**4.1. Main relations**

We assume that the center *O* of a gymnast's waist moves in time according to the elliptic law *x* = *a* sin *ωt*, *y* = *b* cos *ωt* with the amplitudes *a*, *b* and the excitation frequency *ω* > 0, Fig. 12. The equations of motion in the waist-fixed coordinate system take the following form

20 Will-be-set-by-IN-TECH

numerical first approx. second approx.

*θ* calculated from the first order approximate solution (72) and second order

<sup>0</sup> 2\*pi −2

time τ

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

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

**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*.

We assume that the center *O* of a gymnast's waist moves in time according to the elliptic law *x* = *a* sin *ωt*, *y* = *b* cos *ωt* with the amplitudes *a*, *b* and the excitation frequency *ω* > 0, Fig. 12.

There is also a rolling resistance due to the waist deformation (right).

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).

bi-directional motion has been treated numerically and experimentally in [39].

−1.5

−1

angular velocity dθ/dτ

keeping contact with the waist during twirling.

**Figure 11.** Angular velocity ˙

**4.1. Main relations**

−0.5

0

*IC* ¨ *θ* + *k* ˙ *<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*FTR* <sup>−</sup> *d N* sign(˙ *θ*), (76)

$$m(R - r)\ddot{\varphi} = m\left(\ddot{x}\sin\varphi + \ddot{y}\cos\varphi\right) + F\_{\nabla} \tag{77}$$

$$m(\mathcal{R} - r)\dot{\varphi}^2 = N + m\left(\ddot{\mathbf{x}}\cos\varphi - \ddot{\mathbf{y}}\sin\varphi\right) \,\,\,\,\,\tag{78}$$

where *θ* is the rotation angle around center of mass *C*, *IC* = *mR*<sup>2</sup> is the central moment of inertia of the hula-hoop, *ϕ* is the angle between axis *x* and radius *CO*� , *r* is the radius of the 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 transverse directions to the radius *CO*� , where *N* is the normal reaction force of the hula-hoop to the waist. Equations (77) and (78) contain additional inertial forces since the waist-fixed reference system is noninertial.

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

$$\left(\left(\mathbb{R} - r\right)\dot{\boldsymbol{\varphi}} = \mathbb{R}\dot{\boldsymbol{\theta}}\right) \tag{79}$$

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

$$\begin{split} \ddot{\varphi} &+ \frac{k}{2mR^2} \dot{\varphi} + \frac{d}{2R} \text{sign} \dot{\varphi} \left( \dot{\varphi}^2 + \frac{\omega^2 \left( a \sin \omega t \cos \varphi - b \cos \omega t \sin \varphi \right)}{R - r} \right) \\ &+ \frac{\omega^2 \left( a \sin \omega t \sin \varphi + b \cos \omega t \cos \varphi \right)}{2 \left( R - r \right)} = 0. \end{split} \tag{80}$$

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

$$(R - r)\,\dot{\varphi}^2 + \omega^2 \left( a \sin\omega t \cos\varphi - b \cos\omega t \sin\varphi \right) > 0\tag{81}$$

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

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

$$\gamma = \frac{k}{2m\mathbb{R}^2 \omega'}, \quad \delta = \frac{d}{2\mathbb{R}}, \quad \varepsilon = \frac{a-b}{4\left(R-r\right)}, \quad \mu = \frac{a+b}{4\left(R-r\right)},\tag{82}$$

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) and inequality (81) take the form

$$\begin{aligned} \ddot{\boldsymbol{\varrho}} &+ \gamma \dot{\boldsymbol{\varrho}} + \delta \dot{\boldsymbol{\varrho}} \, |\, \dot{\boldsymbol{\varrho}}\rangle + \mu \cos(\boldsymbol{\varrho} - \boldsymbol{\tau}) - 2\mu \delta \, \text{sign}(\dot{\boldsymbol{\varrho}}) \sin(\boldsymbol{\varrho} - \boldsymbol{\tau}) \\ &= \varepsilon \cos(\boldsymbol{\varrho} + \boldsymbol{\tau}) - 2\varepsilon \delta \, \text{sign}(\dot{\boldsymbol{\varrho}}) \sin(\boldsymbol{\varrho} + \boldsymbol{\tau}) \, \end{aligned} \tag{83}$$

$$
\dot{\varphi}^2 - 2\mu \sin(\varphi - \tau) + 2\varepsilon \sin(\varphi + \tau) > 0,\tag{84}
$$

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