Page 185 - Dynamics of Mechanical Systems

0593_C06_fm Page 166 Monday, May 6, 2002 2:28 PM

166 Dynamics of Mechanical Systems

The resultant R of a system of forces S is simply the sum of the forces. That is,

N
R = F (6.3.1)
∑ i
= i 1
Unlike the individual forces of S, the resultant of S is a free vector and is not associated
with any particular point of V.
The moment of a system S of forces about a point O, designated by M S/O or M , is simply
O
the sum of the moments of the individual forces of S about O. That is,

O ∑
M SO = M = N p × F i (6.3.2)
i
i=1

where p is the position vector locating an arbitrary point, typically P , on the line of action
i
i
of F , relative to O (see Figure 6.3.3). Like R, M is also a free vector.
O
i
The resultant of a system of forces is unique. The moment of the system about an
ˆ
arbitrary point O, however, depends upon the location of O. If some other point, say O,
is chosen instead of O, the moment M S/O ˆ or M ˆ is generally different than M . But, this
O
O
raises the question: What is the relationship between M ˆ and M ? To answer this question
O
O
ˆ
consider Figure 6.3.4 depicting a force system S, points O and O, and position vectors p i
ˆ
and , locating a point on the line of action of typical force F , relative to O and O. Then,
ˆ p
i
i
from the definition of Eq. (6.3.2) and M and M ˆ are:
O
O
O ∑
O ∑
M = N p × F and M = N p × F i (6.3.3)
ˆ
ˆ
i
i
i
i=1 i=1
ˆ p
From Figure 6.3.4, we see that p and are related by:
i
i
ˆ
p = ˆ p + OO (6.3.4)
i i

FIGURE 6.3.3 FIGURE 6.3.4
ˆ
Position vector p i for moment calculation. A force system S and points O and O.