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## Über dieses Buch

From the preface: "The present text deals with attitude dynamics and is devoted to satellites of finite size. It begins with a discussion of the inertia moment tensor, Euler's law, Euler's angles, Euler's equations, and Euler's frequencies. After that a thorough treatment of the concept of centre of gravity versus centre of mass is given. After libration has been discussed and gyrodynamics proper has been dealt with, the attitude of the moment-free satellite, including the gyrostat, is studied. Particular attention is paid to the attitude behaviour of torquefree single and dual spinners, and the new collinearity theorems are introduced and explored to predict attitude stability and attitude drift. The derivation of each significant formula is followed by the discussion of a practical sample problem in order to acquaint the student with typical situations, typical results, and typical numerical values. There are numerous problems following each chapter. The most important data and the answers to the problems are compiled in appendices."

## Inhaltsverzeichnis

### 1. The Gyroscope

Abstract
In order to describe the behavior of a gyroscope (or simply gyro), Euler’s angular momentum law in the form
$$M = \dot H$$
(1.1)
is essential. The quantity M is the torque applied to the gyroscope from the outside. The quantity H is the angular momentum (or the moment of momentum) of the gyroscope, and H is its first derivative with respect to time. The torque M and the angular momentum H are taken with respect to the same reference point.
F. P. J. Rimrott

### 2. Center of Gravity

Abstract
The position vector ρC linking the origin of an arbitrary coordinate system and the mass center C of a massive body is defined by
$${\rho_C} = \frac{1} {m}\int\limits_m {\rho dm}$$
(2.1)
F. P. J. Rimrott

### 3. Libration

Abstract
Vibratory motion superimposed upon locked rotatory motion of a satellite is called libration. In order to obtain a basic understanding of the libration behavior of satellites, we shall investigate a few simple special cases. Because it exhibits strong sensitivity to the gravity gradient, among all possible satellite configurations, we select a one-dimensional rod satellite, with an inertia tensor of, say,
$$\left[ I \right] = \left[ {\begin{array}{*{20}{c}} 0 & {amp;0} & {amp;0} \\ 0 & {amp;B} & {amp;0} \\ 0 & {amp;0} & {amp;B} \\ \end{array} } \right]$$
(3.1)
We assume that the satellite is very small, i.e., $$B \ll m\rho_C^2$$.
F. P. J. Rimrott

### 4. Stability of Satellite Attitude in a Central Force Field

Abstract
The subsequent stability analyses are for absolutely rigid satellites of finite size in the gravitational field of a point master. The term gravity gradient stabilization (or GG stabilization) is used frequently by space engineers for satellites stabilized by taking advantage of the gradient of the gravitational field surrounding the master, the gradient of the field being none other than the Kepler force K, which, because of the finite size of the satellite, also exerts a torque on the satellite.
F. P. J. Rimrott

### 5. Torquefree Gyros

Abstract
A torquefree gyro is a gyro upon which no external torque is acting
$$M = 0$$
(5.1)
F. P. J. Rimrott

### 6. Torquefree Axisymmetric Gyros

Abstract
Satellites often have rotational symmetry. The properties and characteristics of torquefree gyros with rotational symmetry are the subject of the subsequent discussion, where it shall be understood that symmetry refers to massive symmetry as expressed by equation (6.1), which does not presuppose geometric symmetry. The equations derived in this chapter apply when the origin of the xyz-coordinate system is either the gyro’s mass center C or its fixed point 0 (Figure 1.2). If mass center C and fixed point 0 coincide, then the gyroscopic motion is not only torquefree but also force-free; if they do not coincide, an external force is present to bring about the mass center’s linear accelerations associated with the gyroscopic motion, among which the centripetal acceleration is most common.
F. P. J. Rimrott

### 7. Deformable Axisymmetric Gyros

Abstract
In the present chapter a near-rigid solid is defined. The concept is then introduced into gyrodynamics. After defining the properties of a suitable floating coordinate system, the latter is used to obtain suitable equations of gyrodynamics for deformable bodies. The concept of a body of constant configuration in a floating coordinate system is introduced. The theory is then demonstrated on typical examples.
F. P. J. Rimrott

### 8. Secular Attitude Drift of a Torquefree Dissipative Axisymmetric Gyro

Abstract
In order to investigate attitude drift and attitude stability of a torquefree solid gyro, we begin with the general case (equation (7.38)). The angular velocity of the phantom body is
$$\omega = \left[ {{e_u}\,{e_v}\,{e_z}} \right]\left[ {\begin{array}{*{20}{c}} {{w_u}} \\ {{w_v}} \\ {{w_z}} \\ \end{array} } \right]$$
(8.1)
and the angular momentum of the deforming body is, according to equation (7.38),
$$H = \left[ {{e_u}\,{e_v}\,{e_z}} \right]\left[ {\begin{array}{*{20}{c}} {A{w_u} + {Y_{uz}}{w_z}} \\ {A{w_v}} \\ {A{w_z}} \\ \end{array} } \right]$$
(8.2)
.
F. P. J. Rimrott

### 9. Despin

Abstract
Satellites are often carried to the insertion point of their orbit on rockets, which spin at a certain rate for reasons of stability.
F. P. J. Rimrott

### 10. Torque about Body-Fixed Axes of an Axisymmetric Gyro

Abstract
Satellites can be, and frequently are, equipped with minirockets for attitude and/or orbit adjustments. By their very nature, these minirockets are body-fixed.
F. P. J. Rimrott

### 11. Rigid Gyrostats

Abstract
A gyrostat, according to Kelvin, consists of two parts: a (typically small) axisymmetric rotor (or momentum wheel, or simply wheel) inside a (large) platform (or carrier). The rotor spins about its axis of symmetry with respect to the platform. The rotor spin consequently does not affect the mass distribution of the gyrostat. Thus the inertia tensor of the gyrostat remains constant, provided of course that a platform and rotor are both rigid bodies.
F. P. J. Rimrott

### 12. Torquefree Dissipative Axisymmetric Gyrostats

Abstract
For a near-rigid rotor in its undeformed state, using Cuvz floating coordinates, the inertia tensor is
$$\left[ {{{I}^{R}}} \right]unde{\mkern 1mu} f = \left[ {\begin{array}{*{20}{c}} {{{A}^{R}}} & {amp;0} & {amp;0} \\ 0 & {amp;{{A}^{R}}} & {amp;0} \\ 0 & {amp;0} & {amp;{{C}^{R}}} \\ \end{array} } \right]$$
(12.1)
F. P. J. Rimrott

### Backmatter

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