Dynamics of a discontinuous coupled electro-mechanical system oscillator with strong irrational nonlinearities and with two outputs

The dynamics of the nonlinear electromechanical device, consisting of a mechanical part with two outputs and an electrical part which acts as the server is strongly investigated in the present work. The mechanical part consists of two nonlinear oscillators with strong irrational nonlinearities having smooth or discontinuous characteristics, where nonlinearity is just due to the inclination of springs, the geometric configuration, which are both elastically coupled. While the electrical part is the φφ6Rayleigh equation. By using the Lagrangian formulation, the model equations are established and used to investigate the equilibrium points and their stabilities. Nest by using the multiple time scales method, the analytical solutions are found both for the case of large amplitude and the weak amplitude, leading to interesting bifurcation sets of the equilibria by varying the control parameters, the inclination angles and driven frequency. Finally, numerical investigations of the exact equation of the system are used to justify the validity of analytical results and to find new phenomena such as chaotic impulses, chaotic bursting and the train of kink signal generations.


Introduction
Nowadays, much more attentions are focused on nonlinear systems useful in the design and modeling of some practical systems, and this in most scientific and technology applications [1,2,3]. To name just a few, Qingjie Cao et al. proposed an archetypal system to investigate transitions from smooth to discontinuous dynamics, which in the smooth regime, bears significant similarities to the Duffing oscillator, exhibiting the standard dynamics governed by the hyperbolic structure associated with the stationary state of the double well, while Dupac et al. [5] modeled an electromagnetically levitated droplet as a three-dimensional system with lumped masses and elastic springs where its nonlinear behaviour was investigated. Next Boubaker et al. [6] investigated the mesoscopic fabric models by employing a discrete massspring approach. Terumichi et al. [7] proposed a mass-spring system attached at the lower end to a time-varying length string and the non-stationary vibrations were investigated. Although certain single autonomous oscillators like the Shua, Colpits and Vander Pol have been proved to have riches dynamics, the driven versions of these oscillators have richer dynamics and this being due to the fact that the time varying excitation introduces one more degree of freedom in the dynamical systems [8]. Let us mention that some of the driven oscillators like the Duffing one have oscillations which vanish when the driven force is removed. In addition, it has been proved that when these oscillators are coupled each to others, the resulting system will have more applications, among which the increasing of the frequency range of signals generated, the increasing of the generated powers as well as the conversion of one form of energy to other, with different dynamics [9]. For e.g. the periodic regular electrical signal can be converted to chaotic mechanical motion and vice versa.
Nature is full of nonlinearities, responsible for a great variety of responses in natural systems. Natural behaviors could either be periodic or chaotic over time and space. Let us mention that all technology innovations are consequences of the development in scientific tools, and this after strong mathematical investigations. New phenomena in nonlinear dynamics were discovered because of chaotic and unpredictable behavior from apparently deterministic systems, including mechanical systems. It is known that the chaotic motion of dynamical systems cannot always be predicted far into the future. Generally, dynamical systems depend on one or several parameters that control the strength of external influences. For example in forced oscillations, the frequency and amplitude of the exciting oscillation are the control parameters. In varying these parameters one may hit critical values at which the behaviors of systems change qualitatively. Critical values of this kind are called bifurcations. Bifurcations play an important role in the development of deterministic chaos. Many mechanical systems, being them single or coupled can be used to see the effects of nonlinear dynamics.
The attention carried on researches in the area of coupled non-linear oscillators has received a great particular attention in recent years [10,11]. This being due to the fact that coupled oscillators provide fundamental models for the dynamics of various physical, electrical, mechanical and biological systems [12,13,14]. The electro-mechanical systems, that is systems with coupled mechanical structures and electrical/electromagnetic circuits, are widely used in mechatronics for a variety of applications (e.g. as actuators, sensors, servo mechanisms, switches,…). Dynamics of such systems have been proved to be possible by means of two mutually coupled differential equations that are the equations describing both the electrical, and the mechanical parts.
In [4], Y W Han, et al. proposed a nonlinear oscillator with a pair of irrational nonlinearities, which lead to the transition from smooth to discontinuous dynamics, this nonlinear oscillator being comprised a lumped mass, linked by a pair of inclined elastic linear springs. As a consequence the nonlinearity obtained was strong and due just to the inclination. In this work, we focus on the elastically coupling of two of the previous models of nonlinear oscillators, both magnetically coupled to nonlinear electrical Rayleigh circuit. As we shall see further, the obtained circuit has richer dynamics and would provide a variety of applications in engineering.

Description of the system model and equations of motion
We consider the dynamic response of an electromechanical instrument consisting of an electrical part described by the 6 Rayleigh equation [8] coupled to the mechanical part as depicted in Fig.1. The mechanical part consists of two simple mechanical systems, elastically coupled through the spring with stiffness . Each mechanical system comprises a lumped mass, ( = 1,2), linked by a pair of inclined elastic springs of stiffness ( = 1,2), capable of resisting both tension and compression and which are pinned to their rigid supports (T). Although each of the inclined springs provides linear restoring resistance, it has been proved that the resulting force has a strong irrational nonlinearity due to the geometric configuration [4]. The electrical part consists of a linear inductor L, a nonlinear capacitor (NC), a nonlinear resistor (NR) in series with electromotive force e(t). The voltage over the NR and the NC can be expressed as where by the dot, we mean the derivative with respect to time t.
being a dimensionless parameter which can be positive or negative, while is a positive dimensionless parameter. These parameters account of nonlinear diode used. 0 and 0 are the nominal angular frequency and charge, respectively. The coupling between the electrical and mechanical parts is made by the electromagnetic transducer where the first coupling equation follows from Faraday's law: A coil of n turns moving at the velocity v with respect to the magnetic flux density B generates an electromotive force (voltage) given by = 2 =̇. The second equation follows from the Lorentz force law: The external force f required to balance the total force of the magnetic field on turns of the conductor is = − 2 = −̇. The kinetic energy of the system is then deduced as: where the over dot is the derivative with respect to time t and where x and y are the displacements of masses 1 and 2 , respectively. The potential energy is: While the dissipative function is expressed as: From the above defined energy components, one can define the Lagrangian given as = − , while the equation of motion can be obtained from the Lagrangian equation leading to the following equation of motion: Let us make the changes of variables and parameters: leading to the following dimensionless equation: The mechanical parts the above equations (7) have already been found in [4], in which in place of ( ) and ( ), they had and , respectively. With = ℎ and = , while = √ℎ 2 + 2 . As one can see, 2 + 2 = 1, meaning that and of the differential equation obtained in [4] could not be treated as independent parameters as it was the case. This is why in order to avoid this misunderstanding, we use here ( ) and ( ) in place of the old and since cos 2 ( ) + sin 2 ( ) = 1 .For the large amplitude oscillations, or for weak value of angles 1 and 2 that is for | ± , equation (7) leads to csgn(x) being the signum function.
For weak amplitude oscillations, Eq. (7) can be expanded to sixth order to give:

Equilibrium points
In order to find the equilibrium points of the system, let us rewrite the set of Eqs. (7) in the vector form as ̇= ( ) + ( ), where = ( ,̇, ,̇, ,). The non-linear flow F represents the motion equations of the non-excited system and E is the external excitation. The equilibrium states are solutions of the set of non-linear algebraic equations ( ) = 0, that is In order to simplify our studies, let us impose 1 = 2 = , leading that 1 = 2 = = (ℎ/ ) and 1 = 1. One can then have at the equilibrium point, 0 = 0 = 0 , 0 being the solution of the following equation: admitting as solutions 0 = 0, and/or 0 = , where R is the solution of the following polynomial equation: The obtained solution is plotted in Fig.2, showing for each value of two solutions with opposite signs. Since 2 > 0, it appears that the third line of Eq.(12) will admit the following solution:  , meaning that the system will admit the combination of 3 fixed points for the first two lines and 3 others for the third line, which give 3 × 3 = 9 equilibrium points.
For 1 > 2 √ 2 , the last line of Eq.(12) has as solution meaning that the system will admit the combination of 3 fixed points for the first two lines and 5 others for the third line, which 3 × 5 = 15 equilibrium points. For these obvious reasons the system will have the multi stability-like behaviour. To prove the above remarks, let us mention that for 1 = 1 and 1 = 2 , the set of Eq.(11) admits the potential energy This potential energy is plotted in Fig.(3) for the case = , Fig.(3) (a), we have the These graphs are in agreement with the above analysis. = and = 1 < √ .

Stability of equilibrium points
The local stability analysis of the above equilibrium points can be determined by investigating the linearized system ̇= , J is the six th order Jacobean matrix defined by = ( ) , leading to: . This jacobean matrix leads to the characteristic equation The analytic determination of the eigenvalues solutions of Eq.(18) leads to quite large analytical expressions, this is why in order to find the parameter bands for which the system is stable, we recall the Routh-hurwitz stability criteria as follows; All parameters of Eq.(18}) must have the same sign, that is the positive sign since the first parameter is positive. That is > 0, = 0,1,2,3,4,5. In addition, The numerical algorithm has been used to compute these eigenvalues, and the results are plotted in Figs.4(a,b), for −1 < 0 = 0 < 1 and −15 < 0 < 15, showing that the system admits some solutions with positive real part, meaning that the system model will admit unbounded or chaotic solutions. As one can see, the equation admits some solution with positive real part, which is the precursor sign of the existence of unbounded or chaotic solutions. The equation will also admit solution with zero real part, the well known Hopf bifurcation.

Analytical solution of the system equation
Amongst all the analytic approaches for the non-linear oscillations, the multiple time scales method (MTSM) has been chosen because it is the most adapted to the study of dynamical systems around resonance frequencies. In this paper, the damping, the coupling, the external excitation and the non-linearity are considered as global first order perturbations. Therefore let us write 1 = 10 , 2 = 20 , 3 = \ 30 , 1 = 10 , 2 = 20 , 1 = 10 , 2 = 20 , 1 = 10 , 2 = 20 0 = . In order to simplify our studies and to approach the solutions with the best accuracy, we will consider first the large amplitude oscillations, and approximated by Eq.(9), and secondly the weak amplitude given by Eq.(11):
Substituting Eqs.(23) into the set of Eqs.(21) and equating coefficients of like powers of $\epsilon$, one obtains the following set of ordinary differential equations: Equations at order 0 , then we have : admitting as solution: At the order 1 , one has: leading by taking into account (25) to , The unknown amplitudes , , = 1,2,3 may now be determined by eliminating secular terms in Eqs.(27). However, because of the high dimensionality of the system, it is impossible to obtain unambiguously a set of secular equations valid for all frequencies. This is why it is necessary to find the solution near the unique possible resonance √2 1 = √2 2 = Ω = , with = 1. Let us then set where 1 , 2 and are are the tuning parameters expressing the quantitative nearness of the resonance frequencies √2 1 , √2 2 Ω with the fixed electric natural frequency = 1. Then, accounting Eqs.(28) into Eq.(31) leads to the following solvability conditions:

Case of the weak amplitude oscillations: smooth case
In order to study this case, let us take 1 and 2 such weak as = 1/2 0 , meaning that 2 ( ) ≈ By finding , and Z as outlined in Eq.(23), one obtains at order 0 the same equation as in (24), but with 1 = 2 = 0, admitting as solution: from where by proceeding as above, one has the following secularity conditions: while the odd lines lead to: This equation is numerically solved and plotted in Fig.8 showing that the form of the obtained graph is different to that obtained in Fig.5. Meaning that the inclination angle could considerably affect the system's dynamics. = . .

Preliminary
In this section, we give numerical simulations results in order to confirm the theoretical results obtained in the previous sections and to find other new dynamics. The interesting problem is to analyze the parameter regions for which the system is regular or chaotic. The system having several parameters, each of them playing different and virtual roles on the system behavior, we will then limit our study by chosen both the driven frequency and inclination angles as control parameters. Otherwise the values of other parameters are fixed as follow: Using then these parameters, the system Eq. (7) is numerically solved by using the fourth order Runge Kutta scheme, with the time step always kept = 10 −2 .

Effect of the driven frequency
To begin, let us mention that there are some tools to measure, predict, and quantify chaotic dynamics in mechanical systems. The queries are how to recognize chaos and predict their occurrence? The answer is that they can be recognized by looking period doubling and homoclinic bifurcations in the systems. They can also be quantified by Lyapunov exponents [15]. While the maximum Lyapunov exponent is positive, the systems remains chaotic. In this subsection we analyze the effects of the driven frequency Ω by keeping constant the other parameters as given in Eq.(44) including the inclination angles which are 1 = 2 = /4. The first results in the such of case is plotted in Fig.(9), showing the regular behavior zones for 0.8 < Ω < 0.8926 and 0.94 < Ω < 0.96. Elsewhere we have the bifurcation bands, with some regular behaviors windows in the bifurcation band. As we can see Fig.(9), (a,b and c) are bifurcation diagrams for variables X,Y and Z, respectively which are in agreement with (d), that is the maximum Lyapunov exponent. In order to qualify the signals generated, we have plotted in Figs. (10,11 and 12), the signal profiles at the left column, the phase portraits at the middle, and the frequency spectrum at the right. In all these figures, the top lines are for variable X, the middles for Y and the bottoms for Z. Figs.(10 and 11) are obtained for Ω = 0.02 and Ω = 0.5, which belong to chaotic zones. The signals generated for X and Y are chaotic impulses, while that generated for Z are chaotic bursting. It is obvious that the bursting or impulses widths are more larger for Ω = 0.02 than Ω = 0.5. As one can see from the last columns of these figures, the spectrum frequency of these figures have two noisily bands, justifying then that the system is chaotic. Figure(12), obtained for Ω = 0.85 chosen in the regular band frequency shows that X and Y are regular impulses signals, while Z is the regular bursting, even thought the bursting width for Z is too weak, but the phase portrait of (12)( 2 ) justifies our conclusion. The spectrum frequency of these figures presents some regular picks.   Figure 11 Left: Solution, Middle: phase portrait, and right: frequency spectrum, obtained for parameters chosen as in Fig.( 9), and for Ω = 0:5.

Effect of the inclination angles = =
Let us analyze in this sub-section the effects of the inclination angles 1 = 2 = , by keeping constant the other parameters as given in Eq(44). In Figs. (13) and (14), we have the bifurcation diagrams for Ω = 0.02, that is for the weak frequency value, and Ω = 0.85, that is for the large frequency value.
In Fig.(13) obtained for weak value of Ω, the regular band is found for the inclination angle 0 ≤ < 0.239, that is for the less inclination. Otherwise for > 0.239, chaos is found, with the appearing of regular behavior windows.
However for large value of Ω as shown in Fig.(14), the regular behavior is found for > 0.62, that is when springs are more inclined. Figure (15) shows the regular train of kink signal obtained for 1 = 2 = = 0.4 belonging to chaotic band of Fig.(13), agreeing the belonging of regular frequency windows inside the chaotic bands. In Fig.(16) we have the periodic signals generations for 1 = 2 = = 1.2 belonging to regular windows.

Discussion
From the above results, it is obvious that: The system is very sensitive for the varying values of the driven frequency Ω and the inclination angles 1 and 2 . The system is also sensitive to nonlinearity coefficients and the amplitude of the driven signal, but this last case is not investigated in the present work.
The system can generate bursting-like signals for very low frequencies, which are regular or chaotic according to the inclination angles.
The system can also generate regular or chaotic periodic kinks for higher frequency, according also to the inclination angles.

Conclusion
In this paper, we have investigated the dynamics of the nonlinear electromechanical device, consisting of the mechanical part with two outputs and the electrical part both couplet magnetically. The mechanical part consisted of two nonlinear elastically coupled oscillators with strong irrational nonlinearities having smooth or discontinuous characteristics, where nonlinearity was due to the geometric configuration, while the electrical part is the 6 -Rayleigh equation. By using the Lagrangian formulation, the model equations have been established and used to investigate the equilibrium points and their stabilities. Next by using the multiple time scales method, the analytical solutions have been found both for the case of large amplitude, and the weak amplitude, respectively leading to an interesting bifurcation sets of the equilibria for varying values of control parameters, that are the inclination angles and driven frequency. Next through numerical investigations, the bifurcation diagrams, the maximum Lyapunov exponents, the frequency spectrums and phase portraits have been plotted to demonstrate the complex transitions of the system to chaos. By plotting the output signals for the electrical and mechanical parts, it appeared that chaotic impulses, chaotic bursting and the train of kink signals were generated by the system, according to the tuning parameters.
From the above obvious results, it appeared that the use of the coupled oscillator constructed in the present work can represent an excellent way to multi-signal generations, allowing their possible applications as hammer, saw or sieve, with two outputs. However, our worry is what would be the effect of the single-impact or vibro-impact energy on the system dynamics, is it possible to insert in the system a supplementary component able to control chaos by annihilating them. The works is this light will constitute our future investigations.