When a rigid body B moves in a reference frame E in such a way that there exists a unit vector e3 whose orientation in both E and B is independent of time t, B is said to have a planar motion relative to E. The angular velocity of B in E was defined in Chapter 4 as

where e is the radian measure of the angle between a line whose orientation is fixed in E and another line whose orientation is fixed in B, both lines being perpendicular to e3 (Fig. 9.2). The angle e is regarded as positive when it can be generated by a rotation of B during which a right-handed screw parallel to e3 and rigidly attached to B advances in the direction of e3.

We have also shown in Chapter 4 that, in planar motion, if d is any vector fixed in reference frame B then

This equation illustrates why angular velocity plays such an important role in mechanics. With this operation, the time derivatives of vectors of constant length can be reduced to algebraic manipulations. Therefore, it is important to extend the concept of angular velocity to three dimensions. What is the definition of angular velocity in three-dimensional motion? The definition presented here is abstract and complex. Nevertheless this definition is at the heart of three-dimensional mechanics. It will become clear later in the section that the angular velocity thus defined satisfies Eqn. 9.4.

Let b1, b2, b3 be a right-handed set of mutually perpendicular unit vectors fixed in a rigid body B; the angular velocity R«B of B in a reference frame R is defined as

RwB = [(Rdbi/dt) • b2]b3 + [(Rdb2/dt) • b3]b1 + [(Rdb3/dt) • b1]b2 (9.5)

where (Rdbj/dt) denotes the ordinary derivative of bj with respect to time in reference frame R. Note that in the determination of angular velocity using this equation, we need to know only the time rate of change of two of the vectors that define the reference frame B. The time derivative of the relation b3 = b1 X b2 can be used to determine the time rate of change of the third unit vector of the reference frame B.

Next, let us take the vector product of R«B and b1:

RwB X bi = [(Rdbi/dt) • b2]b2 - [(Rdb3/dt) • bi]b3

On the other hand, the rule of chain differentiation requires that

(Rdb3/dt) • bi + (Rdbi/dt) • b3 = Rd(b3 • bi)/dt = 0

Thus the vector product of angular velocity and the unit vector vector b1 can be written as

RwB X bi = [(Rdbi/dt) • b2]b2 + [(Rdbi/dt) • b3]b3

The two terms on the right-hand side of this equation are simply the projections of Rdb1/dt on b2 and b3. Because Rdb1/dt cannot have a projection on b1 (otherwise its length would not remain equal to one), it is clear that

Similarly we can show that

Let r be unit vectors fixed in R. The time derivatives of vector P in reference systems R and B are related by the following equation:

RdP/dt = (dRPi/dt) ri + (dRP2/dt) r2 + (dRP3/dt) r3

= (dBPi/dt) bi + BPi(dRbi/dt) + (dBP2/dt) b2 + BP2(dRb2/dt) + (dBP3/dt) b3 + BP3(dRb3/dt)

Substituting Eqns. 9.2 and 9.6 into this equation, we obtain the following:

This is one of the most important equations in mechanics. According to this equation, if P is constant in B, then the time derivative of P in R is simply the vector product of the angular velocity of B with respect to R and the vector P.

Next let us consider the time derivative of P in the inertial reference frame E:

EdP/dt = RdP/dt + EwR X P = BdP/dt + RwB X P + EwR X P = BdP/dt + (RwB + EwR) X P

Thus, one can show that angular velocities can be added according to the following equation:

Figure 9.3. A tennis player preparing to hit a ball. The reference frames Bj are attached to the body segments shown and move with them.

It is straightforward to extend the additivity relationship to the case of successive reference frames in the study of motion (Fig. 9.3):

E«B5 = E«B1 + B1«B2 + B2«B3 + B3«B4 + B4«B5 (9.9)

Example 9.1. Seated Dumbbell Press. Determine the angular velocities of the upper arm and the forearm of a weight lifter during seated dumbbell press (Fig. 9.4). The rotational angles 6 and $ shown in Fig. 9.4 were determined with the use of two cameras. Both angles varied with time:

If B and D denote the reference frames fixed in the upper arm and forearm, respectively, determine the angular velocities E«B, emd, and B«D.

Solution: Using Eqn. 9.3, it is straightforward to show that the following holds:

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