Page 189 - Dynamics of Mechanical Systems

0593_C06_fm Page 170 Monday, May 6, 2002 2:28 PM

170 Dynamics of Mechanical Systems

6.4 Zero Force Systems
If a system of forces has a zero resultant and a zero moment about some point O, it is
called a zero force system. Recall from Eq. (6.3.6) the moment of the system about a point
Ô, distinct from O, is:

ˆ
M = M + OO R (6.4.1)
×
O ˆ O
Thus, if R is zero, M and M are equal. This means that if the resultant is zero, the
O Ô
moment of the force system about all points is the same. Then, if M is also zero, the
O
moment of the force system about all points is zero.
The concept of a zero force system is the basic principle of static analysis: If a mechanical
system is in static equilibrium, the set of all the forces exerted on the system is a zero
force system.

6.5 Couples

Closely related to zero force systems is the concept of a couple. Specifically, a force system
that has a zero resultant but a non-zero moment about some point O is called a couple.
Observe again from Eq. (6.4.1) that if R is zero, the force system has the same moment
about all points. Thus, a couple has the same moment about all points. This moment is
called the torque of the couple.
A couple may have a large number of forces or it may have as few as two forces (see
Figure 6.4.1). If a couple has only two forces it is called a simple couple. If a force system
S has many forces, it is often tedious and time consuming to conduct an analysis of the
effect of S on a physical system. This difficulty can be overcome by the use of equivalent
force systems. Two force systems are said to be equivalent if they have equal resultants and
equal moments about some point.
ˆ
S
Observe from Eq. (6.3.6) that if two force systems (say S and ) have equal resultants
ˆ
R
(R = ) and equal moments about some point O, then they have equal moments about
all points. To see this, let Q be any point distinct from O. Then, from Eq. (6.3.6) we have:
×
ˆ
ˆ
×
ˆ
M SQ = M SO + QO R and M SQ = M SO + QO R (6.5.1)

FIGURE 6.4.1
Couples.