Page 395 - Dynamics of Mechanical Systems

0593_C11_fm Page 376 Monday, May 6, 2002 2:59 PM

376 Dynamics of Mechanical Systems

S S
1 2
C n B S
F 2 1 C 2 S
F C
1 B C
F S
n

FIGURE 11.8.2 FIGURE 11.8.3
Forces transmitted across smooth contacting surfaces A force exerted on a mechanical system by a
of a mechanical system. smooth body with specified motion.

Because F and F are both normal to the contacting surfaces at the point of contact, they
2
1
may be expressed as:
F =− F n and F = F n (11.8.3)
1 1 2 2

where n is a unit vector normal to the contacting surfaces at the point of contact. Then,
by the law of action and reaction (see Reference 11.3), we have:

F = − F or F + F = 0 or F + F = 0 (11.8.4)
1 2 1 2 1 2
Because S and S are in contact at the instant of interest, the relative velocities of the
2
1
contact points C and C in the normal direction are zero. That is,
2
1
( v − ) ⋅= 0 or v ⋅= v ⋅ n (11.8.5)
v
n
n
C 1
C 2
C 1
C 2
˙ q
Then by differentiating with respect to we have:
r
( v − v C 2 ) n ⋅= 0 or v ⋅= C 2 n ⋅ (11.8.6)
n
C 1
C 1
˙ q r
˙ q r
˙ q r
˙ q r
(Observe that n is not a function of the ˙ q r )
The contributions F ˆ of F and F to the generalized forces F are then:
r 1 2 r
⋅
F = F v + F v C 2 (11.8.7)
⋅
ˆ
C 1
r 1 ˙ q r 2 ˙ q r
Then by using Eqs. (11.8.4) and (11.3.6), we have:
(
(
ˆ
F = F ⋅ v C 1 − v C 2 ) = F ⋅ v C 1 − v C 2 ) = 0 (11.8.8)
n
r 1 ˙ q r ˙ q r 1 ˙ q r ˙ q r
Finally, consider the forces exerted on a mechanical system S by a body B which has a
smooth surface and whose motion is specified (that is, known or given) in an inertial
frame R (see Figure 11.8.3). As before, let S have n degrees of freedom with coordinates