Page 450 - Dynamics of Mechanical Systems

0593_C12_fm Page 431 Monday, May 6, 2002 3:11 PM

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 431

By substituting from Eqs. (12.4.4), (12.4.5), and (12.4.6), F becomes:
θ 1
F =− mg(l 2)sin θ − mglsin θ − mglsin θ + M − M − Mglsin θ
θ 1 1 1 1 1 2 1
or

F =−( 52) mglsin θ + M − M − Mglsin θ (12.4.8)
θ 1 1 1 2 1
Similarly, F and F are:
θ 2 θ 3
F =−( 32) mglsin θ + M − M − Mglsin θ (12.4.9)
θ 2 2 2 3 2
and

F =−( 12) mglsin θ + M − Mglsin θ (12.4.10)
θ 3 3 3 3
The kinetic energy K of the system may be expressed as:

K = ( ) m( ) +( ) ( ) +( ) m( ) +( ) ( ) 2
2
2
2
12 I ωω
12 I ωω
v
12
v
12
G 2
G 1
B 2
B 1
(12.4.11)
2
2
12 I ωω
Q
v
12 M v
+(12 m ) ( ) +( ) ( ) +( ) ( ) 2
G 3
B 3
where I is the central moment of inertia of a rod about an axis normal to the rod and given
by:
I = ( )ml 2 (12.4.12)
12
Using Eqs. (12.4.1) and (12.4.2) K becomes (after simplification):
K = ( ) [ 2 ˙ +( ) 3 θ 2 ˙ +( )θ 2 ˙ + 3θ θ cos θ − )
˙ ˙
θ
m ( ) 3 θ
2
7
12
l
1 4 2 13 3 1 2 ( 2 1
+θ θ cos θ − ) + θ θ cos θ − )]
˙˙
θ
˙˙
θ
2 3 ( 3 2 1 3 ( 3 1 (12.4.13)
˙ ˙
2 ˙
+(12 M ) l θ 1 [ 2 + θ 2 ˙ 2 + θ 2 3 + 2θ θ 3 cos θ 2 ( − θ )
1
1 1
˙˙
˙˙
+ 2θθ cos 3 (θ − θ 2) + 2θθ cos 3 (θ − θ 1)]
2 3 3 1
By differentiating in Eq. (12.4.13), we obtain the following terms, useful in Lagrange’s
equations:
˙˙
˙˙
θ
θ
∂K ∂θ = ( )ml 2 [ 3 θ θ sin (θ − ) − θ θ sin (θ − )]
12
1 12 2 1 1 3 1 3 (12.4.14)
[ ˙ ˙ − )]
θ
2
+( ) Ml 2 θθ sin (θ 2 − ) − θ θ sin (θ 1 θ 3
2
12
2
1
3
1
1