Page 518 - Dynamics of Mechanical Systems

0593_C14_fm Page 499 Tuesday, May 7, 2002 6:56 AM

Stability 499

*
where, as before, the quantities with the ( ) are small. Then, Eqs. (14.6.28), (14.6.29), and
(14.6.30) become (after simplification):
( Mr + I 4 mr )θ +− ( Mr − I − 6 mr )φ ψ
2
2
2 ˙ ˙ *
2 ˙˙*
0
11
22
0]
[
2 ˙
−
+− Mgr 2 mgr −( Mr + I − I + 4 mr )φθ * = 0 (14.3.37)
2
2
22
33
( Mr + I + 4 mr )ψ +( 2 Mr + I + 8 mr )φ θ
2
2
2
2 ˙ ˙ *
˙˙ *
22
22
0
−( mr φ + mgr)ψ * = (14.6.38)
2 2 ˙
2
0 0
and
˙˙*
φ= 0 (14.6.39)
To solve these equations, let I = I = (1/2)I = (1/4)Mr , and let Eqs. (14.6.37) and
2
11
33
22
(14.6.38) be written in the forms:
ψ +
a θ + a ˙ * a θ = 0 (14.6.40)
˙˙*
*
1 2 3
and
ψ +
*
˙ *
b ˙˙ * b θ + b ψ = 0 (14.6.41)
1 2 3
where the coefficients a and b (i = 1, 2, 3) are:
i
i
a = ( 45) Mr + 2 a = − ( [ 3 2) Mr + mr φ 0 ]
2 ˙
2
2
1 4 mr , 2 6
−
2 ˙
a =− Mgr 2 mgr − ( [ 5 4) Mr + 4 mr φ 0 ] 2
2
3
b = ( 32) Mr + 2 b = ( [ 5 2) Mr + mr φ 0 ] (14.6.42)
2 ˙
2
2
1 4 mr , 2 8
2 ˙
b =− mr ( [ g r) + 2φ 0]
2
3
Observe that the signs of these coefficients are:
a > 0, a < 0, a < 0
1 2 3
(14.6.43)
b > 0, b > 0, b < 0
1 2 3
Observe further that the a and b (i = 1, 2, 3) may be written in the forms:
i
i
a = , a =− φ ˙ , a =− c − φ 2 ˙
c
c
c
1 1 2 2 0 3 3 4 0
(14.6.44)
b = c , b = φ ˙ b =− c − φ 2 ˙
1 5 2 c 6 0 , 3 7 c 8 0