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On the periodic solutions of slowly spinning gravity gradient system

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Abstract

The paper emphasizes the importance of periodic solutions in the dynamic stability study of an axi-symmetric satellite in presence of gravity gradient torques. Initial conditions for periodic solutions are presented over a range of system parameters for motion in circular and elliptic orbits. The variational stability of periodic solutions is examined using an extension of the Floquet theory to a fourth order system.

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Abbreviations

H :

Hamiltonian

I :

I x /I Y

I x ,I y ,I z :

principal moments of inertia

L :

Lagrangian

0:

center of force

P :

pericenter

R :

distance between satellite center of mass and center of force

S :

satellite center of mass

Tr :

trace of a matrix

e :

orbit eccentricity

t :

time

x, y, z :

principal body coordinates with origin atS, x-along the axis of symmetry

x 0,y 0,z 0 :

orbital reference frame,x 0-normal to orbital plane,Y 0,z 0 along local vertical and horizontal, respectively (Figure 1)

\(\left. \begin{gathered} x_1 ,y_1 ,z_1 ; \hfill \\ x_2 ,y_2 ,z_2 \hfill \\ \end{gathered} \right\}\) :

intermediate coordinate systems

x', y', z' :

inertial coordinate system

Φ:

solution basis

Φi :

ith solution vector

γ, β, α:

modified Eulerian rotations in given order aboutz 0,y 1,x 2 axes, respectively (Figure 1)

θ:

true anomaly

λ i :

ith characteristic multiplier

μ:

gravitational constant

σ:

spin parameter,\([\dot \alpha /\dot \theta ]_{\theta = \beta = \gamma = 0}\)

τ:

period of the motion

i :

initial value

p :

periodic solution

v :

perturbation in variational analysis

References

  • Baker, R. M. L., Jr.: 1960,ARS J. 30 124–6.

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  • Brereton, R. C. and Modi, V. J.: 1967,Proceedings of the XVIIth International Astronautical Congress, Gordon and Breach Inc., New York, 179–92.

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  • Modi, V. J. and Brereton, R. C.: 1969a,AIAA J. 7, 1217–25.

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  • Modi, V. J. and Brereton, R. C.: 1969b,AIAA J. 7, 1465–8.

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  • Modi, V. J. and Neilson, J. E.: Presented at theXXth Congress of the International Astronautical Federation, Mar del Plata, Argentina, October, 1969 to be published in its proceedings.

  • Zlatousov, V. A., Okhotsimsky, D. E., Sarychev, V. A., and Torzhevsky, A. P.: 1964, in H. Görtler (ed.),Proceedings of the XIth International Congress of Applied Mechanics, Springer-Verlag, Berlin, 436–9.

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Author information

Authors and Affiliations

  1. The University of British Columbia, Vancouver, B.C., Canada

    V. J. Modi (Professor) & J. E. Neilson (Graduate Research Assistant)

Authors
  1. V. J. Modi
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  2. J. E. Neilson
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Additional information

Dots and primes indicate differentiation with respect tot and θ, respectively.

The investigation was supported by the National Research Council of Canada, Grant No. A-2181.

at present Scientific Officer, Defence Research Board.

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Modi, V.J., Neilson, J.E. On the periodic solutions of slowly spinning gravity gradient system. Celestial Mechanics 5, 126–143 (1972). https://doi.org/10.1007/BF01229517

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  • DOI: https://doi.org/10.1007/BF01229517

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