in GS was first introduced by the work of Tong et al 4 All the research works on chaotic motion of a GS are concerned only with the attitude chaotic dynamics. 5,6 The Melnikov method mathematically analyses chaos in the Hamiltonian system. However, the exact solution of the heteroclinic orbits is the main drawback of this method.…”