Page 447 - Dynamics of Mechanical Systems

0593_C12_fm Page 428 Monday, May 6, 2002 3:11 PM

428 Dynamics of Mechanical Systems

j

i n 2 n 3
O
P n 3 3 x 3 n 2
1
θ 2
P
2 n x 2
1
x n 1
P 1
3
T
P P P T
1 2 3
FIGURE 12.3.4 FIGURE 12.3.5
Spring-supported particles in a rotating tube. Coordinates of the particles within the tube.
In Section 11.12 (Eq. (11.12.17)), we found the kinetic energy of the system to be:

K = ( ) m x +(l + x ) θ 2 ˙ + ˙ x +(2l + x ) θ 2 ˙
2
2
2
2
˙
12
[ 1 1 2 2
(12.3.25)
2
2 ˙
2 ˙
θ
2
+ ˙ x +(3l + x ) ] ( ) 6 ML θ 2
+ 1
3 3
where, as before, m is the mass of each particle, M is the mass of the tube T, is the natural
length of the springs, and L is the length of the tube T.
The forces doing work in this system are the gravity and spring forces. In Section 11.11
(Eq. (11.11.24)), we found the potential energy of the system to be:
P =− (l + x ) cosθ − mg( l + x )cosθ − mg( l + x )cosθ
mg
2
3
1 2 3
(12.3.26)
− Mg L ( ) 2 cosθ +(1 kx ) 2 2 1 2 x 2 1 2 k x − ) 2
x
1 +( ) ( 2 1 2
k x − ) +( ) ( 3
At this point we could use Eq. (12.3.6) to form the Lagrangian and then use Lagrange’s
equations in the form of Eq. (12.3.5) to obtain the equations of motion. For this system,
however, it might be simpler to use Lagrange’s equations in the form of Eq. (12.3.3), employing
generalized active forces and thus avoiding the differentiation of the potential energy function.
In Section 11.7 (Eqs. (11.7.14) to (11.7.17)), we found the generalized active forces to be:
F = mgcosθ + kx − 2 kx (12.3.27)
x 1 2 1
F = mgcosθ + kx + kx − 2 kx (12.3.28)
x 2 3 1 2
F = mgcosθ − 2 kx + kx (12.3.29)
x 3 3 2
F =− Mg L ( ) 2 sin θ − mg(6l + x + x + )sin θ (12.3.30)
x
θ
2
1
3
Lagrange’s equations are then (Eq. (12.3.3)):
d  ∂ K  ∂ K
,

dt ∂   − q ∂ r = F r r = x x x , 3 ,θ (12.3.31)
1
2

q ˙
r