Nonlinear Dynamics of a Nonlinear Damping Gyroscope and Its Passive Control

)is article deals with the analysis of the eﬀects of passive control on the complex dynamics of a nonlinear damping gyroscope. After modeling the gyroscope dynamics under the inﬂuence of the control force, using the harmonic balance method, the amplitudes of the harmonic oscillations are determined. Subsequently, the Routh–Hurwitz criterion is used to analyze and determine the stability domains of the oscillations. )e inﬂuence of the control force parameters on the amplitude of the oscillations is studied. )e control of chaotic dynamics and the coexistence of gyroscope attractors are performed through bifurcation diagrams, Lyapunov exponents, phase portraits


Introduction
)e gyroscope is a device based on the principle of conservation of angular motion on one, two, or three axes with respect to an inert frame of reference. Gyroscopes are instruments used in a large number of civil and military applications, such as inertial navigation, guidance, and stabilization of platforms [1,2]. )ey provide a measurement of the rotation, of the reference of the device with respect to an inertia reference frame. )e first gyroscopes designed were mechanical.
Due to the moving mechanical parts that compose them, these gyroscopes are bulky, expensive, and require significant maintenance. Among the solutions developed to replace them, optical gyroscopes that do not contain moving parts have been a very advantageous option. Indeed, they offer a longer life, require less maintenance, are more compact and lighter, and better withstand shocks and strong accelerations. )is system reached its full maturity at the end of the seventies, and the first demonstration of its technical competitiveness was its integration in 1978, in the navigation systems of Boeing. )e number of gyroscopes used in the world is now exploding as they equip a growing number of smartphones.
)ey are used to precisely identify the position and orientation of the device in space in aviation for the artificial horizon, the heading maintainer, the coordinator or turn indicator, camera stabilization during a capture disturbed by the movement of the waves, and the pitch of an airplane [3]. One of the goals of technology, in general, and engineering, in particular, is therefore the miniaturization of devices and components. For industrial machines, whose complexity has constantly increased, understanding and diagnosing the vibratory phenomena involved require increasingly detailed simulations of their behavior [4]. In this context, the prediction of damping characteristics and their effects is fundamental for the design of rotating machines in order to provide an accurate idea of safe rotation ranges [5,6]. For this reason, many studies in the field of dynamics have focused on the modeling of dissipative effects, the prediction of critical velocities, responses to unbalance, and finally on the prediction of instability thresholds [7,8]. In general, the damping translates the energy dissipation of the vibratory system [9,10]. It is therefore in the sense of understanding the influence of damping on the dynamics of gyroscopes that Chen modeled the dynamics of a gyroscope subjected to a cubic dissipation force [2]. )e study of the rotating gyroscope under the influence of the nonlinear damping force has made it possible to understand that the gyroscope under given conditions has a complex dynamic [1][2][3][9][10][11][12][13]. Of these behaviors, the most complex are chaotic movements and the phenomenon of multistability [1]. Several works have attempted to propose modes of control or synchronization of the chaotic gyroscopes according to the field of application of the gyroscope and its intended interest [2][3][4][5][6][11][12][13][14][15][16][17]. Multistability is one of the complex phenomena encountered in nonlinear systems. )us, the dynamics of systems exhibiting this multistability phenomenon are difficult to predict with precision. Indeed, for the same value of a parameter of the system for which multistability appears, the system is in several states or at several vibration amplitudes, thus, making the system difficult to control. On the contrary, bistability reflects the coexistence of two attractors, while megastability designates the coexistence of an infinite number of attractors for the same system [18][19][20][21][22][23][24][25].
Many researchers have endeavored to predict and control multistability because of its impact on the performance of systems that exhibit this phenomenon. Among the methods used for the control of complex phenomena in nonlinear dynamics, there are methods which include active, semiactive, passive, and feedback control. For example, recently, the control of multistability phenomenon of the famous hyperchaotic oscillator TNC has been achieved using a linear augmentation control scheme [26]. Similarly, in [27], the control of up to nine coexisting hidden chaotic attractors in a specific domain of the initial condition is successfully realized. )is control is achieved by choosing initially an attractor as a designated survivor using the feedback term method. It is then found that the employed method can also be used to control multistability in a system without an equilibrium point.
From the literature, it should be noted, at least to our knowledge, that the control of vibration amplitude, chaotic dynamics, and coexistence of attractors for rotating gyroscopes has not yet been done using passive control force. However, during its use, the rotating gyroscope may be under the influence of a passive force when it is influenced by vibrating base velocity, for example. )us, it would be necessary to study complex gyroscope dynamics and control them with passive forces [28,29]. It is therefore in this perspective that this work proposes to model the dynamics of the rotating gyroscope by taking into account the passive control forces. It is also a question in this work of researching the influence of the control force on the amplitude of the harmonic vibrations of the gyroscope, on the one hand, and of analyzing the effects of each of the parameters of the passive control force on the chaotic behavior and on the coexistence of attractors phenomenon when the gyroscope is working. )is paper, therefore, makes it possible to identify in which operating condition of the gyroscope the passive control would be effective.
)e paper is structured as follows: In Section 2, we will give the mathematical modeling of the oscillations of the complex dynamics of nonlinear gyroscope and formulate the process of passive control. In Section 3, we will determine the amplitude of the harmonic oscillations. )e stability limits are analyzed, and the effect of the different passive control force parameters is also studied on the amplitude of the harmonic resonance. Section 4 discusses the route to chaos and the effects of controlling force on chaotic dynamic states and the coexistence of attractors. )e conclusion is presented in the last section.

Model and Equation of Oscillations
In this work, we consider the symmetrical gyroscope mounted on a vibrating base [1,2].
)e geometry of the considered problem is illustrated in Figure 1. )e motion of a symmetrical gyroscope mounted on a vibrating base can be described by Euler angles (nutation), Փ (precession), and Ψ (rotation).
)e Lagrangian of the system is given in the following equation [1,2]: where I 1 and I 3 are the polar and equatorial moments of inertia of the symmetrical gyroscope, respectively, M.g is the force of gravity, l is the magnitude of the external excitation disturbance, and Ω is the frequency of the external excitation disturbance. Using Routh's procedure and using the Routhian associated to hidden coordinates Փ et Ψ, the equation governing the dynamics of the gyroscope as a function of the angle θ is given by [1,2]: where € θ � (d 2 θ/dt 2 ) and _ θ � (dθ/dt). Fsin2Ωtcosθ is parametric excitation, C 1 _ θ and C 2 _ θ 2 are, respectively, linear and nonlinear damping, and α 2 ((1 − cosθ) 2 /sin 3 θ) − βsinθ is a nonlinear restoring force. In general, β, C 1 , and C 2 are fixed parameters, while F is the normalized amplitude, Ω is the frequency of the external excitation, and α 2 is proportional to the speed of rotation.
In this work, we suppose that the support on which the gyroscope is fixed vibrates with a speed. )e additional driving force that applies to the gyroscope can be defined by [29,30] where K represents the control gain coefficient and ε � 0, ± 1 indicates the direction of the passive control force F c . In the particular case where parameter ε � 0, the control system is reduced to the original equation governing the movement of the gyroscope. For ε � +1, the drag force follows the positive direction of vibrating base velocity, while directions of vibrating base velocity and forces are opposite when ε � −1.
)e control system can then be written as follows:

Harmonic Vibrations.
Due to the highly nonlinear problem in gyroscopes, the nonlinear terms ((1 − cosθ) 2 /sin 3 θ) and sinθ are extended in power of _ θ [2,4]. So, we get with Still for the same reason of the nonlinearity of the considered system, the differential equation of the control system is analytically solved using the harmonic balance method [30]. )e general solution of equation (4) is where B 1 and B 2 are amplitudes of oscillations. Inserting equation (6) into (4) and assuming that B 1 (t) and B 2 (t) are slowly varying functions with time, then ignoring the second derivatives and the terms of the second order, we obtain the following set of first-order coupled differential equations: Figure 1: Schematic diagram of a symmetrical gyroscope [12,17].

Journal of Control Science and Engineering
are expressions of amplitude and phase, respectively. )e steady-state vibrations are obtained for _ B 1 � 0; and _ B 2 � 0. )e characteristic geometric equation is thus given by From equation (9) after some algebraic manipulations, we have By combining equations (10) and (11), we get  (11) with fixed parameters [1][2][3], )e effect of ε controller on system behavior is shown in Figure 2. )is figure shows that the amplitude of the harmonic resonance is reduced for both directions of the passive control force. We investigated the effects of control force on the resonant state. Figures 3 and 4 show the effects of F, U, and K on the amplitude of the harmonic oscillations, respectively, for ε � 1 and ε � −1. It can be observed that the amplitude of the harmonic oscillations decreases when one of parameters U and K of the passive control force increases with F. We can conclude that our passive control is practical because the amplitude of the oscillations of the complex dynamics of the nonlinear gyroscope is considerably reduced when the appropriate control parameters are varied.

Stability
Analysis. An important question is the stability of the stationary solution in the parameter space. To determine which of the stationary solutions are physically acceptable, their stability properties must be studied [1]. For this purpose, we consider where b 01 and b 02 are the values of B 1 and B 2 at equilibrium states obtained by solving equation (11), and b 1 b 2 are the disturbances. Inserting (13) in the system of equation (9) and retaining only the linear terms, we obtain the system of linearized equations for the disturbance in the form Journal of Control Science and Engineering 7 with b 01 ; b 02 of B 1 ; B 2 obtained at steady state, respectively, whose expressions are as follows: )ereby, Now, searching the solutions of system (17) are and substituting these solutions (18) into equation (17), we have the following characteristic equation: where H and H 0 are functions of system parameters and are given by According to the Routh-Hurwitz criterion, equilibrium states will be unstable if the characteristic equation (19) has, at least, one positive real root. Otherwise, they are stable. )us, the stability domain is given by the equations H < 0 and H 0 > 0.

Control of Chaotic Dynamic
States. )e complex dynamics of the nonlinear gyroscope considered in this work can be chaotic. )is chaotic dynamic does not guarantee stability in the movement of the gyroscope [1]. )is can reduce the performance of the gyroscope. Our goal in this subsection is to reduce or remove this chaotic dynamics by using a passive control force [31]. To achieve this goal, we used the fourth-order Runge-Kutta algorithm [32] to numerically solve equation (3), and the resulting bifurcation diagrams and its corresponding Lyapunov exponents are plotted using the amplitude of the external excitation forces as the bifurcation parameter. For the simulations, the parameters used are α � 10; β � 1; Ω � 2; C 1 � 0.1; C 2 � 0.05; M � 0.1; K � 0.1; g � 10; U � 2.5; the step for the bifurcation parameter is hf � 0.001; the step for time is h � 0.01; the number of iterations is 1000000. )e bifurcation diagram and its corresponding Lyapunov exponent are obtained in the absence of the control force Figures 5(a) and 5(b) for ε � 0. In the presence of the control force, we plotted for the same parameters as Figures 5(a), 5(c), and 5(d) for ε � 1 and Figures 5(e) and 5(f ) for ε � −1. It noticed through these figures that the chaos is accentuated for ε � 1 but totally reduced for ε � −1 [32]. )us, chaos is well controlled when the gyroscope's velocity is in the opposite direction than the passive control force. For example, to verify the predictions of Figure 5, we plotted for F � 45 the corresponding phase portraits and time series of the system for ε � 0, ε � 1, and ε � −1 (see Figure 6). It was noticed that the phase diagrams and the corresponding times stories confirm the dynamics of Figure 5 and the effectiveness of the control force in the direction opposite to that of the fluid, i.e., for ε � −1. Figure 7 shows the effect of the control gain parameter K and the fluid velocity U on the dynamics of the gyroscope for ε � 1. It emerges from the analysis of this figure, that in this direction, the parameter K accentuates the chaotic behavior, while the large values of the fluid velocity U make it possible to eliminate the chaotic dynamics of the gyroscope (see Figure 7(f ), for example). )is important result is easily seen in Figure 8, where it is clearly seen that the parameters K and U have inverse roles on the dynamics of the system when ε � 1. Always on this figure, one can observe that the chaos is totally eliminated in the direction of the fluid velocity when the latter takes the value U � 8. )e study of the effect of K and U on the dynamics of the gyroscope for ε � −1 revealed that K and U are very good control parameters because when U is fixed, the parameter K makes it possible to completely eliminate the chaotic behavior, and for K fixed, the speed U makes it possible to completely eliminate this undesirable behavior for the stability of the gyroscope. )ese different results are illustrated in Figure 9 where for ε � −1, it is noted that chaos is totally eliminated for K � 1 when U is fixed at U � 2.5, while in the same direction, chaos is eliminated for U � 8 when the gain parameter is set to 0.1. )ese results are confirmed by the phase portraits and their time series presented in Figure 10. It follows from these various analyses that the control of the chaotic dynamics of the gyroscope is very effective when it operates in the opposite direction of the fluid velocity under the influence of the passive control force.

Control of the Coexistence of Attractors.
To further support the results of the theoretical investigations in this work, the research and control of the coexistence of attractors in the dynamic behavior of the gyroscope are introduced. Indeed, the phenomenon of coexistence of attractors is very complex and can affect the performance of the gyroscope. It is therefore, for this reason that we seek and propose the control of this complex phenomenon. For this, we study the influence of the passive control force on this behavior using bifurcation diagrams, Lyapunov exponents, and phase portraits. )e bifurcation diagram and the Lyapunov exponents are obtained by choosing F as the bifurcation parameter.
On these diagrams, the curve in blue represents the diagram obtained by varying the amplitude F of the external excitation in the increasing direction, while the results obtained by varying F in the decreasing direction while remaining in the same interval is the curve in red. When the two curves have the same nature and are exactly superimposed (merged) throughout the domain, then the gyroscope takes the same path on the outward as on the return side when parameter F is varied in both directions. If the two curves do not overlap exactly over a given interval of F, then the path followed by the gyroscope in its outward movement is not the same as that which it takes in the return for increasing and decreasing evolutions from F. Attractors are said to coexist. In this condition, if the dynamics of the gyroscope are the same in this domain, we say that attractors of the same nature coexist; otherwise, we speak of the coexistence of attractors of different natures. )us, for the same value of the bifurcation parameter, the vibrations of the gyroscope will have different amplitudes and of different natures depending on the case. It is therefore a very complex phenomenon because, in this state, it would be difficult to identify exactly and uniquely the amplitude and the nature of the vibrations of this gyroscope. )is could have a negative impact on the performance of the rotating gyroscope because it would complicate the precise acceleration of Coriolis when the gyroscope will enter this phase. Figure 11 represents the dynamics of the rotating gyroscope when ε � 0 for F between 30 and 50. )us, we note through this figure the coexistence of periodic attractors of period 1T, of period 2T, of chaotic attractors, and of attractors of period 2T with chaotic attractors. In the presence of the passive control force (Figure 11), we note when ε � 1 for F between 30 and 50, that the coexistence of attractors of period 2T has disappeared giving way to the coexistence of attractors of period 1T with those of period 2T. Similarly, the coexistence of attractors of period 2T with chaotic attractors is eliminated, leaving room for the coexistence of attractors of period 1T with chaotic attractors. For ε � −1, we note that the coexistence of attractors of different natures is totally eliminated and the domain of the coexistence of chaotic attractors is considerably reduced. We can therefore say that the control of the phenomenon of coexistence of attractors is more effective when the control force is applied in the opposite direction of the fluid velocity (ε � −1) than in the same direction (ε � 1). Figure 12 shows the effects of the parameters K and U on the phenomenon of the coexistence of attractors for ε � 1. It is observed that the parameter K does not facilitate the elimination of this phenomenon, whereas for a high value of the velocity U of the fluid, the coexistence of attractors of different natures as well as chaotic attractors is eliminated. We finally obtain, in this case, the coexistence of attractors of period 1T. Finally, Figure 13 represents the effects of the parameters K and U on the phenomenon of the coexistence of attractors for ε � −1. In this case, we note that the parameter K of control and the velocity U of the fluid eliminate the coexistence of the attractors but preserve the coexistence of the attractors of period 1T. Figures 14-16 represent the phase portraits of the system, respectively, for In sum, from the analysis of the results obtained, it is clear that the control force makes it possible to eliminate the coexistence of attractors of different, chaotic natures but preserves the coexistence of attractors of period 1T. We also note that this control is effective for the direction of ε � −1 than in the direction ε � 1. Moreover, the speed of the fluid    Figure 12: Effect of the parameters K and U on the bifurcation diagram for ϵ � 1; α � 10; β � 1; )e diagrams are drawn by sweeping the normalized amplitude F in the ascending direction (blue) and in the opposite direction (red).    makes it possible more to control this complex phenomenon contrary to K.

Conclusions
In this work, we investigated the passive control of complex nonlinear gyroscope dynamics by passive control force. )e model has been described, and the equation of the control system governing the corresponding movement is obtained. Using the method of harmonic balance, we found the amplitude of oscillatory states, the Routh-Hurwitz criterion is used to analyze and determine the stability limits of oscillations. Numerical simulations are used to validate and complete the results obtained by analytical methods. We noted a more robust stability of the complex dynamics of the gyroscope subjected to the passive control force in the negative direction of the fluid velocity than in the positive direction. In both directions, we observe the existence of instability domains. We also noticed the increase or decrease of these instability domains when the control parameters K; U of the control force of the system increase or decrease. )e effect of the control process on chaotic dynamic and on coexistence of attractors was effective with ε � −11 [32]. )e high amplitude of harmonic oscillations, chaotic states, and coexistence of attractors was successfully controlled by the passive control investigated in this work. Advances in the design and manufacture of the rotating machines widely used today in many sectors of industry make it possible to increase both the performance and the efficiency of these machines by operating them in speed ranges increasingly high turnover. However, the forces generated are increasingly important and strongly solicit the overall dynamic behavior of the machines. )us, the results obtained in this work are useful for the study of the control of the dynamics of rotating machines which is more relevant than ever. Precisely, this work has a capital importance in engineering because the results obtained allow to have a stable dynamics for a better performance of the gyroscope.

Data Availability
)e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
)e authors declare that they have no conflicts of interest.