Page 453 - Dynamics of Mechanical Systems

0593_C12_fm Page 434 Monday, May 6, 2002 3:11 PM

434 Dynamics of Mechanical Systems

From Eq. (12.4.4) we see that the partial angular velocities of B are:
i

0 i ≠ j
ωω =  (12.5.4)
i B
 n 3 j
˙ θ j i =
Similarly, from Eq. (12.4.5), we see that the partial velocities of G are:
i
ln θ j j < i

i G
 l
ωω = ( )n2 j = i (12.5.5)

˙ θ j θ j
 0 j > i
From an examination of Eqs. (12.4.8), (12.4.9), and (12.4.10), the generalized forces are:

−
F =− [ N i +( )] mgl sin θ − Mgl sin θ (12.5.6)
12
i θ i i
where M is the mass of Q.
From a generalization of Eq. (12.4.11) the kinetic energy K of the N-rod system is:

K = ( ) ( ) +( ) ( ) +…+( ) ( ) 2
2
2
m v
m v
m v
12
12
12
G 2
G 1
G N
I ) ( ) +( ) ( ) +…+( ) ( ) 2
2
2
+(12 ωω 12 I ωω 12 I ωω (12.5.7)
B N
B 1
B 2
M ) ( ) 2
Q
+(12 v
where as before I is the central moment of inertia of a rod about an axis parallel to n and
3
is given by:
I = (112 m ) l 2 (12.5.8)
Then, by substituting into Eq. (12.5.7) from Eqs. (12.5.1), (12.5.2), and (12.5.3), K becomes:
N
N
K = ( ) ml 2 ∑ ∑ m θθ j (12.5.9)
˙˙
12
ij
i
i=1 j=1
where the coefficients m are:
ij
[
cos θ
1 2 1
m = ( ) + ( 2 N − p)+ ( 2 M m)] ( − θ i)
ij j
(12.5.10)
i ≠ j and p is the larger of i and j
and
ii [
−
m = N i +( ) +( M m)] (12.5.11)
13