Page 383 - Dynamics of Mechanical Systems

0593_C11_fm Page 364 Monday, May 6, 2002 2:59 PM

364 Dynamics of Mechanical Systems

F F F 2
1
B
P P i
P P 2
1
P
N
F
R i
R
F N
FIGURE 11.5.1 FIGURE 11.5.2
A force F applied at a point P of a mechanical A rigid body B with applied forces F i at points
system S. P i .
Observe in Eq. (11.5.1) that the number of generalized forces is the same as the degrees
of freedom of the system. Observe further that if the force F is perpendicular to the partial
velocity v then the generalized force F is zero. Also, if v is zero, F is zero. Finally,
˙ q r r ˙ q r r
observe that the dimensions and units of generalized forces are not necessarily the same
as those of forces. Indeed, the dimensions and units of generalized forces depend directly
upon the dimensions and units of the partial velocity vectors which in turn depend upon
the choice of the coordinate q .
r
Consider next a set of forces F (i = 1,…, N) applied at points P of the system S. The
i i
generalized forces for this set of forces are then obtained by adding the generalized forces
from the individual forces. That is,

r ∑
F = N Fv q r P i ˙ ( r = …, n) (11.5.2)
⋅
1
,
i
i=1
where, as before, n is the number of degrees of freedom, represented by the coordinates q .
r
Finally, consider a rigid body B moving in an inertial frame R as in Figure 11.5.2. Let
forces F (i = 1,…, N) be applied at points P of B as shown. Then, from Eq. 11.5.2, the
i i
generalized forces are:
r ∑
⋅
F = N Fv q r P i ˙ ( r = …, n) (11.5.3)
,
1
i
i=1
where as before n is the number of degrees of freedom represented by the coordinates q .
r
Because in this case we have a rigid body we can obtain a different form of Eq. (11.5.3)
by using the rigidity of the body. Specifically, let Q be an arbitrary point of B. Then, from
Eq. (4.94), we can express the velocity of P in terms of Q as:
i
v = v + ωω × r (11.5.4)
Q
P i
i
where ωω ωω is the angular velocity of B in R, and r locates P relative to Q as in Figure 11.5.3.
i i
By using Eqs. (11.4.5) and (11.4.14), we can differentiate Eq. (11.5.4) with respect to the
˙ q (r = 1,…, n) leading to the expression:
r
v = v + ωω × ( ,… ,n) (11.5.5)
Q
rr = 1
P i
˙ q r ˙ q r ˙ q r i