International Reereed Journal o Engineering and Science (IRJES) ISSN (Online) 39-83X, (Print) 39-8 Volume 6, Issue 7 (July 07), PP.08-0 Active Control and Dynamical Analysis o two Coupled Parametrically Excited Van Der Pol Oscillators * Y.A.Amer,N. M. Ali,Manar M. Dahsha 3,S. M. Ahmed 3 Mathematics Department, Faculty o Science, Zagazig University, Egypt. Mathematics Department, Faculty o Science, Suez canal University, Egypt 3 Mathematics Department, Faculty o Science, El-Arish University, Egypt.. Abstract: The dynamical behavior o two coupled parametrically excited van der pol oscillator is investigated by using perturbation method. Resonance cases were obtained, the worst one has been chosen to be discussed. The stability o the obtained numerical solution is investigated using both phase plane methods and requency response equations. Eect o the dierent parameters on the system behavior is studied numerically. Comparison between the approximate solution and numerical solution is obtained. Keywords: Vibration control, nonlinear oscillation, perturbation technique, Resonance cases, Frequency response curves. I. INTRODUCTION Vibrations at most time are non-desirable, humans suered rom these bad vibrations, so it must be eliminating or at least controlled. The dynamic absorber is the most common methods or reduced the vibrations. Its importance tends to as it is need low coast, and it is a simple operation at one modal requency. El-Badawy and Nayeh [] adopted linear velocity eedback and cubic velocity eedback control laws.yang,cao and Morris[] use Mat lab or applying numerical methods. Amer [3] investigation the coupling o two nonlinear oscillators o the main system and absorber representing ultrasonic cutting process subjected to parametric excitation orce. Non-linearities necessary introduce a whole range o phenomena that are not ound in linear system [4], including jump phenomena, occurrence o multiple solutions, modulation, shit in natural requencies, the generation o combination resonances, evidence o period multiplying biurcations and chaotic motion [5-8]. In these systems the vibrations are needed to be controlled to minimize or eliminating the hazard o damage or destruction. There are two types or vibration control, active and passive control. Pinto and Goncalves [9] investigated the active control o the nonlinear vibration o a simply-supported buckled beam under lateral loading. One o most eective tools o passive control is dynamic absorber or the neutralizer [0]. Nabergoj et al [] studied the stability o auto- parametric resonance in an external excited system. Abdel Haz and Eissa [] study the eect o nonlinear elastomeric torsion absorber to control the vibrations o the crank shat in internal combustion engines, when subject to external excitation torque. Fuller, Elliot and Nelson [3] investigate the active control o vibrations which give many ideas and approaches or controlling chaos. Abe et al. [4] investigate the nonlinear responses o clamped laminated shallow shells with : internal resonance. Eissa et al. [5-6] investigated saturation phenomena in non-linear oscillating systems subjected to multi-parametric and external excitation. Gerald [7] apply the numerical analysis to ind out the solutions or the vibrations problems. Sayed and Kamel [8-9] investigated the eect o dierent controllers on the vibrating system and saturation control o a linear absorber to reduce vibrations due to rotor blade lapping motion. Kamel et al [0] studied the vibration suppression in ultrasonic machining described by non-linear dierential equations via passive controller. Elena et al. [] studied the ormal analysis and description o the steady-state behavior o an electrostatic vibration energy harvester operating in constant-charge mode and using dierent types o electromechanical transducers.orhan and Peter [] investigate the eect o excitation and damping parameters on the super harmonic and primary resonance responses o a slender cantilever beam undergoing lapping motion. In this paper we studied vibration control o a nonlinear system under tuned excitation orce. The method o multiple scale method is applied to obtain the approximate solution o the system. Vibration method is used to reduce the amplitude o vibration at the worst resonance case. The eect o dierent parameters are investigated, the comparison between the numerical solution and approximation solution obtained. II. MATHEMATICAL MODELING The considered system is described by the equations: 3 X ( cos( t )) X ( X Y ) X ( X ay ) X G X () 8 Page
Y ( cos( t )) Y ( X Y ) Y ( bx Y ) Y G Y () 3 where the dots indicate dierentiation with respect to t, X and Y are the generalized coordinate o the plant (main system) and the controller. and are incommensurate undamental requencies, the parametric excitation requencies are and, the constants a and b are o order, and is a small parameter,, are the external excitation orces, G, G are the controller o the main system and the absorber. We can solve equations ()&() analytically using multiple time scale perturbation technique as: 3 X ( t, ) x (T,T ) x (T,T ) x (T,T ) O( ) (3) Y ( t, ) y (T,T ) y (T,T ) y (T,T ) O( ) (4) wheret T 3 t represents a ast time scale characterizing motions with the natural and excitation requencies, and t represents a slow time scales characterizing modulation and phases o both modes o vibration. The times derivatives transorm are reacted in terms o the new time scales as: d D D... (5) dt d dt D D D D... (6) From equations (3) to (6) we have: n 3 ( ; ) ( ) x n ( ) n 0 X t D D O (7) n n 3 (8) n 0 X ( t ; ) ( D D D D ) x O ( ) n 3 ( ; ) ( ) n ( ) n 0 Y t D D y O (9) n n 3 (0) n 0 Y( t ; ) ( D D D D ) y O ( ) Substituting rom equations (3), (4) and (7)-(0) into equations () and () and equating the same power o we have: 0 O( ) : ( D ) x 0 () ( D ) y 0 () O ( ) :( D ) x D D x x cos( t ) x x y ( D x ) 3 x ( D x ) ay ( D x ) G ( D x ) (3) 3 ( D ) y D D y y cos( t ) y x y ( D y ) 3 y ( D y ) bx ( D y ) G ( D y ) (4) 3 O ( ) : ( D ) x D x D D x x cos( t ) 3x x x y y x y ( D x ) ( D x ) x ( D x ) x ( D x ) x x (D x ) ay ( D x ) ay ( D x ) ay y (D x ) 3 G ( D x ) ( D x ) 3 G ( D x ) ( D x ) (5) ( D ) y D y D D y y cos( t ) x y x x y 3 y y ( D y ) ( D y ) b x ( D y ) b x ( D y ) bx x ( D y ) y ( D y ) y ( D y ) y y ( D y ) 9 Page
3 G ( D y ) ( D y ) 3 G ( D y ) ( D y ) (6) The general solution or equations () & () can be written in the orm: i T i T x ( T, T ) A(T,T )e A(T,T )e (7) y ( T, T ) B (T,T )e B (T,T )e (8) i T i T where A and B are unknown complex unctions, which can be determined by imposing the solvability conditions at the next approximation order by eliminating the secular terms, and solving resulting equation gives: 3 i T i ( i ( i ( i ( x H e H e H e H e H e cc (9) 3 4 5 y H e H e H e H e H e cc 3 i T i ( i ( i ( i ( 6 7 8 9 0 where ( Hn, n...0) are complex unction in T and cc denotes the complex conjugate terms. Substituting equations (7),(8), (9), and (0) into equations (5) and (6) the ollowing are obtained, ater eliminating the secular term and solve it : 3i T 5 i T i ( i ( i ( i ( x H e H e H e H e H e H e 3 4 5 6 H e H e H e H e H e i ( i ( i ( i ( i ( 4 7 8 9 0 H e H e H e H e H e i ( 4 i (3 i (3 i (3 i (3 3 4 5 6 H e H e H e H e i ( i ( i ( i ( 7 8 9 30 H e H e H e H e cc i ( i ( i ( i ( 3 3 33 34 y H e H e H e H e H e H e 3i T 5 i T i ( i ( i ( i ( 35 36 37 38 39 40 H e H e H e H e H e i ( i ( i ( i ( i (4 4 4 43 44 45 H e H e H e H e H e i (3 i (4 i ( 3 i ( 3 i (3 50 46 47 48 49 H e H e H e H e i ( i ( i ( i ( 5 5 53 54 H e H e H e H e cc () i ( i ( i ( i ( ) T 55 56 57 58 III. STABILITY ANALSIS Ater studying the dierent resonance numerically to see the worst resonance; one o the worst cases has been chosen to study the system stability. The selected resonance case and.in this case we introduce the detuning parameter according to: and (3) where and are called detuning parameters. Also or stability investigation, the analysis is limited to the second approximation. So our solution is only depend on T andt. Substituting equation (3) into equations (3) and (4) and eliminating the leads to the solvability conditions: 3 3 3 i T i A A A ABB i A i A A ia ABB ig A A Ae 0 (4) (5) 3 3 3 i T i B B B AAB i B i B B ib AAB ig B B Be 0 To analyze the solution o equations (5.30) and (5.3), it is convenient to express A and B in the polar orm as: i ( T) A ( T) a ( T) e i ( T) and B ( T) b( T) e (6) where A, B and, are the steady state amplitudes and phases o the motions respectively. Inserting equation (6) into equations (4) and (5) and equating real and imaginary parts, we obtain: (0) () 0 Page
a ab a 3 G a a sin 3 8 3 3 a ( ) a b a a cos 4 3 b a b b 3G b b sin 4 8 (9) 3 3 b ( ) b a b b cos 4 T and T. where (7) (8) (30) For steady state solutions, a b 0 and the periodic solution at the ixed points corresponding to equations (7)-(30) is given by: 3G a ab sin 4 3 a b cos (3) 4 (3) 3G b a b sin 8 4 3 b a cos (34) 4 Squaring equations (5.4) and (5.4) and summation, yields: (33) 3 3G a a a cos a a cos 4 3 4 3 4 b a cos (35) 3 Similarly, rom equations (33) and (34), we get: 3 3G b a b b a 8 4 4 4 (36) From equations (35) and (36), we have the ollowing cases: Case: a b 0 (the trivial solution). Case: a 0, b 0, in this case, the requency response equation (35) is given by: 3 3G a a 4 4 0 (37) Ater that we have: 3a 4 6 9 6G 9G 0 4 (38) 6 a Page
The solution o algebraic equation (38) has two roots, given by: Case3: a 0, b 0 3a 4 6 8 6G 9G a 4 a (39), in this case the requency response equations (35) and (36) are given by: 3G a a a cos 4 3 3 a a cos 4 3 (40). Linear Solution: To study the stability o the linear solution o the obtained ixed points, let us consider A and B in the polar orms: i T A ( T) p iqe i T, B ( T) p iq e (4) where p, q, p and q are real unctions in T. Substituting equation (4) into the linear parts o equations (4) and (5) and equating the imaginary and real parts, we have the ollowing cases: Case: ( a 0, b 0 ) p p q 0 q p q 0 The stability o the linear solution is obtained rom the zero characteristic equation: 0 Then, we have that: Case: ( a 0, b 0 ), p p q 0 q p q 0 (4) (43) (44) (45) (46) (47) Page
p p q 0 q p q 0 Equations (46)-(49) can be written in the matrix orm: p 0 0 p q 0 0 q p 0 0 p 0 0 q q (48) (49) (50) The stability o a particular ixed point with respect to perturbation proportional to exp( T) is determined by zeros o characteristic equation: 0 0 0 0 0 0 0 0 Ater extract we obtain that: 4 3 r r r r (5) 3 4 0 According to Routh-Huriwitz criterion, the above linear solution is stable i the ollowing are satisied: r 0, r r r 0, r ( r r r ) r r 0, r 0. (53) 3 3 3 4 4. Non-Linear Solution: To determine the stability o the ixed points, one lets: a a a, b b b, ( m,) (54) m m 0 m where a, b and m 0 are solutions o equations (7)-(30) and a, b, m are perturbations which are assumed to be small comparing to a, b and m 0. Substituting equation (54) in to equations (7)-(30) and keeping only the linear terms in a, b, m we obtain:. For the case ( a0, b 0), we have: 0 (5) 3 Page
a 3 9 [ 0 0 sin 0] 8 8 a [ a0 cos 0 ] (55) 9 [ a0 cos ] a [ sin 0 ] a 4 a (56) 0 0 The stability o a given ixed point to a disturbance proportional to exp( t ) is determined by the roots o: 3 9 a0 Ga0 sin0 a0 cos0 8 8 0 9 a0 cos sin0 a0 4 a0 (57) Consequently, a non-trivial solution is stable i and only i the real parts o both eigenvalues o the coeicient matrix (57) are less than zero.. For the practical solution ( a0, b 0), we have: a 3 9 [ 0 0 sin 0] [ 0 0] [ a0 cos 0] 8 a 8 G a a a b b (58) 9 [ a0 b0 cos ] a [ b0 ] b [ sin 0 ] (59) a0 4 a0 a0 b 3 9 [ 0 0] [ 0 0 0 sin 0] [ 0 cos 0] ba b a 8 b 4 ba 8 G b b b (60) 9 [ a 0] a [ 0 cos 0] [ sin 0] 0 4 b b b b0 (6) The stability o a particular ixed point with respect to perturbations proportional to exp( t ) depends on the real parts o the roots o the matrix. Thus, a ixed point given by equations (58)-(6) is asymptotically stable i and only i the real parts o all roots o the matrix are negative. IV. NUMERICAL RESULTS: The nonlinear dynamical system is solved numerically using Rung -Kutta ourth order method by using Maple 6 sotware. At non resonance case (basic case) as shown in Fig., we can see that the steady state amplitude without controller is about 0.08 and with controller is about0.0. 3.) Resonance cases Sub harmonic resonance,, or the main system the phase plane is a limit cycle and its steady state amplitude which is the largest without controller is about 0.9 and about 0.7, which appears in Fig.(3). In Fig (4), Sub harmonic resonance, or the main system the phase plane is a limit cycle and its steady state amplitude without controller is about 0.0004 and about 0.0. 3.) Eect o control The eect o controller appears at Fig.(5) and Fig.(6), in Fig.(5), the amplitude o the main system at Sub harmonic resonance: when, is decreasing to 0.05. Similarly or the Fig.(6), the amplitude o the main system Sub harmonic resonance: when decreasing to 0.0. 3.3) Eect o parameters 4 Page
For the damping coeicient, Fig(7) (a) shows that the steady state amplitude o the main system is monotonic increasing unction. For the parameter, Fig.(7) (b) shows that the steady state amplitude o the main system is monotonic decreasing unction. But that the steady state amplitude o the main system is monotonic increasing unction o the excitation orces,, Fig (7) (c),() shows. 3.4) Frequency response curves The requency response in the second case ( a0, b 0) which represented by equation (37) is a nonlinear algebraic equation solved numerically o the amplitude against the detuning parameter. Fig.(8)(a) shows that the steady state amplitude is monotonic decreasing unction on the natural requency. Fig.(8)(b) shows that the steady state amplitude is a monotonic increasing unction in the non-linear parameter. The excitation orce o the main system is a monotonic increasing unction at the steady state amplitude which appeared on the Fig.(8)(c). The steady state amplitude is a monotonic decreasing unction on the gaing. 5 Page
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Fig.(7), Eects o dierent parameters. 7 Page
Fig.(8): Response curves at ( a0, b 0). 0.35 0.5 Fig.(9): Response curves at ( a0, b 0) 4) Comparison between approximation solution and numerical solution In this subsection we compare between the numerical solutions (which we obtained by using Maple 6 sotware) and the approximation solution (which we obtained by using equations (58) to (6)). All this comparison done in the resonance sub-harmonic case which we choose to be the worst we obtained that there is a good agreement between the two solutions. 8 Page
Plant Amplitude Active Control And Dynamical Analysis o two Coupled Parametrically 0. 0.05 0-0.05-0. 0 00 00 300 400 500 600 700 800 Fig.(0): the comparison between the numerical solution ( )and approximation solution(----------) V. CONCLUSION The dynamical behavior o two coupled parametrically excited van der pol oscillator is investigated by using perturbation method. Resonance cases were obtained, the worst one has been chosen to be discussed which is,.hence the stability o the system and controller is studied using the requency response unctions rom the above study, the ollowing results are concluded:. The steady state o the system without controller is about 0.08 which is considering basic case.. The damping coeicient is monotonic decreasing unction. 3. The natural requency is monotonic decreasing unction. 4. The orces, are monotonic increasing unctions. 5. The numerical solution has a good agreement with the approximation solution. Time REFERENCES: []. El-BadawyA A and Nayeh,A H,Control o a directly excited structural dynamic model o an F-5 tail section, journal o the Franklin Institute 338 (00),33-47. []. Yang,WYY, Cao, W, Chung,T-SMorris,J, Applied Numerical Methods Using Matlab. John Wiley & Sons, Inc; (005). [3]. Amer YA, Hamed YS, Stability and response o ultrasonic cutting via dynamic absorber, Chaos SolitonsFract. 34 () (007)38-345. [4]. Elnaschie MS. Stress, Stability and Chaos. Mc Graw-Hill International Editions (99). Singapore. Copyright Mc Graw Hill United Kingdom (990). [5]. Nayeh AH, Mook RT. Nonlinear oscillations. New York: John Wiley; (979). [6]. Cartmell MP, Lawson J, Perormance enhancement o an auto-parametric vibration absorber by means o computer control. J. sound Vib. 77()(994)73-95. [7]. Woaa P, Fotsin HB, Chedjou JC, Dynamics o two nonlinear coupled oscillators. PhysScr 57(998)95-00. [8]. Nayeh AH, Sanchez NE. Biurcations in a orced sotening doing oscillators. Int j Non-linear Mech 4(6)(989):483-497. [9]. Pinto OC and P.B. Goncalves, Active non-linear control o buckling and vibrations o a lexible buckled beam. Choas, solitons and Fractals 4 (00), 7-39. [0]. Shen, WeiliGuo TY, Torsional vibration control o a shat through active constrained layer damping treatments, j. Vib. Acoust. 9(997) 504-5. []. Nabergoj R, Tondl A, Ving Z. Auto parametric resonance in an externally excited system. Chaos, Solitons&Franctals 4(994) 63-73. []. Abdel Haez HM, Eissa M, Stability and control o non-linear torsional vibrating systems, Faculty o Engineering Alexandria University, Egypt 4 ()(00):343-353. [3]. Fuller CR, Elliott,SJ,Nelson PA, Active Control o Vibration, Academic press, New York,997. [4]. Abe,A,.Kobayashi,YYamada,Y, Nonlinear dynamic behaviors o clamped laminated shallow shells with one-to-one internal resonance. J Sound Vibrat. 304 (007) 957-58. [5]. Eissa M, El-Ganaini W, Hamed YS, Saturation, stability and resonance o nonlinear systems, Phys. A 356(005) 34-358). 9 Page
[6]. Eissa M, El-Ganaini W, Hamed YS, On the saturation phenomena and resonance o non-linear dierential equations, Minnuiya J. Electron. Eng. Res MJEER 5 ()(005)73-84. [7]. Gerald,GF, Applied Numerical Analysis. Reading, MA: Addison-Wesley; (980). [8]. Sayed M, Kamel M, Stability study and control o helicopter blade lapping vibrations, Appl. Math. Model.35:80-873; (0). [9]. Sayed M, Kamel M, : and :3 internal resonance active absorber or non-linear vibrating system. 36(0) 30-33. [0]. Kamel M, El-Ganaini W, Hamed YS, Vibration suppression in ultrasonic machining described by nonlinear dierential equations via passive controller I Appl Math Comput. 9(03) 469-470. []. B. Elena, G. Dimitri, B. Philippe and F. Orla, Steady-State Oscillations in Resonant Electrostatic Vibration Energy Harvesters, IEEE Transactions on Circuits and Systems I, 60 (05) 875-884. []. O. Orhan and J. A. Peter, Nonlinear response o lapping beam to resonant excitations under nonlinear damping, ActaMech (05) -7. Y.A.Amer. "Active Control and Dynamical Analysis o two Coupled Parametrically Excited Van Der Pol Oscillators." International Reereed Journal o Engineering and Science (IRJES) 6.7 (07): 08-0. 0 Page