Dumbbell Routines and Exercises

15.1 Torque-Free Motion

15.2 Response to Torques

Dynamics is the study of the relationship between motion and the forces affecting motion. The study of the dynamics of objects in interplanetary or interstellar space is called astrodynamics and has two major divisions: celestial mechanics and attitude dynamics. Celestial mechanics or orbit dynamics, discussed briefly in Chapter 3, is concerned with the motion of the center of mass of objects in space, whereas attitude dynamics is concerned with the motion about the center of mass. In Part IV, we deal exclusively with this latter category.

Thus far, we have been concerned primarily with determining the orientation of a spacecraft without consideration of its dynamics, or, at least, with an implicit assumption of a specific and accurate dynamic model. However, knowledge of attitude dynamics is necessary for attitude prediction, interpolation, stabilization, and control. In this chapter, which is less quantitative than the remainder of Part IV, we attempt to provide a physical "feel" for attitude motion and environmental torques affecting the attitude. Chapter 16 then develops the more formal mathematical tools used in the study of attitude dynamics and briefly discusses the effect of nonrigidity in spacecraft structure. Free-body (i.e, satellite) motion differs in several important respects from the motion of rigid objects, such as a spinning top, supported in a gravitational field. Thus, the reader should be careful to avoid relying on either intuition or previous analytic experience with common rotating objects supported in some way near the surface of the Earth.

15.1 Torque-Free Motion

We consider first the simplest case of the attitude motion of a completely rigid, rotating object in space free of all external forces or torques. In describing this motion, four fundamental axes or sets of axes are important. Geometrical axes are arbitrarily defined relative to the structure of the spacecraft itself. Thus, the geometrical 2 axis may be defined by some mark on the spacecraft or by an engineering drawing giving its position relative to the structure. This is the reference system which defines the orientation of attitude determination and control hardware and experiments.

The three remaining axis systems are defined by the physics of satellite motion. The angular momentum axis is the axis through the center of mass parallel to the angulaT momentum vector. The instantaneous rotation axis is the axis about which the spacecraft is rotating at any instant; Euler's Theorem (Section 12.1) establishes the existence of this axis. The angular momentum axis and the instantaneous rotation axis are not necessarily the same. For example, consider the rotation of a symmetric dumbbell, as shown in Fig. 15-1. In elementary mechanics, we define the angular momentum, L, of a point mass, m, at position r relative to some arbitrary origin as

L=rxp=rxmv O5"1)

where p is the momentum and v is the velocity of the particle in question. For a collection of n points,

Assume that the dumbbell is rotating with angular velocity a about an axis through the center of mass and perpendicular to the rod joining the masses (Fig. 15-l(a)). Then, L is parallel to <o and the motion is particularly simple because L and a remain parallel as the dumbbell rotates.

However, if the dumbbell is initially rotating about an axis through the center of mass but inclined to the normal to the central rod (Fig. 15-1(b)), L is in the plane defined by a and the two end masses, but L is clearly not parallel to <o. (Use Eq. (15-2) to calculate the angular momentum about the center of mass.) Now the free-space motion is more complex. Because the conservation of angular momentum requires that L remain fixed in inertial space if there are no external torques, the instantaneous axis of rotation, a, must rotate as the dumbbell rotates. Conversely, if <o is fixed in space by some external supports or axes, a torque must be supplied via the supports to change L as the object rotates. (This may .be conveniently demonstrated by constructing models of the two dumbbells in Fig. 15-1 out of Tinkertoys.)

Clearly, the motion about the axis in Fig. 15-l(a) is simpler than that in Fig. 15-l(b). Thus, the motion of the dumbbell leads us to define as the third physical axis system, preferred axes about which the motion is particularly simple. Specifically, a principal axis is any axis, P, such that the resulting angular momentum is parallel to P when the spacecraft rotates about P. Therefore, for rotation about a principal axis, L is parallel to <o, or

ROTATION AXIS, w

ROTATION AXIS, w

where Ip is a constant of proportionality called the principal moment of inertia. Because the magnitude of the angular velocity is defined by u = v/r, where r is the rotation radius, Eqs. (15-1) through (15-3) imply that for a principal axis and a collection of point masses,

¡= l where r, is the perpendicular distance of m, from the principal axis. For rotation about nonprincipal axes the motion is more complex, and Eq. (J 5-3) does not hold.

The form of Eq. (15-2) shows that whenever the mass of an object is symmetrically distributed about an axis (i.e., if the mass distribution remains identical after rotating the object 360/N deg about the specified axis, where N is any integer greater that 1*), the angular momentum generated by rotation about the symmetry axis will be parallel to that axis. Thus, any axis of symmetry is a principal axis. In addition, we will show in Chapter 16 that any object, no matter how asymmetric, has three mutually perpendicular principal axes defined by Eq. (15-3).

The sets of axes above may be used to define three types of attitude motion called pure rotation, coning, and nutation. Pure rotation is the limiting case in which the rotation axis, a principal axis, and a geometrical axis are all parallel or antiparallel, as shown in Fig. 15-2(a). Clearly, the angular momentum vector will lie along this sar..c axir. These four axes will remain parallel as the object rotates.

Coning is rotation for which a geometrical axis is not parallel to a principal axis. If the principal and rotation axes are still parallel, the physical motion of the object is precisely the same as pure rotation. However, the "misalignment" of the geometrical axis (which may be intentional) causes this axis to rotate in inertial space about the angular momentum vector, as shown in Fig. 15-2(b). Coning is

* In this case, die mass distribution consists of N symmetrically distributed groups of mass points. If L does not lie on the axis of symmetry, then for N >2 it must lie closer to, or farther away from, one group; however, this is impossible because all of the mass points contribute equally to L. For N=2, the mass distribution has the form m(x,y,z)=m(- x, -y,z\ where 2 is the symmetry axis. Therefore, any x at y components of L cancel when summed and L must lie along the z axis.

associated with a coordinate system misalignment rather than a physical misalignment and can be eliminated by a coordinate transformation if the orientation of the principal axes in the body of the spacecraft is known precisely.

Finally, nutation* is rotational motion for which the instantaneous rotation axis is not aligned with a principal axis, as illustrated in Fig. 15-2(c). In this case, the angular momentum vector, which remains fixed in space, will not be aligned with either of the other physical axes. Both P and to rotate about L. P is fixed in the spacecraft because it is defined by the spacecraft mass distribution irrespective of the object's overall orientation. Neither L nor <o is fixed in the spacecraft. <o rotates both in the spacecraft and in inertial space, while L rotates in the spacecraft but is fixed in inertial space. The angle between P and L is a measure of the magnitude of the nutation, called the nutation angle, 9. Nutation and coning can occur together, in which case none of the four axis systems is parallel or antiparallel.

We now describe the simple case in which two of the three principal moments of inertia are equal; that is, we assume /, = /2 ^ I3. Although this is an idealization for any real spacecraft, it is a good approximation for many spacecraft which possess some degree of cylindrical symmetry. In this case, the angular momentum vector, L, the instantaneous rotation axis, <o, and the P3 principal axis are coplanar and the latter two axes rotate uniformly about L. The body rotates at a constant velocity about the principal axis, P3, as P3 rotates about L and the nutation angle remains constant. (Because P3 is a spacecraft-fixed axis and is moving in inertial space, it cannot be the instantaneous rotation axis.)

As shown in Chapter 16, the spacecraft inertial spin rate, u, about die instantaneous rotation axis (when /, = /j) can be written in terms of components along P3 and L as:

Because P3 and L are not orthogonal, the amplitude of u is given by u2 = u2 + uf + 2wpu,cos9 (15-6)

where the nutation angle, 9, is the angle between P and L; the inertial nutation rate, <o,, is the rotation of P3 about L relative to an inertial frame of reference; and the body nutation rate, ap, is the rotation rate of any point, R, fixed in the body (e.g., a geometrical axis) about P3 relative to the orientation of L. Figure 15-3 shows a view looking "down" on the motion of the axes when 9 is small. Here <o, and <o are the rotation rates of lines LP3 and P3R, respectively, relative to inertial space, and is the rotation rate of P3R relative to P3L. The component angular velocities in Eqs.

By resolving <o in Eq. (15-5) into components along P3 and orthogonal to P3 and then using Eq. (15-7), we may obtain an expression for the angle, f, between P3 and

* This definition of notation is in keeping with common spacecraft usage and differs from that used in classical mechanics for describing, say, the motion of a spinning top. In the latter case, nutation refers to the vertical wobble of the spin axis as it moves slowly around the gravitational field vector.

Fig. 15-3. Position of Principal Axis, P3 and Arbitrary Spacecraft Reference Axis R, at Times I, and f2 for a Nutating Spacecraft With Small Values of 0 and J. bt = t2- i,.

to, as follows:

Fig. 15-3. Position of Principal Axis, P3 and Arbitrary Spacecraft Reference Axis R, at Times I, and f2 for a Nutating Spacecraft With Small Values of 0 and J. bt = t2- i,.

to, as follows:

To obtain a physical feel for the motion described by Eqs. (15-5) through (15-8), we note that in inertia) space, <o rotates about L on a cone of half-cone angle (9-f) called the space cone, as illustrated in Fig. 15-4 for It>I3. Similarly, <o maintains a fixed angle, with P3 and, therefore, rotates about P3 on a cone called the body cone. Because <o is the instantaneous rotation axis, the body is instantaneously at rest along the to axis as <o moves about L. Therefore, we may visualize the motion of the spacecraft as the body cone rolling without slipping on the space cone. The space cone is fixed in space and the body cone is fixed in the spacecraft.

Figure 15-4 is correct only for objects, such as a tall cylinder, for which lt is greater than J3. In this case, Eq. (15-7) implies that up and a, have the same sign.* If /3 is greater than /„ as is the case for a thin disk, up and to, have opposite signs and the space cone lies inside the body cone, as shown in Fig. 15-5. The sign of up is difficult to visualize, since to is measured relative to the line joining the axes of the two cones. (Refer to Fig. 15-3.) If we look down on the cones from above, in both Figs. 15-4 and 15-5, P3 is moving counterclockwise about L. In Fig. 15-4, the dot on the edge of the body cone is moving toward <o and, therefore, is also rotating counterclockwise. In Fig. 15-5, the dot on the edge of the body cone is moving counterclockwise in inertial space, but the <o axis is moving counterclockwise more quickly. Therefore, relative to the P3—L—<o plane, the dot is moving clockwise and up has the opposite sign of to. If /, = I2 = I3, the space cone reduces to a line, <0^ = 0, and the spacecraft rotates uniformly about L. In this case, any axis is a principal axis.

Figure 15-6 illustrates the motion in inertial space of an arbitrary point, R

* The terms prolate and oblate are commonly used for /, > /3 and l3 > I,, respectively; these terms refer to the shape of the energy ellipsoid, which is introduced in Section 15.2.

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The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.

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