Velocity Analysis of Rigid-Body Motions

In this chapter the angular velocity of a rigid-body motion is introduced as a skew-symmetric tensor, its linear vector invariant being defined as the angular-velocity vector of the given motion. The linear relations between the angular-velocity vector and the time rates of change of the natural, the linear, and the quadratic invariants of the rotation tensor are derived. The relation between the angular-velocity vector and the time-rate of change of the quadratic invariants—Euler’s parameters—of the rotation tensor have been reported previously, e.g., in Wittenburg (1977) and Kane, Likins, and Levinson (1983). A comprehensive study of the relations between the first and second time derivatives of the Euler parameters and the angular-velocity and angular-acceleration vectors was reported by Nikravesh, Wehage. and Kwon (1985). Apart from these, the other relations are derived for the first time in invariant form. A preliminary derivation of the relation between the angular-velocity vector and the time rate of change of the linear invariants was first introduced in Angeles (1985). Spring (1986) includes a table showing some of the results that are derived here. Furthermore, a theorem related to the velocity distribution in a rigid body, paralleling that of Chasles’ of Chapter 2, is proven. Next, thė Theorem of Aronhold-Kennedy, pertaining to the relative motion of three rigid bodies, is proven. Additional theorems related to the velocity distribution throughout a moving rigid body are presented and proven, and the concept of twist of a rigid body is introduced. Finally, the problem of determining the angular velocity of a rigid-body motion from point-velocity data is discussed, and compatibility equations which the given data should verify, are derived.