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On the periodic solutions and resonance of spinning satellites in near-circular orbits

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Abstract

Attitude motion of spinning axisymmetric satellites in presence of gravity-gradient and solar radiation pressure torques is studied analytically. The approximate closed-form solution developed for the nonlinear, nonautonomous, coupled fourth-order system proves to be an excellent tool in locating periodic solutions of the system in both circular and noncircular orbits. The variational stability of the periodic motion is examined using the Floquet theory. The resonance analysis suggests the existence of critical combinations of system parameters leading to large amplitude oscillations.

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Abbreviations

A 0 :

2Ce/n 21 , Equation (3a)

A 1 :

C(1+3e)(1−n 22 −l 1)/(1−k 21 )(1−k 22 ), Equation (3a)

A 2 :

2Ce(n 22 +2l 1−4)/(4−k 21 )(4−k 22 ), Equation (3a)

B 1 :

C(1+3e)(1−n 21 −l 2)/(1−k 21 )(1−k 22 ), Equation (3b)

B 2 :

2Ce(n 21 +2l 2−4)/(4−k 21 )(4−k 22 ), Equation (3b)

C :

solar parameter, 2R 3 p p 0 rlε(1−τ+ϱ/3)/μI y

C p :

center of pressure of the satellite

G :

solar aspect ratio, (πr/2l)(1−τ−ϱ)/(1−τ+ϱ/3)

I :

inertia parameter,I x /I y

I x ,I y ,I z :

principal moments of inertia of the satellite

O :

center of force

P :

pericenter

R :

distance between the satellite center of mass and the center of force

S :

satellite center of mass

e :

orbit eccentricity

i :

inclination of the orbital plane from the ecliptic

i, j, k :

unit vectors alongx, y, z axes, respectively

k i :

characteristic frequencies of linear systemi=1, 2

l, r :

length and radius of the cylindrical satellite, respectively

l 1 :

I(σ+1)(1+2e)−2, Equation (2a)

l 2 :

I(σ+1)(1+2e)−2, Equation (2b)

n 21 :

3I−4+I(σ+1)(1+2e), Equation (2a)

n 22 :

I(σ+1)(1+2e)−1, Equation (2b)

p 0 :

solar radiation pressure

u :

unit vector in the direction of the Sun,u i i+u j j+u k k

u i :

cos ϕ(sin γ cos β cos η+sin β sin η)+sin ϕ{cosi(sin γ cos β sin η−sin β cos η)−sini cos β cos γ}

u j :

cos ϕ cos γ cos η+sin ϕ(cosi cos γ sin η+sini sin γ)

u k :

cos ϕ(sin γ sin β cos η−cos β sin η)+sin ϕ{cosi(sin γ sin β sin η+cos β cos η)−sini sin β cos γ}

x, y, z :

principal body coordinates

x′, y′, z′ :

inertial coordinates

x 0,y 0,z 0 :

rotating coordinate system withx 0 normal to the orbital plane andy 0 along the local vertical

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

intermediate body coordinates resulting from rotations γ and β aboutz 0 andy 1 axes, respectively

Ω:

angle between the vernal equinox and the line of nodesNN′

α, β, γ:

librational angles

α i :

(k 2 i −n 21 )/l 1 k i ,i=1, 2, Equation (3b)

ε:

distance between the center of pressure and the center of mass of the satellite

η:

ω+θ

θ:

orbital angle from perigee (true anomaly)

λ i :

characteristic multipliers of the variational Equations (18),i=1, 2, 3, 4

ϱ, τ:

reflectivity and transmissibility of the satellite surface, respectively

σ:

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

ϕ:

solar aspect angle

ψ i :

k i θ+β i (θ),i=1,2

ω:

angle between the line of apsides and the line of nodes

ω i :

frequencies of nonlinear oscillations,i=1, 2

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Authors and Affiliations

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

    V. J. Modi (Professor, Department of Mechanical Engineering) & K. C. Pande (Post Doctoral-Fellow, Department of Mechanical Engineering)

Authors
  1. V. J. Modi
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  2. K. C. Pande
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Additional information

Primes indicate differentiation with respect to the orbital angle θ. The subscript 0 denotes initial condition.

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Modi, V.J., Pande, K.C. On the periodic solutions and resonance of spinning satellites in near-circular orbits. Celestial Mechanics 11, 195–212 (1975). https://doi.org/10.1007/BF01230545

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  • DOI: https://doi.org/10.1007/BF01230545

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