Page 380 - Dynamics of Mechanical Systems

0593_C11_fm Page 361 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 361

Let B be a part of a mechanical system S which has n degrees of freedom represented
by n generalized coordinates q (r = 1,…, n). Specifically, let the movement of B in an
r
inertial reference frame R be determined by the coordinates q and time t. Then, if c is
r
fixed in B, the derivative of c in R is:

R n
dc = c ∂ + ∑ c ∂ q ˙
dt t ∂ q ∂ r (11.4.10)
r=1 r
By following the exact same procedure as in Section 4.5 in the development of Eq. (11.4.8)
we can readily show that the partial derivatives ∂c/∂t and ∂c/∂q may be expressed as:
r
∂∂ = ωω × c and ∂∂q = ωω × c (11.4.11)
t
c
c
t r q r ˙
where ωω ωω and ωω ˙ q r are defined as:
t
 ∂n   ∂n   ∂n 
ωω =  t ∂ 2 ⋅n 3  n 1 +   t ∂ 3 ⋅nn 2 +   t ∂ 1 ⋅n 2  n 3 (11.4.12)




t
1
 ∂n   ∂n   ∂n 
= 2 ⋅n n + 3 ⋅nn + 1 ⋅n n (11.4.13)
ωω ˙ q r   q ∂ r 3   1   q ∂ r 1   2   q ∂ r 2   3
where, as in Eq. (11.4.9) n , n , and n are mutually perpendicular unit vectors fixed in B,
2
3
1
and the partial derivatives are all computed in R. Then, by substituting from Eq. (11.4.11)
into (11.4.10), dc/dt becomes:
R
r )
R n  n 
dc = ωω × c + ∑ ( ωω × c ˙ q = ωω t ∑ ω q ×
+
˙
dt t q r ˙   q r ˙ r  c (11.4.14)
r=1 r=1
Recalling that c is an arbitrary vector fixed in B and by comparing Eqs. (11.4.8) and (11.4.14)
we see that the angular velocity of B in R may be expressed as:
n
=
ωωωω + ∑ ωω q ˙ (11.4.15)
t q r ˙ r
r =1
The vectors ω and ωω ˙ q r are called partial angular velocity vectors of B in R. As with the
t
partial velocity vectors of Eq. (11.4.3), the partial angular velocity vectors may be thought
of as being base vectors for the movement of B in the n-dimensional space of the q .
r
Partial velocity and partial angular velocity vectors are remarkably easy to evaluate:
from Eqs. (11.4.5) and (11.4.15) we see that the partial velocity and partial angular velocity
vectors v and ωω are simply the coefficients of the ˙ q . Specifically, we see from Eq.
˙ q r ˙ q r r
(11.4.15) that analogous to Eq. (11.4.4):

∂
ωω =∂ ωω ˙ q (11.4.16)
˙ q r r
To illustrate the ease of evaluation of these vectors, consider first the motion of a particle
P in a three-dimensional inertia frame R as in Figure 11.4.1. Let p locate P relative to a