By Yanzhu Liu
Attitude dynamics is the theoretical foundation of angle regulate of spacecrafts in aerospace engineering. With the advance of nonlinear dynamics, chaos in spacecraft angle dynamics has drawn nice cognizance because the 1990's. the matter of the predictability and controllability of the chaotic perspective movement of a spacecraft has a pragmatic importance in astronautic technological know-how. This booklet goals to summarize easy recommendations, major methods, and up to date development during this region. It makes a speciality of the examine paintings of the writer and different chinese language scientists during this box, supplying new equipment and viewpoints within the research of spacecraft angle movement, in addition to new mathematical types, with sure engineering backgrounds, for extra analysis.
Professor Yanzhu Liu was once the Director of the Institute of Engineering Mechanics, Shanghai Jiao Tong collage, China. Dr. Liqun Chen is a Professor on the division of Mechanics, Shanghai college, China.
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Additional info for Chaos in Attitude Dynamics of Spacecraft
12. The phase trajectories have singularities at ϕ = nʌ 2 (n = 0,1,2," ) . The types of the singularities depend on the sign of B − A. In the case of B > A, the singularity is a center when n is even, or a saddle when n is odd. The result is opposite when B < A. 17) requires that the principal axis directed to the Earth should correspond to the minimal inertia moment of the satellite. 18) an analytical solution of Eq. 22) where Tc = 2ʌ a 3 μ is the period of circular orbit and K (k ) is the first complete elliptic integral.
23) In more general case of an elliptic orbit, the parameters r and ων in Eq. 13) are functions of the true anomaly ν as shown in Eqs. 24) To replace the argument t by ν in Eq. 25) Substitution of Eqs. 25) into Eqs. 26) where κ = 3( B − A) C . Zlatoustov et al. (1964) analyzed the oscillation determined by nonlinear Eq. 26) with periodic coefficients and provided a stability diagram in the 21 Chaos in Attitude Dynamics of Spacecraft parameter plane (e, κ ) (Fig. 13) . It was shown that when e = 0 the entire domain of κ > 0 is stable, and afterwards the stable area is reduced with increasing eccentricity.
16) Direct computation of the first two terms on the right hand of Eq. 19) Using Eqs. 22) Based on Eqs. 24) The existence of simple zeros of the Melnikov function %(t0 ) indicates that the displacement d (t0, t0) vanishes. At a simple zero tz, %(tz) d N (t0 , t0 ) = 0, but d %(tz)/dt0 ≠ 0. In this case, the stable and unstable manifolds are to intersect transversely to form a transversal homoclinic point. 25) If the unperturbed system is Hamiltonian, then tr(D f) = 0. 4 Interruptions The Melnikov theory is of considerable significance because it can be applied to check in specific systems whether the stable and unstable manifolds intersect transversely or not, by a direct calculation of the approximate distance between these manifolds.