**3. Elliptically excited pendulum**

10 Will-be-set-by-IN-TECH

**Figure 5.** Absolute values |*b*| of relative rotational velocities are shown with different colors on the plane of parameters *ε* and *ω* at the damping *β* = 0.05. The correspondence between the colors and values is shown by the color bar on the right. Approximate boundaries for rotations are drawn with bold dashed

**Figure 6.** Maximal Lyapunov's exponents are shown on the plane of parameters *ε* and *ω* at the damping *β* = 0.05. The correspondence between the colors and values is shown by the color bar on the right, where white color distinguishes zero maximal Lyapunov's exponent which corresponds to regular

example at *ω* = 0.67; see Fig. 7(b). We can see the change of the system dynamics in its route to chaos along *ω* = 0.5 in the bifurcation diagram shown in Fig. 7(a), where red points denote rotations with mean angular velocity equal to one excitation frequency (|*b*| = 1) and green points denote those equal to two excitation frequencies (|*b*| = 2). The domain with the most complex regular dynamics is surrounded by the red rectangle, where the system can have

Basins of attractions in Fig. 8 have been plotted using program Dynamics [21]. These basins track the changes of the system dynamics in its route to chaos along *ω* = 0.67. In Fig. 8(a) the oscillatory attractor (limit cycle) coexists with stationary attractor (lower vertical position of PPVL). In Fig. 8(b) we can see the first emergence of two rotational attractors with counterrotations. This picture is in a good agreement with condition (18) for existence of rotational solutions |*b*| = 1, see Fig. 5. Closer to the boundary of chaotic region in Fig. 8(c) only stationary and rotational attractors remain. Note that the basins of rotational attractors are

regime. Positive Lyapunovs' exponents characterize chaotic motions.

coexisting oscillations, rotations and rotations-oscillations.

line (for |*b*| = 1) and bold solid line (for |*b*| = 2).

Elliptically excited pendulum (EEP) is a mathematical pendulum in the vertical plane whose pivot oscillates not only vertically but also horizontally with *π*/2 phase shift, so that the pivot has elliptical trajectory, see Fig. 9. EEP is a natural generalization of pendulum with vertically vibrating pivot that is one of the most studied classical systems with parametric excitation. It is often referred to simply as *parametric pendulum*, see e.g. [9, 22–27] and references therein. Stability and dynamics of EEP have been studied analytically and numerically in [28–30]. Approximate oscillatory and rotational solutions for EEP are the common examples in literature [31–34] on asymptotic methods. Sometimes EEP is presented in a slightly more general model of unbalanced rotor [31–33], where the phase shift between vertical and horizontal oscillations of the pivot can differ from *π*/2. EEP is also a special case of generally excited pendulum in [35]. The usual assumption for approximate solution in the literature is the smallness of dimensionless damping and pivot oscillation amplitudes in the EEP's equation of motion. We could find only one paper [36], where oscillations of EEP with high damping and yet small relative excitation were studied.

In this section we study rotations of EEP with not small excitation amplitudes and with both small and not small linear damping. Our analysis uses the exact solutions for EEP with the absence of gravity and with equal excitation amplitudes, when elliptical trajectory of the pivot

**Figure 9.** Scheme of the elliptically excited mathematical pendulum of length *l*. The pivot of the pendulum moves along the elliptic trajectory (dashed line) with semiaxes *X* and *Y* in the uniform

It is assumed that the pivot of the pendulum moves according to the periodic law

where *X*, *Y*, and Ω are the amplitudes and frequency of the excitation.

<sup>2</sup> *<sup>l</sup>* , *<sup>μ</sup>* <sup>=</sup> *<sup>Y</sup>* <sup>+</sup> *<sup>X</sup>*

**3.2. Exact rotational solution when** *ε* = 0 **and** *ω* = 0

*<sup>ε</sup>* <sup>=</sup> *<sup>Y</sup>* <sup>−</sup> *<sup>X</sup>*

¨ *θ* + *β* ˙

We introduce new time *τ* = Ω*t* and the following dimensionless parameters

With this notation equation (36) with substituted (37) in it takes the following form

where we use the formula *Y* cos(Ω *t*) sin(*θ*) + *X* sin(Ω *t*) cos(*θ*) = *<sup>Y</sup>*+*<sup>X</sup>*

where *l* is the distance between the pivot and the concentrated mass *m*; *c* is the viscous damping coefficient; *θ* is the angle of the pendulum deviation from the vertical position; *t* is time; *g* is gravitational acceleration at the angle *δ* with respect to the negative direction of

<sup>2</sup> *<sup>l</sup>* <sup>&</sup>gt; 0, *<sup>ω</sup>* <sup>=</sup> <sup>1</sup>

<sup>2</sup> sin(<sup>Ω</sup> *<sup>t</sup>* <sup>+</sup> *<sup>θ</sup>*) <sup>−</sup> *<sup>Y</sup>*−*<sup>X</sup>*

Conditions *ε* = *ω* = 0 mean that we find the mode of rotation for the circular excitation *X* = *Y* with absence of gravity *g* = 0. In this case, we call equation (39) the *unperturbed equation*

<sup>1</sup> Note that this formula excludes the generalization *y* = *Y* cos(Ω*t* + Φ) which is considered e.g. in the model of unbalanced rotor [31–33]. Instead of Φ we introduce the angle *δ* of deviation of gravitational acceleration *g* from the

*x* = *X* sin(Ω*t*), *y* = *Y* cos(Ω*t*), (37)

, *<sup>β</sup>* <sup>=</sup> *<sup>c</sup>*

Dynamics of a Pendulum of Variable Length and Similar Problems 81

*m l*<sup>2</sup> <sup>Ω</sup> . (38)

Ω *g l*

*<sup>θ</sup>* <sup>+</sup> *<sup>μ</sup>* sin(*<sup>τ</sup>* <sup>+</sup> *<sup>θ</sup>*) <sup>=</sup> *<sup>ε</sup>* sin(*<sup>τ</sup>* <sup>−</sup> *<sup>θ</sup>*) <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> sin(*<sup>θ</sup>* <sup>+</sup> *<sup>δ</sup>*), (39)

<sup>1</sup> Here the upper dot denotes differentiation with respect to the new time

gravitational field *g*.

<sup>2</sup> sin(Ω *t* − *θ*).

vertical direction.

*τ*.

the axis *y*.

**Figure 8.** Basins of attractions in Poincare section for different excitation amplitude *ε* at the same frequency *ω* = 0.67 and damping *β* = 0.05. � marks period-two oscillational attractors, � marks period-one rotational attractors, marks period-two rotational attractors, × marks fixed points.

becomes circular. When there is no gravity the model of EEP coincides with that of hula-hoop, see section 4. The material of the section is based on the paper [5].

### **3.1. Main relations**

Equation of EEP's motion can be derived with the use of angular momentum alteration theorem, see [30]

$$\operatorname{Im} l^2 \frac{d^2 \theta}{dt^2} + c \frac{d\theta}{dt} + \operatorname{ml} \left( g \cos(\delta) - \frac{d^2 y(t)}{dt^2} \right) \sin(\theta)$$

$$+ \operatorname{ml} \left( g \sin(\delta) - \frac{d^2 x(t)}{dt^2} \right) \cos(\theta) = 0,\tag{36}$$

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*ε* = 0.05 *ε* = 0.06

*ε* = 0.45 *ε* = 0.5

becomes circular. When there is no gravity the model of EEP coincides with that of hula-hoop,

Equation of EEP's motion can be derived with the use of angular momentum alteration

*<sup>g</sup>* cos(*δ*) <sup>−</sup> *<sup>d</sup>*2*y*(*t*)

*<sup>g</sup>* sin(*δ*) <sup>−</sup> *<sup>d</sup>*2*x*(*t*)

*dt*<sup>2</sup>

*dt*<sup>2</sup>

 sin(*θ*)

cos(*θ*) = 0, (36)

+ *m l* 

**Figure 8.** Basins of attractions in Poincare section for different excitation amplitude *ε* at the same frequency *ω* = 0.67 and damping *β* = 0.05. � marks period-two oscillational attractors, � marks period-one rotational attractors, marks period-two rotational attractors, × marks fixed points.

see section 4. The material of the section is based on the paper [5].

*dθ dt* <sup>+</sup> *m l*

*m l*<sup>2</sup> *<sup>d</sup>*2*<sup>θ</sup> dt*<sup>2</sup> <sup>+</sup> *<sup>c</sup>*

**3.1. Main relations**

theorem, see [30]

**Figure 9.** Scheme of the elliptically excited mathematical pendulum of length *l*. The pivot of the pendulum moves along the elliptic trajectory (dashed line) with semiaxes *X* and *Y* in the uniform gravitational field *g*.

where *l* is the distance between the pivot and the concentrated mass *m*; *c* is the viscous damping coefficient; *θ* is the angle of the pendulum deviation from the vertical position; *t* is time; *g* is gravitational acceleration at the angle *δ* with respect to the negative direction of the axis *y*.

It is assumed that the pivot of the pendulum moves according to the periodic law

$$\mathbf{x} = X \sin(\Omega t), \quad \mathbf{y} = Y \cos(\Omega t), \tag{37}$$

where *X*, *Y*, and Ω are the amplitudes and frequency of the excitation.

We introduce new time *τ* = Ω*t* and the following dimensionless parameters

$$
\varepsilon = \frac{Y - X}{2l}, \quad \mu = \frac{Y + X}{2l} > 0, \quad \omega = \frac{1}{\Omega} \sqrt{\frac{\mathfrak{g}}{l}}, \quad \beta = \frac{c}{m l^2 \Omega}. \tag{38}
$$

With this notation equation (36) with substituted (37) in it takes the following form

$$
\ddot{\theta} + \beta \dot{\theta} + \mu \sin(\pi + \theta) = \varepsilon \sin(\pi - \theta) - \omega^2 \sin(\theta + \delta),
\tag{39}
$$

where we use the formula *Y* cos(Ω *t*) sin(*θ*) + *X* sin(Ω *t*) cos(*θ*) = *<sup>Y</sup>*+*<sup>X</sup>* <sup>2</sup> sin(<sup>Ω</sup> *<sup>t</sup>* <sup>+</sup> *<sup>θ</sup>*) <sup>−</sup> *<sup>Y</sup>*−*<sup>X</sup>* <sup>2</sup> sin(Ω *t* − *θ*). <sup>1</sup> Here the upper dot denotes differentiation with respect to the new time *τ*.
