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0593_C11_fm Page 355 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 355

Z
Y
P(x,y,z)

P(x,y)

r R Y

0 θ X
X
FIGURE 11.2.5 FIGURE 11.2.6
A particle P moving in a plane. A particle moving in a Cartesian reference frame.

The number of degrees of freedom of a mechanical system is the number of coordinates
of the system if it were unrestricted minus the number of constraint equations. For
example, if a particle P moves relative to a Cartesian reference frame R as in Figure 11.2.6,
then it has, if unrestricted, three degrees of freedom. If, however, P is restricted to move
in a plane (say, a plane parallel to the X–Z plane), then P is constrained, and its constraint
may be described by a single constraint equation of the form:

y = constant (11.2.3)

Hence, in this case there are three minus one, or two, degrees of freedom.
To further illustrate these concepts, consider the system of two particles P and P at
2
1
opposite ends of a light rod (a “dumbbell”) as in Figure 11.2.7. In a three-dimensional
space, the two particles with unrestricted motion require six coordinates to specify their
positions. Let these coordinates be (x , y , z ) and (x , y , z ) as shown in Figure 11.2.7.
1
2
1
2
1
2
Now, for the particles to remain at opposite ends of the rod, the distance between them
must be maintained at the constant value , the rod length. That is,
x − ) +( y − ) +( z − ) = l 2 (11.2.4)
1 (
2
2
2
z
x
y
2
1
2
2
1
This is a single constraint equation; thus, the system has a net of five degrees of freedom.
Z
P (x ,y ,z )
1 1 1 1

P (x ,y ,z )
2 2 2 2
R Y
FIGURE 11.2.7 X
Particles at opposite ends of a light rod.