Abstract
The orbital perturbations induced by the librational motion and flexural oscillations are studied for satellites having large flexible appendages. Using a Lagrangian procedure, the equations for coupled motion are derived for a satellite having an arbitrary number of appendages in the nominal orbital plane and two flexible members normal to it. The formulation enables one to study the influence of flexibility on both the orbital and attitude motions. The orbital coordinates are expanded as perturbation series in ε=(l/a 0)2,l anda 0 being a characteristic length of the satellite and unperturbed semi-major axis of the orbit, respectively. The first order perturbation equations are solved in terms of elastic deformations and librational angles using the WKBJ method in conjunction with the variation of parameter technique. Existence of secular perturbations is noted for certain librational flexural motions. Three specific examples, Alouette II, Radio Astronomy Explorer and Tethered Orbiting Interferometer, are considered subsequently and their possible secular drifts estimated.
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Abbreviations
- A ij, Bij :
-
coefficients in the eigenfunction expansion ofv i andw i respectively, Equation (10)
- C k, Dk :
-
constants, Equation (21)
- EI i :
-
flexural rigidity of theith appendage
- E(u0):
-
μ 2(1+e 0 cosu 0)2 h 30
- F(u0):
-
perturbation function, Equation (17b)
- F ξ,F η,F ζ :
-
functions of librational angles and flexural displacements, Equation (11i)
- F ξ,F η,F ζ :
-
F ξ,F η,F ζ with change of independent variable fromt tou 0
- I xx, Iyy, Izz :
-
principal moments of inertia of the undeformed satellite
- [J i]:
-
inertia dyadic of the deformedith appendage
- [J d]:
-
inertia dyadic of the deformed satellite
- M :
-
mass of the satellite
- P R, Pu :
-
functions of librational angles and flexural displacements, Equation (15d) and (15e), respectively
- R c :
-
magnitude ofR c
- R c0, R1 :
-
unperturbed value and first order perturbation ofR c, respectively
- R c ,R 0 :
-
position vectors of the c.m. of the deformed and undeformed satellite, respectively
- T :
-
kinetic energy of the satellite
- U :
-
potential energy of the satellite
- U e, Ug :
-
elastic and gravitational potential energy, respectively
- X, Y, Z :
-
orbital co-ordinate axes, located at the c.m. of the deformed satellite
- Y 1(u0), Y2(u0):
-
functions ofu 0, Equation (18b) and (18c), respectively
- a :
-
semi-major axis
- a 0 :
-
unperturbed value ofa
- e :
-
eccentricity
- e 0 :
-
unperturbed value ofe
- h 0 :
-
unperturbed angular momentum per unit mass of the satellite
- i :
-
inclination of the orbital plane to the ecliptic
- i, j, k :
-
unit vectors alongx (or ξ),y (or η) andz (or ζ) axes, respectively
- l :
-
characteristic length of the satellite
- l i :
-
length of theith appendage
- [l i]:
-
matrix of direction cosines ofx i, vi andw i
- l ξ,l η,l ζ :
-
direction cosines ofR c
- m 0, mi :
-
mass of the main body andith appendage, respectively
- p 2i :
-
\(\left( {\frac{{1 - s^2 }}{2}} \right)\left\{ {\left( {\frac{{\partial v_i }}{{\partial s}}} \right)^2 + \left( {\frac{{\partial w_i }}{{\partial s}}} \right)^2 } \right\}\)
- q m, Qm :
-
generalized co-ordinate and force, respectively
- r 1 :
-
R 1/Rc0
- r :
-
position vector of an element of the body referred toxyz axes
- r u :
-
position vector of an element after deformation, referred to ξηζ axes
- r c :
-
x c i+y c j+z c k, position vector of the c.m. of the deformed body referred toxyz axes
- s :
-
x i/li
- t :
-
time
- u :
-
true anomaly
- u 0, u1 :
-
unperturbed value and the first order perturbation ofu, respectively
- u :
-
elastic displacement vector
- u c :
-
u−r c
- \(\dot u_{cr}\) :
-
velocity of an element relative to ξηζ axes
- v i, wi :
-
flexural deformations
- x, y, z :
-
body co-ordinate axes with origin at the c.m. of the undeformed satellite
- x i :
-
distance of an element of theith appendage from the root
- Φ j :
-
jth eigenfunction (normalized) of a cantilever
- Ω:
-
angle between the line of nodes and vernal equinox
- Ω ξ,Ω η,Ω ζ :
-
components of nondimensionalized angular velocity of the satellite
- α, β, γ:
-
pitch (spin), yaw and roll, respectively
- αi :
-
nominal inclination of theith appendage in the orbital plane
- \(\dot \delta _j\) :
-
\(\int\limits_0^1 {\Phi _j (s) ds}\)
- ε:
-
small parameter, (l/a 0)2
- λj :
-
jth eigenvalue of a cantilever
- μ:
-
gravitational constant
- μ jk :
-
constant, Equation (11j)
- ξ, η, ζ:
-
body co-ordinate axes with origin at the c.m. of the deformed satellite
- ω:
-
(ω ξ i +ω η j +ω ζ k), angular velocity of the satellite
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Misra, A.K., Modi, V.J. The influence of satellite flexibility on orbital motion. Celestial Mechanics 17, 145–165 (1978). https://doi.org/10.1007/BF01371327
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DOI: https://doi.org/10.1007/BF01371327