Page 520 - Dynamics of Mechanical Systems

0593_C14_fm Page 501 Tuesday, May 7, 2002 6:56 AM

Stability 501

where A, B, and C are (see Eqs. (14.6.43)):

A = a b = c c
11 1 5

B = a b + b a − a b (14.6.54)
31 3 1 2 2

or

c c + c c − c c − φ 2 ˙
B =−( 35 c c ) +( 2 6 c c ) 0 (14.6.55)
1 7 4 5 1 8
and

C = a b = c c + φ 2 ˙ + c c φ 4 ˙ (14.6.56)
33 c c +( 3 8 c c ) 0 48 0
3 7
4 7
Finally, by solving Eq. (14.6.53) for λ , we have:
2
/
B
λ = −± (B 2 − 4AC ) 12 (14.6.57)
2
2A
*
Observe from Eq. (14.6.46) that the solutions θ and ψ for the motion following the
*
disturbance will be bounded (that is, the motion is stable) if λ is either negative or
imaginary. Expressed another way, the motion is stable if λ does not have any positive
2
real part. Thus, from Eq. (14.6.57) we see that λ will be negative, and there will be
stability if:

/
2
B A > 0 and either 4 AC > B 2 or [ B − 4 AC] 12 < B (14.6.58)
The first of these conditions is satisfied if A and B have the same signs. From Eq. (14.6.54)
we see that A is positive because c and c are positive. Therefore, the first inequality in
1
5
Eq. (14.6.58) is satisfied if B is positive. From Eq. (14.6.55), we see that B will be positive if:
cc − c c − c c > 0 (14.6.59)
26 4 5 1 8
and

1 8)
˙ 2
1 7) (cc
φ > (cc + c c − c c − c c (14.6.60)
0 35 2 6 4 5
Because each of the c (i = 1,…, 8) is positive (see Eq. (14.6.45)), the inequality of Eq. (14.6.60)
i
will be satisfied if the spin rate φ ˙ 0 is sufficiently large and the inequality of Eq. (14.6.59)
is satisfied. That is, the first inequality of Eq. (14.6.58) will be satisfied if φ ˙ 0 is sufficiently
large and if:

cc − c c − c c > 0 (14.6.61)
26 4 5 1 8