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Twirling of Hula-hoop: New Results
A. O. Belyakov 1,2 A. P. Seyranian,2

1 2

Vienna University of Technology

Lomonosov Moscow State University

25 July 2011


Literature
Caughey T.K., Hula-hoop: an example of heteroparametric excitation, American J. Physics, 1960. 28(2). pp. 104­109. Bogolyubov, N. N. and Mitropolsky, Yu. A. Asymptotic Methods in the Theory of Nonlinear Oscillations, Nauka, Moscow, 1961. I.I. Blekhman, Vibrational Mechanics, Fizmatlit, Moscow, 1994. A.O. Belyakov and A.P. Seyranian, The hula-hoop problem, Doklady Physics, 2010, Vol. 55, No. 2, pp. 99­104.


Formulation of the problem
­ angle about the mass center C IC = mR 2 ­ central moment of inertia m and R ­ mass and radius of the hula-hoop k ­ coefficient of viscous friction r ­ radius of the waist FT ­ friction force N ­ reaction force ­ angle between x and CO

x = a sin t , IC m m ¨ +k (R - (R -

y = b cos t

= -FT R r ) = m (x sin + y cos ) + FT ¨ ¨ ¨ r ) 2 = m (x cos - y sin ) + N ¨ ¨

(R - r ) = R ­ non-slippage condition, N > 0 ­ non-separability condition


Equation of motion and non-separability condition
+ ¨ k 2 (a sin t sin + b cos t cos ) + =0 2mR 2 2 (R - r )

N = m (R - r ) 2 + m 2 (a sin t cos - b cos t sin ) > 0 we use simple trigonometric relations a sin t sin + b cos t cos = a sin t cos - b cos t sin =
a+b 2 a+b 2

cos( t - ) - sin( t - ) +

a -b 2 a -b 2

cos( t + ), sin( t + ),

we introduce new time = t and dimensionless parameters = k , 2mR 2 µ= a+b > 0, 4(R - r ) = a-b 0, 4(R - r ) (1) (2)

+ + µ cos( - ) = cos( + ) ¨ 2 - 2µ sin( - ) + 2 cos( + ) > 0,


Exact solution of the unperturbed equation
The unperturbed equation ( = 0) + + µ cos( - ) = 0 ¨ has the exact solutions = + 0 if | | µ, where constants 0 mod 2 are defined from + µ cos 0 = 0. Asymptotic stability conditions > 0, µ sin 0 < 0, (4) (3)

yield 0 < < µ. Inseparability condition (2) takes the form 1 - 2µ sin 0 > 0 The rotation is asymptotically stable and inseparable with 0 = - arccos(- /µ), while unstable rotation with 0 = arccos(- /µ) is inseparable only if µ < 1/4 + 2 . (5)


Approximate solution ( = 0)
We represent the solution as the series = + 0 + 1 ( ) + . . . 0 : :
1

+ µ cos 0 = 0, 1 + 1 - µ sin(0 ) 1 = cos(0 + 2 ), ¨

(6) (7)

where we take 0 = - arccos(- /µ) corresponding to the stable solution of the unperturbed system. Thus, (7) takes the form 1 + 1 + ¨ µ2 - 2 1 = cos(0 + 2 ) (8)

and has the unique periodic solution 1 ( ) = C sin(0 + 2 ) + D cos(0 + 2 ) -4+ µ2 - 2 , D= 2 2 2 C= 2 2 2 2
3 +µ -8 µ - +16 3 +µ -8

(9)
2

where

µ - 2 +16


Stability analysis
We add a small perturbation = + u to the true solution of (1) and linearize (1) w.r.t. u obtaining the Mathieu-Hill equation with damping u + u + ¨ µ2 - 2 + (2 ) u = 0, (10)

where = ( C + 1) sin(2 + 0 ) + D cos(2 + 0 ) + O (). Stability condition (absence of parametric resonance in (10)) < 2 ( C + 1)2 + 2 D
2

+ o ()

(11)

is also the stability condition for the original equation (1) according to the Lyapunov's theorem.


Non-separability condition in the first approximation

Solution
= - arccos -

µ

+ C sin(0 + 2 ) + D cos(0 + 2 ) + o ()

We substitute it into condition (2) that the hula-hoop is not separated from the waist 2 - 2µ sin( - ) + 2 cos( + ) > 0, which is guaranteed for all if the following inequality is satisfied < 1 + 2 µ2 - 2
2

µ2 + 3 2 - 8 µ2 - 2 + 16 µ2 + 8 2 - 12 µ2 - 2 + 36

+ o ().


Comparison between approximate and numerical solutions
1.1 numer. approx.

angular velocity

= 0.2 µ = 1.2 = 1.0 a = 1.4 2(R - r ) b = 1.0 2(R - r )

1

0.9 0

time

2*pi

6 -

Angular velocity first approximation (solid line) numerical simulation (circles).


If µ and are also small of order then both direct and reverse rotation are possible


Approximate solution when µ and are small
We represent the solution as the series = + 0 + 1 ( ) + . . . We introduce new not small parameters µ = µ/, ~ First adjustment 1 is defined by the equation: = /. ~ (13) (12)

1 = cos( + 0 + ) - µ cos( + 0 - ) - ¨ ~ ~

By equating separately constant and oscillation terms we find that solutions exist only if angular velocity {1, 0, -1}. With = 1 we get the first order adjustment 1 1 ( ) = - cos(0 + 2 ) , 4 cos 0 = - µ


Stability and non-separability conditions with small µ
Approximate solution with = 1
= + 0 - cos (0 + 2 ) + o (), 4 cos 0 = - µ (14)

Stability conditions in first approximation
µ

0 < < µ,

sin 0 < 0 0 = - arccos -

mod 2

Condition that the hula-hoop is not separated from the waist
< 1 + 2 µ2 - 2 + o (). 3 (15)


Stability and non-separability conditions with small µ
Approximate solution with = -1
= - + 0 + µ cos (0 - 2 ) + o (), 4 cos 0 = - (16)

Stability conditions in first approximation
0 < < , sin 0 > 0 0 = arccos - mod 2

Condition that the hula-hoop is not separated from the waist
µ< 1 + 2 2 - 2 + o (). 3 (17)


Condition of coexistence of direct and reverse rotations

Condition of coexistence direct and reverse rotations 0 < < min{, µ} (18)

is obtained from combination of 0 < < µ (for direct rotation) and 0 < < (for reverse rotation), where both parameters and µ are supposed to be small. In physical variables (18) takes the form 0 < 2k R -r < a - |b | R 2m (19)

i.e. the trajectory of the waist center should be sufficiently prolate.


Angular velocity of direct rotation
numer. approx. 1.2 angular velocity

= 0.6 µ = 0.8 = 0.5 a = 1.4 2(R - r ) b = 0.2 2(R - r )

1

0.8 0 2*pi

time

Angular velocity first approximation (solid line) numerical simulation (circles).

6

-


Angular velocity of reverse rotation
-0.6 numer. approx.

= 0.6 µ = 0.8 = 0.5 a = 1.4 2(R - r ) b = 0.2 2(R - r )

-1.4 0

angular velocity

-1

time

2*pi

Angular velocity first approximation (solid line) numerical simulation (circles).

6

-


Conclusion
Exact solutions for the hula-hoop under a circular excitation are obtained and their stability is studied Approximate solutions for an elliptic excitation are found The non-separability condition of the hula-hoop from the waist of a gymnast during rotation is derived The coexisting rotations for the direct and reverse rotations of the hula-hoop are analyzed The analytical solutions are compared with the results of numerical simulation A.P. Seyranian and A.O. Belyakov, How to twirl a hula hoop, American Journal of Physics, 2011, Vol. 79, Issue 7, pp. 712­715.