where the vectors p, are resolved into components along the principal axes in Eqs. (16-40) and (16-41). Equations (16-41) must have balancing positive and negative contributions in the sums on the right-hand sides. Thus, principal axes can be thought of intuitively as axes around which the mass is symmetrically distributed. In particular, any axis of rotational symmetry of the mass distribution is a principal axis. Equation (16-40) shows that /„ I2, and /3 are all nonnegative, and thus det/ = /,/2/3 > 0. The determinant is zero only if all the p, are collinear, i.e., if all the mass is along a mathematical straight line. Thus, for real objects det/ > 0, and, because the determinant is invariant under a change of coordinate system (see Appendix C), this holds in all coordinate systems. Consequently, in any coordinate system, the moment of inertia tensor has an inverse, /"', which is also a 3x3 matrix*. We can thus write Eqs. (16-34) and (16-35) as
respectively. According to Eq. (16-43), a surface of constant energy is an ellipsoid in L,, Lj, Lj space, as was discussed in Section 15.2. Note that Eq. (16-35) similarly
'The moment-of-inertia tensor is not an orthogonal matrix, so / ~1 is not equal to /T.
defines an energy ellipsoid in u„ w2, w3 space, with semiaxis lengths \2Ek/1, , y2Ek/12 , and ylEk//3 . This ellipsoid may be used for qualitative discussions of rotational motion, but it will not be considered in this work. According to Whittaker [1937, page 124], "The existence of principal axes was discovered by Euler, Mem. de Berl., 1750, 1758, and by J. A. Segner, Specimen Th. Turbinem, 1755. The momental ellipsoid was introduced by Cauchy in 1827, Exerc. de math. I, p. 93."
The basic equation of attitude dynamics relates the time derivative of the angular momentum vector, dL/d/, to the applied torque, N. This relation was introduced in Section 15.2, and Eq. (15-9) gives dL/dt in inertial coordinates. In this section, we consider the time derivatives of the components of L along spacecraft-fixed axes, because the moment of inertia tensor of a rigid body is most conveniently expressed along these axes. Combining Eqs. (15-9), (16-34), and (16-18) gives*
d/ dt v / where the torque vector, N, is defined as
i = i and <o is the instantaneous angular velocity vector discussed in Section 16.1. The force, F,, on the /th mass consists of two parts: an externally applied force, Ff", and an internal force consisting of the sum of the forces, ty, exerted by the other masses (the cohesive forces of the rigid body):
Each pair of masses contributes two terms to the second sum, r, X f ^ and r} X tj,. By Newton's third law of motion, - f^, so the contribution to the sum of each pair of masses is (r, -r/)Xfl^. If the line of action of the force between each pair of masses is parallel to the vector between the masses, r, — ry., the cross product vanishes, and the net torque, N, is equal to the torque due to external forces alone. This is always assumed to be the case in spacecraft applications. Some forces, most notably magnetic forces between moving charges, violate this condition, so that the
'The moment-of-inertia tensor of a rigid body is constant This is not the case when flexibility effects (Section 16.4) or fuel expenditure (Section 17.4) are considered.
w rate of. change of mechanical angular momentum is not equal to the external torque. In the case of electromagnetic forces, this difference can be ascribed to the angular momentum of the electromagnetic field, but this is negligible for spacecraft dynamics problems.
Equation (16-44) is the fundamental equation of rigid body dynamics. The presence of the a XL term on the right side means that L, and hence a, is not constant in the spacecraft frame, even if N=0. The resulting motion is called nutation, and is discussed qualitatively in Section 15.1. Rotational motion without nutation occurs only if <o and L are parallel, that is, only if the rotation is about a principal axis of the rigid body.
Substituting Eq. (16-34) or Eq. (16-42) into Eq. (16-44) gives
respectively. These equations can be written out in component form, but no insight is gained by it, except when the vector quantities are referred to the principal axis coordinate system. In the principal axis system, Eq. (16-48) has the components:
/j"d7-^3 +(A(16"50c) and Eq. (16-49) has the components:
Equations (16-44), (16-48), and (16-49) and their component forms Eqs. (16-50) and (16-51) are alternative formulations of Euler's equations of motion.
A spacecraft equipped with reaction or momentum wheels is not a rigid body, but the dynamic equations derived above can still be used, with one minor modification. When wheels are present, the total angular momentum of the spacecraft, including the wheels, is
where the moment of inertia tensor I includes the mass of the wheels and the vector h is the net angular momentum due to the rotation of the wheels relative to the spacecraft. The inverse 6f Eq. (16-52) is
Substituting Eq. (16-32) or (16-53) into Eq. (16-44) gives
or dL df
respectively. For numerical calculations the second form is sometimes preferable because it does not involve the derivatives of the wheel angular momenta. The derivative term in Eq. (16-54) has a natural physical interpretation, however. The quantity dh/df is the net torque applied to the wheels by the spacecraft body; so, by Newton's third law of motion, -dh/dt is the torque applied to the spacecraft body by the wheels. Writing Eqs. (16-54) and (16-55) in component form in the principal axis system yields equations similar to Eqs. (16-50) and (16-51).
Euler's equations of motion can be used to discuss the stability of rotation about a principal axis of a rigid spacecraft. Let P3 be the nominal spin axis, so that w, and to2 are much smaller than w3. Let us also assume that the applied torques are negligible. Then the right side of Eq. (16-50c) is approximately zero, and w3 is approximately constant. Taking the time derivative of Eq. (16-50a), multiplying by 12, and substituting Eq. (16-50b) gives
If (I2 - /3X/3 - /1) < 0, then «>>1 will be bounded and have sinusoidal time dependence with frequency VCz- AOCi - AO/Ci^) u3> however, if (/2-/3)(/3-/i)>0, then <o, will increase exponentially. Thus, the motion is stable if /3 is either the largest or the smallest of the principal moments of inertia, and unstable if I3 is the intermediate moment of inertia. This can be seen in the form of the paths of the angular momentum vector in the body shown in Fig. 15-14; the loci in the neighborhood of the principal axes of largest and smallest moment of inertia are elliptical closed curves, but the loci passing near the third principal axis go completely, around the angular momentum sphere. Equation (16-56) only establishes the stability over short time intervals; over longer time intervals, energy dissipation effects cause rotational motion about the axis of smallest moment of inertia to be unstable, too, as discussed in Sections 15.2, 17.3, and 18.4.
We now turn to a discussion of the solutions of the kinematic and dynamic equations of motion presented in the previous section. These equations must be solved simultaneously because, in general, the torque N depends on the spacecraft attitude. Numerical integration methods and approximate closed-form solution methods for the general case are discussed in Section 17.1.
16.2 Motion of a Rigid Spacecraft
If N is independent of the attitude, the dynamic equations can be solved separately for the instantaneous angular velocity to, which can then be used to solve the kinematic equations. A special case for which analytic solutions are available is the N = 0 case, which is treated in Sections 16.2.1 and 16.2.2. These solutions are intrinsically interesting and furnish a useful approximation for the motion when the torques are small. Sections 16.2.1 and 16.2.2 provide an analytic counterpart to the qualitative discussion of attitude motion in Section 15.1. They also provide the starting point for the variation-of-parameters formulation of attitude dynamics presented in Section 16.2.3.
16.2.1 Torque-Free Motion—Dynamic Equations
The vector quantities in this section will be resolved along spacecraft principal axes to simplify the equations of motion. If two of the principal moments of inertia are equal, we shall take these to be /, and 12; this is referred to as the axial symmetry case. If no two moments of inertia are equal, we shall denote the intermediate moment by I2. In this case, the labeling of /, and I3 will be fixed by the following convention: if L2<2J2Ek, we take /3</2</,; and if L2>2J2Ek, we label the principal axes so that /,<J2<J3. If L2 = 2J2Ek, either labeling can be used. With this convention, L2 always lies between 212Ek and 213Ek. The two limits can be visualized by considering the loci of the angular momentum vector on the angular momentum^ sphere shown in Fig. 15-14. Motion with L2 = 2I3Ek is pure rotation about the P3 axis, that is, nutation-free motion. Motion with L2=2J2Ek, on the other hand, means that L lies on one of the loci passing throught the axis of intermediate symmetry, P2. In the axial symmetry case, this locus is the equator of the angular momentum sphere relative to the P3 axis. With the convention adopted here and the to3>0 convention adopted below, then, L will lie on the P3 side of the L2=212Ek loci, which is the upper hemisphere in the axial symmetry case and a smaller surface in the asymmetric case.
When the body is axially symmetric, we define the transverse moment of inertia
1T=1X = 12 (16-57) In this case, Euler's equations, Eq. (16-50a) through (16-50c) simplify to dto.
Equation (16-58c) shows that to3 is a constant.* We choose the sense of the P3 axis so that to3> 0. Differentiating Eq. (16-58a) with respect to t, multiplying by 1T, and substituting Eq. (16-58b) yields
*For spherically symmetric spacecraft, It = I2=ly, and Eq. (16-58) shows immediately that a is a constant vector in the body. This also follows from the fact that If 'L in the spherical symmetry case. Because L is constant in inertia! coordinates, a must be constant also. Then, because uXL=0, Eq. (16-44) shows that L, and hence a, is constant in the body reference system.
which has the solution t0| = <or cos<o (/ — /|)
In this equation, <or is the maximum value of <o„ /, is some time at which <o, attains its maximum value, and
is the body nutation rate introduced in Section 15.1.* The derivation of Eq. (16-59a) is analogous to that of Eq. (16-56), but Eq. (16-59a) is exact if /, = /2, whereas Eq. (16-56) is an approximation based on the smallness of <o, and <o2 relative to <o3. Combining Eqs. (16-58a) and (16-59a) gives to2= -<oT sin «,,(/-<,) (16-59c)
Equations (16-59a, c) show that wr = («J + io^)l/2, so <or is the magnitude of the component of the angular velocity perpendicular to the symmetry axis and is called the transverse angular velocity.
By using the addition formulas for the sine and cosine, we can rewrite Eqs. (16-59) in terms of the components (<o0l,<o02,<o03) of «„, the initial value of <o in the body frame. Thus,
It is often useful to express <o03 and uT in terms of the magnitude of the angular momentum vector and the rotational kinetic energy. In the axial symmetry case, Eqs. (16-38) and (16-39) give
so we have
'Equation (16-66) shows that this definition is equivalent to that of Eq. (15-7).
Equations (16-59) can be written in vector form as tt = «03P3 + «7.[cosu/1(/-/,)P| -sin«^/-/,)^] (16-64)
Then the angular momentum vector is
L= /3w03P3+ /rw7-[cosw/,(/-/l)P1-sinw/,(/-/l)P2] (16-65)
Thus, the body nutation rate, up, and the inertial nutation rate, u„ defined by Eq. (16-59b) and u,=L/It (16-67a)
agree with the quantities introduced in Section 15.1. It is also clear that cos 0=L3/L = /3w03/L (16-67b)
where 9 is the angle between L and P3. Thus, h-13
These equations form the basis for the discussion of nutation in the axial symmetry case given in Section 15.1.
The solutions to Euler's dynamic equations of motion in the asymmetric case, /, /2, cannot be written in terms of trigonometric functions. Instead, they involve the Jacobian elliptic Junctions [Milne-Thomson, 1965; Neville, 1951; Byrd and Friedman, 1971]. These solutions, found by Jacobi , are discussed by Synge and Griffith , MacMillan, , Thomson , and Morton, et ai, . The angular velocity components in body principal coordinates are given by w,=wImcn(4>|w) (16-69a)
where cn, sn, and dn are the Jacobian elliptic functions with argument
As in the axial symmetry case, /, is a time at which ut=uim. The maximum values
*Many authors use the modulus, k = m1/2, rather than the parameter. We follow the notation of Milne-Thomson  and Neville .
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