**4.2. Exact solutions**

When the waist center moves along a circle (*a* = *b*, i.e. *ε* = 0) equation (83) takes the following form

$$\left|\ddot{\varphi} + \gamma \dot{\varphi} + \delta \dot{\varphi}\right| \left|\dot{\varphi}\right| + \mu \cos(\varphi - \tau) - 2\mu \delta \operatorname{sign}(\dot{\varphi}) \sin(\varphi - \tau) = 0\tag{85}$$

and has the exact solution [6]

$$
\varphi = \tau + \varphi\_0 \tag{86}
$$

*4.2.2. Condition for hula-hoop's contact with the waist*

(84) which takes the form

the condition *γδ* < 1/4.

to twirl a hula-hoop!

**4.3. Approximate solutions**

the following chain of equations

It has a unique periodic solution

exact solution *ϕs*(*τ*) of (83) can be expressed in a series

*ε*

*ε*

where constants *C* and *D* are defined as follows

*C* = <sup>2</sup>*<sup>γ</sup> μ*<sup>2</sup>+3*γ*<sup>2</sup>−8

*ϕ*¨1 + *γϕ*˙ <sup>1</sup> +

Taking solution of equation (95) for *μ* > 0

Let us verify for the exact solutions (86), (88) the condition of twirling without losing contact

Thus, stable solution (86), (91) provides asymptotically stable twirling of the hula-hoop with the constant angular velocity *ω* without losing contact with the waist of the gymnast under

Without rolling resistance *δ* = 0 condition (93) is always satisfied for the stable solution (86), (91). While for unstable solution (86), (92) we have *<sup>μ</sup>* sin *<sup>ϕ</sup>*<sup>0</sup> <sup>=</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup> so condition (93) holds only if *<sup>μ</sup>* <sup>&</sup>lt; 1/4 <sup>+</sup> *<sup>γ</sup>*2. The phase *<sup>ϕ</sup>*<sup>0</sup> of the stable solution belongs to the interval [−*π*, −*π*/2] mod 2*π*, and for vanishing damping *γ* → +0 the phase tends to −*π*/2. Below we will show that this phase inequality also holds for the approximate solutions. This is how

Let us find approximate solutions for the case of close but not equal amplitudes *a* ≈ *b*. For the sake of simplicity from now on we will keep *δ* = 0 and assume that *a* ≥ |*b*| which means *ε* ≥ 0, *μ* ≥ 0. Taking *ε* as a small parameter we apply perturbation method assuming that the

After substitution of series (94) in (83) and grouping the terms by equal powers of *ε* we derive

corresponding to the stable unperturbed solution (86), (91) we write equation (96) as

<sup>√</sup>*μ*<sup>2</sup>−*γ*<sup>2</sup>+<sup>16</sup> , *<sup>D</sup>* <sup>=</sup> <sup>−</sup>4<sup>+</sup>

*ϕs*(*τ*) = *τ* + *ϕ*<sup>0</sup> + *εϕ*1(*τ*) + *o*(*ε*). (94)

<sup>0</sup> : *γ* + *μ* cos(*ϕ*0) = 0 (95)

<sup>1</sup> : *<sup>ϕ</sup>*¨1 <sup>+</sup> *γϕ*˙ <sup>1</sup> <sup>−</sup> *<sup>μ</sup>* sin(*ϕ*0)*ϕ*<sup>1</sup> <sup>=</sup> cos(*ϕ*<sup>0</sup> <sup>+</sup> <sup>2</sup>*τ*) (96)

*ϕ*<sup>0</sup> = − arccos(−*γ*/*μ*) + 2*πn*, *n* = 1, 2, . . . (97)

*ϕ*1(*τ*) = *C* sin(2*τ* + *ϕ*0) + *D* cos(2*τ* + *ϕ*0), (99)

*μ*<sup>2</sup>+3*γ*<sup>2</sup>−8

<sup>√</sup>*μ*<sup>2</sup>−*γ*<sup>2</sup>

<sup>√</sup>*μ*<sup>2</sup>−*γ*<sup>2</sup>+<sup>16</sup> . (100)

*<sup>μ</sup>*<sup>2</sup> − *<sup>γ</sup>*<sup>2</sup> *<sup>ϕ</sup>*<sup>1</sup> = cos(*ϕ*<sup>0</sup> + <sup>2</sup>*τ*). (98)

1 2

2*δ* (*γ* + *δ*) <sup>1</sup> <sup>+</sup> <sup>4</sup>*δ*<sup>2</sup> .

. (93)

Dynamics of a Pendulum of Variable Length and Similar Problems 91

*μ* sin *ϕ*<sup>0</sup> <

The second stability condition in (90) can be rewritten with the use of (87) as follows

*μ* sin *ϕ*<sup>0</sup> <

with the constant initial phase *ϕ*<sup>0</sup> given by the equation

$$
\gamma + \delta + \mu \cos \varphi\_0 - 2\mu \delta \sin \varphi\_0 = 0. \tag{87}
$$

Therefore, solution (86) exists only under the condition |*γ* + *δ*|≤|*μ*| √ 1 + 4*δ*2, so we find from equation (87)

$$\varphi\_0 + \arccos\left(\frac{1}{\sqrt{1+4\delta^2}}\right) = \pm \arccos\left(-\frac{\gamma+\delta}{\mu\sqrt{1+4\delta^2}}\right) + 2\pi n, \quad n = 0, 1, 2, \dots \tag{88}$$

provided that *μ* �= 0. Solutions (86), (88) correspond to the rotation of the hula-hoop with the constant angular velocity equal to the excitation frequency *ω*.

### *4.2.1. Stability of the exact solutions*

Let us investigate the stability of the obtained solutions. For this purpose we take the angle *ϕ* in the form *ϕ* = *τ* + *ϕ*<sup>0</sup> + *η*(*τ*) where *η*(*τ*) is a small quantity, and substitute it into equation (85). Taking linearization with respect to *η* and with the use of (87) we obtain a linear equation

$$
\ddot{\eta} + \left(\gamma + 2\delta\right)\dot{\eta} - \mu \left(\sin\varphi\_0 + 2\delta\cos\varphi\_0\right)\eta = 0.\tag{89}
$$

According to Lyapunov's theorem on the stability based on a linear approximation [17] solution (86), (88) is asymptotically stable if all the eigenvalues of linearized equation (89) have negative real parts. From Routh-Hurwitz criterion [17] we obtain the stability conditions as

$$
\gamma + 2\delta > 0, \quad \mu \left(\sin \varphi\_0 + 2\delta \cos \varphi\_0\right) < 0. \tag{90}
$$

Without loss of generality we assume *μ* > 0 since the case *μ* < 0 can be reduced to the previous one by the time transformation *τ*� = *τ* + *π* in equation (83). The second condition in (90) can be written as sin(*ϕ*<sup>0</sup> <sup>+</sup> arccos(1/<sup>√</sup> 1 + 4*δ*2)) < 0. Thus, from conditions (90), relation (88) and due to the assumption *μ* > 0 we find that for 0 < *γ* + 2*δ* < *δ* + *μ* √ 1 + 4*δ*<sup>2</sup> solution (86) with

$$\varphi\_0 = -\arccos\left(-\frac{\gamma + \delta}{\mu\sqrt{1 + 4\delta^2}}\right) - \arccos\left(\frac{1}{\sqrt{1 + 4\delta^2}}\right) + 2\pi n, \quad n = 0, 1, 2, \dots \tag{91}$$

is asymptotically stable, and solution (86) with

$$\varphi\_0 = \arccos\left(-\frac{\gamma + \delta}{\mu\sqrt{1 + 4\delta^2}}\right) - \arccos\left(\frac{1}{\sqrt{1 + 4\delta^2}}\right) + 2\pi n, \quad n = 0, 1, 2, \dots \tag{92}$$

is unstable.

### *4.2.2. Condition for hula-hoop's contact with the waist*

22 Will-be-set-by-IN-TECH

When the waist center moves along a circle (*a* = *b*, i.e. *ε* = 0) equation (83) takes the following

*ϕ*¨ + *γϕ*˙ + *δϕ*˙ |*ϕ*˙| + *μ* cos(*ϕ* − *τ*) − 2*μδ* sign(*ϕ*˙) sin(*ϕ* − *τ*) = 0 (85)

*ϕ* = *τ* + *ϕ*<sup>0</sup> (86)

√

1 + 4*δ*2, so we find from

+ 2*πn*, *n* = 0, 1, 2, . . . (88)

*γ* + *δ* + *μ* cos *ϕ*<sup>0</sup> − 2*μδ* sin *ϕ*<sup>0</sup> = 0. (87)

*η*¨ + (*γ* + 2*δ*) *η*˙ − *μ* (sin *ϕ*<sup>0</sup> + 2*δ* cos *ϕ*0) *η* = 0. (89)

*γ* + 2*δ* > 0, *μ* (sin *ϕ*<sup>0</sup> + 2*δ* cos *ϕ*0) < 0. (90)

1 + 4*δ*2)) < 0. Thus, from conditions (90), relation

√

+ 2*πn*, *n* = 0, 1, 2, . . . (91)

+ 2*πn*, *n* = 0, 1, 2, . . . (92)

1 + 4*δ*<sup>2</sup> solution

**4.2. Exact solutions**

and has the exact solution [6]

*ϕ*<sup>0</sup> + arccos

*4.2.1. Stability of the exact solutions*

(90) can be written as sin(*ϕ*<sup>0</sup> <sup>+</sup> arccos(1/<sup>√</sup>

is asymptotically stable, and solution (86) with

<sup>−</sup> *<sup>γ</sup>* <sup>+</sup> *<sup>δ</sup> μ* √ 1 + 4*δ*<sup>2</sup>

<sup>−</sup> *<sup>γ</sup>* <sup>+</sup> *<sup>δ</sup> μ* √ 1 + 4*δ*<sup>2</sup>

*ϕ*<sup>0</sup> = − arccos

*ϕ*<sup>0</sup> = arccos

with the constant initial phase *ϕ*<sup>0</sup> given by the equation

 1 √ 1 + 4*δ*<sup>2</sup>

Therefore, solution (86) exists only under the condition |*γ* + *δ*|≤|*μ*|

= ± arccos

provided that *μ* �= 0. Solutions (86), (88) correspond to the rotation of the hula-hoop with the

Let us investigate the stability of the obtained solutions. For this purpose we take the angle *ϕ* in the form *ϕ* = *τ* + *ϕ*<sup>0</sup> + *η*(*τ*) where *η*(*τ*) is a small quantity, and substitute it into equation (85). Taking linearization with respect to *η* and with the use of (87) we obtain a linear equation

According to Lyapunov's theorem on the stability based on a linear approximation [17] solution (86), (88) is asymptotically stable if all the eigenvalues of linearized equation (89) have negative real parts. From Routh-Hurwitz criterion [17] we obtain the stability conditions

Without loss of generality we assume *μ* > 0 since the case *μ* < 0 can be reduced to the previous one by the time transformation *τ*� = *τ* + *π* in equation (83). The second condition in

(88) and due to the assumption *μ* > 0 we find that for 0 < *γ* + 2*δ* < *δ* + *μ*

− arccos

− arccos

 1 √ 1 + 4*δ*<sup>2</sup>

 1 <sup>√</sup><sup>1</sup> <sup>+</sup> <sup>4</sup>*δ*<sup>2</sup>

<sup>−</sup> *<sup>γ</sup>* <sup>+</sup> *<sup>δ</sup> μ* √ 1 + 4*δ*<sup>2</sup>

constant angular velocity equal to the excitation frequency *ω*.

form

equation (87)

as

(86) with

is unstable.

Let us verify for the exact solutions (86), (88) the condition of twirling without losing contact (84) which takes the form

$$
\mu \sin \varphi\_0 < \frac{1}{2}. \tag{93}
$$

The second stability condition in (90) can be rewritten with the use of (87) as follows

$$
\mu \sin \varrho\_0 < \frac{2\delta \left(\gamma + \delta\right)}{1 + 4\delta^2}.
$$

Thus, stable solution (86), (91) provides asymptotically stable twirling of the hula-hoop with the constant angular velocity *ω* without losing contact with the waist of the gymnast under the condition *γδ* < 1/4.

Without rolling resistance *δ* = 0 condition (93) is always satisfied for the stable solution (86), (91). While for unstable solution (86), (92) we have *<sup>μ</sup>* sin *<sup>ϕ</sup>*<sup>0</sup> <sup>=</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup> so condition (93) holds only if *<sup>μ</sup>* <sup>&</sup>lt; 1/4 <sup>+</sup> *<sup>γ</sup>*2. The phase *<sup>ϕ</sup>*<sup>0</sup> of the stable solution belongs to the interval [−*π*, −*π*/2] mod 2*π*, and for vanishing damping *γ* → +0 the phase tends to −*π*/2. Below we will show that this phase inequality also holds for the approximate solutions. This is how to twirl a hula-hoop!

### **4.3. Approximate solutions**

Let us find approximate solutions for the case of close but not equal amplitudes *a* ≈ *b*. For the sake of simplicity from now on we will keep *δ* = 0 and assume that *a* ≥ |*b*| which means *ε* ≥ 0, *μ* ≥ 0. Taking *ε* as a small parameter we apply perturbation method assuming that the exact solution *ϕs*(*τ*) of (83) can be expressed in a series

$$
\varphi\_s(\tau) = \tau + \varphi\_0 + \varepsilon \varphi\_1(\tau) + o(\varepsilon). \tag{94}
$$

After substitution of series (94) in (83) and grouping the terms by equal powers of *ε* we derive the following chain of equations

$$
\varepsilon^0 \colon \quad \gamma + \mu \cos(\varphi\_0) = 0 \tag{95}
$$

$$\varepsilon^1: \qquad \ddot{\varphi}\_1 + \gamma \dot{\varphi}\_1 - \mu \sin(\varphi\_0) \varphi\_1 = \cos(\varphi\_0 + 2\tau) \tag{96}$$

Taking solution of equation (95) for *μ* > 0

$$\varphi\_0 = -\arccos(-\gamma/\mu) + 2\pi n, \quad n = 1, 2, \dots \tag{97}$$

corresponding to the stable unperturbed solution (86), (91) we write equation (96) as

$$
\ddot{\varphi}\_1 + \gamma \dot{\varphi}\_1 + \sqrt{\mu^2 - \gamma^2} \,\varphi\_1 = \cos(\varphi\_0 + 2\tau) \,. \tag{98}
$$

It has a unique periodic solution

$$\varphi\_1(\tau) = \mathbb{C}\sin(2\tau + \varphi\_0) + D\cos(2\tau + \varphi\_0) \, \, \, \, \, \, \tag{99}$$

where constants *C* and *D* are defined as follows

$$C = \frac{2\gamma}{\mu^2 + 3\gamma^2 - 8\sqrt{\mu^2 - \gamma^2} + 16}, \ D = \frac{-4 + \sqrt{\mu^2 - \gamma^2}}{\mu^2 + 3\gamma^2 - 8\sqrt{\mu^2 - \gamma^2} + 16}. \tag{100}$$

**Figure 13.** Comparison between the approximate analytical (solid line) and numerical (circles) results for parameters: *μ* = 1.2, *ε* = 0.2, and *γ* = 1. Initial conditions for numerical calculations are *<sup>θ</sup>*(0) = <sup>−</sup>2.5389, ˙ *θ*(0) = 0.90802.

We see that the approximate solution *ϕ*(*τ*) = *τ* + *ϕ*<sup>0</sup> + *εϕ*1(*τ*) with (97), (99), (100) differs from the exact solution (86), (91) of the unperturbed system by small vibrating terms of frequency 2, see (99). Note that the approximate solutions were obtained with the assumption that the excitation amplitudes and damping are not small.

### *4.3.1. Stability of the approximate solutions*

To find the stability conditions for solution (94), (97), (99) we take a small variation to the solution of (94) *ϕ* = *ϕ<sup>s</sup>* + *u* and substitute this expression into (83). After linearization with respect to *u* and keeping only terms of first order we obtain a linear equation

$$\ddot{\boldsymbol{\mu}} + \gamma \dot{\boldsymbol{\mu}} + \left( -\mu \left( \sin \varphi\_0 + \varepsilon \varphi\_1 \cos \varphi\_0 \right) + \varepsilon \sin(2\tau + \varphi\_0) \right) \boldsymbol{\mu} = \mathbf{0},\tag{101}$$

where *ϕ*<sup>0</sup> is given by expression (97). Equation (101) can be written in the form of damped Mathieu-Hill equation as

$$
\ddot{u} + \gamma \dot{u} + \left(p + \varepsilon \Phi(2\tau)\right) u = 0,\tag{102}
$$

*4.3.3. Comparison with numerical simulations*

**4.4. Small excitation amplitudes and damping**

*γ*/*ε* and assume that the solution has the form

are not small.

bounded.

following equations

*4.4.1. Clockwise rotation*

*ε*

*ε*

the first approximation the solution

and holds true for sufficiently small *ε*.

*<sup>ϕ</sup>*1(*τ*) = <sup>−</sup><sup>1</sup>

in (109) is the special case of (99), (100) when *μ* = *γ* = 0.

In Fig. 13 the approximate analytical solution is presented and compared with the results of numerical simulation for the case when the excitation parameter *μ* and damping coefficient *γ*

It is interesting to consider the case when the excitation amplitudes and damping coefficient are small having the same order as *ε*. Then we introduce new parameters *μ*˜ = *μ*/*ε* and *γ*˜ =

where *ρ* is the angular velocity of rotation, and the functions *ϕ*0(*τ*), *ϕ*1(*τ*) are supposed to be

We substitute expression (105) into (83) and equating terms of the same powers of *ε* obtain the

From equation (106) we get that function *ϕ*0(*τ*) can remain bounded only if it is constant (*ϕ*0(*τ*) ≡ *ϕ*<sup>0</sup> = *const*). Then equation (107) can have bounded solutions *ϕ*1(*τ*) only when *ρ* takes the values -1, 0, 1. Thus, besides clockwise rotation we have also counterclockwise rotation *ρ* = −1, and no rotational solution *ρ* = 0. The letter is not interesting, so we omit it.

For clockwise rotation *ρ* = 1, see Fig. 14 a), from equations (105), (106), and (107) we obtain in

where function *ϕ*1(*τ*) is found up to the addition of a constant, which we have set to zero for determinacy. Thus, we let only *ϕ*<sup>0</sup> contain a constant term of the solution. The first expression

To verify the stability conditions for solution (108) we use damped Mathieu-Hill equation (102) and for the case of small damping and excitation amplitudes get *γ* > 0, sin *ϕ*<sup>0</sup> < 0, see [17]. These conditions are similar with inequalities (90) derived for the undisturbed exact

solution. Thus, the stable solution (108) with *ϕ*<sup>0</sup> = − arccos(−*γ*/*μ*) + 2*πn* exists for

For solution (108) condition (84) of keeping contact in the first approximation reads

1 + 2

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

*ε* <

<sup>4</sup> cos(*ϕ*<sup>0</sup> <sup>+</sup> <sup>2</sup>*τ*), cos *<sup>ϕ</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup> *<sup>γ</sup>*

*ϕ*(*τ*) = *ρτ* + *ϕ*0(*τ*) + *εϕ*1(*τ*) + *o*(*ε*), (105)

Dynamics of a Pendulum of Variable Length and Similar Problems 93

*ϕ*∗(*τ*) = *τ* + *ϕ*<sup>0</sup> + *εϕ*1(*τ*), (108)

0 < *γ* < *μ*. (110)

<sup>3</sup> , (111)

*<sup>μ</sup>* , (109)

<sup>0</sup> :*ϕ*¨0 = 0 (106)

<sup>1</sup> :*ϕ*¨1 <sup>=</sup> cos(*ϕ*<sup>0</sup> <sup>+</sup> *<sup>τ</sup>* <sup>+</sup> *ρτ*) <sup>−</sup> *<sup>μ</sup>*˜ cos(*ϕ*<sup>0</sup> <sup>−</sup> *<sup>τ</sup>* <sup>+</sup> *ρτ*) <sup>−</sup> *<sup>γ</sup>*˜ *<sup>ϕ</sup>*˙ <sup>0</sup> <sup>−</sup> *γρ*˜ . (107)

where *<sup>p</sup>* <sup>=</sup> <sup>−</sup>*<sup>μ</sup>* sin *<sup>ϕ</sup>*<sup>0</sup> <sup>=</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2, <sup>Φ</sup>(2*τ*) = (*γ<sup>C</sup>* <sup>+</sup> <sup>1</sup>) sin(2*<sup>τ</sup>* <sup>+</sup> *<sup>ϕ</sup>*0) + *<sup>γ</sup><sup>D</sup>* cos(2*<sup>τ</sup>* <sup>+</sup> *<sup>ϕ</sup>*0). Then the stability condition (absence of parametric resonance at all frequencies √*p*) is given by the inequalities [17]

$$\varepsilon < \frac{2\gamma}{\sqrt{(\gamma \mathbb{C} + 1)^2 + \gamma^2 D^2}} \tag{103}$$

with *C* and *D* defined in (100). This is the inequality to the problem parameters *γ*, *ε* and *μ*.

### *4.3.2. Condition for hula-hoop's contact with the waist*

The condition of twirling without losing contact (84) takes the following form

$$\varepsilon < \frac{1 + 2\sqrt{\mu^2 - \gamma^2}}{2} \sqrt{\frac{\mu^2 + 3\gamma^2 - 8\sqrt{\mu^2 - \gamma^2} + 16}{\mu^2 + 8\gamma^2 - 12\sqrt{\mu^2 - \gamma^2} + 36}}. \tag{104}$$

Conditions (103), (104) imply restrictions to *ε*, i.e. how much the elliptic trajectory of the waist center differs from the circle.

## *4.3.3. Comparison with numerical simulations*

24 Will-be-set-by-IN-TECH

time τ

**Figure 13.** Comparison between the approximate analytical (solid line) and numerical (circles) results

We see that the approximate solution *ϕ*(*τ*) = *τ* + *ϕ*<sup>0</sup> + *εϕ*1(*τ*) with (97), (99), (100) differs from the exact solution (86), (91) of the unperturbed system by small vibrating terms of frequency 2, see (99). Note that the approximate solutions were obtained with the assumption that the

To find the stability conditions for solution (94), (97), (99) we take a small variation to the solution of (94) *ϕ* = *ϕ<sup>s</sup>* + *u* and substitute this expression into (83). After linearization with

where *ϕ*<sup>0</sup> is given by expression (97). Equation (101) can be written in the form of damped

where *<sup>p</sup>* <sup>=</sup> <sup>−</sup>*<sup>μ</sup>* sin *<sup>ϕ</sup>*<sup>0</sup> <sup>=</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2, <sup>Φ</sup>(2*τ*) = (*γ<sup>C</sup>* <sup>+</sup> <sup>1</sup>) sin(2*<sup>τ</sup>* <sup>+</sup> *<sup>ϕ</sup>*0) + *<sup>γ</sup><sup>D</sup>* cos(2*<sup>τ</sup>* <sup>+</sup> *<sup>ϕ</sup>*0). Then the stability condition (absence of parametric resonance at all frequencies √*p*) is given by the

with *C* and *D* defined in (100). This is the inequality to the problem parameters *γ*, *ε* and *μ*.

2*γ*

*<sup>μ</sup>*<sup>2</sup> + <sup>3</sup>*γ*<sup>2</sup> − <sup>8</sup>

Conditions (103), (104) imply restrictions to *ε*, i.e. how much the elliptic trajectory of the waist

*u*¨ + *γu*˙ + (−*μ* (sin *ϕ*<sup>0</sup> + *εϕ*<sup>1</sup> cos *ϕ*0) + *ε* sin(2*τ* + *ϕ*0)) *u* = 0 , (101)

*u*¨ + *γu*˙ + (*p* + *ε*Φ(2*τ*)) *u* = 0 , (102)

(*γ<sup>C</sup>* <sup>+</sup> <sup>1</sup>)<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2*D*<sup>2</sup> (103)

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup> <sup>+</sup> <sup>16</sup>

*<sup>μ</sup>*<sup>2</sup> <sup>+</sup> <sup>8</sup>*γ*<sup>2</sup> <sup>−</sup> <sup>12</sup>*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup> <sup>+</sup> <sup>36</sup> . (104)

for parameters: *μ* = 1.2, *ε* = 0.2, and *γ* = 1. Initial conditions for numerical calculations are

respect to *u* and keeping only terms of first order we obtain a linear equation

*ε* <

The condition of twirling without losing contact (84) takes the following form

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup> 2

2π

0 0.9

1.0

angular velocity dθ/dτ

*θ*(0) = 0.90802.

*4.3.1. Stability of the approximate solutions*

Mathieu-Hill equation as

inequalities [17]

excitation amplitudes and damping are not small.

*4.3.2. Condition for hula-hoop's contact with the waist*

1 + 2

*ε* <

center differs from the circle.

*<sup>θ</sup>*(0) = <sup>−</sup>2.5389, ˙

1.1

In Fig. 13 the approximate analytical solution is presented and compared with the results of numerical simulation for the case when the excitation parameter *μ* and damping coefficient *γ* are not small.

### **4.4. Small excitation amplitudes and damping**

It is interesting to consider the case when the excitation amplitudes and damping coefficient are small having the same order as *ε*. Then we introduce new parameters *μ*˜ = *μ*/*ε* and *γ*˜ = *γ*/*ε* and assume that the solution has the form

$$
\varphi(\tau) = \rho \tau + \varphi\_0(\tau) + \varepsilon \varphi\_1(\tau) + o(\varepsilon),
\tag{105}
$$

where *ρ* is the angular velocity of rotation, and the functions *ϕ*0(*τ*), *ϕ*1(*τ*) are supposed to be bounded.

We substitute expression (105) into (83) and equating terms of the same powers of *ε* obtain the following equations

$$
\varepsilon^0 \colon \ddot{\varphi}\_0 = 0 \tag{106}
$$

$$\varepsilon^1 \colon \ddot{\varphi}\_1 = \cos(\varphi\_0 + \tau + \rho \tau) - \ddot{\mu}\cos(\varphi\_0 - \tau + \rho \tau) - \tilde{\gamma}\dot{\varphi}\_0 - \tilde{\gamma}\rho. \tag{107}$$

From equation (106) we get that function *ϕ*0(*τ*) can remain bounded only if it is constant (*ϕ*0(*τ*) ≡ *ϕ*<sup>0</sup> = *const*). Then equation (107) can have bounded solutions *ϕ*1(*τ*) only when *ρ* takes the values -1, 0, 1. Thus, besides clockwise rotation we have also counterclockwise rotation *ρ* = −1, and no rotational solution *ρ* = 0. The letter is not interesting, so we omit it.

### *4.4.1. Clockwise rotation*

For clockwise rotation *ρ* = 1, see Fig. 14 a), from equations (105), (106), and (107) we obtain in the first approximation the solution

$$
\varphi\_\* (\tau) = \tau + \varphi\_0 + \varepsilon \varphi\_1 (\tau),
\tag{108}
$$

$$\varphi\_1(\tau) = -\frac{1}{4}\cos(\varphi\_0 + 2\tau), \quad \cos\varphi\_0 = -\frac{\gamma}{\mu'} \tag{109}$$

where function *ϕ*1(*τ*) is found up to the addition of a constant, which we have set to zero for determinacy. Thus, we let only *ϕ*<sup>0</sup> contain a constant term of the solution. The first expression in (109) is the special case of (99), (100) when *μ* = *γ* = 0.

To verify the stability conditions for solution (108) we use damped Mathieu-Hill equation (102) and for the case of small damping and excitation amplitudes get *γ* > 0, sin *ϕ*<sup>0</sup> < 0, see [17]. These conditions are similar with inequalities (90) derived for the undisturbed exact solution. Thus, the stable solution (108) with *ϕ*<sup>0</sup> = − arccos(−*γ*/*μ*) + 2*πn* exists for

$$0 < \gamma < \mu. \tag{110}$$

For solution (108) condition (84) of keeping contact in the first approximation reads

$$
\varepsilon < \frac{1 + 2\sqrt{\mu^2 - \gamma^2}}{3},
\tag{111}
$$

and holds true for sufficiently small *ε*.

**Figure 14.** Stable twirling of the hula-hoop for the cases: a) direct twirling b) inverse twirling.

### *4.4.2. Counterclockwise rotation*

For counterclockwise rotation *ρ* = −1 in Fig. 14 b) we obtain in the first approximation the solution

$$
\varphi\_\*(\tau) = -\tau + \varphi\_0 + \frac{\mu}{4} \cos(\varphi\_0 - 2\tau), \quad \cos \varphi\_0 = -\frac{\gamma}{\varepsilon'} \tag{112}
$$

with the stability conditions *γ* > 0, sin *ϕ*<sup>0</sup> > 0. Thus, the stable counterclockwise rotation (112) with *ϕ*<sup>0</sup> = arccos(−*γ*/*ε*) + 2*πn* exists for

$$0 < \gamma < \varepsilon. \tag{113}$$

0

**5. Conclusions**

calculations are *<sup>θ</sup>*(0) = <sup>−</sup>1.7301, ˙

time τ

pendulum motion were plotted and analyzed.

2π 0

*θ*(0) = 0.98403 (left) and *θ*(0) = 2.4696, ˙

In section 2 we showed that the pendulum with periodically varying length exhibits diversity of behavior types. We recognized that the analytical stability boundaries of the vertical position of the pendulum and the frequency-response curve for limit cycles are in a good agreement with the numerical results. The second resonance zone appeared to be empty. The stability conditions of limit cycles are derived based on direct use of Lyapunov's theorem on stability of periodic solutions. We found numerically regular rotation, oscillation, and rotation-oscillation regimes with various periods and mean angular velocities of the pendulum including high-speed rotations and rotations with fractional relative velocities (it is rotation-oscillation regime when the pendulum makes regular sequence of rotations in both directions). We derived analytically the conditions for existence of regular rotation and oscillation regimes which agree with the numerical results. Domains for chaotic motions are found and analyzed numerically in the parameter space via calculation of Lyapunov exponents and bifurcation diagrams. Basins of attractions of different regimes of the

In section 3 we studied the planar rotational motion of the pendulum with the pivot oscillating both vertically and horizontally when the trajectory of the pivot is an ellipse close to a circle. The analysis of motion was based on the exact rotational solutions in the case of circular pivot trajectory and zero gravity. The conditions for existence and stability of such solutions were derived. Assuming that the amplitudes of excitations are not small while the pivot trajectory has small ellipticity the approximate solutions were found both for large and small linear damping. Comparison between approximate and numerical solutions was made for different values of the damping parameter demonstrating good accuracy of the method involved.

Finally, in section 4 we assumed that the waist of a sportsman twirling a hula hoop is a circle and its center moves along an elliptic trajectory close to a circle. We studied the system with both small and not small linear viscous damping as well as with some rolling resistance. For the case of the circular trajectory, two families of the exact solutions were obtained, similar to those in section 3. Both of them correspond to twirling of the hula-hoop with a

**Figure 15.** Comparison between the approximate analytical (solid line) and numerical (circles) results for small excitation amplitudes and damping coefficient for clockwise (left) and counterclockwise (right)

rotations, for parameters: *μ* = 5/32, *ε* = 1/32, and *γ* = 1/40. Initial conditions for numerical

−1.05

time τ

Dynamics of a Pendulum of Variable Length and Similar Problems 95

*θ*(0) = −0.95212 (right).

2π

−1.00

angular velocity dθ/dτ

−0.95

0.99

1.00

angular velocity dθ/dτ

1.01

For this case condition (84) takes the form similar to (111) and holds true for sufficiently small *μ*

$$
\mu < \frac{1 + 2\sqrt{\varepsilon^2 - \gamma^2}}{3}.\tag{114}
$$

### *4.4.3. Coexistence of clockwise and counterclockwise rotations*

It follows from conditions (110), (113) that stable clockwise and counterclockwise rotations (108), (109) and (112) coexist if the following conditions are satisfied

$$0 < \gamma < \min\{\varepsilon, \mu\}. \tag{115}$$

Conditions (115) in physical variables take the form

$$0 < 2k \frac{R - r}{R^2 \omega m} < a - |b|\_{\prime} \tag{116}$$

meaning that the trajectory of the waist should be sufficiently prolate. Coexisting clockwise and counterclockwise rotations are illustrated in Fig. 14.

### *4.4.4. Comparison with numerical simulations*

In Fig. 15 the approximate analytical solutions for rotations in both directions are presented and compared with the results of numerical simulation for the case of small excitation parameters *μ*, *ε* and the damping coefficient *γ*. The values of *μ*, *ε* correspond to the dimensional parameters *a* = 15*cm*, *b* = 10*cm*, *r* = 10*cm*, *R* = 50*cm*.

**Figure 15.** Comparison between the approximate analytical (solid line) and numerical (circles) results for small excitation amplitudes and damping coefficient for clockwise (left) and counterclockwise (right) rotations, for parameters: *μ* = 5/32, *ε* = 1/32, and *γ* = 1/40. Initial conditions for numerical calculations are *<sup>θ</sup>*(0) = <sup>−</sup>1.7301, ˙ *θ*(0) = 0.98403 (left) and *θ*(0) = 2.4696, ˙ *θ*(0) = −0.95212 (right).
