Page 302 - Dynamics of Mechanical Systems

0593_C09_fm Page 283 Monday, May 6, 2002 2:50 PM

Principles of Impulse and Momentum 283

FIGURE 9.4.2 FIGURE 9.4.3
A set S of moving particles. A rigid body B with mass m, mass center G, and
reference point Q.
(i = 1,…, N). Again, let Q be an arbitrary reference point. Then, as in Eq. (9.4.3), the angular
momentum of B about Q in R is defined as:

N
BQ ∑
A = p × m i v P i (9.4.4)
i
i=1
where p locates P relative to Q.
i
i
Observe that because P and mass center G are both fixed in B their velocities in R are
i
related by the expression (see Eq. (4.9.4)):
v = v + ωω × r i (9.4.5)
G
P i
where ωω ωω is the angular velocity of B in R and where r locates P relative to G (see Figure
i
i
9.4.3). Observe further from Figure 9.4.3 that p and r are related by the expression:
i
i
+
p = QG r (9.4.6)
i i
By substituting from Eqs. (9.4.5) and (9.4.6) into (9.4.4) A B/Q becomes:
N
∑ i( QG r ) ×( v + ωω × r )
+
=
BQ
G
A m i i
i=1
N
N
= ∑ m i QG v + ∑ m i QG ×(ωω × r )
×
G
i
i=1 i=1
N
N
r ×(ωω
+ ∑ m ii v + ∑ m ii × r )
r ×
G
i
i=1 i=1
(9.4.7)
 N     N  

×
r 
= ∑ m i QG v + QG × ωω ×  ∑ m ii
G
i  =1     i=1   
 N  N
+ ∑ m r  × v G + ∑ m r ×(ωω × ) r i
ii
ii
 i =1  = i 1
N
×
= MQG v G +++ ∑ m r ×(ωω × ) r i
00
ii
= i 1