**5. Conclusions**

26 Will-be-set-by-IN-TECH

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

*<sup>ϕ</sup>*∗(*τ*) = <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>ϕ</sup>*<sup>0</sup> <sup>+</sup> *<sup>μ</sup>*

(112) with *ϕ*<sup>0</sup> = arccos(−*γ*/*ε*) + 2*πn* exists for

*4.4.3. Coexistence of clockwise and counterclockwise rotations*

Conditions (115) in physical variables take the form

and counterclockwise rotations are illustrated in Fig. 14.

dimensional parameters *a* = 15*cm*, *b* = 10*cm*, *r* = 10*cm*, *R* = 50*cm*.

*4.4.4. Comparison with numerical simulations*

(108), (109) and (112) coexist if the following conditions are satisfied

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

with the stability conditions *γ* > 0, sin *ϕ*<sup>0</sup> > 0. Thus, the stable counterclockwise rotation

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

It follows from conditions (110), (113) that stable clockwise and counterclockwise rotations

meaning that the trajectory of the waist should be sufficiently prolate. Coexisting clockwise

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

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

1 + 2

*μ* <

<sup>0</sup> <sup>&</sup>lt; <sup>2</sup>*<sup>k</sup> <sup>R</sup>* <sup>−</sup> *<sup>r</sup> R*2*ωm*

<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>*

*ε*

<sup>3</sup> . (114)

< *a* − |*b*|, (116)

0 < *γ* < *ε*. (113)

0 < *γ* < min{*ε*, *μ*}. (115)

, (112)

*4.4.2. Counterclockwise rotation*

solution

*μ*

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 pendulum motion were plotted and analyzed.

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 constant angular speed equal to the speed of the excitation. We showed that one family of the solutions is stable, while the other one is unstable. These exact solutions allowed us to obtain the approximate solutions for the case of an elliptic trajectory of the waist. An interesting effect of inverse twirling was described when the waist moves in opposite direction to the hula-hoop rotation. It is shown that the approximate analytical solutions agree with the results of numerical simulation.

Averaging operator is defined as follows

operators

�·� = lim *T*→∞

*f uk* =

Hence, we can write recurrent expressions

*u*<sup>3</sup> = 

*F*<sup>4</sup> = 

*u*<sup>4</sup> = 

*F*<sup>5</sup> = 

> *u*<sup>5</sup> =

+ 1 <sup>2</sup> *<sup>f</sup>*1*u*2,2 <sup>+</sup>

+ 1 <sup>2</sup> *<sup>f</sup>*1*u*2,2 <sup>+</sup>

*n* ∑ *i*=1

*u*<sup>2</sup> = { *f*<sup>2</sup> + *f*1*u*<sup>1</sup> − *u*1*F*1} + *U*2, *F*<sup>3</sup> =

*f*<sup>3</sup> + *f*1*u*<sup>2</sup> + *f*2*u*<sup>1</sup> +

− *u*1*F*<sup>3</sup> − *u*2*F*<sup>2</sup> − *u*3*F*<sup>1</sup>

1 <sup>2</sup> *<sup>f</sup>*3*u*1,1 <sup>+</sup>

> 1 <sup>2</sup> *<sup>f</sup>*3*u*1,1 <sup>+</sup>

− *u*1*F*<sup>4</sup> − *u*2*F*<sup>3</sup> − *u*3*F*<sup>2</sup> − *u*4*F*<sup>1</sup>

1 *T T* 0

We also define integral operator {·} with the following expression

{ *f*(*x*, *τ*)} =

*∂ f ∂xi ui*

and so on, where *i* and *j* are the indices of vector components placed in *u<sup>i</sup>*

·|*x*=*Xd<sup>τ</sup>* <sup>=</sup> <sup>1</sup>

which is such an antiderivative that satisfies the condition �{ *f*(*x*, *τ*)}� = {�*f*(*x*, *τ*)�} = 0. Latter condition is necessary to obviate an ambiguity. We define following vector product

not to confuse it with smallness order indices *k* and *m*; *n* is the length of vectors *f* , *uk* and *um*.

*<sup>k</sup>*, *f uk*,*<sup>m</sup>* =

1

*f*<sup>4</sup> + *f*1*u*<sup>3</sup> + *f*2*u*<sup>2</sup> + *f*3*u*<sup>1</sup> + *f*1*u*1,2 +

*f*<sup>4</sup> + *f*1*u*<sup>3</sup> + *f*2*u*<sup>2</sup> + *f*3*u*<sup>1</sup> + *f*1*u*1,2 +

*f*<sup>5</sup> + *f*1*u*<sup>4</sup> + *f*2*u*<sup>3</sup> + *f*3*u*<sup>2</sup> + *f*4*u*<sup>1</sup> + *f*1*u*1,3 + *f*2*u*1,2

<sup>2</sup> *<sup>f</sup>*1*u*1,1,2 <sup>+</sup>

<sup>2</sup> *<sup>f</sup>*1*u*1,1,2 <sup>+</sup>

*f*<sup>5</sup> + *f*1*u*<sup>4</sup> + *f*2*u*<sup>3</sup> + *f*3*u*<sup>2</sup> + *f*4*u*<sup>1</sup> + *f*1*u*1,3 + *f*2*u*1,2

1

1

2*π*

(*f*(*x*, *τ*) − �*f*(*x*, *τ*)�) *dτ*,

*n* ∑ *i*,*j*=1

*u*1(*X*, *t*) = { *f*1(*x*, *t*)} + *U*1(*X*), *F*2(*X*) = �*f*2(*x*, *t*) + *f*1(*x*, *t*)*u*1(*X*, *t*)�, (121)

<sup>2</sup> *<sup>f</sup>*1*u*1,1 <sup>−</sup> *<sup>u</sup>*1*F*<sup>2</sup> <sup>−</sup> *<sup>u</sup>*2*F*<sup>1</sup>

1

1 <sup>2</sup> *<sup>f</sup>*2*u*1,1 <sup>+</sup>

1

1

<sup>6</sup> *<sup>f</sup>*2*u*1,1,1 <sup>+</sup>

<sup>6</sup> *<sup>f</sup>*2*u*1,1,1 <sup>+</sup>

*∂*<sup>2</sup> *f ∂xi∂xj*

*ui kuj <sup>m</sup>*, ...

*F*1(*X*) = �*f*1(*x*, *t*)�, (120)

1 <sup>2</sup> *<sup>f</sup>*1*u*1,1

*f*<sup>3</sup> + *f*1*u*<sup>2</sup> + *f*2*u*<sup>1</sup> +

<sup>2</sup> *<sup>f</sup>*2*u*1,1 <sup>+</sup>

1 <sup>6</sup> *<sup>f</sup>*1*u*1,1,1

1 <sup>6</sup> *<sup>f</sup>*1*u*1,1,1

+ *U*4, (125)

1

1

<sup>24</sup> *<sup>f</sup>*1*u*1,1,1,1

<sup>24</sup> *<sup>f</sup>*1*u*1,1,1,1

+ *U*5, (127)

*<sup>k</sup>* and *<sup>u</sup><sup>j</sup>*

+ *U*3, (123)

*<sup>m</sup>* on the top

, (122)

, (124)

, (126)

 2*π* 0

·|*x*=*Xdτ*.

Dynamics of a Pendulum of Variable Length and Similar Problems 97
