Page 503 - Dynamics of Mechanical Systems

0593_C14_fm Page 484 Tuesday, May 7, 2002 6:56 AM

484 Dynamics of Mechanical Systems

2
If Ω > g/r, a small disturbance about the equilibrium position may be expressed as:
=
θθ + θ * (14.3.11)
3
Then, to the first order in θ , sinθ and cosθ and may be approximated as:
*
sinθ = ( θ * sinθ + θ * cosθ
sin θ + ) =
3 3 3
and

(
cosθ = cos θ + ) = cosθ − θ * sinθ 3 (14.3.12)
θ
*
3
3
By substituting from Eqs. (14.3.11) and (14.3.12) into (14.3.1) we obtain:
]
θ + Ω 2 ( [ sin θ − cos θ ) +(gr ) cosθ θ *
2
2
˙˙*
3
3
3
(14.3.13)
= Ω 2 sinθ cosθ −(gr ) sinθ
3 3 3
Because the right side of Eq. (14.3.13) is a constant, the stability (or instability) of the
equilibrium position is determined by the sign of the coefficient of θ . Specifically, the
*
2
2
equilibrium position is stable if the term [Ω (sin θ – cos θ ) + (g/r)cosθ ] is positive. If
2
3
3
3
the term is negative, the equilibrium position is unstable.
From the definition of θ in Eq. (14.3.3), we see that:
3
cosθ = grΩ 2 , cos θ = g r Ω 4
2
2
2
3 3
and
sin θ =− cos θ =−(gr Ω 4 ) (14.3.14)
2
2
2
2
1
1
3 3
*
Hence, the coefficient of θ in Eq. (14.3.13) becomes:
Ω sin θ − Ω cos θ +(gr ) cosθ = [ 2 2 Ω )]
4
2
2
2
2
2
Ω 1−(g r
3 3 3
− Ω (gr 2 Ω ) +(gr 2 Ω ) (14.3.15)
2
4
2
2
2
2
= Ω 1−( [ gr Ω ) ]
2
2
Therefore, the equilibrium position is stable if [1 – (g/rΩ ) ] is positive or if:
2 2
2
Ω > gr (14.3.16)
Comparing the inequality of Eq. (14.3.16) with that of Eq. (14.3.7) we see that they are
opposite. Also, as noted earlier, we see that Eq. (14.3.16) is a necessary condition for the
existence of the equilibrium position θ = θ = cos (g/rΩ ). That is, for slow tube rotation
–1
2
3