**3.4. Approximate rotational solutions when** *<sup>ε</sup>* <sup>≈</sup> <sup>0</sup>**,** *<sup>ω</sup>* <sup>∼</sup> <sup>√</sup>*ε***, and** *<sup>β</sup>* <sup>∼</sup> <sup>√</sup>*<sup>ε</sup>*

One can see in (38) that assumptions *<sup>ω</sup>* <sup>∼</sup> *<sup>β</sup>* <sup>∼</sup> <sup>√</sup>*<sup>ε</sup>* are valid for the high frequency of excitation <sup>Ω</sup> <sup>∼</sup> 1/√*<sup>ε</sup>* with other parameters being of order 1. Another option is small gravity *<sup>g</sup>* <sup>∼</sup> *<sup>ε</sup>* along with small ratio *<sup>c</sup>*/*<sup>m</sup>* <sup>∼</sup> <sup>√</sup>*ε*.

After change of variable *<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> <sup>√</sup>*εϑ* equation (39) takes the following form

$$\ddot{\theta} + \mu \theta - \tilde{\beta} = \mu \left( \theta - \frac{\sin\left(\sqrt{\varepsilon}\theta\right)}{\sqrt{\varepsilon}} \right) - \sqrt{\varepsilon} \dot{\theta} \dot{\theta} + \sqrt{\varepsilon} \sin\left(2\tau - \sqrt{\varepsilon}\theta\right) + \sqrt{\varepsilon} w \sin\left(\tau - \sqrt{\varepsilon}\theta - \delta\right) \tag{61}$$

with small right-hand side, where we denote *β*˜ = *β*/ <sup>√</sup>*<sup>ε</sup>* and as in the previous section *<sup>w</sup>* <sup>=</sup> *<sup>ω</sup>*2/*ε*. With zero right-hand side equation (61) *<sup>ϑ</sup>*¨ <sup>+</sup> *μϑ* <sup>−</sup> *<sup>β</sup>*˜ <sup>=</sup> 0 would describe harmonic oscillations about *β*˜/*μ* value with frequency √*μ*. After Taylor's expansion of sines in the right-hand side of (61) about *ϑ* = 0 we obtain the following equation

$$\begin{aligned} \ddot{\theta} &+ \left( \mu + \varepsilon \cos(2\tau) + \varepsilon w \cos(\tau - \delta) \right) \theta - \tilde{\beta} \\ &= \sqrt{\varepsilon} \left( \sin(2\tau) + w \sin(\tau - \delta) - \tilde{\beta} \dot{\theta} \right) + \varepsilon \mu \frac{\theta^3}{6} + o(\varepsilon), \end{aligned} \tag{62}$$

which describes oscillator with both basic and parametric excitations. To solve equation (62) we use the method of averaging [9, 18, 19]. For that purpose we write (62) in the *standard form* of first order differential equations with small right-hand sides. First, we use *Poincaré variables <sup>q</sup>* and *<sup>ψ</sup>* defined via the following solution of *generating system <sup>ϑ</sup>*¨ <sup>+</sup> *μϑ* <sup>−</sup> *<sup>β</sup>*˜ <sup>=</sup> 0 which is (62) with *ε* = 0

$$
\theta = \frac{\vec{\not\theta}}{\mu} + q \cos(\psi), \quad \dot{\theta} = -\sqrt{\mu}q \sin(\psi). \tag{63}
$$

In Poincaré variables equation (62) becomes a system of first order differential equations

$$\dot{q} = -\frac{\sin\psi}{\sqrt{\mu}} f(\tau, q, \psi), \quad \dot{\psi} = \sqrt{\mu} - \frac{\cos\psi}{q\sqrt{\mu}} f(\tau, q, \psi), \tag{64}$$

#### 18 Will-be-set-by-IN-TECH 86 Nonlinearity, Bifurcation and Chaos – Theory and Applications Dynamics of a Pendulum of Variable Length and Similar Problems <sup>19</sup>

where small function *<sup>f</sup>*(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*) = <sup>√</sup>*εf*1(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*) + *<sup>ε</sup>f*2(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*) + *<sup>o</sup>*(*ε*) is the right hand side of (61), where

$$f\_1(\tau, \eta, \psi) = \sin(2\tau) + w\sin(\tau - \delta) + \tilde{\beta}\eta\sqrt{\mu}\sin(\psi),\tag{65}$$

$$f\_2(\tau, q, \psi) = -\left(\cos(2\tau) + w\cos(\tau - \delta)\right) \left(\frac{\tilde{\mathcal{B}}}{\mu} + q\cos(\psi)\right) + \frac{\mu}{6} \left(\frac{\tilde{\mathcal{B}}}{\mu} + q\cos(\psi)\right)^3,\tag{66}$$

meaning that *f*(*τ*, *q*, *ψ*) = *O*( <sup>√</sup>*ε*). Our next assumption is that <sup>√</sup>*<sup>μ</sup>* <sup>−</sup> <sup>1</sup> <sup>=</sup> *<sup>O</sup>*( <sup>√</sup>*ε*) which means that excitation frequency is close to the first resonant frequency of basic excitation component sin(*τ* − *δ*) and to the first resonant frequency of parametric excitation component cos(2*τ*) in equation (62). Thus, system (64) is transformed by *ψ* = *ζ* + *τ* to the standard form

$$\dot{\eta} = -\frac{1}{\sqrt{\mu}} \sin(\zeta + \tau) f(\tau, q, \zeta + \tau) \,. \tag{67}$$

which does not contain higher harmonics observed numerically. That is why we need to

In the second approximation averaged equations can be obtained as described in Appendix. Stationary solutions (*Q*˙ = 0, *Z*˙ = 0) of (69), (70) in the second approximation can be found numerically or with the absence of gravity (*ω* = 0) analytically. Solution of system (67), (68)

<sup>2</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>Z</sup>*{2} <sup>−</sup> *<sup>δ</sup>*

<sup>2</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>Z</sup>*{2} <sup>−</sup> *<sup>δ</sup>*

*<sup>β</sup> <sup>Q</sup>*{2} <sup>4</sup> sin

 + 1 <sup>3</sup> sin

<sup>√</sup>*ε*), (73)

Dynamics of a Pendulum of Variable Length and Similar Problems 87

 + 1 <sup>3</sup> cos

<sup>√</sup>*ε*). (74)

*<sup>Z</sup>*{2} <sup>+</sup> *<sup>τ</sup>*

+ *o*(*ε*). (75)

<sup>3</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>Z</sup>*{2}

<sup>3</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>Z</sup>*{2}

√*ε*

proceed to the second order approximation.

in the second approximation is the following, see (119),

*<sup>τ</sup>* <sup>−</sup> *<sup>Z</sup>*{2}

<sup>2</sup>*<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*Z*{2}

 + *o*(

results in the approximate solution of the original equation (39)

 cos

<sup>2</sup>*<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*Z*{2}

*μ*

*<sup>ω</sup>*<sup>2</sup> sin(*<sup>τ</sup>* <sup>−</sup> *<sup>δ</sup>*) 4

 + *w* <sup>2</sup> sin

 + *o*(

*<sup>τ</sup>* <sup>−</sup> *<sup>Z</sup>*{2}

 + *w* <sup>2</sup> cos

<sup>+</sup> <sup>√</sup>*εQ*{2} cos(*Z*{2} <sup>+</sup> *<sup>τ</sup>*) + <sup>√</sup>*<sup>ε</sup>*

<sup>√</sup>*<sup>μ</sup>* <sup>−</sup> *<sup>ε</sup>* sin(2*τ*) 3 √*μ*

Substitution of these expressions into (63) yields the second order approximate solution of (61), which after changes of variable *<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> <sup>√</sup>*εϑ* and parameters *<sup>w</sup>* <sup>=</sup> *<sup>ω</sup>*2/*ε*, *<sup>β</sup>*˜ <sup>=</sup> *<sup>β</sup>*/

Agreement of solution (75) with the numerical experiment is shown in Fig. 11. We see that the amplitude of angular velocity oscillations is much higher than that for not small *β* in Fig. 10.

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

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

*3.4.2. Second order approximation*

√*ε* 2 √*μ* <sup>−</sup> sin

*<sup>β</sup>*˜ *<sup>Q</sup>*{2} <sup>4</sup> sin

> 2 <sup>√</sup>*μQ*{2}

√*ε*

*<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>β</sup>*

+

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

elliptic trajectory close to a circle.

hard brittle materials, see [33].

*<sup>q</sup>* <sup>=</sup> *<sup>Q</sup>*{2} <sup>+</sup>

<sup>+</sup>√*<sup>ε</sup>*

*<sup>ζ</sup>* <sup>=</sup> *<sup>Z</sup>*{2} <sup>+</sup>

<sup>+</sup>√*<sup>ε</sup> β*˜ <sup>4</sup> cos

$$\dot{\zeta} = \sqrt{\mu} - 1 - \frac{1}{q\sqrt{\mu}} \cos(\zeta + \tau) \, f(\tau, q, \zeta + \tau) \,. \tag{68}$$

with small right-hand side, where new slow variable *ζ* is often referred to as *phase mismatch*.

In the second approximation so called *averaged equations* can be obtained from the system of equations (67) and (68) as follows, see (121) in the Appendix,

$$\begin{split} \dot{Q} &= -\sqrt{\varepsilon} \left( \frac{w}{2\sqrt{\mu}} \cos(Z + \delta) + \frac{\tilde{\beta}}{2} Q \right) \\ &+ \varepsilon \left( \frac{w\tilde{\beta}}{8\sqrt{\mu}} \left( \frac{4}{\mu} - 1 \right) \sin(Z + \delta) + \frac{\sin(2Z)}{4\sqrt{\mu}} Q \right) + o(\varepsilon), \\ \dot{Z} &= \sqrt{\mu} - 1 + \frac{\sqrt{\varepsilon}w}{2\sqrt{\mu}Q} \sin(Z + \delta) \\ &+ \varepsilon \left( \frac{w\tilde{\beta}}{8\sqrt{\mu}Q} \left( \frac{4}{\mu} - 1 \right) \cos(Z + \delta) - \frac{\tilde{\beta}^2}{8} \left( \frac{2}{\mu\sqrt{\mu}} + 1 \right) + \frac{\cos(2Z)}{4\sqrt{\mu}} - \frac{\sqrt{\mu}}{16} Q^2 \right) + o(\varepsilon), \end{split} \tag{69}$$

where *Q* and *Z* are the *averaged variables* corresponding to *q* and *ζ*.

### *3.4.1. First order approximation*

Stationary solutions (*Q*˙ = 0, *Z*˙ = 0) of (69)-(70) in the first approximation are the following

$$\mathbb{Q}\_{\{1\}}^2 = \frac{\omega^2/\varepsilon}{\mu \left(4(\sqrt{\mu}-1)^2 + \beta^2\right)}, \quad \mathbb{Z}\_{\{1\}} = \arctan\left(\frac{2(\mu-1)}{\beta}\right) - \delta + 2\pi k,\tag{71}$$

where we have substituted back *w* = *ω*2/*ε* and *β*˜ = *β*/ <sup>√</sup>*ε*. Symbol arctan stands for the principal value of the function on the interval from 0 to *<sup>π</sup>*. Note that the phase *<sup>Z</sup>*{1} is determined to within 2*<sup>π</sup>* rather than *<sup>π</sup>*, since the functions sin(*Z*{1}) and cos(*Z*{1}) obtained from equations (69) and (70) determine *<sup>Z</sup>*{1} up to an additive term 2*πk*. Solution of system (67)-(68) in the first approximation is *<sup>q</sup>* <sup>=</sup> *<sup>Q</sup>*{1} <sup>+</sup> *<sup>o</sup>*(1), *<sup>ζ</sup>* <sup>=</sup> *<sup>Z</sup>*{1} <sup>+</sup> *<sup>o</sup>*(1) so the solution of (39) is the following

$$\theta = -\tau + \frac{\beta}{\mu} + \sqrt{\varepsilon}Q\_{\{1\}}\cos(Z\_{\{1\}} + \tau) + o(\sqrt{\varepsilon}),\tag{72}$$

which does not contain higher harmonics observed numerically. That is why we need to proceed to the second order approximation.

### *3.4.2. Second order approximation*

18 Will-be-set-by-IN-TECH

where small function *<sup>f</sup>*(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*) = <sup>√</sup>*εf*1(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*) + *<sup>ε</sup>f*2(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*) + *<sup>o</sup>*(*ε*) is the right hand side of

 *β*˜ *μ*

that excitation frequency is close to the first resonant frequency of basic excitation component sin(*τ* − *δ*) and to the first resonant frequency of parametric excitation component cos(2*τ*) in

with small right-hand side, where new slow variable *ζ* is often referred to as *phase mismatch*. In the second approximation so called *averaged equations* can be obtained from the system of

> sin(2*Z*) 4 √*μ Q*

> > 8

Stationary solutions (*Q*˙ = 0, *Z*˙ = 0) of (69)-(70) in the first approximation are the following

, *<sup>Z</sup>*{1} <sup>=</sup> arctan

<sup>+</sup> <sup>√</sup>*εQ*{1} cos(*Z*{1} <sup>+</sup> *<sup>τ</sup>*) + *<sup>o</sup>*(

principal value of the function on the interval from 0 to *<sup>π</sup>*. Note that the phase *<sup>Z</sup>*{1} is determined to within 2*<sup>π</sup>* rather than *<sup>π</sup>*, since the functions sin(*Z*{1}) and cos(*Z*{1}) obtained from equations (69) and (70) determine *<sup>Z</sup>*{1} up to an additive term 2*πk*. Solution of system (67)-(68) in the first approximation is *<sup>q</sup>* <sup>=</sup> *<sup>Q</sup>*{1} <sup>+</sup> *<sup>o</sup>*(1), *<sup>ζ</sup>* <sup>=</sup> *<sup>Z</sup>*{1} <sup>+</sup> *<sup>o</sup>*(1) so the solution of (39)

 2 *μ* √*μ* + 1 +

equation (62). Thus, system (64) is transformed by *ψ* = *ζ* + *τ* to the standard form

*q* √*μ*

2 *Q* 

cos(*<sup>Z</sup>* <sup>+</sup> *<sup>δ</sup>*) <sup>−</sup> *<sup>β</sup>*˜2

sin(*Z* + *δ*) +

where *Q* and *Z* are the *averaged variables* corresponding to *q* and *ζ*.

<sup>√</sup>*<sup>μ</sup>* <sup>−</sup> <sup>1</sup>)<sup>2</sup> <sup>+</sup> *<sup>β</sup>*2)

where we have substituted back *w* = *ω*2/*ε* and *β*˜ = *β*/

*<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>β</sup>*

*μ*

<sup>√</sup>*ε*). Our next assumption is that <sup>√</sup>*<sup>μ</sup>* <sup>−</sup> <sup>1</sup> <sup>=</sup> *<sup>O</sup>*(

+ *q* cos(*ψ*)

 <sup>+</sup> *<sup>μ</sup>* 6 *β*˜ *μ*

sin(*ζ* + *τ*) *f*(*τ*, *q*, *ζ* + *τ*), (67)

cos(*ζ* + *τ*) *f*(*τ*, *q*, *ζ* + *τ*), (68)

+ *o*(*ε*), (69)

√*μ* <sup>16</sup> *<sup>Q</sup>*<sup>2</sup> 

+ *o*(*ε*), (70)

− *δ* + 2*πk*, (71)

<sup>√</sup>*ε*. Symbol arctan stands for the

<sup>√</sup>*ε*), (72)

cos(2*Z*) 4 √*<sup>μ</sup>* −

<sup>2</sup>(*<sup>μ</sup>* <sup>−</sup> <sup>1</sup>) *β*

√*<sup>μ</sup>* sin(*ψ*), (65)

+ *q* cos(*ψ*)

3

<sup>√</sup>*ε*) which means

, (66)

*<sup>f</sup>*1(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*) = sin(2*τ*) <sup>+</sup> *<sup>w</sup>* sin(*<sup>τ</sup>* <sup>−</sup> *<sup>δ</sup>*) <sup>+</sup> *<sup>β</sup>*˜*<sup>q</sup>*

*f*2(*τ*, *q*, *ψ*) = − (cos(2*τ*) + *w* cos(*τ* − *δ*))

*<sup>q</sup>*˙ <sup>=</sup> <sup>−</sup> <sup>1</sup> √*μ*

*<sup>ζ</sup>* <sup>=</sup> <sup>√</sup>*<sup>μ</sup>* <sup>−</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

equations (67) and (68) as follows, see (121) in the Appendix,

cos(*<sup>Z</sup>* <sup>+</sup> *<sup>δ</sup>*) <sup>+</sup> *<sup>β</sup>*˜

<sup>√</sup>*μ<sup>Q</sup>* sin(*<sup>Z</sup>* <sup>+</sup> *<sup>δ</sup>*)

˙

(61), where

meaning that *f*(*τ*, *q*, *ψ*) = *O*(

*<sup>Q</sup>*˙ <sup>=</sup> <sup>−</sup>√*<sup>ε</sup>*

+ *ε*

+ *ε*

is the following

 *w* 2 √*μ*

> 4 *<sup>μ</sup>* <sup>−</sup> <sup>1</sup>

> > <sup>√</sup>*ε<sup>w</sup>* 2

{1} <sup>=</sup> *<sup>ω</sup>*2/*<sup>ε</sup> μ* (4(

 4 *<sup>μ</sup>* <sup>−</sup> <sup>1</sup> 

 *wβ*˜ 8 √*μ*

 *wβ*˜ 8 √*μQ*

*3.4.1. First order approximation*

*Q*2

*<sup>Z</sup>*˙ <sup>=</sup> <sup>√</sup>*<sup>μ</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup>

In the second approximation averaged equations can be obtained as described in Appendix. Stationary solutions (*Q*˙ = 0, *Z*˙ = 0) of (69), (70) in the second approximation can be found numerically or with the absence of gravity (*ω* = 0) analytically. Solution of system (67), (68) in the second approximation is the following, see (119),

$$q = Q\_{\{2\}} + \frac{\sqrt{\varepsilon}}{2\sqrt{\mu}} \left( -\sin\left(\tau - Z\_{\{2\}}\right) + \frac{w}{2}\sin\left(2\tau + Z\_{\{2\}} - \delta\right) + \frac{1}{3}\sin\left(3\tau + Z\_{\{2\}}\right) \right)$$

$$+ \sqrt{\varepsilon}\frac{\tilde{\mathbb{P}}\,Q\_{\{2\}}}{4}\sin\left(2\tau + 2Z\_{\{2\}}\right) + o(\sqrt{\varepsilon}),\tag{73}$$

$$\zeta = Z\_{\{2\}} + \frac{\sqrt{\varepsilon}}{2\sqrt{\mu}\,Q\_{\{2\}}} \left( \cos\left(\tau - Z\_{\{2\}}\right) + \frac{w}{2}\cos\left(2\tau + Z\_{\{2\}} - \delta\right) + \frac{1}{3}\cos\left(3\tau + Z\_{\{2\}}\right) \right)$$

$$+ \sqrt{\varepsilon}\frac{\tilde{\mathbb{P}}}{4}\cos\left(2\tau + 2Z\_{\{2\}}\right) + o(\sqrt{\varepsilon}).\tag{74}$$

Substitution of these expressions into (63) yields the second order approximate solution of (61), which after changes of variable *<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> <sup>√</sup>*εϑ* and parameters *<sup>w</sup>* <sup>=</sup> *<sup>ω</sup>*2/*ε*, *<sup>β</sup>*˜ <sup>=</sup> *<sup>β</sup>*/ √*ε* results in the approximate solution of the original equation (39)

$$\theta = -\tau + \frac{\beta}{\mu} + \sqrt{\varepsilon}Q\_{\{2\}}\cos(Z\_{\{2\}} + \tau) + \sqrt{\varepsilon}\frac{\beta^{\dagger}Q\_{\{2\}}}{4}\sin\left(Z\_{\{2\}} + \tau\right)$$

$$+ \frac{\omega^{2}\sin(\tau - \delta)}{4\sqrt{\mu}} - \frac{\varepsilon\sin(2\tau)}{3\sqrt{\mu}} + o(\varepsilon). \tag{75}$$

Agreement of solution (75) with the numerical experiment is shown in Fig. 11. We see that the amplitude of angular velocity oscillations is much higher than that for not small *β* in Fig. 10.
