Page 375 - Dynamics of Mechanical Systems

0593_C11_fm Page 356 Monday, May 6, 2002 2:59 PM

356 Dynamics of Mechanical Systems

Y P B
P (x ,y ) 1
1
1
1
P
2
p
p 2 P
1 3
P (x ,y ) R
2 2 2 O p
3
0 X

FIGURE 11.2.8 FIGURE 11.2.9
A dumbbell moving in the X–Y plane. A rigid body B moving in a reference frame R.

If the movement of the dumbbell system is further restricted to the X–Y plane, additional
constraints occur, as represented by the equations:

z = 0 and z = 0 (11.2.5)
1 2
These expressions together with Eq. (11.2.4) then form three constraint equations, leaving
the system with six minus three, or three, degrees of freedom. These degrees of freedom
might be represented by either the parameters (x , y , θ) or (x , y , θ) as shown in Figure
1
1
2
2
11.2.8.
As a final illustration of these ideas consider a rigid body B composed of N particles P i
(i = 1,…, N) moving in a reference frame R as in Figure 11.2.9. The rigidity of B requires
that the distances between the respective particles are maintained at constant values.
Suppose, for example, that P , P , and P are noncollinear points. Let p , p , and p locate
1
2
2
3
3
1
P , P , and P relative to O in R. Then, the respective distances between these particles are
1
3
2
maintained by the equations:
1 ( 2 1 ( 2 2 ( 2 2
2
2
p
p − ) = d , p − ) = d , p − ) = d 3 (11.2.6)
p
p
1
3
2
3
2
where the distances d , d , and d are constants.
3
2
1
P , P , and P thus form a rigid triangle. The other particles of B are then maintained in
1
2
3
fixed positions relative to the triangle of P , P , and P by the expressions:
2
1
3
i (
i (
i (
2
2
2
p − ) = d , p − ) = f , p − ) = g ( i = 4, …, N) (11.2.7)
2
2
2
p
p
p
3
i
1
i
2
i
Equations (11.2.6) and (11.2.7) form 3 + 3(N – 3) or 3N – 6 constraint equations. If the
particles of B are unrestricted in their movement, 3N coordinates would be required to
specify their position in R. Hence, the number of degrees of freedom n of the rigid body is:
6
n = 3 N −(3 N − ) = 6 (11.2.8)