The time delayed feedback control to suppress the vibration of the autoparametric dynamical system

The response of a dynamical system of two-degree-of-freedom with parametrically excited pendulum is solved and studied. The delayed feedback control is applied to suppress or stabilize the vibration of the system. The case of 1:2 sub-harmonic resonances between pendulum and primary system is studied; the method of multiple scales is applied to obtain second-order approximations of the response of the system. It is shown that the delayed feedback control can be used to suppress the vibration or stabilize the system when the saturation control is invalid, the vibration of the system can be suppressed by the delayed feedback control. The effect of delay on the suppression is discussed; the vibration of the system can be suppressed at some values of the delay.


INTRODUCTION
In the last years, many papers have been devoted to the control of resonantly forced systems in various engineering fields.In passive vibration absorbers a physical device is connected with the primary structure, while in the case of active absorbers the device is replaced by a control system of sensors, actuators and filters.Active control of mechanical and structural vibrations is superior to passive control, because the former is more flexible in many aspects.
Periodically forced nonlinear systems under delay control have been analyzed by Plaut and Hsieh (1987) in the case of nonlinear structural vibrations with a time delay in damping.The studying of an approach for implementing an active nonlinear vibration absorber (El-Bassiouny, 2005).The strategy exploits the saturation phenomenon that is exhibited by multi-degree-of-freedom systems with cubic nonlinearities possessing one-to-one internal resonance.The proposed technique consists of introducing a second-order controller and coupling it to the plant through a sensor and an actuator, where both the feedback and control signals are cubic.The vibration and chaos control of nonlinear torsion al vibrating systems is studied (El-Bassiouny, 2006).
The resonance, stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback is investigated in (Naik and Singru, 2011).The primary, super harmonic and sub harmonic resonances of a harmonically excited nonlinear quarter-car model with time delayed active control are investigated.They focused on the influence of time delay in the system.This naturally gives rise to a delay differential equation model of the system.It was found that proper selection of timedelay showed optimum dynamical behavior.The modeling and optimal active control with time delay dynamics of a strongly nonlinear beam.They investigated the control by a sandwich beam and one using piezoelectric absorber (Nana Nbendjo et al., 2003, 2009).
The effect of the time delay on the non-linear control of a beam when subjected to multi-external excitation forces is determined, multiple scale perturbation method is applied to obtain the solution up to the second order approximations (El-Gohary and El-Ganaini, 2011).Using the delayed feedback control and saturation control to suppress the vibration of the dynamical system under external force is studied (Zhao and Xu, 2012).The nonlinear dynamical analysis on four semi-active dynamic vibration absorbers with time delay is solved (Yongjun and Mehdi, 2013).Vibration reduction, stability and resonance of a dynamical system excited by external and parametric excitations via time-delay absorber is studied (Sherif and Sayed, 2014).The dynamical system of a twin-tail aircraft, which is described by two coupled second order nonlinear differential equations having both quadratic and cubic nonlinearities, solved and controlled (Amer et al., 2009).The effect of different controllers on the vibrating system and the saturation control of a linear absorber to reduce vibrations due to rotor blade flapping motion is obtained (Sayed andKamel, 2011, 2012).studied the primary resonance, stability and design methodology of a piecewise bilinear system under cubic velocity feedback control with a designed time-delay are investigated through combining multi-scale perturbation method with Fourier expansion, the effects of time delay on dynamics behaviors are explored (Gao and Chen, 2013).
The nonlinear analysis of time-delay position feedback control of container cranes studied (Nayfeh and Baumann, 2008).Stabilizability of the turning process subjected to a digital proportional-derivative controller is analyzed, the governing equation involves a term with continuous-time point delay due to the regenerative effect and terms with piecewise-constant arguments due to the zero-order hold of the digital control (Lehotzky et al., 2014).The dynamical system with time varying stiffness subjected to multi external forces studied.The system is written as two degree of freedom consists of the main system and absorber.The multiple time scale perturbation method is applied to get the approximate solution up to the third approximation.The stability of the system at the simultaneous primary resonance is investigated using both frequency response equations and phase-plane methods (Amer and Ahmed, 2014).The application of the renormalization group (RG) methods to the delayed differential equation is determined by analyzing the Mathieu equation with time delay feedback, get the amplitude and phase equations, and then obtain the approximate solutions by solving the corresponding R G equations (Wu and Xu, 2013).
In this paper, the delayed feedback control is applied to suppress the vibration of the system.The equation of motion and the perturbation analysis and the stability of the equilibrium solutions are given.All possible resonance cases are extracted and investigated at this approximation order.The stability of the system is studied using the phase plane and frequency response curves.The analytical solutions and numerical simulations of the delayed feedback control are presented and compared.

EQUATIONS OF MOTION
A two degree-of-freedom dynamic vibration absorber system with a parametrically excited pendulum is described in (Song et al., 2003).In the present paper, a position feedback with delay is introduced into the model to control the vibration of the system and the model is shown in Figure 1.The system consists of the mass M, the linear spring with stiffness k, and the viscous damper presented by coefficient c.The second system is a simple pendulum of mass m p hinged at the system.The distance between the supporting point and the center of gravity of the pendulum is .
lJ and c  represent the inertia moment with respect to the supporting point and the damping coefficient of the pendulum respectively.
The system is excited directly by a harmonic force where x the displacement of is , M  is the angle of rotation of the pendulum, and , ( be noted that the delayed feedback disappears in (1) and (2) when 0,   and (1) and (2) are identical to Song et al. (2003).Thus it is easy to observe effects of the delayed feedback on vibration suppression performance when 0.

 
Firstly, a set of dimensionless (or normalized) variables are defined as:   1) and (2) can be written as the following nondimensional forms by dropping the asterisk for convenience. where Equations ( 1) and ( 2) are the nonlinear delayed differential equation of the forced vibration system.The non-autonomous differential equations can be transformed into autonomous differential equations by the method of average or the multiple scales.The method of multiple scales is used to obtain the approximate analytical solutions.The bifurcation problem is investigated based on the new autonomous differential equations given thus.

PERTURBATION ANALYSIS
Since  are very small, here it is set  4) are written as: The method of multiple scales is employed to seek second order approximate solutions of Equations ( 5) and ( 6) in the following form: The derivatives with respect to time are expressed in terms of the new scales as: Where , 0,1.
Substituting Equations ( 7) to(10) into Equations ( 5) and ( 6) and equating coefficients of like powers of  yield that: The general solution of Equations ( 11) and ( 12) can be expressed as: Where A and B are arbitrary functions at this level of approximation, cc denotes complex conjugate.The external excitation and the delayed feedback are expressed in complex forms.
A  can be expanded in a Taylor series (2002) under the assumption that the product of time delay and the small parameter  is small compared to unity.

  
2) Sub-harmonic resonance: .            5) Simultaneous resonance: any combination of above resonance cases is considered as simultaneous resonance.

STABILITY ANALYSIS
From numerically studying the different resonance cases, we find that the worst is resonance case is the simultaneous resonance case Substituting Equation (28) into Equations ( 22) and ( 23) and setting the coefficients of the secular terms to zero Amer and Soleman 493 yield the solvability conditions given by: 30)   We express the complex function , AB in the polar form as Where 1 , b, a  and 2  are real-valued functions.
Substituting Equation (31) into Equations ( 29) and ( 30) and separating real and imaginary parts yields: Where . Then it follows from Equations ( 32) to (35) that the steady state solutions are given by: From Equations ( 36) to (39), we have the following cases: Case 1. 0 a  and 0 b  : in this case, the frequency response equation are given by the following equations: Case 2: 0 a  and 0 b  : in this case, the frequency response equation is given by:

Linear solution
Now to the stability of the linear solution of the obtained fixed let us consider A and B in the forms p , p , q and 2 q are real values and considering 42) into the linear parts of Equations ( 29) and ( 30) and separating real and imaginary parts, the following system of equations are obtained: For the solution ( 0 a  and 0 b  ), we get: The stability of the linear solution in this case is obtained from the zero characteristic equation After extract we obtain that Where According to the Routh-Huriwitz criterion, the above linear solution is stable if the following are satisfied: When Conditions (49) are not satisfied, the initial equilibrium solution is unstable, and bifurcations may occur.But if Conditions (49) are not satisfied.Conditions ( 49) imply that all the eigenvalues of the Equation ( 47) have negative real parts.When Conditions (49) are not satisfied, this is not the case.First, we prove that zero cannot be a simple eigenvalue of the Equation ( 47).If zero is an eigenvalue of the Equation ( 47), then it is not simple.In fact, suppose zero is an eigenvalue of the Equation ( 47), then it follows

Regard
A as a variable, if Equation (50) has a real solution, it follows, the discriminate of quadratic Equation (50), 0.  On the other hand,   .One choice of the parameters that satisfy these conditions is:

(ii) Double zero and a pair of purely imaginary eigenvalues
Taking the following parameter values: 49) has a double zero and two imaginary eigenvalues

 
For the solution ( 0 a  and 0 b  ), we get: (51) (52) The stability of the linear solution is obtained from the zero characteristic equation The linear solution is stable in this case if and only if 5 0 r  , and otherwise it is unstable.

Non-linear solution
To determine the stability of the fixed points, one lets Substituting Equation (54) into Equations ( 32) to (35), using Equations ( 36) to (39) and keeping only the linear terms in ,, m ab we obtain: For the solution ( 0 a  and 0 b  ), we get: The stability of a particular fixed point with respect to perturbations proportional to exp( ) t  depends on the real parts of the roots of the matrix.Thus, a fixed point given by Equations ( 55) to ( 58) is asymptotically stable if and only if the real parts of all roots of the matrix are negative.
For the solution ( 0 a  and 0 b  ), we get: The stability of a given fixed point to a disturbance proportional to exp( ) t  is determined by the roots of: Consequently, a non-trivial solution is stable if and only if the real parts of both eigenvalues of the coefficient matrix Equation ( 61) are less than zero.

NUMERICAL RESULTS
Runge-kutta fourth order method has been conducted to determine the numerical solution of the given system.Figure 2 illustrates the response and phase-plane for the non-resonant system at some practical values of the equations parameters without time delay control.

Resonance cases
we see that the amplitude increasing at the resonance cases and the worst case is the simultaneous resonance case when 2 , 2 ,       which the amplitudes are increased to about 175% compared with the basic case shown in Figure 3, which means that the system need s to reduced the amplitude of vibration or controlled.
Figure 4 shows the effects of time delayed, from this figure we can see that the delayed is effect at the regions 1.1 2.2   , 3.5 4.5   and 5.9 6.9   , otherwise not affect.Figure 5, illustrates system at the worst case is the simultaneous resonance case when 2 , 2 ,       with time delay control the amplitude of vibrations is very minimum , which means that the time delay control is very effective.

Effect of parameters
The amplitude of the system is monotonic decreasing function of the coefficient gain G 1 more increasing of G 1 leads to saturation phenomena as shown in Figure 6a.The amplitude is a monotonic increasing function of the excitation amplitude of the coefficient G 3 .But more increasing of coefficient G 3 leads to very high amplitude and the system becomes unstable as shown in Figure 6b

Response curves
The frequency response Equations ( 40) and ( 41

Comparison between numerical solution and approximation solution
Now, we get good agreement of the approximate solution obtained from frequency response equation and the

Figure 1 .
Figure 1.A model describing the autoparametric dynamical vibration absorber with delayed feedback. 2 is introduced into the equations used for bookkeeping only.A fast scale is characterized in 0 Tt  with the motion at  and a slow scale in

Figure 4 .
Figure 4. Effects of time delayed ( ) on the system.
) are nonlinear algebraic equations of , ab .These equations are solved numerically as shown in Figures7 and 8.From Case 1 where 0, 0 ab  : Figure7, shown that the steady state amplitudes of the system are monotonic decreasing