journal of vibration and control - german university in...

http://jvc.sagepub.comJournal of Vibration and Control

DOI: 10.1177/1077546307076283 2007; 13; 403 Journal of Vibration and Control

Ayman A. El-Badawy and Tarek N. Nasr El-Deen Quadratic Nonlinear Control of a Self-excited Oscillator

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Quadratic Nonlinear Control of a Self-excitedOscillator

AYMAN A. EL-BADAWYDepartment of Engineering Design and Production Technology, The German University in Cairo,Cairo, Egypt ([email protected])

TAREK N. NASR EL-DEENMechanical Engineering Department, Al-Azhar University, Cairo, Egypt

(Received 30 September 2005� accepted 16 November 2006)

Abstract: The objective of this work is to demonstrate the feasibility of the use of the nonlinear saturation-based control concept to suppress self-excited vibrations by means of active nonlinear feedback control.The authors use the van der Pol oscillator as the working model for a self-excited system. A saturationphenomenon is induced by tuning the frequency of the under-damped second-order absorber to one-half thatof the primary system. Although the authors conclude that we can achieve some level of performance withthis control technique, we question its robustness due to the rich dynamics introduced by the controller.

Key words: Saturation, self-excited system, quadratic absorber.

1. INTRODUCTION

The force acting on a vibrating system is usually external to the system and independent ofits motion. However, there are systems for which the exciting force is a function of some partof the state of the system, such as its displacement, velocity, and acceleration. Such systemsare called self-excited oscillators, because the force that sustains the motion is created orcontrolled by the motion itself. In many cases, a self-excited system possesses negative lineardamping and positive nonlinear damping. A positive damping force is directed opposite tothe velocity and a negative damping force is directed along the velocity.

Self-excited oscillations resulting from a form of negative damping occur in many phys-ical systems. The instability of rotating shafts, the flutter of wings and turbine blades, theflow-induced vibration of pipes, the aerodynamically induced motion of bridges, the gallop-ing of cables, and the chattering of machine tools are all typical examples of self-excitedvibrations. A number of textbooks (see, e.g., Nayfeh and Mook, 1979� Nayfeh, 1981), dealwith self-excited systems and the methods used to investigate them. Thus, it would be ofmuch interest to develop new ways of reducing the persistent oscillations of such systems.

Parametric vibrations are induced by a varying system parameter (e.g. stiffness). On theother hand, autoparametric vibration (self parametric) is characterized by an internal cou-pling involving at least two modes. From a mathematical point of view, this coupling isinduced by nonlinear terms in the equations of motion of the combined system. Physically

Journal of Vibration and Control, 13(4): 403–414, 2007 DOI: 10.1177/1077546307076283

��2007 SAGE Publications



404 A. A. EL-BADAWY and T. N. N. EL-DEEN

speaking, an autoparametric system consists of two parts: A main system and a secondarysystem. When the main system is externally excited, the secondary system is parametri-cally excited as a result of the variation of its stiffness by the response of the main system.In other words, a two-mode interaction occurs when the main system exhibits a forced re-sponse which, in turn, drives the secondary system into parametric resonance. In this case,energy is transferred from one part of the combined system to the other. This energy transferdepends on the type of nonlinearities and the damping forces, and can be partial or complete,depending on the system parameters.

In the case of quadratic nonlinearities and free vibrations, the energy transfer is completewhen the natural frequency of the main system is exactly twice the natural frequency of thesecondary system� this case is referred to as a two-to-one internal resonance, (Nayfeh andMook, 1979� Nayfeh, 2000). In the case of forced vibrations, the response of the main systemcan be suppressed. When the main system is excited at a frequency near its natural frequency(i.e., primary resonance), the response of the main system will have the same frequency asthe excitation, and the response amplitude will increase linearly with the amplitude of excita-tion. However, above a critical value, the response of the main system saturates and all of theadditional energy from the exciter is channeled to the secondary system. This is referred to asthe saturation phenomenon and has been developed as a nonlinear control strategy by Oueiniet al. (Oueini et al., 1998, 1999� Oueini and Nayfeh, 2000). A similar control strategy ofself-excited vibrations was implemented by Verhulst (2005), who quenched undesirable vi-brations by energy absorption through embedding the oscillator in an autoparametric systemby coupling to a damped oscillator.

Hall et al. (2001) applied the nonlinear saturation phenomenon to suppress the flutter ofa wing in numerical simulations. They illustrated the concept by means of an example with arather flexible, large-span wing of the type found on such vehicles as high endurance aircraftand sailplanes. They concluded that to control the amplitude response, they need to activelyadapt the controller frequency because of the nonlinear nature of the systems.

In this article, the authors will evaluate the method of saturation-based control developedby Oueini and Nayfeh (2000) for suppression of the large-amplitude vibrations of a vander Pol self-excited system. We will evaluate the feasibility of the control law for variousparameter combinations.

2. MATHEMATICAL MODEL

Considering a van der Pol oscillator, (Nayfeh, 1981), coupled with a saturation-based con-troller (Oueini et al., 1998), we obtain the following system:

�x � �21x � �1 �x � �3x2 �x � �1 y2 (1)

�y � �22 y � ��c �y � �2xy� (2)

Here, x and y denote the generalized coordinates of the oscillator and absorber respec-tively, and �1 and �2 are their linear undamped natural frequencies. The overdot representsdifferentiation with respect to time t. For the energy dissipation oscillator, we incorporatethe negative linear damping term �1 �x , and the positive nonlinear damping term �3x2 �x . Forthe absorber, we incorporate a positive linear damping term �c �y. Finally, the two quadratic



QUADRATIC NONLINEAR CONTROL OF A SELF-EXCITED OSCILLATOR 405

coupling terms �1 y2 and �2xy are necessary for the energy exchange between the oscillatorand absorber to take place. The authors will study these equations using both numericaland analytical techniques. To begin the analytic studies, we must first scale the governingequations. The scaled equations are

�x � �21x � ��1 �x � ��3x2 �x � ��1 y2 (3)

�y � �22 y � ��c �y � ��2xy (4)

where � is a small bookkeeping parameter. To express the nearness of �1 to 2�2, we write

�1 � 2�2 � �� (5)

We attack the problem using the method of multiple scales (Nayfeh, 1981), and assumethat x and y can be approximated by the following expansions:

x � x0 �T0� T1� �x1 �T0� T1

y � y0 �T0� T1� �y1 �T0� T1 �

Where T0 = t and T1 = �t. Substituting the above expansions into the scaled equations andequating coefficients of like powers of �, we obtain the following systems of differentialequations, which are solved in cascade:

O(1):

D20 x0 � �2

1x0 � 0 (6)

D20 y0 � �2

2 y0 � 0 (7)

O(�):

D20 x1 � �2

1x1 � �2D0 D1x0 � �1 D0x0 � �3x20 D0x0 � �1 y2

0 (8)

D20 y1 � �2

2 y1 � �2D0 D1 y0 � �c D0 y0 � �2x0 y0� (9)

Where Dn � Tn

.The solution to problem O(1) is given by

x0 � A �T1 ei�1T0 � �Ae�i�1T0 (10)

y0 � B �T1 ei�2T0 � �Be�i�2T0 (11)

where A (T1) and B(T1) are arbitrary complex-valued functions of T1. Substituting thissolution into (8) and (9), we obtain




D20 x1 � �2

1x1 � ��2i�1 D1 A � i�1�1 A � i�3�1 A2 �A � �1 B2e�i�T1�

ei�1T0

� N ST � cc (12)

D20 y1 � �2

2 y1 � ��2i�2 D1 B � i�c�2 B � �2 A �Bei�T1�

ei�2T0 � N ST � cc (13)

where NST stands for the nonsecular terms and cc stands for the complex conjugate of thepreceding terms. In order to obtain a uniform expansion up to first order, the terms in brack-ets must vanish. This condition yields the modulation equations for the complex-valuedamplitudes A and B:

2i�1 A � i�1�1 A � i�3�1 A2 �A � �1 B2e�i�T1 (14)

2i�2 B � �i�c�2 B � �2 A �Bei�T1 � (15)

Where primes are derivatives with respect to the slow time scale T1. Expressing (14) and(15) in terms of polar coordinates gives

A � 1

2aei� (16)

B � 1

2bei� (17)

where a, b, á and â are unknown real-valued functions of T1. Substituting these equationsfor A and B into (16) and (17), we obtain

i�1

�ia� � a

�ei� � 1

2i�1�1aei� � 1

8i�3�1a3ei� � 1

4�1b2e2i�e�i�T1 (18)

i�2

�ib� � b

�ei� � �1

2i�c�2bei� � 1

4�2abei��ei�T1 � (19)

Separating the real and imaginary parts of (18) and (19), we obtain the following systemof real-valued modulation equations:

a � 1

2�1a � 1

8�3a3 � �1b2

4�1sin ��T1 � � � 2� (20)

b � �1

2�cb� �2ab

4�2sin ��T1 � � � 2� (21)

a� � ��1b2

4�1cos ��T1 � � � 2� (22)

b� � ��2ab

4�2cos ��T1 � � � 2� � (23)




These modulation equations represent a non-autonomous system� to obtain an equivalentautonomous system, we define

� �T1 � � � 2�� (24)

Thus, the modulation equations are reduced from a system of four-first order differentialequations into the following set of three equations:

a � 1

2�1a � 1

8�3a3 � �1b2

4�1sin (25)

b � �1

2�cb� �2ab

4�2sin (26)

ab � ab� ��2a2

2�2� �1b2

4�1

�b cos � (27)

The equilibrium solution (fixed points) is obtained by setting a = b = = 0, giving

1

2�1a � 1

8�3a3 � �1b2

4�1sin � 0 (28)

�1

2�cb� �2ab

4�2sin � 0 (29)

ab� ��2a2

2�2� �1b2

4�1

�b cos � 0� (30)

From equations (28) to (30), we obtain the following simplified equations for a, b and , which are the response amplitudes of the plant and controller, and a measure of the phasebetween the responses, respectively.

�22�

23

�a2�3 � 4

��2

2�2c�

23 � 2�2

2

��1 � �c

��3

� �a2�2

� 16��1 � �c

� ��2

2

��1 � �c

�� 2�22�

2c�3

�a2

� 4�22�

2c

�16��1 � �c

�2 � 64� 2�� 0 (31)

b2 � �1�2

�2�1�c

��1 � 1

4�3a2

�a2 (32)

sin � 2�c�2

�2a� (33)

Since (31) is a cubic equation in a2, we expect up to three real solutions for the plantresponse, excluding the trivial one. Numerically, we can determine the roots of the cubicequation (the a values) and then solve for b and from (32) and (33).




We can also try to solve (28) to (30) analytically. Here, to minimize the math, the authorsconsider only the case in which the controller is perfectly tuned (i.e., � = 0). Again, thereare two primary cases: The controller is deactivated, and b = 0, or the controller is activated,and b � 0. When the controller is activated, (29) requires that a � 0 when �c � 0. When �= 0, the four possible solutions are

(i) a � 0 b � 0 arbitrary

(ii) a ��

4�1

�3

� 12

b � 0 arbitrary

(iii) a � 2�2�c

�2b �

�1�2�c

�4�1 � �3a2

��1�2

12

� �

2

(iv) a ��

4�1 � 8�c

�3

� 12

b ��

2�1�2

�2�1

� 12

a � sin�1

�2�2�c

�2a

�Solution (i) is the trivial solution, solution (ii) corresponds to the uncontrolled response, so-lution (iii) is the saturated solution, and solution (iv) is another controlled solution. Solution(iii) is identified as the saturated solution because the amplitude of the plant response a isindependent of the original system parameters �1, �1, and �3, and so it will not change evenif these are varied. The amplitude a of solution (iii) is specified by the controller damping �c

and the controller feedback �2, and can, thus, be made as small as desired. The additional en-ergy that would normally be dissipated by the plant response is channeled into the controllerthrough the two-to-one internal resonance (i.e., through the quadratic coupling terms).

Both controlled solutions exist only in certain regions of the parameter space. Solution(iii) exists when �3 � �32 (condition 1) where

�32 � �1�22

�22�

2c

and solution (iv) exists only when �1 � 2�c (condition 2). The next section considers theinfluence of both �3 and �c on the response behavior of the system.

3. STABILITY ANALYSIS

To determine the stability of the solutions, we introduce a small perturbation to the fixedpoints:

a � a0 � �� (34)

b � b0 � �b (35)

� 0 � � � (36)




Then, we substitute the above expansions into equations (25) to (27), linearize with respect tothe perturbations �a, �b, and � , and obtain the following equations to describe the evolutionof the perturbations:

�a ��

1

2�1 � 3

8�3a2

0

��a �

�2�1

4�1b0 sin 0

��b �

��1

4�1b2

0 cos 0

�� (37)

�b ��2

4�2b0 sin 0

��a �

�1

2�c � �2

4�2a0 sin 0

��b

��2

4�2a0b0 cos 0

�� (38)

a0b0 ��

�� 4�2

4�2a0 cos 0

�b0

��a

��

a0� ��2

2�2a2

0 �3�1

4�1b2

0

�cos 0

��b

��

�2

2�2a2

0 ��1

4�1b2

0

�b0 sin 0

�� (39)

We determine the stability of the four solutions by substituting each into equations (37) to(39) and calculating whether the perturbations grow or decay over time. For solution (i) wefind that

��a � 1

2�1�a

��b � �1

2�c�b

and hence the solution is unstable because a perturbation will grow. Next, we considersolution (ii):

��a � ��1�a

��b � �

1

2�c � �2

4�2

�4�1

�3

� 12

sin 0

�b

0 � 2�2

�2

��1

�3

�cos 0�b

but �b � 0, therefore the third equation indicates that cos 0 � 0 and we have

��a � ��1�a (40)




��b � �

1

2�c � �2

4�2

�4�1

�3

� 12

�b� (41)

Thus, solution (ii) is stable when �3 � �32.The stability of the other solutions was determined numerically by calculating the eigen-

values of equations (37) to (39) (Nayfeh, 1995). This is necessary because some of the equa-tions do not uncouple, and we obtain a cubic polynomial for the eigenvalues that is difficultto solve in closed form. If the real part of every eigenvalue is negative, the correspondingequilibrium solution is asymptotically stable. If the real part of any of the eigenvalues ispositive, the corresponding equilibrium solution is unstable.

Figures 1 and 2 present the effect of tuning the absorber’s frequency away from exactlyhalf the oscillator frequency on the amplitude response of both the oscillator and the absorberfor two different gains. The dashed lines denote unstable equilibrium solutions and the solidlines denote stable equilibrium solutions. For both the oscillator and absorber responses,there exists one response that corresponds to the trivial solution and another response thatis close to the trivial solution. Both of these responses are unstable. The third and fourthamplitude responses are similar but one is the mirror image of the other. By slowly increas-ing the value of the frequency � from –8 to 8 the authors encounter supercritical pitchforkbifurcation PF1 at about � = –4 for the third solution. Then we encounter a reverse pitch-fork bifurcation PF2 for the fourth solution. For the cases of PF1 and PF2, we have onebranch of stable fixed points on one side of the bifurcation that becomes unstable due to thebifurcation and another created stable branch whose amplitude of response monotonicallydecreases for the oscillator and increases for the absorber. Thus it is clear that at � = 0 thereis only one stable solution, while for any other � there are two stable solutions that coexistand can lead to jumps in the amplitude response of both the oscillator and the absorber. Itis clear from this conclusion that this is not a robust method of control since many potentialdisturbances exist that can change the value of � , leading to an unpredictable response. Fig-ure 2 shows the same behavior for the amplitude responses of the oscillator and the absorberas the feedback gain increases to a value of five instead of one. The only difference is theresponse of the absorber, which is reduced by a factor proportional to the gain. Thus no realimprovements in the oscillator response occur, and the choice of gain can be decided uponbased on the practical implementation of the absorber circuit and the actuator used.

Next, the authors examine the influence of the nonlinear damping �3 on the systemresponse. Figures 3 and 4 show the four different solutions that were determined analyticallyearlier for the case of � = 0. Again, solid lines represent stable solutions while dashed linesrepresent unstable ones. It is clear from the figures that only solution (iv) is stable, and thatas the nonlinear damping coefficient increases, the response amplitude decreases for boththe system and the absorber. The only difference between Figures 3 and 4 is the value of thelinear damping factor of the system �1. We can see that the response amplitude of Figure 4is less than that of Figure 3, which is to be expected because less energy is needed (due tolower value of �1) for the flow of energy between the two modes to occur.

Plots of the fixed-point solutions as a function of �c are shown in Figure 5. The shadedarea represents the region where the Hopf-bifurcation solutions exist. Reverse pitchforkbifurcations exist at �c of about 4.4, after which only solution (ii) exists, obeying condition(1). Solution (iv) exists only for values of �c � 0.6, thus abiding by condition (2).




Figure 1. Fixed-point solutions and their stability as a function of the frequency detuning � when �2 =30, �1 = 4�1, �2 = 120, �1 = 0.02 �1, �3 = 1, and �c = 0.005. Solid lines represent stable solutions,dashed lines represent unstable solutions.

Figure 2. Fixed-point solutions and their stability as a function of the frequency detuning � when �2 =30, �1 = 20�1, �2 = 120, �1 = 0.02�1, �3 = 1, and �c = 0.005. Solid lines represent stable solutions,dashed lines represent unstable solutions.




Figure 3. Fixed-point solutions and their stability as a function of the nonlinear damping coefficient �3 at� = 0 when �2 = 30, �1 = 240, �2 = 120, �1 = 1.2, and �c = 0.005. Solid lines represent stable solutions,dashed lines represent unstable solutions.

Figure 4. Fixed-point solutions and their stability as a function of the nonlinear damping coefficient �3 at� = 0 when �2 = 30, �1 = 240, �2 = 120, �1 = 0.6, and �c = 0.005. Solid lines represent stable solutions,dashed lines represent unstable solutions.




Figure 5. Fixed-point solutions as a function of the controller damping coefficient �c at � = 0 when �2

= 30, �1 = 240, �2 = 120, �1 = 1.2, and �3 = 1. Solid lines represent stable solutions, dashed linesrepresent unstable solutions, and the shaded area is the region of Hopf bifurcated solutions.

Figure 6. Fixed-point solutions and their stability as a function of the nonlinear damping coefficient �3 at� = 0 when �2 = 30, �1 = 240, �2 = 120, �1 = 1.2, and �c = 2. Solid lines represent stable solutions,dashed lines represent unstable solutions.




Finally, in Figure 6, and using the results from Figure 5, the controller damping, �c,is chosen (�c = 2) to insure that the system operates in the region where solution (ii) isunstable, solution (iii) is stable and is the desired saturated solution, and solution (iv) doesnot exist. The feedback gain (�2) can be chosen to reduce the amplitude of the oscillationsto any desired value and is only limited by the available control authority of the actuator.Again, solid lines represent stable solutions while dashed lines represent unstable ones.

4. CONCLUSION

From this analysis of the nonlinear feedback control implementation, we conclude that it ispossible to obtain system responses that are independent of the original system parameterswhen the controller is tuned to exactly half the natural frequency of the plant. This is thesaturated solution. Second, the amplitude of the controlled response is never larger than theamplitude of the uncontrolled system, although the improvement is sometimes small. Third,the addition of a saturation-based controller to a self-excited system may add an abundance ofinteresting dynamics to the system which, brings into question the robustness of that controlstrategy.

REFERENCES

Hall, B. D., Mook, D. T., Nayfeh, A. H., and Preidikman, S., 2001, “Novel strategy for suppressing the flutteroscillations of aircraft wings,” AIAA Journal 39, 1843–1850.

Nayfeh, A. H., 1981, Introduction to Perturbation Techniques, Wiley, NY.Nayfeh, A. H. and Balachardran, B., 1995, Applied Nonlinear Dynamics, Wiley, NY.Nayfeh, A. H., 2000, Nonlinear Interactions, Wiley, NY.Nayfeh, A. H. and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, NY.Oueini, S. S. and Nayfeh, A. H., 2000, “Analysis and application of a nonlinear vibration absorber,” Nonlinear

Dynamics 6(7), 999–1016.Oueini, S. S., Nayfeh, A. H., and Pratt, J. R., 1998, “A nonlinear vibration absorber for flexible structures,” Nonlin-

ear Dynamics 15, 259–282.Oueini, S. S., Nayfeh, A. H., and Pratt, J. R., 1999, “A review of development and implementation of an active

nonlinear vibration absorber,” Archive of Applied Mechanics 69, 585–620.Verhulst, F., 2005, “Quenching of self-excited vibrations,” Journal of Engineering Mathematics 53, 349–358.



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