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0593_C11_fm Page 373 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 373

or

F = mgcosθ − 2 kx + kx (11.7.16)
x 3 3 2
and
i + ) O G
⋅
⋅
⋅
⋅
F = ( O x O y j v − Mg j v + C 1 nv ˙ Q 1 + C 2 nv ˙ Q 2
θ
2
θ
θ
θ ˙
2
θ ˙
θ [
⋅
+C 3 nv ˙ Q 3 + −mg j+ ( k x 2 − x 1 n ) 1 − kx 1 1 1 n ] ⋅ v ˙ P 1
n − C
θ
2
2
[ n ) n ) P 2
+−mg j+ ( k x 3 − x 2 1 − ( k x 2 − x 1 1 − − C n 2 ] ⋅v ˙ θ
2
+− [ mgj − kx n 1 − ( k x 3 − )n 1 − C n 2 ] ⋅v ˙ θ P 3
x
3
2
3
( )
=− Mg L 2 sinθ + (l + ) + (2l + ) + (3l + )
C
x
x
C
x
0
C
3
1
1
3
2
2
− (
− (l + )sinθ − (l + ) − mg (2l + ) sinθ C 2l + )
x
C
x
mg
x
x
C
2
1
1
1
2
2
3 (
(
− mg 3l + ) sinθ − C 3l + )
x
x
3
3
or
(
( )
x
F =−Mg L 2 sin θ − mg 6l + x 1 + x 2 + ) sin θ (11.7.17)
θ
3
Observe that neither the pin forces nor the contact forces exerted across the smooth
surface of T appear in the expressions for the generalized forces. The pin forces are absent
because they are exerted at the point O, which has zero velocity and zero partial velocities.
The contact forces across the smooth surface of T do not appear because they are perpen-
dicular to the partial velocities of their points of application. In the following section, we
will explore and identify conditions that will always lead to the elimination of force
components from the generalized forces.
Finally, observe that the force components appearing in the generalized forces are from
gravitational and spring forces. This means that we could have obtained the generalized
forces using Eqs. (11.6.5) and (11.6.7). Specifically, the contribution F ˆ to the generalized
r
forces from the gravitational forces are:
ˆ
∂
F =− mg h ∂ q (11.7.18)
r i i r
where m is the mass of particle P and h is the elevation of P above an arbitrary reference
i
i
i
i
level. If we take the reference level to be the elevation of the pin O, we find the h (i = 1,
i
2, 3) to be (see Figure 11.7.6):
h =− + ) cosθ
(l
x
1
1
2 +
h =−( l x ) cosθ (11.7.19)
2
2
3 +
h =−( l x ) cosθ
3
3