Page 522 - Dynamics of Mechanical Systems

0593_C14_fm Page 503 Tuesday, May 7, 2002 6:56 AM

Stability 503

Note that in the development of these equations we have neglected all products of small
*
quantities (those with a [ ]). Also, we have approximated trigonometric functions as:

=
sinθθ * , cosθ = 1 , sinψ =− ψ * , cosψ =−1 (14.6.70)
The last of these makes the sum (1 + cosψ) vanish, leading to the simple forms of Eqs.
(14.6.67), (14.6.68), and (14.6.69).
After further simplification, we see that Eqs. (14.6.67) and (14.6.68) may be written in
the forms:

a θ + a ψ + ˆ a θ = 0 (14.6.71)
*
ˆ ˙˙ *
ˆ ˙ *
1
2
3
and
ˆ
ˆ
b ψ + b θ + b ψ = 0 (14.6.72)
*
ˆ ˙ *
˙˙ *
1 2 3
where the a and b (i = 1, 2, 3) are:
i
i
ˆ a = 5 , a = − 6φ ˙ , a = −( 4 g r) − 5φ ˙ 2
ˆ
ˆ
1 2 0 3 0
and
ˆ
ˆ
ˆ
b = 3 , b = 5φ ˙ , b = ( 2 mg Mr) (14.6.73)
1 2 0 3
Following the procedure of the foregoing case we can solve Eqs. (14.6.67) and (14.6.68)
*
by letting θ and ψ have the forms:
*
*
*
θ = Θ e λt and ψ = Ψ e λt (14.6.74)
*
*
Then, Eqs. (14.6.67) and (14.6.68) become:
ˆ a λ Θ * λ t ˆ a λΨ * λ t ˆ a Θ * λ t 0 (14.6.75)
e =
e +
e +
2
1 2 3
and
ˆ
ˆ
ˆ
b λ Ψ * λ t b λΘ * λ t b Ψ * λ t 0 (14.6.76)
e +
e +
e =
2
1 2 3
or
( ˆ a λ + )Θ * + ˆ a λΨ * = 0 (14.6.77)
2
ˆ a
2
3
1
and
b λΘ + ( b λ + ) *
ˆ
ˆ
ˆ
b Ψ =
2
*
2 1 3 0 (14.6.78)