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0593_C11_fm Page 389 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 389

*
F = v G φ ˙ ⋅F * + ωω φ ˙ ⋅T *
φ
˙ ˙
=− mr sin ( ˙˙ + ˙˙ θ φθcos θ)
2
θψ φsin + 2
−( mr ) 4 sin ( 2 φsin + 2 θφcos θ)
˙ ˙
˙˙
θ
2
θψ ˙˙ + 2
(11.10.54)
−( mr ) 4 cos ( ˙˙ θ ψθ)
˙ ˙
2
θφcos + 2
[
˙ ˙
θ
˙˙
θ
θ
= mr −( ) ψ ˙˙ sin −( ) φsin θ −( ) φθsin cos θ
2
2
32
52
32
˙˙
−(14 φ ) cos 2 θ +( ) 2 ˙ ˙ θ]
ψθcos
1
and
F =− ( ˙˙ + ˙˙ θ φθcos θ)
˙ ˙
2
*
mr ψφsin + 2
ψ
)(
˙ ˙
− mr ( 2 42 ψ ˙˙ + 2 ˙˙ θ θφcos θ) (11.10.55)
φsin + 2
[
˙ ˙
=− mr ( ) ψ ˙˙ +( ) φsin +( ) θφcos θ ]
θ
˙˙
2
32
52
32
As an aside, we can readily develop the generalized applied (or active) forces for this
system. Indeed, the only applied forces are gravity and contact forces, and of these only
the gravity (or weight) forces contribute to the generalized forces. (The contact forces do
not contribute because they are applied at a point of zero velocity.)
The weight forces may be represented by a single vertical force W passing through G
given by:
W =−mg N =− ( sinθ n +cosθ n ) (11.10.56)
mg
3 2 3
Then, from Eq. (11.10.43), the generalized forces are:
θ
F = mgsin , F = , F = 0 (11.10.57)
0
φ
θ
x
11.11 Potential Energy
In elementary mechanics, potential energy is often defined as the “ability to do work.”
While this is an intuitively satisfying concept it requires further development to be com-
putationally useful. To this end, we will define potential energy to be a scalar function of
the generalized coordinates which when differentiated with respect to one of the coordi-
nates produces the negative of the generalized force for that coordinate. Specifically, we
define potential energy P(q ) as the function such that:
r
F = −∂ ∂ q ( r = …, n) (11.11.1)
D
1,
P
r
r