Page 195 - Dynamics of Mechanical Systems

0593_C06_fm Page 176 Monday, May 6, 2002 2:28 PM

176 Dynamics of Mechanical Systems

To summarize: if S is a given force system, and if O is an arbitrarily chosen point, then
*
*
there exists an equivalent force system S consisting of a single force F passing through
*
a point Q together with a couple with torque M where F and M are:
*
*
*
F = R(resultant of S ) (6.6.15)
*
and
n n (
M = ( M ⋅ ) n = R R)
*
O R R R (6.6.16)
and where Q is located relative to O by the expression:
*
OQ = ( R M ) 2
×
O R (6.6.17)
*
The magnitude of M will be smaller (or, at most, equal to) the magnitude of the torque
of any other equivalent force–couple system.
*
Observe that F is parallel to M . This equivalent force system thus has the characteristics
*
of a push or a pull with a twist about the line of action of the push or pull. Such a system
is called a wrench (although it might be better be described as a screwdriver).
As an example, consider again the force system on the box as in Figures 6.3.6 and 6.5.1
and reproduced in Figure 6.6.2. Recall from Eqs. (6.3.14) and (6.3.15) that the resultant and
the moment of the system about point O are:

R = 3 n + 24 n + 24 n lb (6.6.18)
1 2 3
and

M = 160 n − 30 n − 168 n ftlb (6.6.19)
O 1 2 3
Then, from Eqs. (6.6.15) and (6.6.16), the wrench equivalent to the force system consists
*
*
*
of the single force F and couple with the torque M where F and M are:
*
F = R = 3 n + 24 n + 24 n lb (6.6.20)
*
1 2 3

FIGURE 6.6.2
Force system applied to a box.