Page 81 - Dynamics of Mechanical Systems

0593_C03_fm Page 62 Monday, May 6, 2002 2:03 PM

62 Dynamics of Mechanical Systems

Consider again Figure 3.4.2. Let r be the position vector locating P relative to Q. Then,
from the rules of vector addition, we have:

+
−
=
pq r οορρ r = p q (3.4.4)
By differentiating, we have:

d dt = dp dt dq dt = V − V − V P Q (3.4.5)
−
P
/
Q
r
Therefore, we see that the relative velocity V P/Q is simply the derivative of the position
vector locating P with respect to Q. Hence, V P/Q may be interpreted as the movement of
P as seen by an observer moving with Q.
From an analytical perspective, Eq. (3.4.3) is generally written in the form:
V = V + V P Q (3.4.6)
P
Q
/
Thus, if the velocity of Q is known and if the velocity of P relative to Q is known, then
the velocity of P may be calculated.
The same concepts hold for accelerations. Specifically, by differentiating in Eqs. (3.4.5)
and (3.4.6), we have:

/
Q
P
P
/
a PQ = a − a Q or a = a + a PQ (3.4.7)
P
Q
where a P/Q is called the acceleration of P relative to Q in R, and a and a are called absolute
accelerations of P and Q in R.
Example 3.4.1: Relative Velocity of Two Particles of a Rigid Body
Consider a rigid body B moving in a reference frame R, and let P and Q be particles of B
represented by points P and Q. Let P be located relative to Q by the vector r given by:

r = 2 n − 3 n + 7 n ft (3.4.8)
1 2 3
where n , n , and n are unit vectors fixed in B as in Figure 3.4.3. Let the angular velocity
1
3
2
of B in R be given as:
ωω= −2n + 3n − 4n rad sec (3.4.9)
1 2 3

P
B
n r
3

n n 2
1 Q
R
FIGURE 3.4.3
A body B with particles P and Q.