Page 379 - Dynamics of Mechanical Systems

0593_C11_fm Page 360 Monday, May 6, 2002 2:59 PM

360 Dynamics of Mechanical Systems

Thus, Eq. (11.4.2) may be expressed in the form:

n
v = v + ∑ v q ˙ (11.4.5)
t q r ˙ r
r =1
From Eq. (11.4.2), we see that the partial rate of change of the velocity v with respect to
q is:
s
∂ 
n
∂ 
n
∂v = ∂  ∂p + ∑ ∂p q ˙  = ∂  p  ∑ ∂  p q ˙
+
∂ 
q
∂q ∂q   ∂t ∂q r   ∂  t  ∂  ∂q   q  r
s s r =1 r s r =1 r s
(11.4.6)
∂ 
= d  p  = d v q s ˙

∂ 
dt  q s dt
Because the partial velocity vectors are analogous to base vectors or to unit vectors, it
is useful to compute the projections of forces along these vectors. These force projections
are called generalized forces. For example, the projections of inertia forces along partial
velocity vectors are called generalized inertia forces.
To further develop and illustrate these concepts, recall that inertia forces are proportional
to accelerations; hence, it is helpful to consider the projection of the acceleration a of a
particle P along the partial velocities v . Thus, we have:
˙ q r
av ⋅ = d v v ⋅ = d ( v v ⋅ ) −⋅ d ( )
v
v
dt dt dt
˙ q r ˙ q r ˙ q r ˙ q r
2
= 1 d   ∂ v   −⋅ ∂ v (11.4.7)
v
∂ 
2 dt  q ˙ r ∂q r
2
= 1 d   ∂ v   − 1 ∂ v 2
∂ 
2 dt  q ˙ r 2 ∂q r
Observe in Eq. (11.4.7) that the common factor (1/2)v in each term is proportional to
2
kinetic energy. Indeed, Eq. (11.4.7) forms a basis for the development of Lagrange’s equa-
tions that use kinetic energy. We will use it for this purpose in subsequent sections.
Consider next a rigid body B. Recall that in Section 4.5 we saw that, if c is a vector fixed
in B, the derivative of c in a reference frame R is ((see Eq. (4.5.2)):

R
dc = ωω ×
dt c (11.4.8)
where ωω ωω is the angular velocity of B in R defined as (see Eq. (4.5.1)):

R
R
R
 dn   dn   dn 
ωω=  2 ⋅n  n +  3 ⋅nn +  1 ⋅n  n (11.4.9)

 dt 3  1  dt 1  2  dt 2  3
where n , n , and n are mutually perpendicular unit vectors fixed in B.
1
3
2