Position and Velocity Time Delay for Suppression Vibrations of a Hybrid
Rayleigh-Van der Pol-Duffing Oscillator

In this paper, we used time delay feedback to minimize the vibrations of a hybrid Rayleigh–van der Pol–Duffing oscillator. This system is a one-degree-offreedom containing the cubic and fifth nonlinear terms and an external force. We applied the multiple scales method to get the solution from first approximation. Graphically and numerically, we studied the system before and after adding time delay feedback at the primary resonance case ( ffi !). We used MATLAB program to simulate the efficacy of different parameters and the time delay on the main system.


Introduction
The Duffing oscillator is used as a main type model for different engineering and physical problems such that electric circuit, oscillation of plasma, optical stability and the buckled beam [1][2][3][4][5][6]. Wen et al. [7] presented two kinds of Van der Pol oscillator containing fractional order terms. The averaging method used for obtaining the approximation solution. The additional stiffness coefficient is almost zero and the additional damping coefficient damping is almost the maximum value when the two kinds of Van der Pol fractional existed. The vibrations of Van der Pol oscillator are suppressed by using the nonlinear time delayed feedback controller and the effectiveness of the feedback gain on bifurcation point is studied numerically [8]. In [9], the dynamic stable and unstable behavior of the ring of coupled Van der Pol oscillators are discussed numerically also, the amplitude of the oscillator increased if the stability conditions are not satisfied. The Homotopy analysis method is used to obtain the analytically solution for the first time of a single-well, double-well and double-hump van der pol-Duffing oscillator [10]. Soleman et al. [11], utilized the position time delayed feedback control to restrain the auto parametric dynamical system vibrations. The Rayleigh equation with a cubic nonlinearity oscillator is presented in [12] and is studied for the following cases: positive linear and cubic coefficients, positive linear and negative cubic coefficients and negative linear and positive cubic coefficients. [13][14][15], modified and studied the bifurcation of Van der Pol-Duffing-Rayleigh oscillator. Of great importance to restrained the vibrations of Van der Pol oscillator. One of the important kinds of controllers is the time delay control.
The time delay control is used for suppression the nonlinear beam vibrations in [16] and they deduced that, the vibrations could be reduced for some values of time delay, which called the vibration suppression region. The positive position feedback controller is adjusted with the time delay to minimize the horizontal vibration of a magnetically levitated body and to control the vibration of a forced and self-exited nonlinear beam [17,18]. The time delay control is used to suppress the vibrations of many dynamical systems such that, Stainless-steel beam, Helicopter blade flapping and Duffing oscillator [19][20][21]. In this article, the vibrations of a hybrid Rayleigh-Van der Pol-Duffing oscillator exciting by external forces are suppressed by using position and velocity time delay feedback controllers. Numerically, we simulated the behavior of the system without and with time delay controllers. We used MATLAB program to simulate the efficacy of different parameters and the time delay on the main system. The influences of some chosen coefficients are illustrated numerically and analytically. The rapprochement between numeric and analytic solution is offered.

Mathematical Formulation
The one-degree-of-freedom of a hybrid Rayleigh-Van der Pol-Duffing oscillator presented in [15] as: We used the position and velocity time delay to minimize the vibrations of a hybrid Rayleigh-Van der Pol-Duffing oscillator subjected to an external force as the following:

Perturbation Analysis
We used the multiple scales method [22,23] to obtain the solutions of Eq. (2) up to the first approximation: The first and second derivatives take the forms: For the first approximation solution, we performed a two time scales T r ¼ e r t such that ðr ¼ 0; 1Þ. The Oðe 0 Þ: OðeÞ: Eq. (6) is a homogenous differential equation of second order its solution takes the form: Denote that A is a complex function in T 1 . The complex conjugate parts collected in the term c.c. From Eq. (8), we have By using Taylor expansion, we get the following form of A s 1 and A s 2 : For computation the right hand sides of Eq. (7), we will use Eqs. (8)-(10) so that, For the particular solution of Eq. (13) be bounded, we will remove the secular terms such that, where M @ ð@ ¼ 1; …; 5Þ offering complex functions in T 1 are defined in the "Appendix". From the first approximation, there is only one resonance case, which is the Primary resonance ffi x.

Equilibrium Solution of a Fixed Point
While in movement to evolve the steady state solution's stability, start with the following procedures: Inserting Eq. (29) into Eqs. (24) and (25) then, we obtained the following system: For the above system's solution be stable, the real parts of its Eigen-values must be negative.

Numerical Illustration
Numerically, we applied Runge-Kutta 4 th (RK-4) order method using MATLAB program to solve the differential equation of the main system after using the time delay feedback controller. This study occurs at the worst resonance case (Primary resonance) by the following values of parameters: Fig. 1 clarifies the amplitude of the uncontrolled main system, which equal 1.5. The influence of the main system parameters (damping coefficient l and nonlinearities coefficients k; g; b; d; and h) has been presented on Fig. 2. From this figure, we note that, the amplitude of the main system is monotonic decreasing in the damping coefficient l and nonlinearities coefficients k; g; b; d and but monotonic increasing in the nonlinear coefficient h. More increasing of the damping coefficient l leads to saturation phenomena and the amplitude value equal to 0.9 so that, the system might be need a control. After using time delay feedback controller, the main system amplitude reduced to reach 0.09 as represented on Fig. 3 this means that, the effectiveness of the controller (E a = amplitude without control/amplitude with) equal 17. Eq. (28) solved numerically to obtain the graphical solution for the amplitude via the detuning parameter ðrÞ which, presented by one peak. The response curve of the amplitude-delay s 1 at s 2 ¼ 0 and s 2 ¼ 0:05 for different values of c 1 was shown in Fig. 4. From this figure, we can see that, for small values of c 1 the vibration suppression region (is the region at which the amplitude-delay's response curves demonstrates stable solution) increased. The response curve of the amplitude-delay s 2 at s 1 ¼ 0 and s 1 ¼ 0:03 was shown in Fig. 5 for different values of c 1 . For s 1 ¼ 0 we can notice that, the vibration suppression region increased for small values of c 2 but at s 1 ¼ 0:03, the vibration suppression region increased for large values of c 2 .
The response curves of the main system a against the detuning parameter r is presented for s 1 ¼ 0:03; s 2 ¼ 0:05 such that the solid line expresses the stable solution of Eq. (28), while the dash one expresses the unstable solution of the same equation as shown in Fig. 6a. For large values of the external force, the main system's amplitude increase also as notice in Fig. 6b. The amplitude increasing and shift to right for small values of the time delay displacement's feedback gain c 1 and the real part of all Eigen-Values is negative so, the solution is stable for small values of c 1 as illustrated in Fig. 6c, and this is consistent with Fig. 4. Fig. 6d shows that, the main system's amplitude is monotonic decreasing function on the time delay velocity's feedback gains c 2 and the solution is stable for large values of c 2 , and this is  Fig. 5b. For natural frequency x, the main system's amplitude is monotonic decreasing function and shifted to right as shown in Fig. 6e. Fig. 7 presents the response of the main system a against the external force f before and after control, from this figure we can see that, the effectiveness of the time delay control for suppression the vibrations of the main system. The Eqs. (24) and (25) solved analytically and presented graphically by (---) lines which be in agreement with the numerical solution of Eq. (1) as shown in Fig. 8. From Fig. 9, there is a    (  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬  ‫ـ‬ ) and the perturbation analysis (------) for the controlled system good agreement between the frequency response curves (FRC) which given by the sold line and the numerical solution of Eq. (1) using (RK-4) that marked by green circles.

Conclusion
Time delay control has been illustrated for the primary resonance case ( ffi x) of the hybrid Rayleighvan der Pol-Duffing oscillator. The solution of the nonlinear system from the first approximation is obtained applying the method of multiple scales. We success to reduce the vibrations of the hybrid Rayleigh-van der Pol-Duffing oscillator from 1.5 to 0.09 by using Time delay control.
The study divulged that: 1. For increasing the value of external excitation leads to increasing in the system amplitude. Funding Statement: The author(s) received no specific funding for this study.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.