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F

www.elsevier.com/locate/cnsns

3

R. Yamapi

4 a

5 6 7

a,*

PR OO

Nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions

2

, F.M. Moukam Kakmeni b, J.B. Chabi Orou

c

Department of Physics, Faculty of Sciences, University of Douala, P.O. Box 24157, Douala, Cameroon b Department of Physics, Faculty of Sciences, University of Bue´a, P.O. Box 63, Bue´a, Cameroon c Institut de Mathe´matiques et de Sciences Physiques (I.M.S.P.), B. P. 613 Porto-Novo, Be´nin

ED

Received 14 February 2005; received in revised form 11 May 2005; accepted 11 May 2005

RE

This paper deals with the nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions, described by an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators. Firstly, the amplitudes of the sub- and super-harmonic oscillations for the resonant states are obtained and discussed using the multiple time scales method. The equations of motion are solved numerically using the Runge–Kutta algorithm. It is found that chaotic and periodic orbit coexist in the electromechanical system depending on the set of initial conditions. Secondly, the problem of synchronization dynamics of two coupled electromechanical systems both in the regular and chaotic states is also investigated, and estimation of the coupling coefficient under which synchronization and no-synchronization take place is made.
2005 Elsevier B.V. All rights reserved.

UN C

21 PACS: 05.45.Xt; 85.25+j

OR

11 12 13 14 15 16 17 18 19 20

CT

10 Abstract

22 Keywords: Electromechanical systems; Sub- and super-harmonic oscillations; Synchronization 23

*

Corresponding author. Tel.: +237 932 93 76; fax: +237 222 62 75. E-mail addresses: [email protected] (R. Yamapi), [email protected] (F.M. Moukam Kakmeni), [email protected] (J.B. Chabi Orou). 1007-5704/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.05.003

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24 1. Introduction

OR

RE

CT

ED

PR OO

F

In recent years, the dynamics of coupled systems including Duffing, Van der pol and Rayleigh equations have received a great attention in the scientific community. This, due to the fact that such coupled nonlinear oscillators described various physical, electrical, mechanical, electromechanical and biological systems, and can exhibit various types of behaviors [1–7]. Recent contributions on the study of the behavior of such coupled systems have been focussed on a nonlinear electromechanical system with multiple functions described by the Duffing electrical oscillator magnetically coupled to linear mechanical oscillators [7]. The amplitude of the harmonic oscillatory states and their stability boundaries have been found using respectively the harmonic balance method and the Floquet theory [1]. The chaotic state has been obtained from numerical simulation of the equations of motion. The indicators used are the bifurcation diagram and the Lyapunov exponent. The effects of the number of linear mechanical oscillators on the frequency-response and the chaotic states have been discussed and it appears that for some set of physical parameters, the hysteresis and jump phenomena disappear with the increase of the number of linear mechanical oscillators. Depending on the amplitude of the external excitation and when the frequency of the voltage source is very large or very small as compare of the values of the natural frequencies of the oscillators, the effects of the excitation will be small unless this amplitude is hard. The consequence is the generation of higher or super-harmonic and sub-harmonic oscillations where the amplitude of various orders resonant states [1] can be found and established using the multiple time scales method. This paper deals with the determination of such oscillatory states which are important in the field of nonlinear oscillations, and they frequently occur in various branches of electromechanical engineering and physical sciences [8,9]. The paper is organized as follows. In Section 2, the electromechanical system with multiple functions and its equations of motion are presented. Using the multiple time scales method, the amplitudes of sub- and super-harmonic oscillatory states of the model are found in Section 3. The effects of linear mechanical oscillators on the behavior of the model are analyzed analytically and it appears that for some set of physical parameters, the hysteresis and jump phenomena are hardly affected with the increase of the number of linear mechanical oscillators. Different dynamical states are also identified in the electromechanical system depending on the set of the initial conditions. Section 4 deals with the synchronization states of coupled electromechanical systems with 25 functions. One performers analytically and numerically the calculations to obtain for the identification of the transition boundaries of the synchronization process which appeared in the coupled models. A conclusion is given in Section 5.

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25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

57 2. The electromechanical system 58 59 60 61 62 63

As described in Ref. [7], the electromechanical system with multiple functions is schematically represented in Fig. 1. It consists of an electrical part coupled magnetically to a mechanical part governed by n linear mechanical oscillators. The coupling between both parts is realized through the electromechanical force due to a permanent magnet. The electrical part of the system consists of a resistor R, an inductor L, the condenser C with nonlinear characteristic [7] and a sinusoidal voltage source, all connected in series, while the mechanical part is composed of n mobile beams

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s B

3

spring k1

C stone

N

R x1

B magnet

rod T1

e(t)

PR OO

coupling magnet cool

F

mobile beam (m1)

rod Tn

L

mobile beam (mn) magnet

s B

stone N s

B

xn

spring kn

ED

Fig. 1. Schema of the electromechanical system with multiple functions.

i¼1

ð1Þ

OR

€xi þ ci x_ i þ w2i xi  ki1 x_ ¼ 0; .. .

RE

€x1 þ c1 x_ 1 þ w21 x1  k11 x_ ¼ 0; .. .

CT

64 which can move respectively along the ~ xi ði ¼ 1; . . . ; nÞ axis on both sides. The rods Ti are bound 65 to mobile beams with springs of constants ki. The motion of the entire electromechanical system is 66 governed by the following n + 1 nondimensional coupled nonlinear differential equations n X €x þ c_x þ x þ bx3 þ ki x_ i ¼ E0 cos wt;

€xn þ cn x_ n þ w2n xn  kn1 x_ ¼ 0;

where the dots over the quantities denote differentiation with respect to time, the variables x and xi denote respectively the instantaneous electrical charge of the condenser and the displacement of the ith mobile beams of the mechanical part. The electromechanical system is described by the system consisting of an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators.

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70 71 72 73

74 3. Nonlinear dynamics

75 3.1. The multiple time scales method 76 The multiple time scales method has been chosen because it is more indicated to find and estab77 lish the amplitudes of various orders of resonant states [1]. For this method, one have seen an 78 asymptotic expansion of the solutions of Eqs. (1) in the following form:

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x ¼ x01 ðT o ; T 2 Þ þ 3 x03 ðT o ; T 2 Þ þ    ; xi ¼ xi1 ðT o ; T 2 Þ þ 3 xi3 ðT o ; T 2 Þ þ   

F

where the independent variables or time scales To = t and T2 = 2t are respectively the fast scale (associated to the unperturbed system) and the slow scale (associated to the amplitude and phase modulations induced by the global first order perturbation).  is a small dimensionless parameter. In this paper, the damping and the coupling coefficients are considered as global second order perturbations, then one writes c = 2co, ci = 2coi, ki = 2koi and ki1 = 2koi1 (with i = 1, 2, . . ., n). The amplitude E0 is taken at the order E0 = E to indicate that E0 is hard. Inserting these expansions in Eq. (1) and equating coefficients of like powers of , one obtains Order ,

PR OO

81 82 83 84 85 86 87 88

D2o x01 þ x01 ¼ E cos wT o ; D2o x11 þ w21 x11 ¼ 0; .. .

ð3Þ

ED

D2o xi1 þ w2i xi1 ¼ 0; .. . D2o xn1 þ w2n xn1 ¼ 0. Order 3,

CT

92

ð2Þ

ði ¼ 1; 2; 3; . . . ; nÞ;

D2o x03 þ x03 ¼ 2Do D2 x01  co Do x01  bx301 

n X

koi Do xi1 ;

RE

i¼1

D2o x13 þ w21 x13 ¼ 2Do D2 x11  co1 Do x11 þ ko11 Do x01 ; .. . .. .

þ

w2i xi3

ð4Þ

¼ 2Do D2 xi1  ci Do xi1 þ koi1 Do x01 ;

OR

D2o xi3

D2o xn3 þ w2n xn3 ¼ 2Do D2 xn1  cn Do xn1 þ kon1 Do x01 ;

UN C

96 where Do ¼ oTo o ; D2 ¼ oTo 2 . The general solutions of Eqs. (3) can be expressed as x01 ¼ AðT 2 Þ expðjT o Þ þ K expðjwT o Þ þ AðT 2 Þ expðjT o Þ þ K expðjwT o Þ; x11 ¼ A1 ðT 2 Þ expðjw2 T o Þ þ A1 ðT 2 Þ expðjw2 T o Þ; .. . xi1 ¼ Ai ðT 2 Þ expðjwiþ1 T o Þ þ Ai ðT 2 Þ expðjwiþ1 T o Þ; .. . xn1 ¼ An ðT 2 Þ expðjwnþ1 T o Þ þ An ðT 2 Þ expðjwnþ1 T o Þ;

ð5Þ

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99 where the over-bar represents the complex conjugate, j2 = 1 and 1 E . 2 1  w2 102 Substituting x01 and xi1 into Eq. (4) yields D2o x03 þ x03 ¼ ð2jA0 þ jco A þ 3bA2 A1 þ 4bK2 AÞ expðjT o Þ

PR OO

þ ðjco wK þ 4bKAA þ 2K3 bÞ expðjwT o Þ  bA3 expð3jT o Þ

F

K¼

 3bA2 K expðjð2 þ wÞT o Þ  3bAK2 expðjð2w þ 1ÞT o Þ  3bK2 A expðjð1  2wÞT o Þ  3bKA2 expðjð2  wÞT o Þ n X 3 wi koi Ai expðjwi T o Þ þ C.C.;  K b expð3jwT o Þ  j i¼1

D2o x13

þ

w21 x13

¼

jw1 ð2A01

þ co1 A1 Þ expðjw1 T o Þ þ jko11 A expðjT o Þ þ jwko11 K expðjwT o Þ þ C.C.;

ED

.. .

ð6Þ

CT

D2o xi3 þ w2i xi3 ¼ ji ð2A0i þ coi Ai Þ expðjwi T o Þ þ jkoi1 A expðjT o Þ þ jwkoi1 K expðjwT o Þ þ C.C.; .. . D2o xn3 þ w2n xn3 ¼ jwn ð2A0n þ con An Þ expðjwn T o Þ þ jkon1 A expðjT o Þ þ jwkon1 K expðjwT o Þ þ C.C.;

OR

RE

where the prime (on A, Ai, i = 1, 2, . . ., n) denotes the differentiation with respect to T2 and C.C. stands for the complex conjugate of the previous terms. As was mentioned before, the unknown amplitudes A and Ai may now be determined by eliminating secular terms in Eq. (6). However, due to the high dimension of the system, it is impossible to obtain a set of secular equations valid from all frequencies. According to the different values of the frequencies w and wi relatively to the natural frequency of the Duffing electrical oscillator (which is one), it comes two types of interesting resonant structures: the first one is the super-harmonic resonant state wi = 1 and 3w = 1, and the second one corresponds to the sub-harmonic resonant state w = 3 with wi = 1. Therefore, we restrict our attention in the following subsection to the case of sub- and super-harmonic resonances.

UN C

106 107 108 109 110 111 112 113 114 115

116 3.2. Amplitude of the super-harmonic resonant state 117 One considers the case where the electrical Duffing oscillator enters in super-harmonic reso118 nance with the external excitation, that is 3w = 1 + 2r and assuming that wi ¼ 1 þ 2 ri ;

ð7Þ

122 where r and ri are the detuning parameters indicating the accuracy of the resonances. The secular 123 producing terms in Eqs. (6) must be eliminated and the solvability conditions are defined as

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2jA0 þ jco A þ 4bK2 A þ 3bA2 A þ j

n X

wi koi Ai expðjri T 2 Þ þ K3 b expðjrT 2 Þ ¼ 0;

i¼1

 .. .

w1 ð2A01

 .. .

wi ð2A0i

þ co1 A1 Þ expðjr1 T 2 Þ þ ko11 A ¼ 0;

F

ð8Þ

 wn ð2A0n þ con An Þ expðjrn T 2 Þ þ kon1 A ¼ 0. 127 One expresses A(T2) and Ai(T2), in the polar form:

PR OO

þ coi Ai Þ expðjri T 2 Þ þ koi1 A ¼ 0;

UN C

OR

RE

CT

ED

1 AðT 2 Þ ¼ aðT 2 Þ expðjbðT 2 ÞÞ; 2 ð9Þ 1 Ai ðT 2 Þ ¼ ai ðT 2 Þ expðjbi ðT 2 ÞÞ; 2 131 where a, ai and b, bi are respectively the amplitudes and the phases of the oscillations. Substituting 132 Eqs. (9) into Eqs. (8), this yields after separating real and imaginary parts the following set of first 133 order differential equations: n 3 3 1X ba  ab0 þ 2bK2 a þ bK3 cos d  wi koi ai sin di ¼ 0; 8 2 i n 1 1X co a þ a0 þ bK3 sin d þ wi koi ai cos di ¼ 0; 2 2 i   1 0 0 a1 b1 cos d1 þ a1 þ co1 a1 sin d1 ¼ 0; 2   1 1 w1 a01 þ co1 a1 cos d1  w1 b01 a1 sin d1  ko11 a ¼ 0; 2 2 .. . ð10Þ   1 ai b0i cos di þ a0i þ coi ai sin di ¼ 0; 2   1 1 wi a0i þ coi ai cos di  wi b0i ai sin di  koi1 a ¼ 0; 2 2 .. .   1 0 0 an bn cos dn þ an þ con an sin dn ¼ 0; 2   1 1 wn a0n þ con an cos dn  wn b0n an sin dn  kon1 a ¼ 0; 2 2

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142 where F n ¼ 2bK2  r 

n 1X koi koi1 ðr  ri Þ ; 2 i¼1 4ðr  ri Þ2 þ c2oi

n X 1 koi koi1 coi . Gn ¼ co  2 2 2 i¼1 4ðr  ri Þ þ coi

ð11Þ

PR OO

9 2 6 3 b asp þ bF n a4sp þ ðF 2n þ G2n Þa2sp  b2 K6 ¼ 0; 64 4 k2oi1 a2sp 2 ; aiðspÞ ¼ 2 2 wi ðcoi þ 4ðr  ri Þ2 Þ

F

137 where d = rT2  b and di = riT2 + bi  b. Since it is particularly interested in studying the steady138 state responses, one must have a0 ¼ a0i ¼ 0 and d0 ¼ d0i ¼ 0. Thus b 0 = r and b0i ¼ r  ri . Eliminat139 ing d, di from Eqs. (10), we obtain the following set of nonlinear equations:

1 nko1 ko11 ðr  ri Þ ; 2 4ðr  r1 Þ2 þ c2o1

1 nko1 ko11 coi Gn Gidn ¼ co þ . 2 4ðr  r1 Þ2 þ c2o1

CT

F n ¼ F idn ¼ 2b2 K2  r 

ED

147 When the linear mechanical oscillators are identical, Mn and Nn take the following expressions:

RE

152 Thus, in the case of super-harmonic resonances, the motion of the n + 1 oscillators are coupled 153 and described by xðtÞ ¼ asp cosð3wt þ dÞ þ K cos wt þ Oð3 Þ; xi ðtÞ ¼ aiðspÞ cosð3wt þ di  dÞ þ Oð3 Þ.

OR

Using the Newton–Raphson algorithm, the amplitudes a and ai are plotted as a function of the detuning parameter r, since ri = 0 (exact internal resonance) will be considered throughout the paper. With the appropriate set of parameters E = 0.5, co = 0.01; co1 = 1.2, ko1 = 0.12, ko11 = 0.2, b = 0.6, ri = 0, the super-harmonic frequency–response curves obtained are presented in Fig. 2(i, ii) for several value of n and show the well-known hysteresis phenomena. The effects of the number of linear mechanical oscillators on the behavior of the model are observed (see Fig. 2(i, ii)) and it appears that with the above set of parameters, the hysteresis phenomenon hardly affects with the increasing of linear mechanical oscillators. In Fig. 2(iii, iv), one finds the effects of the amplitude E on the frequency–response curves and it appears that the hysteresis phenomenon disappears in the considering r-band when E increases. With the form of solutions (12), one finds through analytical investigations that the super-harmonic response is the periodic solutions is the period of the external excitation). This is confirmed by the numerical with T3 -period (T ¼ 2p w simulation as it shown in Fig. 3 where the temporal evolution of the linear mechanical solution is plotted. This temporal variation xk(t) is the T3 -harmonic oscillations. All numerical simulations of the equations of motion in this paper all carried out using the fourth-order Runge–Kutta algorithm.

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157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172

ð12Þ

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1.6 1.4

n=1 n=10 n=20

(i)

1.2

n=1 n=10 n=20

(ii) 0.15

1 0.1

ai

0.6 0.4

F

a 0.8

0

0 –0.4

–0.2

0

0.2

σ

0.4

0.6

–0.4 0.25

E=0.5 E=1.5 E=5.0

3.5 3

(iii)

(iv) 0.2

2.5

0.15

2

ai

a

0.1

1.5 1

–0.2

0

σ

0.2

0.4

0.6

E=0.5 E=1.5 E=5.0

ED

0.05

0.5 0 –2

PR OO

0.05

0.2

0

–1

0

1

σ

2

3

–2

–1

0

σ

1

2

3

0.03

T/3

OR

0.02

RE

CT

Fig. 2. (i, ii) Effects of the number of linear mechanical oscillators on the super-harmonic frequency–response curves a(r) and ai(r) (i = 1, . . ., n) with E = 0.5. (iii, iv) Effects of the amplitude E on the super-harmonic frequency–response curves a(r) and ai(r). The parameters used are:  = 0.001; ko1 = 0.12; ko11 = 0.2; co = 0.01; co1 = 1.2; b = 0.6; ri = 0 and n = 1.

x(t)

0.01 0

UN C

–0.01 –0.02

–0.03 5000

5005

5010

5015

5020

5025

5030

Time (s)

Fig. 3. Super-harmonic response curves xi(t) showing the T/3-periodic motion with the parameters of Fig. 2 and r = 2 and n = 25.

173 The stability of the super-harmonic steady-state motions can be determined by investigating the 174 nature of the super-harmonic steady-state solutions of Eqs. (10). To accomplish this, we let

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a ¼ as þ ap ; ai ¼ ais þ aip ;

d ¼ ds þ dp ;

9

ð13Þ

di ¼ dis þ dip ;

PR OO

F

178 where as and ais are the amplitudes of the super-harmonic steady-state solutions and ds and dis its 179 phases. Substituting (13) into (10), expending for small ap, aip, dp and dip, and keeping linear terms 180 in ap, dp, aip and dip, we obtain the following n + 2 set of first order differential equations n X a_ p ¼ P1 ap þ P2 dp þ ðP2i dip þ P1i aip Þ; i¼1

d_ p ¼ P2 ap þ P4 dp þ

n X

ðP4i dip þ P3i aip Þ;

i¼1

a_ 1p ¼ C11 a1p þ C21 d1p þ C31 ap ; d_ 1p ¼ C41 a1p þ C51 d1p þ C61 ap þ C71 dp ;

a_ ip ¼ C1i aip þ C2i dip þ C3i ap ; d_ ip ¼ C4i aip þ C5i dip þ C6i ap þ C7i dp þ Co5i

CT

RE

a_ np ¼ C1n anp þ C2n dnp þ C3n ap ;

d_ np ¼ C4n anp þ C5n dnp þ C6n ap þ C7n dp þ Co5n

n X

ðP4j djp þ P3j ajp Þ;

j6¼n

OR

where the parameters Pi, Pij and Cij are given in Appendix A. Then the stability of the super-harmonic steady-state solutions depends on the eigenvalues S of the coefficient matrix on the righthand sides of (14). But due to the order of this matrix (2n + 2 · 2n + 2), its difficult to find the eigenvalue equation, we restrict our analyze to the case of one function (n = 1) and Eq. (14) become n X ðP2i dip þ P1i aip Þ; a_ p ¼ P1 ap þ P2 dp þ

UN C

184 185 186 187 188

ð14Þ

n X ðP4j djp þ P3j ajp Þ; j6¼i

.. .

ED

.. .

i¼1

d_ p ¼ P2 ap þ P4 dp þ

n X

ðP4i dip þ P3i aip Þ;

i¼1

a_ 1p ¼ C11 a1p þ C21 d1p þ C31 ap ; d_ 1p ¼ C41 a1p þ C51 d1p þ C61 ap þ C71 dp

ð15Þ

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191 Using the above equations, one can obtain the following eigenvalue equation S 4 þ Q1 S 3 þ Q2 S 2 þ Q4 ¼ 0;

ð16Þ

195 where the coefficients Qi are given in Appendix B. The super-harmonic steady-state motions are 196 stable under the following conditions:

F

Ca ¼ Q4 > 0;

ED

and unstable otherwise. The stability boundary curves of the super-harmonic oscillatory solutions in the as  r plane are drawn in Fig. 4. One sees that for a given value of the amplitude of the external excitation E, a point M(as, r) will fixed on the stability chart and can be moved along the five following regions (I), (II), (III), (IV) and (V) as it appears in Fig. 4. Hence, by virtue of the stability conditions (17), the super-harmonic oscillatory solutions becomes unstable if the point M lies in the unstable regions. The unstable portions are the regions (I), (II), (III), and (IV) in Fig. 4. We remind that the super-harmonic oscillatory solutions are stable if both Ca and Cb are positive, and shall now consider each unstable region in detail. When the point M lies in the regions (I) and (III), Ca is negative and Cb positive, then the stability conditions are not satisfied. In the region (II), we have Ca < 0 and Cb < 0 while in the region (IV), Ca is positive and Cb negative. Our investigations shown that the stability portion is the region (V), since both Ca and Cb are positive. To summarize the above consideration, we mention that the stability portion of the super-harmonic oscillatory solutions is the region in which the super-harmonic oscillation is sustained.

CT

200 201 202 203 204 205 206 207 208 209 210 211 212 213

ð17Þ

PR OO

Cb ¼ Q1 ðQ2 Q3  Q1 Q4 Þ  Q23 > 0

RE

214 3.3. Amplitude of the sub-harmonic resonant state

OR

215 To analyze the sub-harmonic resonances, we set w = 3 + 2ro. ro is another detuning parameter 216 indicating the accuracy of the sub-harmonic resonances. Eliminating in Eq. (6) the terms that pro217 duce secular terms in xo3, xi3 and considering the expressions given by Eq. (7), we have

5

UN C

4.5

(III)

4

asp

3.5 3

(II)

(I)

2.5 2 (V): Stable portions

1.5 1 (IV )

0.5 0 –1

0

1

2

σ

3

4

5

Fig. 4. Stability boundaries of super-harmonic response curves with the parameters of Fig. 2.

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2jA0 þ jco A þ 4bK2 A þ 3bA2 A þ j

n X

11

 2 expðjro T 2 Þ ¼ 0; wi koi Ai expðjri T 2 Þ þ 3KbA

i¼1

 w1 ð2A01 þ co1 A1 Þ expðjr1 T 2 Þ þ ko11 A ¼ 0; .. .  wi ð2A0i þ coi Ai Þ expðjri T 2 Þ þ koi1 A ¼ 0; .. .  wn ð2A0n þ con An Þ expðjrn T 2 Þ þ kon1 A ¼ 0.

PR OO

F

ð18Þ

UN C

OR

RE

CT

ED

220 Again, introducing the polar notations for A, Ai and separating real and imaginary parts, we ob221 tain the following first order differential equations: n 3 3 3 1X ba  ab0 þ 2bK2 a þ bKa2 cos do  wi koi ai sin di ¼ 0; 8 4 2 i¼1 n 1 3 1X 0 2 wi koi ai cos di ¼ 0; c a þ a þ bKa sin do þ 2 o 4 2 i¼1   1 0 0 a1 b1 cos d1 þ a1 þ co1 a1 sin d1 ¼ 0; 2   1 1 w1 a01 þ co1 a1 cos d1  w1 b01 a1 sin d1  ko11 a ¼ 0; 2 2 .. . ð19Þ   1 0 0 ai bi cos di þ ai þ coi ai sin di ¼ 0; 2   1 1 wi a0i þ coi ai cos di  wi b0i ai sin di  koi1 a ¼ 0; 2 2 .. .   1 0 0 an bn cos dn þ an þ con an sin dn ¼ 0; 2   1 1 wn a0n þ con an cos dn  wn b0n an sin dn  kon1 a ¼ 0. 2 2 225 where do = roT2  3b and di = riT2 + bi  b. For the steady-state responses, one has b0 ¼ r3o and 226 b0i ¼ r3o  ri . Eliminating do and di from Eq. (19), one obtains the following set of nonlinear 227 equations:   9 2 6 3 9 2 2 4 b asb þ bM n  b K asb þ ðM 2n þ N 2n Þa2sb ¼ 0; 64 4 16 ð20Þ 2 2 k a i a2iðsbÞ ¼ h  oi1 sb2 w2i 4 r3o  ri þ c2oi

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231 with

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  n koi koi1 r3o  ri ro X M n ¼ 2bK   ; ro 2 3 þ c2oi i¼1 4 3  ri 2

PR OO

F

n 1 1X koi koi1 coi N n ¼ co  . ro  2 2 i¼1 4  ri 2 þ c2oi 3

234 In the case of identical linear mechanical oscillators, Mn and Nn take the following form:   ro nko1 ko11 r3o  r1 2 id M n ¼ M n ¼ 2bK    ;  3 4 ro  r1 2 þ c2 o1

3

1 1 nko1 ko11 co1 N n ¼ N idn ¼ co  .   2 2 4 ro  r1 2 þ c2 o1 3

244 with 9b2 ; 64

3 9 bo ¼ bM n  b2 K2 ; 4 16

ð22Þ

co ¼ M 2n þ N 2n .

OR

ao ¼

ð21Þ

RE

CT

240 which is quadratic in asb. Its solutions are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bo b2o  4ao co a2sb ¼ 2ao

ED

237 Eqs. (20) show that either asb = 0 or   9 2 4 3 9 2 2 2 b asb þ bM n  b K asb þ M 2n þ N 2n ¼ 0; 64 4 16

247 One finds that nontrivial sub-harmonic oscillations occur only when b2o P 4ao co . This condition 248 demands that

252 where

UN C

ac K4 þ bc K2 þ cn P 0;

81 4 ac ¼ b; 256

27 bc ¼ b4 ; 16

ð23Þ

"  # n koi koi1 r3o  ri 27 3 ro X 9  b2 N 2n . cn ¼  b   ro 2 2 32 16 3 þ coi i¼1 4 3  ri

255 It follows from Eqs. (23) that, for a given ro, the boundary of the region where nontrivial sub-har256 monic solutions in the Kro-plane can exist is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bc b2c  4ac cn K2 ¼ . ð24Þ 2ac

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OR

RE

CT

ED

PR OO

In the exact internal resonances, one finds in Fig. 5 the behaviors of the amplitudes asb and aisb when the detuning parameter ro and the amplitude E are varied. The effects of the number of the linear mechanical oscillators on the sub-harmonic frequency–response curves (resp. amplitude–response curves) and the boundary of the region where nontrivial solutions can exist are observed. It is found that the behaviors of the electromechanical system are hardly affected and this boundary increases when the number of linear mechanical oscillators n varies from (n = 1), roc = 0.9 to (n = 20), roc = 2.4176 while for the case a(E) and ai(E), this boundary decreases for the increase of n from Ec = 11.4 (n = 1) to Ec = 3.21 (n = 20). With the form of solutions (25), one also finds through analytical investigations that the sub-harmonic response is the periodic solutions with 3T-period. This is confirmed by the numerical simulation as it shown in Fig. 6 where the temporal evolution of the linear mechanical solutions is plotted, this response xk(t) is

UN C

264 265 266 267 268 269 270 271 272 273 274

F

260 In the sub-harmonic resonant state, the motion of the electromechanical system is described by   1 wt þ do þ K cos wt þ Oð3 Þ; xðtÞ ¼ asb cos 3   ð25Þ 1 3 wt þ di  do þ Oð Þ. xi ðtÞ ¼ aiðsbÞ cos 3

Fig. 5. (i, ii) Effects of the number of linear mechanical oscillators on the sub-harmonic frequency–response curves a(r) and ai(r) (i = 1, . . ., n), E = 0.5. (iii, iv) Effects of the number of linear mechanical oscillators on the sub-harmonic amplitude–response curves a(E) and ai(E) with r = 2.5. The parameters used are:  = 0.001; ko1 = 0.3; ko11 = 0.25; co = 0.2; co1 = 0.5; b = 0.6; ri = 0.

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3T

0.02

F

x(t)

0.01

–0.01 –0.02

–0.03 5000

5005

5010

5015

5020

Time(s)

PR OO

0

5025

5030

ED

Fig. 6. Sub-harmonic response curves xi(t) showing the 3T-periodic motions with the parameters defined in Fig. 4 and n = 25 and r = 2. 25

20

CT

n=20

E

15

n=10

5

RE

10

n=1

0.2

0.4

OR

0 0

0.6

0.8

1

σ0

1.2

1.4

1.6

1.8

2

Fig. 7. Boundary of the existence of sub-harmonic oscillations with the parameters defined in Fig. 4 in the (E, r)-plane.

UN C

275 the 3T-harmonic oscillations. This investigations enable us to find the variation of the boundary 276 of the existence of sub-harmonic oscillations (below the curve) in the E–r plane and the results are 277 provided in Fig. 7. 278 3.4. Coexistence of chaotic and periodic orbits 279 280 281 282 283

One finds in this subsection different dynamical states which appear in the nonlinear electromechanical system depending to the set of initial conditions. For this aim, we numerically solve the equations of motion (1) and plot the resulting Poincare´ cross-section and phase portrait with the following set of parameters: c = ci = 0.2; ki = 0.01; ki1 = 0.25; wi = 1; b = 1.0; E0 = 25 and w = 0.8 where i = 1, 2, . . ., 25 since one considers the electromechanical system with 25 functions. For the

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15

1.5

(i)

1

(ii)

5

dxi/dt

0

0

F

dx/dt

0.5

–0.5 –1 –10 –5

0

5

x

–1.5 –2

(iii)

(iv)

6

1.2

5 4

1

ED

dxi/dt

dx/dt

–1

1.4

7

PR OO

–5

0

1

2

0.3

0.4

0.5

xi

0.8

3

–3

x

–2.5

–2

0.6 0.1

CT

2 –3.5

0.2

xi

OR

judicious choice of the initial conditions, one finds that the nonlinear electromechanical system with 25 functions is able to exhibit two degenerates attractors with the above mentioned set of parameters. The first one shown in Fig. 8 is obtained with the initial conditions Ip: ðxð0Þ; x_ ð0ÞÞ ¼ ð0.0; 0.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ which is the periodic orbit. The second one, obtained with the initial conditions Ic: ðxð0Þ; x_ ð0ÞÞ ¼ ð5.5; 2.0Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.01; 0.01Þ is the chaotic attractor as it appears in Fig. 9. With the above two sets of initial conditions, a bifurcation diagram is drawn (see Fig. 10) when the amplitude E0 varies, and one finds around E0 = 25, two types of degenerate attractors have appeared depending to the set on initial conditions.

UN C

284 285 286 287 288 289 290 291 292

RE

Fig. 8. (i, ii) Phase portrait and (iii, iv) Poincare´ cross-section showing the periodic orbit with the parameters c = 0.2; c1 = 0.2; k1 = 0.01; k11 = 0.25; w1 = 1.0; b = 1.0; E = 25; w = 0.8; n = 25 and the initial conditions Ip: ðxð0Þ; x_ ð0ÞÞ ¼ ð0.0; 0.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ (i = 1, . . ., 25).

293 4. Synchronization of two coupled nonlinear electromechanical systems 294 Today great and extensive interest in synchronization problems is displayed, spanning quite dif295 ferent fields of science. The phenomenon of synchronization is extremely wide spread in nature as 296 well as in the realm technology. The fact that, various objects seek to achieve order and harmony 297 in their behavior, which is a characteristic of synchronization, seems to be a manifestation of the

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1.5 (i)

1

(ii)

5

0

0

F

dxi /dt

dx/dt

0.5

–0.5 –1 –10 –5

0

–1.5 –2

5

–1

x 1.2

7.5 7

(iii)

1.1 1

6

dxi /dt

5.5

0

1

xi

2

0.8

5

0.7

4.5 –3.2

–3

x

–2.8

–2.6

–2.4

0.6 0.1

CT

4 –3.4

0.9

(iv)

ED

dx/dt

6.5

PR OO

–5

0.15

0.2

xi

0.25

0.3

0.35

OR

natural tendency of self-organization existing in nature [10]. Considerable attention paid to such topics is due to the potential applications of synchronization in communication engineering (using chaos to mask the information bearing signal) [11–14], in biology, chemistry and medicine the phenomenon of synchronization is of interest to those studying rhythms, electric rhythms of the brain in particular, cardiac rhythms, wave reactions in chemistry. In industry, synchronization is also used in the power generation to ensure exact coincidence of frequencies of several alternating current generator operating in parallel mode for common loading. In medecine, the effect of synchronization has found wide application in the construction of various vibro-technical devices. In this section, one considers the problem of the synchronization dynamics of two coupled electromechanical systems with a large number of functions, both to the regular and chaotic states.

UN C

298 299 300 301 302 303 304 305 306 307

RE

Fig. 9. (i, ii) Phase portrait and (iii, iv) Poincare´ cross-section showing the chaotic orbit with the parameters defined in Fig. 7 and the initial conditions Ic: ðxð0Þ; x_ ð0ÞÞ ¼ ð5.5; 2.0Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.01; 0.01Þ (i = 1, . . ., 25).

308 4.1. Statement of the problem and variational equations 309 310 311 312 313 314

4.1.1. Statement of the problem In the regular regime, for some sets of physical parameter and due to the nonlinearity, the response of the system to the external excitation shows the well-known hysteresis phenomena with two stable harmonic oscillations and different amplitudes. Each harmonic state has its own basin of attraction in the space of initial conditions. Consequently, if two systems are launched with different initial conditions belonging to different basins of attraction, they will finally circulate on

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17

(i)

x

4 3.5

4.5

3

3.5 3 24.5

20

22

E0

24

25

26

4.5

(ii)

x

4 4.5 4

3.5

3.5 3 24.5

22

20

18

28

25

26

E 0 24

30

25.5

28

ED

3

25.5

PR OO

2.5 18

F

4

30

Fig. 10. Bifurcation diagrams showing the coordinate x versus E0 with the parameters defined in Fig. 7 and the following initial conditions: Ip for (i) and Ic for (ii).

8

.

OR

6

RE

CT

315 different orbits. The goal of the synchronization in this case is to call one of the system (slave) 316 from its orbit to that of the other system (master) as it appears in Fig. 11. 317 The study of the transition boundaries and the derivation of the characteristics of the synchro318 nization process of two coupled electromechanical systems with 25 functions will be considered

4

Transition

slave system

*

0

UN C

dx/dt

2

master system

–2

–4 –6

–8 –3

–2

–1

0 x

1

2

3

Fig. 11. The two basins of attractions with the initial conditions: I aA : ðxð0Þ; x_ ð0ÞÞ ¼ ð0.0; 0.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ (i = 1, . . ., 25) with tin-line and I cA : ðxð0Þ; x_ ð0ÞÞ ¼ ð5.0; 5.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ (i = 1, . . ., 25) with fat-line. The parameters used are: c = 0.01; c1 = 0.1; k1 = 0.2; k11 = 0.4; w1 = 1; b = 0.95; w = 3; E = 1.05.

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i¼1

PR OO

€x1 þ c1 x_ 1 þ w21 x1  k11 x_ ¼ 0; .. . €xn þ cn x_ n þ w2n xn  kn1 x_ ¼ 0; n X € ki u_ i ¼ E0 cos wt  Kðu  xÞH ðt  T 0 Þ; u þ cu_ þ u þ bu3 þ

F

319 here. The master system is described by the components x and xi while the slave system has the 320 corresponding components u and ui. The coupling between the master and the slave systems is 321 carried out as n X €x þ c_x þ x þ bx3 þ ki x_ i ¼ E0 cos wt;

i¼1

ED

€ u1 þ c1 u_ 1 þ w21 u1  k11 u_ ¼ 0; .. . € un þ cn u_ n þ w2n un  kn1 u_ ¼ 0;

ð26Þ

where K is the feedback coupling synchronization coefficient, T0, the onset time of the synchronization and H(a) is the Heaviside function defined by H(a) = 0 for a < 0 and H(a) = 1 for a P 0. The schematic circuit of the two coupled identical nonlinear electromechanical systems with a unidirectionally homogenous coupling element is realized as it is shown in Ref. [5]. The two electromechanical systems, namely the master and slave systems, are coupled by a linear condenser and a buffer. The buffer acts on a signal-driving element that isolates the master system variable from the slave system variable, thereby providing a one-way coupling or unidirectional coupling. In the absence of the buffer, the system represents two identical self-sustained models coupled by a common condenser, when both the master and the slave systems will mutually affect each other.

334 335 336 337 338

4.1.2. Nonlinear variational equations From the instant t > T0, the system of coupled electromechanical systems changes its configuration and becomes physically interesting only so long as the dynamics of the slave electromechanical system are stable. The determination of the range of K for which the synchronization process is achieved is equivalent to the boundedness of  and i defined as

UN C

OR

RE

CT

324 325 326 327 328 329 330 331 332 333

eðtÞ ¼ uðtÞ  xðtÞ;

ei ðtÞ ¼ ui ðtÞ  xi ðtÞ;

ð27Þ

i ¼ 1; 2; . . . ; 25.

342 The variables e and ei are the measure of the relative nearness of the slave to the master and obey 343 the following nonlinear variational equations n X €eðtÞ þ c_eðtÞ þ ð1 þ K þ 3bx2s ÞeðtÞ þ 3bxs eðtÞ2 þ beðtÞ3 þ ki e_ i ðtÞ ¼ 0; ð28Þ i¼1 €ei ðtÞ þ ci e_ i ðtÞ þ x2i ei ðtÞ  ki1 e_ ðtÞ ¼ 0

for i ¼ 1; 2; . . . ; 25.

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346 347 348 349 350

19

Synchronization is achieved when the variations of the variables  and i go to zero as the time t increases or is less than a given precision, on the other hand, the synchronization is not achieved and the variations  and i diverge. The behavior of  and i depend on K, and the form of the master (xs, x1s, . . ., xns). The master time evolution in the harmonic states can be described as i ¼ 1; 2; . . . ; 25;

PR OO

xis ðtÞ ¼ Ai cosðxt  /Þ;

F

xs ðtÞ ¼ A cosðxt  /Þ;

ð29Þ

354 where A and Ai are the amplitudes while / and /i are the phases. As reported in Ref. [7], the 355 amplitudes A and Ai are the solutions of the following nonlinear equations 9 2 6 3 b A þ bF m A4 þ ðF 2m þ Gm ÞA2  E20 ¼ 0; 16 2 358 where

(

Di ¼ ðx2i  x2 Þ þ x2 c2i ;

n X ki ki1 xðx2i  x2 Þ ; Di i¼1

Gm ¼ cx þ

€e þ c_e þ ðX21 þ g cosð2wt  2/ÞÞe þ

n X

ki e_ i þ 3bA cosðwt  /Þe2 þ be3 ¼ 0;

€ei þ ci e_ i þ

w2i i

 kk1 e_ ¼ 0;

368 where

i ¼ 1; 2; . . . ; n;

3 g ¼ bA2 . 2

UN C

X21 ¼ 1 þ K þ g;

ð31Þ

OR

i¼1

371 372 373 374 375 376 377 378

n X ki ki1 ci x3 . Di i¼1

Using the Newton–Raphson algorithm, it appears that the corresponding two real positive physical solutions of Eqs. (29) with the parameters c = 0.01; c1 = 0.1; k1 = 0.2; k11 = 0.4; w1 = 1; b = 0.95; w = 3; E0 = 1.05, are Aa = 0.186 and Ac = 2.97. With the expressions (24), Eqs. (29) yield to the following parametric equations:

RE

361 362 363 364

ð30Þ

CT

F m ¼ 1  x2 

i ¼ 1; 2; . . . ; 25;

ED

/ ¼ tan1

) Gm ; F m þ 34 bA2

wki1 Ai ¼ pffiffiffiffiffi A; Di

From the expression of X21 , one finds that if K < 1  32 bA2 , e(t) and ei(t) will grow indefinitely leading the slave electromechanical system to continuously drift away from it original basins of attraction. In this case the feedback coupling is dangerous. Since it continuously adds energy to the slave electromechanical system. The appropriate tool to investigate the stability of the nonlinear parametric equations (31) is the Floquet theory [1] or the method of multiple time scales [1], but due to the complexity in solving analytically this Eq. (31), we will use in the following section the numerical simulation to find the range of K for the stability of the synchronization process of the two coupled nonlinear electromechanical systems.

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379 4.2. Results of the numerical simulations

RE

CT

ED

PR OO

F

4.2.1. Synchronization of the regular states As one has mentioned before, our aim now is to use the numerical simulation for determining the range of the appropriate coupling coefficient which enable to call one of the electromechanical system from its orbit to that of the other electromechanical system. The coupled equation (27) are solved numerically and determined the range of K in which the variables e and ei tend to zero, since the synchronization process is achieved. It can be noticed that the stability of the synchronization process means that the deviations e(t) and ei(t) tend to zero as the time increases, and the synchronization process is achieved. In the unstable region of K, e(t) and ei(t) show a bounded oscillatory behavior (possible synchronization) or grow indefinitely (no synchronization phenomena). One has found through a direct numerical simulation of Eq. (27) the appropriate range of K and the following results are observed. The master and slave electromechanical systems are initially launched with the initial conditions ðxð0Þ; x_ ð0ÞÞ ¼ ð0.0; 0.01Þ and ðxi ð0Þ; x_ i ð0ÞÞ ¼ ð0.0; 0.0Þ _ (i = 1, . . ., 25), and ðuð0Þ; uð0ÞÞ ¼ ð5.0; 5.01Þ and ðui ð0Þ; u_ i ð0ÞÞ ¼ ð0.0; 0.0Þ, (i = 1, . . ., 25) respectively. These set of initial conditions lead to periodic oscillation with amplitudes approximately equal to Aa and Ac respectively. Here the slave system is forced to come from the orbit Aa to the orbit Ac. We just inverse the initial conditions for the case Ac to Aa. In this analysis, the synchronization is launched at T0 = 500 and the coefficient K is varied until synchronization is achieved. It appears through this numerical simulation of Eqs. (27) that the slave transition from Ac to Aa requires that K 2 [3.1; 3.9] [ [4.4; 4.8] [ [5.1; +1[, while for the slave transition Aa to Ac, one has K 2 [0.92; 3,3] [ [4.05; +1[. The synchronization time Ts, defined as Ts = ts  T0 is computed following the time trajectory of the slave system relative to that of the master, ts is the time instant at which the two trajectories are close enough to be considered as synchronized. Let us look now for the behavior of the condition of the synchronization process between the master and slave electromechanical systems. Synchronization is achieved with the following synchronization condition: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 25 25 u X X 2 2 2 2 ð32Þ ðui  xi Þ þ ðu_ i  x_ i Þ < h; jdðtÞj ¼ tðu  xÞ þ ðu_  x_ Þ þ

OR

380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404

i¼1

where h is the synchronization precision or tolerance. For these investigations, the value of the precision is h = 1010 for the slave transition from Ac to Aa, while in the reverse case, the synchronization process with this precision was not possible. In fact, with the tolerance h = 104, one finds that the synchronization process is achieved with the range of K defined before. This is due to the fact that it is easy to move from the orbit with small amplitude to the one with high amplitude, than in the reverse case. This situation can be understood by the fact that we have Aa < Ac. We remind that this synchronization has enabled us to identify different bifurcation mechanisms which appear in the coupled system. To illustrate the above results, let us evaluate the temporal variation of the quantity     25 25 X X   ðui  xi Þ þ ðu_ i  x_ i Þ pðtÞ ¼ ðu  xÞ þ ðu_  x_ Þ þ   i¼1 i¼1

UN C

407 408 409 410 411 412 413 414 415

i¼1

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50

(i)

(iii)

p(t)

0

0

–50 400

500

600

700

PR OO

F

p(t)

21

–50 400

800

500

Time (s) 50

50

700

800

p(t)

(iv)

500

600

700

0

ED

0

–50 400

800

500

CT

p(t)

(ii)

–50 400

600

Time (s)

Time (s)

600

700

800

Time (s)

RE

Fig. 12. Temporal variation of p(t) versus time for the slave transition from periodic orbit Aa to the periodic orbit Ac with the value of K chosen both in the unstable region ((i) K = 1 and (ii) K = 3.5) and stable region ((iii) K = 2 and (iv) K = 5).

OR

418 with the value of the coupling coefficient K both chose in the unstable and stable regions of the 419 synchronization process. The results are shown in Fig. 12 for the variation of Ts versus K and 420 in Fig. 13 for the variation of p(t), and one finds that in the stable domain of K, p(t) decreases 3500

UN C

3000

Ts

2500

2000

1500

1000 0

1

2

3

4

5

6

7

8

9

K

Fig. 13. Synchronization time Ts versus the coupling coefficient K for the slave transitions from the orbit Aa to the orbit A c.

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421 and tend to zero when t evolves (see Fig. 13(iii, iv)), while in the unstable region of K, p(t) presents 422 the bounded oscillatory behavior (see Fig. 13) or grows to infinity (see Fig. 13(i, ii)) or exhibits the 423 chaotic behavior (see Fig. 13(i)).

ED

PR OO

F

4.2.2. Synchronizing chaotic orbit (respectively periodic orbit) on the periodic orbit (respectively chaotic orbit) Now one considers the two electromechanical systems with the following parameters c = 0.2; c1 = 0.2; k1 = 0.01; k11 = 0.25; w1 = 1.0; b = 1.0; E = 25; w = 0.8. This set of parameters leads to the following approximated amplitude A = 3.18. In this case, the model can exhibit the chaotic and periodic behavior, depending on the initial conditions as we have shown in Figs. 7 and 8. If two electromechanical systems with the above set of parameters are launched with the following initial conditions Ip and Ic, they will finally circulate on the periodic and chaotic orbits, respectively. The objective of the synchronization is to call the slave system from the periodic orbit (respectively chaotic orbit) to follow the master system to the chaotic orbit (periodic orbit). The numerical simulation of the nonlinear equations (22) indicates that, the slave transition from the periodic orbit to the chaotic orbit occurs for K 2 [2.85; 2.75] [ [2.55; 2.15] [ [1.55; 2.25] [ [4.4; +1[, while to the reverse transition, one needs K 2 [2.15; 1.4] [ [0.75; 1.40] [ [3.9; +1[. In Fig. 14, the temporal variation of p(t) is presented for the slave transition from the chaotic orbit to the periodic one 40

60

(i)

(iii)

CT

40

20

P(t)

0 –20 –40

450

500

50

UN C

(ii)

450

500

Time (s)

500

600

700

Time (s) 30

(iv)

20 10

0

–50 400

–40 400

600

OR

Time (s)

550

0

–20

P(t)

–60 400

RE

P(t)

20

P(t)

424 425 426 427 428 429 430 431 432 433 434 435 436 437

0 –10 –20

550

600

–30 400

500

600

700

Time (s)

Fig. 14. Temporal variation of p(t) versus time for the slave transition from chaotic orbit to the periodic orbit with the value of K chosen both in the unstable region ((i) K = 3.2 and (ii) K = 3.5) and stable region ((iii) K = 2 and (iv) K = 4.6).

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23

2500

2000

F

Ts

1500

500

0 –4

–2

0

2

4

6

8

K

10

PR OO

1000

12

14

16

Fig. 15. Synchronization time Ts versus the coupling coefficient K for the slave transitions from the chaotic orbit to the periodic orbit.

ED

with the value of K chosen both in the unstable and stable regions of K. The temporal variations of p(t) versus time for the slave transitions from the chaotic orbit to the periodic orbit with the values of K chosen both in the unstable region (K = 3.2; 3.5) and the stable domain (K = 2.1; 4.6) is shown in Fig. 14, and we find that p(t) in the stable region of K versus the time increases while in the unstable region, p(t) shows the chaotic behaviors. Fig. 15 shows the variation of the synchronization time Ts versus the coupling coefficient K for the slave transition from the chaotic orbit to the periodic orbit.

CT

438 439 440 441 442 443 444

OR

In this paper, the dynamics and synchronization of coupled electromechanical systems with multiple functions in series is studied. Using the multiple time scale method, attention is focussed on the amplitude of sub- and super-harmonic resonances. It is observed that the number of linear mechanical oscillators can be increased in order to destroy the hysteresis phenomena in the system. This is very important in electromechanical system where the hysteresis phenomena must be destroyed in order to avoid distortions. On the other hand, uni-directional coupling have been applied to study the synchronization of the master and slave system both in the regular and chaotic states. These phenomena of synchronization or nonsynchronization is proved by direct numerical simulations of the coupled system. The threshold value of the coupling parameter K for the phenomena (synchronization or nonsynchronization) to be observed is determined from the nonlinear variational equation defining the synchronization error dynamics.

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445 5. Conclusion

457 Appendix A 458

The coefficients of Eqs. (14) are

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CT

Di ¼ xi ais sin2 dis  xi ais cos dis ; xi ais sin dis C01i ¼ ; Di ais cos dis ; C02i ¼ Di xi C03i ¼ ; Di sin dis C04i ¼ ; Di xi ais sin2 dis þ xi ais cos dis 0 C5i ¼ ; a s Di 1 C11i ¼ ðr  ri Þ cos dis þ c0i sin dis ; 2 1 1 C2i ¼ ais ðr  ri Þ sin dis þ c0i ais cos dis ; 2 1 C13i ¼ xi ðr  ri Þ sin dis þ c0i cos dis ; 2 1 1 C4i ¼ xi ais ðr  ri Þ cos dis  c0i sin dis ; 2 C1i ¼ C01i C11i þ C02i C13i ; C2i ¼ C01i C12i þ C02i C14i ; 1 C3i ¼  C02i k0i1 ; 2 1 0 1 C4i ¼ C3i C1i þ C04i C13i  C05i xi k0i sin dis ; 2 1 0 1 0 1 C5i ¼ C3i C2i þ C4i C4i  C05i xi k0i ais cos dis ; 2  1 0 2 0 9 2 C6i ¼  C4i k0i1 þ C5i bas  r þ 2bK ; 2 8 C7i ¼ C05i bK3 sin ds ; 1 P1 ¼  c0 ; 2 P2 ¼ bK3 cos ds ;   1 9 2 2 ba  r þ 2bK ; P3 ¼  as 8 s 1 P4 ¼ bK3 sin ds ; as P1i ¼ xi k0i cos dis ; P2i ¼ xi k0i ais sin dis ; 1 P3i ¼  xi k0i sin dis ; 2 1 P4i ¼  xi k0i ais cos dis ; 2

F

R. Yamapi et al. / Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx

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ð33Þ

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463 Appendix B 464

The coefficients Qi of Eqs. (16) are Q2 ¼ P2 P3 þ P4 ðP1 þ C11 þ C51 Þ þ P1 C11 þ C51 ðP1 þ C11 Þ  C61 P21  C41 C21  C31 P11 þ C71 P41 ;

PR OO

Q3 ¼ P2 P3 ðC11 þ C51 Þ  P2 C61 P41  P2 C31 P31  P4 P1 C11

F

Q1 ¼ P4  P1  C11  C51 ;

 P4 C51 ðC11 þ P1 Þ  P1 C11 C51 þ P4 C61 P21 þ P4 C41 C21 þ P4 C31 P11  C31 C41 P21  C61 P11 C21 þ C61 C11 P21 þ P1 C41 C21 þ C31 P11 C51  C71 P31 C21  C71 P3 P21 þ C71 P41 ðP1 þ C11 Þ; Q4 ¼ P2 P3 C11 C51  P41 P2 C31 C41  P2 C61 P31 C21

þ P2 C61 C11 P41 þ P2 P3 C41 C21 þ P2 C31 P31 C51 þ P4 P1 C11 C51

ED

þ P4 C31 C41 P21 þ P4 C61 C21 P11  P4 C61 P21 C11  P4 C61 C11 P1  P4 C31 P11 C51 þ C71 P1 P31 C21 þ C71 P3 P21 C11 þ C71 C31 P41 P11

CT

 C71 C31 P31 P21  C71 C11 P41 P1  C71 C21 P3 P11 .

467 References

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[1] Nayfeh AH, Mook DT. Nonlinear oscillations. New York: Wiley-Interscience; 1979. [2] Awrejcewicz J. Bifurcation and chaos in coupled oscillators. Singapore: World Scientific; 1991. [3] El-Bassiony AF, Eissa M. Response of three-degree-of-freedom with cubic non-linearities to harmonic excitations. Phys Scr 1999;59:183–94. [4] Kozlov AK, Sushchik MM, Molkov YaI, Kuznetsov AS. Bistable phase synchronization and chaos in a system of coupled Van der Pol–Duffing oscillators. Int J Bifurc Chaos 1999;9:2271–7. [5] Yamapi R, Woafo P. Dynamics and synchronization of coupled self-sutained electromechanical devices. J Sound Vibr, in press. [6] Yamapi R, Bowong S. Dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity. Commun Nonlinear Sci Numer Simul, in press. [7] Woafo P, Yamapi R, Chabi Orou JB. Dynamics of a nonlinear electromechanical system with multiple functions in series. Commun Nonlinear Sci Numer Simul 2005;10:229–51. [8] Stocker JJ. Nonlinear vibration. New York: Interscience Publishers; 1950. [9] Hayashi C. J Appl Phys 1953;24:521. [10] Blekhman II. Synchronization in nature and techniques. Moskou: Nanka; 1981. [11] Pecora LM, Caroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:812–24. [12] Oppenheim AV, Wornell GW, Isabelle SH, Cuoma K. Signal processing in the context of chaotic signals. Proceedings of the international conference on acoustic, speech and signal processing, vol. 4. New York: IEEE; 1992. p. 117–20. [13] Kovarev LJ, Halle KS, Eckert K, Parlitz U, Chua LO. Experimental demonstration of secure communications via chaotic synchronization. Int J Bifurc Chaos 1992;2:709–13. [14] Perez G, Cerdeira HA. Extracting messages masked by chaos. Phys Rev Lett 1995;74:1970–3.

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