Abstract

This work analyzes the chaotic dynamics and the chaotic synchronization and their control in the complex dynamics of a rotating gyroscope modeled following Euler angles using the Lagrange approach. It is obtained for appropriate conditions the chaotic dynamics and its control using the four order Runge-Kutta algorithm. By the backstepping method, the chaotic synchronization conditions of two gyroscopes are obtained by building a Lyapunov function and numerical simulations. The study also pointed out that the first integrals of the moments of inertia of the gyroscope influence the chaotic dynamics and the chaotic synchronization. The analysis of the effects of the amplitudes and frequencies of this excitation makes it possible to find the best areas where the control and synchronization are effective.

1. Introduction

Research in the field of gyroscope dynamics dates back to the 19th century. Gyroscopic movement constitutes a relevant and very interesting subject which has inspired a great deal of researches. Theses researches interest in the gyroscope stems largely from its numerous applications in mechanical, optical and micro-electro-mechanical devices where it functions variously as an attitude heading reference system, a gyrocompass, an inertial measurement unit, and in inertial navigational aid systems. Previous works on nonlinear gyroscopes considered bifurcation and chaos in a harmonically excited rate gyro and in a symmetric gyro with linear-plus-cubic damping [1-5] ¹ [6-10] ^{10 11, 12, 13, 14, 15, 16, 17}. It should be noted that the concept of the chaotic movement of gyroscopes was not evolved until 1981. This explains the recent studies undertaken in the domain by Ge et al who analyzed the nonlinear state of a symmetrical heavy weight displayed by a gyroscope mounted on a base vibrating ^{9, 10, 11}. It emerges from their study that the motion of a symmetrical gyroscope has already been the subject of a previous study. However, their study did not take damping into account. From their results, we also retain that a symmetrical gyroscope presents regular and chaotic movements. Therefore, it emerges from the analysis that the use of the chaotic method turns out to be a new method. The presence of chaotic behavior for a linear system may or may not be beneficial for that system and therefore it can be researched or avoided depending on the field of application of the system. Thus, the chaotic dynamics of the system can be enhanced or avoided or controlled so as to improve the performance of the dynamic system ^{18, 19, 20, 21, 22, 23, 24 25, 26, 27, 28, 29, 30 31, 32, 33, 34, 35, 36, 37}.

The synchronization of systems has also been widely discussed in the literature ^{38, 39, 40, 41, 42, 43 44, 45, 46, 47, 48 49, 50, 51, 52, 53 54, 55, 56, 57} due to its applications in the fields of finances ^{48, 49, 50, 51, 52, 53}, of communication ^{42, 43}, of electronic ^{54, 55}, of biochemistry ⁵⁶ and of naval and aerospace navigation ⁵⁷. For several years, synchronization between two chaotic systems seemed impossible because of their sensitivity to initial conditions. But thanks to scientific work this illusion has become a reality and several types of synchronizations have been studied and different methods have been proposed ^{38, 39, 40, 41, 42 43, 44, 45, 46, 47, 48 49, 50, 51, 52, 53 54, 55, 56, 57}. With the work of Yamada and Fujisaka ⁵⁸, a local approach to synchronization was used and subsequently Afraimovich et al. ⁵⁹ developed important concepts related to chaotic synchronization. Later, Pecora and Carroll ⁴⁷ defined chaotic synchronization known as identical synchronization, developed on the basis of coupled chaotic circuits, with one called master and the other slave. Among the approaches proposed in the literature ^{38, 39, 40, 41, 42, 43 44, 45, 46, 47, 48, 49, 50 51, 52, 53, 54, 55 56, 57, 58, 59} to synchronize identical, non-identical, uncertain, hyper-chaotic chaotic systems, excitation oscillators, etc., we can cite synchronization methods based on: nonlinear control, adaptive control, sliding mode control, active control, backstepping control, state feedback control, etc. The backstepping method is a recursive method which is based on the choice of a Lyapunov function with the necessary controller design. There are several advantages in this method:

- It presents a systematic procedure for the selection of the controller;

- It can be applied to different chaotic systems: identical, different, hyper-chaotic systems, with external excitation;

- It offers the possibility of performing synchronization with a single controller whatever either the size of the systems to be synchronized;

- The calculated controller does not contain a derivative which offers simplicity in the implementation of the algorithm.

Indeed, according to the work of Z.-M. Ge and J.-K. Lee ⁵⁷ the gyroscope can be under the influence of periodic, biharmonic, etc. excitation. This is one of the reasons that motivated the work of K.S. Oyeleke et al. ¹⁶ dealing with parametric vibrational resonance in a gyroscope driven by dual-frequency forces. In their work, they have shown by suitable methods that the gyroscope can experience vibrational resonance when under the influence of a two-frequency force. Motivated by the work above, the present work aims to determine and analyze the conditions under which a rotating gyroscope with more than one degree of freedom can undergo the chaotic dynamics, the synchronization and then study the influence of a biharmonic excitation on these phenomena. For this, we first model the dynamics of a gyroscope following Euler's angles. Subsequently, using numerical simulations through bifurcation diagrams, Lyapunov exponent and phase portraits, the chaotic dynamic and the chaotic synchronization are rigorously analyzed. Then by the analysis of the effects of the frequency and the amplitude of the excitation, we studied the possibilities of eliminating or accentuating the chaotic dynamic and the chaotic synchronization for the gyroscope.

The paper is structured as follows: section 2 gives the mathematical modeling of the dynamics of a gyroscope. Section 3 deals with the chaotic synchronization and the influences of a biharmonic excitation on the synchronization of a gyroscope. Finally, we provide a conclusion in the last section.

2. A symmetric Gyroscope System Dynamic Equation

The geometry of the considered problem is illustrated in Figure 1. The motion of a symmetrical gyroscope mounted on a vibrating base can be described by Euler angles (nutation), (precession) and (rotation).

The Lagrangian of the system is ^{12, 17, 19}:

(1)

where and are the polar and equatorial moments of inertia of the symmetrical gyroscope respectively; is the force of gravity; is the distance between the center of gravity and O; is the magnitude of the external excitation disturbance, and is the frequency of the external excitation disturbance.

The dissipative force is also assumed to be in the form ^{12, 17, 19}:

(2)

According to the Lagrange approach, we write:

(3)

Thus, along the axis , we obtain

(4)

Subsequently, we assume ⁵⁷ , , , (), and only first term is considered for the Fourier series and we set .

Thus, we have:

(5)

Along the axis , we obtain:

(6)

Along the axis , we obtain:

(7)

Using the two first momentum integrals (), and after some algebraic transformations, the full dynamics of the gyroscope is given by:

(8)

and

(9)

Where,

, , , , , ,

with the gyroscope's spin velocity.

In the remain of this work, we set ; and . It follows that the complete gyroscope dynamics studied in this work are modeled by:

(10)

Equation (10) characterizes the dynamics of the gyroscope subjected to periodic excitation ⁵⁷. In the present work, we consider that the excitation force is biharmonic. This amounts to consider that are non-zero. Thus, the dynamics of the gyroscope are governed by:

(11)

with

3. Chaotic Synchronization by the Backstepping Method

In this section, we study the influence of biharmonic excitation on the chaotic synchronization of the gyroscope. For this, we consider the master and slave systems defined respectively by

(12)

and

(13)

To achieve the chaotic synchronization of these two gyroscopes, we use the so-called backstepping method. So, we define synchronization errors as follows:

(14)

In this case, we have:

(15)

So,

(16)

(17)

(18)

First backstepping step

We define the backstepping variables and the virtual command in the following way:

(19)

In the present study, we choose the virtual command . Thus, the Lyapunov function is defined as follows:

(20)

From (16) and (17) we have:

(21)

We choose the control law as follows:

(22)

Then, becomes

or ,

So,

Second backstepping step

We define the second Lyapunov function as follows:

(23)

We then obtain:

(24)

We choose the control law as:

(25)

In this case, becomes:

(26)

Returning to the error variables, we have:

(27)

The chosen Lyapunov function is positive definite () and its derivative is definite semi-negative (). Then the synchronization of the two systems is thus achieved.

Proof of synchronization

The system of error variables is:

(28)

with the fixed point

Then the only fixed point of the system is

Nature and stability of the fixed point

The Jacobian matrix of the system is

Thus, we have:

The discriminant of this equation is . Then the equation admits a double solution .

Conclusion: the eigenvalue of the system is negative. Then the fixed point is a stable attractor node. The system of error variables is globally asymptotically stable. Hence the synchronization of the two master and slave systems is efficient.

To ensure the accuracy of the analytical calculations, we performed the numerical simulations using the Runge-Kutta algorithm of order 4. The parameters and the initial conditions used to have the chaotic phase portrait of Figure 2 which served as a reference for synchronization. The simulation results are represented in Figure 3 and Figure 4. Figure 3 shows the temporary evolution of state variables , ,, , and . When the controller is activated on the date , the trajectories of the different state variables coincide, i.e. coincides with , with and with . Figure 4 shows the temporary evolution of synchronization errors. When the controllers are activated on the date , the various errors to effectively converge towards 0. Thus, synchronization between the two identical gyroscopes is achieved.

We now study the effect of the control parameters , , and on the synchronization and the results are represented in Figure 5, Figure 6, Figure 7, Figure 8 respectively. The analysis of Figure 5 shows that the different synchronization errors converge towards zero before for an increase in the values of . There is therefore a decrease in the synchronization time when the values of increase. The same observation is made in Figure 7 for the parameter . The analysis of Figure 6 shows that the various synchronization errors converge towards zero after the date t=50s for an increase in the values of . There is therefore an increase in the synchronization time when the values of increase. Figure 8 also shows an increase in the synchronization time when the values of increase. By fixing the values of the other parameters used in Figure 7 and Figure 8 synchronization becomes impossible for values of .

4. Conclusions

In this work, the effects of biharmonic excitation force on the chaotic synchronization of a symmetric gyroscope with cubic damping have been studied according to Euler angles. To achieve this, the complete dynamics of the two degree of freedom gyroscope under biharmonic excitation are modeled. Also, the conditions for the existence of chaotic dynamics and the chaotic synchronization are obtained numerically. The effects of all gyroscope parameters on its behavior are thoroughly studied. It appears that the parameters and influence the chaotic synchronization of the gyroscope. Likewise, the results of this study revealed that certain parameters of the gyroscope can accelerate or delay their chaotic synchronization. These different results obtained in this work prove that the dynamics and the efficiency of the gyroscope are not only strongly influenced by its internal parameters but also by the nature and characteristics of its external excitation. For example, if the support on which the gyroscope is fixed is moving, this could affect the performance of the latter and their synchronization. For the continuation of this work, it would be good to study the vibrational resonance of the gyroscope and to physically realize the synchronization of several gyroscopes fixed on a vibrating table. This could allow the gyratory movement of several machines to be synchronized.

Author Contribution Statement

G. F. Pomalegni and J. M. Aguessivognon contributed to the modeling, to the various calculations, to the drafting of the paper

C. H. Miwadinou contributed to the modeling, the various calculations and the analyzes of the results of the paper

A. V. Monwanou contributed to the analysis of the results and supervised this work.

Data Availability Statement

All data are included in text directly.

Declarations

Conflict of interest The authors declare that there are no conflicts of interest.

References