Page 509 - Dynamics of Mechanical Systems

0593_C14_fm Page 490 Tuesday, May 7, 2002 6:56 AM

490 Dynamics of Mechanical Systems

3
Thus, if the angular speed of D exceeds gr , D will remain erect and continue to roll
3
in a straight line. If the angular speed of D is less than gr , the motion is unstable. D
will wobble and eventually fall.

Case 2: Rolling in a Circle
φ
˙
Next, suppose D is rolling in a circle with uniform speed such that θ, , and ˙ ψ are (see
Figure 14.5.1):

˙
=
˙
˙
θθ , φ = φ , ψ = ˙ ψ (14.5.14)
0 0 0
By inspection of the governing equations, we see that Eqs. (14.5.2) and (14.5.3) are then
identically satisfied and that Eq. (14.5.1) becomes:
( 4gr)sinθ 0 + ψ φ cosθ 0 + 5φ 2 0 sinθ 0 cosθ 0 = 0 (14.5.15)
6 ˙ ˙
0 0
Equation (14.5.15) provides a relationship between θ , φ ˙ 0 , and ˙ ψ 0 . By inspection of
0
Figure 14.5.1 we see that if φ ˙ 0 and ˙ ψ 0 are positive, then Eq. (14.5.15) requires that θ be
0
negative. That is, the disk will lean toward the interior of the circle on which it rolls.
To test the stability of this motion, let the disk encounter a small disturbance such that
˙
φ
θ, , and ˙ ψ have the forms:
=
˙
θθ + θ , φ = φ + φ , ψ = ˙ ψ * (14.5.16)
˙
* ˙
ψ + ˙
˙
*
0 0 0
where, as before, the quantities with the ( ) are small. Then, sinθ and cosθ are:
*
sinθ = ( θ * sinθ + θ * cosθ
sin θ + ) =
0 0 0 (14.5.17)
and
(
θ
cosθ = cos θ + ) = cosθ − θ * sinθ 0 (14.5.18)
*
0
0
By substituting from Eqs. (14.5.16), (14.5.17), and (14.5.18) into Eqs. (14.5.1), (14.5.2), and
(14.5.3) and by neglecting quadratic (second) and higher order terms in the ( ) terms we
*
obtain:
( 4gr)sinθ 0 + ψ φ cosθ 0 + 5φ 2 ˙ 0 sinθ 0 cosθ 0 ( 4gr) cosθ 0
θ
6 ˙ ˙
+
*
0 0
− 5θ + ψ φ cosθ + ψ φ cosθ − ψ φ θ * sinθ (14.5.19)
6 ˙ ˙
6 ˙ ˙
*
˙˙*
6 ˙ ˙ *
0 0 0 0 0 0 0
2 ˙
+ 10φφ sinθ cosθ + 5φ θ * cos θ − 5φ θ sin θ= 0
2 ˙
2
2
˙˙ *
* *
0 0 0 0 0 0 0
ψ +
˙˙*
3 ˙˙ * 3φ sinθ + 5φ θ cosθ = 0 (14.5.20)
˙ ˙ *
0 0 0