International Journal of Non-Linear Mechanics 38 (2003) 531–541

Linear feedback and parametric controls of vibrations and chaotic escape in a
6 potential R. Tchoukuegno, B.R. Nana Nbendjo, P. Woafo ∗ Laboratoire de Mecanique, Faculte des Sciences, Universite de Yaounde I, B.P 812 Yaounde, Cameroon Received 3 January 2001; accepted 1 August 2001

Abstract The control of vibration amplitude and chaotic escape of an harmonically excited particle in a single well
6 potential is considered. The linear feedback and parametric control strategies are used. The control e3ciency on amplitude is found by analysing the behaviour of the amplitude of the controlled system as compared to that of the uncontrolled system. The conditions for inhibition of the chaotic escape are obtained by means of the Melnikov method. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Linear and parametric controls; Vibrations; Chaotic escape

1. Introduction In recent years, considerable e8orts have been devoted to the control of vibrating structures in various 9elds of fundamental and applied sciences. Two major aims in the scope of the researchers are: reduce the amplitude of vibrations, inhibit chaos and escape from potential well. Applications are well encountered in structural mechanics (see Refs. [1–5] and references therein). Among various types of control strategies, the active feedback control usually developed by means of electromagnetic force or by a mechanical device [2–5] and the parametric time modulation of one parameter of the structure (see for instance Refs. [6 –8]) are well known. In Refs. [3,4], the authors studied the e8ects of the active feedback control strategy on the stability, the bifurcation structures and the chaotic responses on structures with non-linear soft spring (the
4 model). In the autonomous case, they derived the bifurcation criteria in a space where the parameters are the structural damping, the control gain and speed parameters. For structures submitted to a harmonic forcing, they used the Melnikov method to predict in the parameters space the limits separating domain of regular basins from domain where the basin geometry is fractal. Another important system encountered in physics and in structural mechanics is the vibrating structure with the
6 potential [9 –11] described by the following di8erential equation: q@ + a0 q˙ + !02 q + cq3 + dq5 = f0 cos(wt); ∗

Corresponding author. E-mail address: [email protected] (P. Woafo). 0020-7462/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 0 8 1 - 6

(1)

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Fig. 1. (a) 6 potential with a single well (!0 = 1; c = − 0:5; d = − 0:05). (b) Phase diagram of a 6 potential with a single well.

where !0 and a0 are, respectively, the natural frequency and the damping coe3cient of the structure, c and d are the other parameters of the potential while f0 and ! stand, respectively, for the amplitude and the frequency of the external force, t is the time and the dot over q stands for the time derivative. In Ref. [12], we considered the dynamics of this model by deriving, using the multiple time scales method, the harmonic, sub- and superharmonic oscillatory states of the model. Using the Melnikov theory, we also derived the analytical criteria for the occurrence of transverse intersections on the surface of homoclinic and heteroclinic orbits both for three potential well case and a single potential well case. The case of the potential with a single well at q = 0 (Fig. 1) is particularly interesting since it can lead to catastrophic failure resulting from the escape from the potential well. Consequently, it can be desirable

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to control the dynamics of such a system both by reducing the amplitude of vibrations and modifying the criteria for the appearance of the chaotic escape. This constitutes the aim of this paper. We use the aforementioned two types of control strategy. The values of the system parameters used in the paper are a0 = 0:05; !0 = 1; c = − 0:5 and d = − 0:05. In Section 2, we consider the model with the linear feedback control strategy. The e3ciency of the control on the amplitude is analysed in Section 2.1 both in the linear and the non-linear cases. In Section 2.2, we derive the e8ects of the control parameter on the Melnikov criterion for the occurrence of the fractal basin geometry. Section 3 which follows the same structure as Section 2 deals with the parametric control strategy. The results of the numerical simulation of the di8erential equations of the model are obtained for comparison or as complement to the analytical results. We conclude in Section 4. 2. Linear feedback control The system with a linear feedback control is described by the following equations: q@ + a0 q˙ + !02 q + cq3 + dq5 + z = f0 cos(!t);

(2a)

z˙ + z = q;

(2b)

where z is the control force variable,  the control speed parameter and the control gain parameter (see Refs. [3,4]). 2.1. E7ects of the control on the amplitude of vibrations In the linear limit (c = d = 0), the amplitude of the harmonic oscillations of the controlled system is given by Ac =

f0 (!2 + 2 ) : [((!02 − !2 )(!2 + 2 ) + 2 )2 + (!  − a0 !(!2 + 2 ))2 ]1=2

(3)

Comparing this to the amplitude Anc of the vibrations of the uncontrolled system, we have that Ac ¡ Anc if the control gain parameter is as follows:

∈] − ∞; −2(!02 − !2 ) +

2!2 a0 [∪]0; +∞[ 

for ! ∈]0;


!0 [ a0 = + 1

and

∈ ] − ∞; 0[∪] − 2(!02 − !2 ) +

2!2 a0 ; + ∞[ 

for ! ∈]


!0 ; +∞[: a0 = + 1

(4)

For instance when ! = 1; ∈ ] − ∞; 0[ ∪ ]0:02525; +∞[. In the non-linear case, the amplitude of the harmonic oscillations satis9es the following non-linear algebraic equation:     

2 3

2 9 2 5 25 2 10 15 6 2 2 d Ac + cdA8c + c + d !02 − !2 + 2 c A + ! − ! + A4c c 0 64 16 16 4  + !2 2  2 + !2  2  2 

2

! 2 2 + !0 − ! + 2 + −a0 ! + 2 A2c − f02 = 0: (5)  + !2  + !2

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Fig. 2. E8ects of the control gain parameter on the amplitude of vibrations.

Fig. 3. E8ects of the control speed parameter  on the amplitudes of vibrations with ! = 1; f0 = 0:12 and = − 0:5.

In Fig. 2, we have plotted (for ! = 1; f0 = 0:12 and  = 3:96) the variations (solid line) of the amplitude Ac computed from Eq. (5) versus the control gain parameter . The solid horizontal line is the value of the amplitude without control. It is found that Ac ¡ Anc for ∈ ] − ∞; 0[ ∪ ]0:37; +∞[. A domain, which is comparable with that of the linear limit with the same parameters. In Fig. 2, we have also plotted with thick line the amplitude obtained from a direct numerical simulation of the di8erential equations (2). It is found that Ac ¡ Anc for a restricted domain of de9ned as ∈ ] − 1; 0[. For ¡ − 1 and ¿ 0, the motion grows inde9nitely indicating the instability of the controlled structure. In Figs. 3 and 4, we have plotted, for a 9xed value of chosen in the range where the control is assumed e8ective, the variations of Ac as a function of the control speed parameter for two values of !. These 9gures show that the frequency ! of the external excitation also a8ects the control e8ectiveness. Indeed, in Fig. 3, it is found that with ! = 1, Ac () ¡ Anc () whatever be the value of , while in Fig. 4 with ! = 0:5, the opposite happens. These 9gures are obtained from both analytical and numerical simulations. This dependence of the e3ciency of the control on  and ! also appears in the analytical equation (4) for the linear model. From the above critical value of  = 2:45, it is found that the controlled system is unstable (see Fig. 4).

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Fig. 4. E8ects of the control speed parameter  on the amplitude of vibrations for ! = 0:5; f0 = 0:12 and = − 0:5.

2.2. E7ects of the control on the Melnikov criterion for chaos In Ref. [12], we derived the condition for the appearance of chaotic escape in the uncontrolled structure using the Melnikov theory. In fact, the Melnikov theory helps to de9ne the conditions for the existence of the so-called transverse intersection points between perturbed and unperturbed separatrices. One would like to know how the control strategy a8ects the Melnikov criterion or in what range of the control parameters the heteroclinic chaos in our model could be inhibited. To deal with such a question, let us express the dynamical structure in the form U = g0 (U˙ ) + gp (U; t);

(6)

where U = (q; p = q; ˙ z) is the state vector, g0 = (p; −!02 q − cq3 − dq5 ; −z + q) and gp = (0; −a0 p − z +f0 cos(!t); 0). The hamiltonian part of Eq. (6) (the term gp (U; t) discarded) possesses a heteroclinic orbit connecting the unstable points of the potential. This orbit is de9ned as q= √

±X sinh(Yt)

2(1 + (1 − 2 ) sinh2 (Yt))1=2

p= √ and

±XY cosh(Yt)

2(1 + (1 − 2 ) sinh2 (Yt))3=2



z=

;

t

−∞

 q exp(s) ds exp(−t);

(7a) (7b)

(7c)


where X 2 = q12 (3 +  2 ); Y = q12 ((−d=2)( 2 + 1))1=2 ;  =  2=3( 2 + 1);  = q2 =q1 , q1 is the unstable equilibrium point and q2 the modulus of the complex root of the equation !02 + cq2 + dq4 : With Eqs. (6) and (7), the Melnikov function M (t0 ) [13,14] can be evaluated. After some algebra, we get

M (t0 ) = A cos(!t0 ) − B − C;

(8)

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where X!f0 ; Y (1 − 2 ) sinh ! 2Y    1+ a0 X 2 Y 32 − 1 (1 + 32 ) ln + ; B= 8 1 − 2 2 1− A=

C = k(); with X 2 2 Y k() = 2



+∞

−∞

exp(−!) cosh(Y!) (1 + (1 − 2 ) sinh2 Y!)3=2



!

−∞

exp(s) sinh(Ys) ds (1 + (1 − 2 ) sinh2 Ys)1=2



d!:

From Eq. (8), we get the condition for the appearance of the Melnikov chaos which is given by the equation   2    Y (1 − 2 ) sinh( w 3 − 1 (1 + 32 ) 1+ 2Y ) a0 XY ln + +

k() : (9) f0 ¿ w 8 1 − 2 2 1− Therefore, the critical value of f0 is proportional to the control gain parameter and depends non-linearly on the control speed parameter through k(). After some mathematical transformations (see the appendix), we have integrated numerically the quantity k(). With the set of parameters used in this paper and  = 3:96, we obtain k() = − 1:2363. It is interesting to know whether the good ranges of control parameter for the reduction of the amplitude of vibrations is safe from chaos. To deal with this problem, we have looked for the fractality of the basin of attraction by solving numerically Eq. (2) for pertaining to the restricted limits where the controlled amplitude is less than the uncontrolled one. From the analytical calculations, it is found that inhibition of chaos occurs when belongs to the following domain:

∈ ] − ∞; 1 [ ∪ ] 2 ; +∞[ with

1 =

(A − B) k()

and

2 = −

(A + B) ; k()

which for the parameters used give 1 = − 0:401 and 2 = 2:22. Our numerical results are reported in Fig. 5 for f0 = 0:12. It is found that for 6 − 0:9 and ¿ − 0:4, the basins of attraction are fractal and show a regular shape for ∈ [ − 0:9; −0:4]. This domain is included in that of the reduction of amplitude (see Fig. 2). It is also interesting to note the evolution and deformation of the basin of attraction as varies. 3. Parametric control The philosophy of the parametric control is to add small time modulations of one parameter of the physical structure. The periodic modulations which are frequently used will be applied on the natural frequency of our model. Then the parametrically controlled structure is described by the following equation: q@ + a0 q˙ + !02 [1 + " cos(#t + )]q + cq3 + dq5 = f0 cos(!t);

(10)

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Fig. 5. Basins of attraction for various values of ;  = 3:96; f0 = 0:12 and ! = 1: (a) = − 0:95, (b) = − 0:9, (c) = − 0:7, (d) = − 0:5, (e) = − 0:4, (f) = − 0:3, (g) = 0.

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Fig. 6. E8ects of the phase  of the parametric perturbation on the amplitude of vibration (linear case) with " = 0:2.

where "; # and  are the amplitude, the frequency and the phase of the parametric modulation, respectively. We restrict ourselves to the case # = 2! for mathematical convenience. The problems to tackle are the same as in Section 2. In the linear limit, the amplitude of the harmonic response of the structure is Ac =

f0 (((!02 − !2 ) − ("=2) cos())2 + (a0 ! + ("=2) sin())2 )1=2 (!2 − !2 )2 − ("2 =4) cos2 () + (a0 ! + ("=2) sin())2 : 0

(11)

The variations of Ac versus  is reported in Fig. 6 with ! = 1; f0 = 0:12 and " = 0:2. The horizontal line is the amplitude of the uncontrolled structure. It appears that the phase  plays an important role in the controlled structure. In the general form (11), it is di3cult to derive practical expressions for " and  for which the control is achieved. However, for particular values of , we can express the good values of " in terms of the parameters of the structure. For for  = 0 (mod ), the e3cient values of √ instance, √ " for Ac ¡Anc belong to the intervals " ∈ ] − ∞; − 3a0 =2[ ∪ ] 3a0 =2; +∞[. For  = =2 (mod 2), we have " ∈ ] − ∞; −4a0 [ ∪ ]4a0 ; +∞[ and for  = 3=2 (mod 2), " ∈ ] − ∞; 0[ ∪ ]2a0 !; +∞[. In the non-linear limit, the amplitude Ac satis9es the non-linear algebraic equation 2 4 20 ( 25 64 ) d Ac +

3 18 125 512 cd Ac

2 + ( 45 16 c d$ +

25 2 2 32 % d

3 + ( 125 128 $d +

+ ( 54 d$ +

2 5 + ( 15 8 dc% + 3$c( 4 d$ +

9 2 16 c )

+ ( 94 c2 $2 + ( 52 d$ + 98 c2 )%2 −

225 2 2 512 c d

2 2 2 16 135 3 + ( 15 16 ) c d )Ac + ( 128 c d +

14 225 2 2 64 c d $)Ac

12 9 2 2 16 c ) )Ac

−

10 25 2 2 64 d f0 )Ac

2 8 15 16 cdf0 )Ac

+ (3$c%2 − ( 54 d$ +

+ (%4 − ( 32 c$ − 34 c" cos())f02 )A4c − (K 2 + $2 − $" cos() +

9 2 16 c "2 4

− 58 " cos())f02 )A6c

cos2 )A2c f02 = 0;

(12)

where $ = !02 − !2 ; %2 = $2 − ("2 =4) cos2  + K 2 and K = a0 ! + ("=2) sin(). A trivial solution is Ac = 0. The variation of the non-trivial solution versus  is plotted in Fig. 7(a). The solid horizontal line corresponds to the amplitude of the uncontrolled system. The control is e3cient for  ∈ [1:1309; 4:68]. We have complemented the analytical results by solving Eq. (10) numerically. Fig. 7(a) shows that the range of the control e3ciency obtained from numerical simulation is shorter

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Fig. 7. (a) E8ects of  on the amplitude of vibrations in the non-linear model (analytical and numerical results) with ˙ = 0. (b) E8ects f0 = 0:12; # = 2!; ! = 1 and " = 0:2. Initial conditions for the numerical simulation are q(0) = 0 and q(0) of  on the amplitude of vibrations (numerical results) with f0 = 0:12; # = 2!; ! = 1 and " = 0:2 (initial conditions q(0) = 0:5 q(0) ˙ = 0). (c) E8ects of " on the amplitude of vibrations (numerical result) for three values of ;  = 0; 3:5814 ˙ = 0). and 4, with f0 = 0:12; # = 2!; ! = 1 and " = 0:2 (initial conditions q(0) = 0:5; q(0)

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( ∈ [2:733; 4:147]) than that obtained from Eq. (12). The numerical curve of Fig. 7(a) is obtained with the initial conditions q(0) = q(0) ˙ = 0. In Fig. 7(b), we have plotted Ac versus  but with the initial conditions q(0) = 0:5 and q(0) ˙ = 0. We 9nd ranges of , where the motion is unbounded contrary to what is observed in Fig. 7(a). This indicates that, in some ranges of , the 9nal state behaviour depends on the initial conditions. The variations of Ac versus " are plotted in Fig. 7(c) for some selected values of . It is found that the good ranges of " for the control e3ciency as well as the ranges of unbounded motion (ranges where Ac does not exist) depend on . In the case  = 0, it is found that Ac increases with " until " = 0:32, where unbounded motion occurs. This unstable range lasts till " = 0:73 and an oscillatory behaviour comes again until " = 0:91 above which the system becomes totally unstable. We have also analysed the e8ects of the parametric control on the Melnikov criterion. Here, the Melnikov function is given by M (t0 ) = − E cos(!t0 ) + F sin(2!t0 + ) − G;

(13)

where E=

X!f0 ; Y (1 − 2 ) sinh (!=2Y )

G=

a0 X 2 Y 8



F=

!02 "X 2 ! sin(#=2Y cosh−1 (1 + 2 =1 − 2 )) ; 2Y (1 − 2 ) sinh (!=Y )

  32 − 1 (1 + 32 ) 1+ ln + 1 − 2 2 1−

while X and  have been de9ned in Eq. (7). It is di3cult to obtain the condition for the Melnikov function (13) to have zeros for any " and . From Ref. [6], it is found that for  = (2n + 1)=2 (n being an integer), the inhibition of chaos takes place for   G ; (15) " ¿ R 1 − E 2wf0 sinh(#=2Y ) . w02 X# sinh(w=2Y ) sin(#=2Y arch(1 + 2 =1 − 2 )) With the values of the parameters used in this paper, we get " ¿ 1:196.

where R =

4. Conclusion In this paper, we have analysed the question of the control of vibrations and chaotic escape of a harmonically excited particle moving in a 6 potential. The case of the potential which can lead to unbounded motion has been considered. The linear feedback and the parametric control strategies have been used. For each type of control, conditions for the reduction of the amplitude of vibrations and those of the inhibition of the chaotic escape have been derived using, respectively, the harmonic balance method and the Melnikov theory for chaos. The analytical results have been complemented and compared with the results of numerical simulations of the di8erential equations of motions of the controlled structure. The model can 9nd applications in engineering and fundamental physics.

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Appendix A X 2 2 Y k() = 2



+∞

−∞

exp(−!) cosh(Y!) (1 + (1 − 2 ) sinh2 Y!)3=2



!

−∞

exp(s) sinh(Ys) ds (1 + (1 − 2 ) sinh2 Ys)1=2



d!

let  = exp( !) ; - = exp( s); a = 1 − 2 ; therefore,     2X 2 2 ∞ 2 + 1 -(-2 − 1) k() = d- d 3=2 2 2 1=2 Y (4 + a(2 − 1)2 )3=2 0 0 2 .a((4- + a(- − 1) )) = =( − 1) and - = p=(p − 1), then the expression of k() is reduced to    (2p − 1)p(=Y −1) (1 − p)−(1+=Y ) 2X 2 2 1 00 ( ) (1−=Y ) (1 − )(1+=Y ) dp d ; k() = Y 03=2 ( ) 01=2 (p) 0 0

we assumed that

where 00 (x) = 2x2 − 2x + 1; 0(x) = 4x2 (x − 1)2 + a(2x − 1)2 . References [1] T. Aida, K. Kawazoe, S. Toda, Vibration control of plates by plate-type dynamic vibration absorbers, J. Vib. Acoust. 117 (1995) 332–338. [2] Y. Okada, K. Matsuda, H. Hashitani, Self-sensing active vibration control using the moving-coil-type actuator, J. Vib. Acoust. 117 (1995) 411–415. [3] K. Hackl, C.Y. Yang, A.H.-D. Cheng, Stability, bifurcation and chaos of non-linear structures with control—I. Autonomous case, Int. J. Non-Linear Mech. 28 (1993) 441–454. [4] A.H.D. Cheng, C.Y. Yang, K. Hackl, M.J. Chajes, Stability, bifurcation and chaos of non-linear structures with control—II. Non autonomous case, Int. J. Non-Linear Mech. 28 (1993) 549–565. [5] Chyuan-Yow Tseng, Pi-Cheng Tung, Dynamics of a Pexible beam with active non-linear magnetic force, J. Vib. Acoust. 120 (1998) 39–46. [6] R. Lima, M. Pettini, Suppression of chaos by resonant parametric perturbations, Phys. Rev. A 41 (1990) 726–733. [7] L. Fronzoni, M. Gioncodo, Experimental evidence of suppression of chaos by resonant parametric perturbations, Phys. Rev. A 43 (1991) 6483–6487. [8] R. Chacon, F. Balibra, M.A. LopRez, Inhibition of chaotic escape from a potential well using small parametric modulations, J. Math. Phys. 37 (1996) 5518–5523. [9] M. Debnath, A.R. Chowdhury, Period doubling and hysteresis in a periodically forced, damped anharmonic oscillation, Phys. Rev. A 44 (1991) 1049–1060. [10] S. Lenci, A.M. Tarantino, Chaotic dynamics of an elastic beam resting on a wrinkler-type soil, Chaos Solitons Fractals 10 (1996) 1601–1614. [11] S. Lenci, G. Menditto, A.M. Tarantino, Homoclinic and heteroclinic bifurcations in the non linear dynamics of beam resting on an elastic substrate, Int. J. Non-Linear Mech. 34 (1999) 615–632. [12] R. Tchoukuegno, B.R. Nana Nbendjo, P. Woafo, Resonant oscillations and fractal basin boundaries of a particle in a 6 potential, Physica A, in press. [13] V.K. Melnikov, On the stability of the center for some periodic perturbations, Trans. Moskow Math. Soc. 12 (1963) 1–57. [14] J. Guckenheimer, P.J. Holmes, Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector 9elds, Springer, Berlin, 1983.