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0593_C11_fm Page 377 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 377

q (r = 1,…, n). Let the points of contact between B and S be C and C , and let n be a unit
S
B
r
vector normal to the contacting surfaces at the point of contact.
Let F represent the forces exerted by B on S at C . Then, because the surface of B is
B
smooth, F may be expressed as:
F = F n (11.8.9)

Because B and S are in contact at C and C at the instant of interest, we have:
S
B
v ( C B − v ) ⋅= 0 or v C B n ⋅= v ⋅ n (11.8.10)
n
C s
C s
Then, by differentiating with respect to ˙ q , we have:
r
v C B n ⋅= v ⋅ n (11.8.11)
C s
˙ q r ˙ q r
where, as before, n is not a function of the ˙ q . Because the motion of B is specified, the
r
velocity of C is also independent of the q . Hence, we have:
r
B
= 0 (11.8.12)
C B
v ˙ q r
Thus, we also have:

n
v ⋅= 0 (11.8.13)
C s
The contributions F ˆ of F to the generalized forces F are then:
r r
ˆ
F =⋅Fv C s = F ⋅nv C s = 0 (11.8.14)
r ˙ q r ˙ q r
Forces that do not contribute to the generalized forces are sometimes called nonworking
because they are analogous to forces that do no work (see Section 10.2). A principal
advantage of the procedures of generalized mechanics is that nonworking forces (which
usually are of little interest) do not enter the analysis and thus can be ignored at the onset.

11.9 Generalized Forces: Inertia (Passive) Forces

As with applied forces, we can also introduce and define generalized inertia forces.
Specifically, a generalized inertia force is defined as the projection of an inertia force along
a partial velocity vector. Consider, for example, a particle P, having mass m and moving
in an inertial reference frame R as in Figure 11.9.1. From Eq. (8.2.5), we recall that the
*
inertia force F on P is:
*
F =−m a (11.9.1)