**4. Twirling of a hula-hoop**

A hula-hoop is a popular toy – a thin hoop that is twirled around the waist, limbs or neck. In recent decades it is widely used as an implement for fitness and gymnastic performances .<sup>2</sup> To twirl a hula-hoop the waist of a gymnast carries out a periodic motion in the horizontal plane. For the sake of simplicity we consider the two-dimensional problem disregarding the vertical motion of the hula-hoop. We assume that the waist is a circle and its center moves along an elliptic trajectory close to a circle.

Previously considered was the simple case in which a hula-hoop is treated as a pendulum with the pivot oscillating along a line, see [37, 38]. The stationary rotations of a hula-hoop

<sup>2</sup> The same model lies in the basis of some industrial machinery such as vibrating cone crushers designed for crushing hard brittle materials, see [33].

The equations of motion in the waist-fixed coordinate system take the following form

*<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*FTR* <sup>−</sup> *d N* sign(˙

where *θ* is the rotation angle around center of mass *C*, *IC* = *mR*<sup>2</sup> is the central moment of

waist, *m* and *R* are the mass and radius of the hula-hoop. Equation (76) describes change of angular momentum due to linear viscous damping with coefficient *k*, rolling drag (rolling resistance) with coefficient *d*, and the tangential friction force *FT* between the waist and the hoop. Equations (77) and (78) describe the motion of the hula-hoop in the longitudinal and

to the waist. Equations (77) and (78) contain additional inertial forces since the waist-fixed

(*<sup>R</sup>* <sup>−</sup> *<sup>r</sup>*) *<sup>ϕ</sup>*˙ <sup>=</sup> *<sup>R</sup>* ˙

We exclude from equations (76) and (77) the force *FT* and with relation (79) obtain the equation

which means that the hula-hoop during its motion keeps contact with the waist of the

<sup>2</sup>*R*, *<sup>ε</sup>* <sup>=</sup> *<sup>a</sup>* <sup>−</sup> *<sup>b</sup>*

where *γ* and *δ* are the damping and rolling resistance coefficients, *μ* and *ε* are the excitation parameters. Relation between *μ* and *ε* determines the form of ellipse – the trajectory of the waist center. For *ε* = *μ* the trajectory is a line, and for *ε* = 0 it is a circle. Then equation (80)

*ϕ*¨ +*γϕ*˙ + *δϕ*˙ |*ϕ*˙| + *μ* cos(*ϕ* − *τ*) − 2*μδ* sign(*ϕ*˙) sin(*ϕ* − *τ*)

4 (*R* − *r*)

= *ε* cos(*ϕ* + *τ*) − 2*εδ* sign(*ϕ*˙) sin(*ϕ* + *τ*), (83) *<sup>ϕ</sup>*˙ <sup>2</sup> <sup>−</sup>2*<sup>μ</sup>* sin(*<sup>ϕ</sup>* <sup>−</sup> *<sup>τ</sup>*) + <sup>2</sup>*<sup>ε</sup>* sin(*<sup>ϕ</sup>* <sup>+</sup> *<sup>τ</sup>*) <sup>&</sup>gt; 0 , (84)

Assuming that slipping at the point of contact is absent we obtain the kinematic relation

<sup>2</sup> (*<sup>R</sup>* <sup>−</sup> *<sup>r</sup>*) <sup>=</sup> 0.

*m*(*R* − *r*)*ϕ*¨ = *m* (*x*¨ sin *ϕ* + *y*¨ cos *ϕ*) + *FT* , (77)

<sup>2</sup> <sup>=</sup> *<sup>N</sup>* <sup>+</sup> *<sup>m</sup>* (*x*¨ cos *<sup>ϕ</sup>* <sup>−</sup> *<sup>y</sup>*¨ sin *<sup>ϕ</sup>*) , (78)

Dynamics of a Pendulum of Variable Length and Similar Problems 89

, where *N* is the normal reaction force of the hula-hoop

*<sup>ω</sup>*<sup>2</sup> (*<sup>a</sup>* sin *<sup>ω</sup><sup>t</sup>* cos *<sup>ϕ</sup>* <sup>−</sup> *<sup>b</sup>* cos *<sup>ω</sup><sup>t</sup>* sin *<sup>ϕ</sup>*) *R* − *r*

<sup>2</sup> <sup>+</sup> *<sup>ω</sup>*<sup>2</sup> (*<sup>a</sup>* sin *<sup>ω</sup><sup>t</sup>* cos *<sup>ϕ</sup>* <sup>−</sup> *<sup>b</sup>* cos *<sup>ω</sup><sup>t</sup>* sin *<sup>ϕ</sup>*) <sup>&</sup>gt; <sup>0</sup> (81)

, *<sup>μ</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *<sup>b</sup>*

*θ* . (79)

<sup>4</sup> (*<sup>R</sup>* <sup>−</sup> *<sup>r</sup>*) , (82)

(80)

*θ*), (76)

, *r* is the radius of the

*IC* ¨ *θ* + *k* ˙

*m*(*R* − *r*)*ϕ*˙

transverse directions to the radius *CO*�

reference system is noninertial.

*ϕ*¨ +

+

*k* <sup>2</sup>*mR*<sup>2</sup> *<sup>ϕ</sup>*˙ <sup>+</sup>

*d* <sup>2</sup>*<sup>R</sup>* sign*ϕ*˙

(*R* − *r*) *ϕ*˙

*<sup>γ</sup>* <sup>=</sup> *<sup>k</sup>*

and inequality (81) take the form

 *ϕ*˙ <sup>2</sup> +

From equation (78) we find the normal force and imply the condition *N* > 0 as

*ω*<sup>2</sup> (*a* sin *ωt* sin *ϕ* + *b* cos *ωt* cos *ϕ*)

We introduce new time *τ* = *ωt* and non-dimensional parameters

<sup>2</sup>*mR*2*<sup>ω</sup>* , *<sup>δ</sup>* <sup>=</sup> *<sup>d</sup>*

where the dot means differentiation with respect to the time *τ*.

of motion

gymnast.

inertia of the hula-hoop, *ϕ* is the angle between axis *x* and radius *CO*�

**Figure 11.** Angular velocity ˙ *θ* calculated from the first order approximate solution (72) and second order approximate solution (75) compared with the results of numerical simulations in the case of small damping *β*. Parameters: *δ* = 0, *μ* = 1, *ω* = 0.3, *ε* = 0.2, *β* = 0.01. Steady state averaged variables *Q* and *Z* are given by expressions in (71) for the first approximation while for the second approximation they are obtained numerically (*Q* = 2.0348, *Z* = 2.6838) from the second order averaged equations (69), (70).

excited in two directions have been studied by an approximate method of separate motions in [33]. The similar problem of the spinner mounted loosely on a pivot with a prescribed bi-directional motion has been treated numerically and experimentally in [39].

Here we derive the exact solutions in the case of a circular trajectory of the waist center and approximate solutions in the case of an elliptic trajectory. We also check the condition of keeping contact with the waist during twirling.

**Figure 12.** A hula-hoop with the radius *R* twirling with the angle *ϕ* around a circular waist (shaded) with the radius *r*. The center *O* of the waist moves along the elliptic curve *x* = *a* sin *ωt*, *y* = *b* cos *ωt* with the fixed center *O*. The hula-hoop acts on the waist with normal force *N* and tangential friction force *FT*. There is also a rolling resistance due to the waist deformation (right).
