Computational Dynamics, 3rd Edition

Chapter 8. SPECIAL TOPICS IN DYNAMICS

In this chapter, several topics in dynamics are presented. In the first section, the use of Euler angles to study the gyroscopic motion is discussed. In the following sections, several alternative methods for defining the orientations of the rigid bodies in space are presented. In Section 2, the Rodriguez formula, which is expressed in terms of the angle of rotation and a unit vector along the axis of rotation, is presented. Euler parameters, which are widely used in general-purpose multibody computer programs to avoid the singularities associated with Euler angles, are introduced in Section 3. Rodriguez parameters are discussed in Section 4 for the sake of completeness. Euler parameters can be considered as an example of the quaternions, which are introduced in Section 5. In Section 6, the problem of nonimpulsive contact between rigid bodies is discussed. A method for the stability and eigenvalue analysis of constrained multibody system is presented in Section 7.

GYROSCOPES AND EULER ANGLES

The study of the gyroscopic motion is one of the most interesting problems in spatial dynamics. This problem occurs when the orientation of the axis of rotation of a rigid body changes. The gyroscope shown in Fig. 1 consists of a rotor that spins about its axis of rotational symmetry Z³ which is mounted on a ring called the inner gimbal. As shown in the figure, the rotor is free to rotate about its axis of symmetry relative to the inner gimbal, and the inner ...