Active control with delay of vibration and chaos in a ...

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Chaos, Solitons and Fractals 18 (2003) 345–353 www.elsevier.com/locate/chaos

Active control with delay of vibration and chaos in a double-well Duﬃng oscillator B.R. Nana Nbendjo, R. Tchoukuegno, P. Woafo

*

Laboratoire de m ecanique, Facult e des sciences, Universit e de Yaound e I, B.P 812 Yaound e, Cameroun Accepted 2 January 2003

Abstract This paper deals with the active control of vibration, snap-through instability and horseshoes chaos in a bistable Duﬃng oscillator. We determine the range of control parameters which leads to a good control. The eﬀect of time-delay between the detection of vibration and action of the control is particularly investigated. Ó 2003 Elsevier Science Ltd. All rights reserved.

1. Introduction In recent years, considerable eﬀorts have been devoted to the control of linear and nonlinear vibrating structures. Among the control strategies, the active control plays a particular role [1–7]. Ref. [1] gives a thorough review of the description and main results of the active control. In Ref. [2], the authors showed that time-delay can lead to the instability of the control process in linear structures. While the Refs. [3,4] considered the behaviour of the catastrophic single well Duﬃng oscillator under active control, Refs. [5,6] extended the study to a Duﬃng oscillator with a quintic nonlinear term leading to various conﬁgurations of the potential including a catastrophic single well potential, a catastrophic double-well potential and a tristable noncatastrophic potential. In Ref. [7], the authors considered the control of nonlinear oscillations of buckled beams by applying concentrated moments at suitable points and using a nonlinear optimal control method. The main results are that the control scheme mitigates the eﬀects of the dynamic loading on the vibration amplitude and prevent dangerous instability phenomena, with a load-carrying capacity of buckled beams increasing with the degree of nonlinearity of the control force. The studies of Refs. [3–7], which considered nonlinear structures, did not take into account the eﬀects of the inevitable time-delay between the detection of the structureÕs motion and the restoring action of the control. However, as quoted here before, Ref. [2] considered such a problem in linear structures and showed that time-delay can even lead to the instability of the whole structure. Thus, it is of interest to analyze the same problem for structures with nonlinear dynamics. This is one of the aims of this paper. The nonlinear model considered is the Duﬃng oscillator with two wells. The interest devoted to this model is due to the following facts. Firstly, to our knowledge, the recent studies of active control have not considered it. The control strategy always applied to this model is the parametric control [8,9]. Secondly, this bistable model describes many situations in physical and mechanical sciences. In particular, it describes the dynamics of the fundamental mode of a buckled beam undergoing forced lateral vibrations in the situation where the axial compressive load is greater than the ﬁrst EulerÕs load [10–12]. The organisation of the paper is as follows. In Section 2, the model is presented and the eﬀects of the control on the harmonic vibrations and snap-through instability of the structure are considered. Section 3 deals with the problem of inhibition of Smale-horseshoe chaos in the model. Particular emphasis will be paid to the eﬀects of the time-delay. Concluding remarks come in the last section.

*

Corresponding author. E-mail addresses: [email protected], [email protected] (P. Woafo).

0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(02)00681-1

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2. Model, control of harmonic vibrations and snap-through instability 2.1. The model The double-well Duﬃng oscillator subjected to the action of an external additive harmonic excitation is described by the following equations dV ðxÞ ð1Þ ¼ f0 cosðxtÞ; dx where k is the damping coeﬃcient, f0 and x are the amplitude and frequency of the external excitation. The potential V ðxÞ is given by €x þ k_x þ

1 1 V ðxÞ ¼ ax2 þ cx4 : ð2Þ 2 4 This equation well describes the dynamics of the amplitude of the fundamental mode of an Euler beam with simple supported ends [10–12]. When the axial load is greater than the ﬁrst EulerÕs load, the coeﬃcient a is negative while c is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ positive. In this case, the structure presents two degenerate stable states xs ¼ a=c and two homoclinic orbits connecting the unstable point x ¼ 0 to itself. Then, under the action of a sinusoidal load, the beam undergoes various types of motion including strange attractor and multiple stability around the degenerate states [10–13]. Because snapthrough instability and chaos can occur in the structure when the external parameters f0 and x vary, it is desirable to add to the structure, a controlling device. We use here an active piezo-electric controller [1,14] or a structural tendon control [2]. In this case, it can be shown that the amplitude of the fundamental mode of the beam under control is described by the following set of equations €xðtÞ þ k_xðtÞ þ axðtÞ þ cx3 ðtÞ czðtÞ ¼ f0 cosðxtÞ;

ð3aÞ

z_ ðtÞ þ azðtÞ ¼ axðt tx Þ;

ð3bÞ

where z is the control force variable, a the control speed parameter, c the control gain parameter and tx the time-delay (see Refs. [2,14]. In considering Ref. [14], we assume that the passive inductance can be neglected). In the rest of the paper we use a ¼ 1, c ¼ 0:5, k ¼ 0:2 and x ¼ 1. 2.2. Control of vibrations and snap-through instability To determine the amplitude of harmonic vibration of the system under control, we use the harmonic balance method. In this spirit, we set x ¼ A0 þ A cosðxt þ uÞ:

ð4Þ

Inserting this expression in Eq. (3) and equating the constants and the coeﬃcients of sin xt and cos xt, we ﬁnd that the amplitude A and the component A0 obey the following nonlinear algebraic equations 225 2 6 15 4 2 ac c A þ cA x þ 2a 3c þ 2 ða cos xt x sin xt Þ x x 16 2 a þ x2 ( 2 2 ) ac ac 2 2 x þ 2a 3c þ 2 ða cos xtx x sin xtx Þ þ kx þ 2 ðx cos xtx þ a sin xtx Þ f02 ¼ 0 þA a þ x2 a þ x2 ð5aÞ and A0 ¼

c a 32 cA2 c

1=2 :

ð5bÞ

To determine analytically the domain in the parameters space where the control of amplitude is eﬃcient, we proceed as follows. In the boundary of such a domain, the amplitude of the controlled system ðAc Þ is equal to that of the uncontrolled system ðAuc Þ: Ac ¼ Auc ;

ð6Þ

where Ac satisﬁes Eq. (5a) and Auc satisﬁes the same equation but with c ¼ 0. With this condition, we ﬁnd that at the boundary, both Auc and Ac equal to

B.R. Nana Nbendjo et al. / Chaos, Solitons and Fractals 18 (2003) 345–353

347

Fig. 1. Boundary criterion for the eﬀectiveness of the control of amplitude in the (c, f0 ) plane for a ¼ 1.

Ab ¼

1=2 2 ; ½nðn 2x2 4aÞ þ gðg þ 2kxÞ 15cn

ð7Þ

where n ¼ 3c g¼

ac ða cosðxtx Þ x sinðxtx ÞÞ; a2 þ x2

ac ðx sinðxtx Þ þ a sinðxtx ÞÞ: a2 þ x2

Inserting Eq. (7) in Eq. (5a), the boundary separating the domain where the control is eﬃcient (reduction of the amplitude of vibration) to the domain where it is not eﬃcient is given by f02 ¼

225 2 6 15 4 2 c Ab þ cAb ðx þ 2aÞ þ A2b ððx2 þ 2aÞ2 þ k2 x2 Þ: 16 2

ð8Þ

In Fig. 1, we have plotted this boundary in the ðc; f0 Þ plane when tx ¼ 0. This critical value of f0 increases with c. The domain located below the curve is where the control is eﬃcient in reducing the amplitude of vibration. One of the interesting phenomena which occur in a double-well system is its escape from the motion around its equilibrium state xs or xs to a large amplitude motion. Called the snap-trough instability in the context of mechanical structures, it corresponds to a change of sign of the structure curvature, thus leading to its inversion or overturning and is dangerous. It is thus of interest to know how the control parameters aﬀect the critical value of the load f0 for such a phenomena to take place. Analytically, the critical value of f0 can be computed from the fact that snap-trough instability occurs when the structure, initially at xs or xs , crosses the potential barrier at x ¼ 0. Thus from Eq. (5), the critical value f0cs is obtained by replacing in Eq. (5a), A2 by A2 ¼

2ðc aÞ : 5c

The variations of f0cs versus c and tx are reported in Fig. 2 along with the results of a direct numerical simulation of Eq. (3) using the fourth-order Runge–Kutta algorithm. f0cs is an increasing function of c and a periodic function of tx . Thus, we ﬁnd that with a good choice of time-delay, a better protection of the structure from overturning can be obtained. However for some values of time-delay, the control is aﬀected in the bad direction.

3. Control of Melnikov chaos In Ref. [10], Holmes used the Melnikov theory [15] to derive in the parameters space the condition for the occurrence of horseshoe chaos in the bistable system studied in this paper. Later on, the condition of the parametric suppression of

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(a)

(b) Fig. 2. Critical forcing amplitude f0cs for the snap-through instability as a function of c (a) with tx ¼ 0 and time-delay (b) with c ¼ 1. The squares of (a) stand for the results of the numerical simulation of Eq. (3).

this type of chaos was established in Refs. [8,9]. In fact, the Melnikov theory deﬁnes the condition for the appearance of the so-called transverse intersection points between the perturbed and the unperturbed separatrices or the appearance of the fractality on the basin of attraction. Our interest here is to ﬁnd how the parameters and the time-delay of the active control strategy aﬀect the Melnikov condition for chaos or in what range of the control parameters and time-delay, the Melnikov or Smale-horseshoe chaos could be inhibited in the bistable system. To deal with such a problem, let us express, the system of Eq. (3) in the form dU ¼ F ðU Þ þ GðU ; tÞ; dt

ð9Þ

where U ðx; v ¼ x_ ; zÞ is the state vector of the system at time t. F is the nonlinear vector ﬁeld given by F ¼ ½v; axðtÞ cx3 ðtÞ; azðtÞ þ axðt tx Þ; and GðU ; tÞ is the perturbation vector given by G ¼ ð0; kvðtÞ þ czðtÞ þ f0 cos xt; 0Þ: The hamiltonian part of Eq. (9) (the term GðU ; tÞ discarded) possesses two homoclinic orbits U0 ðtÞ connecting the unstable point x ¼ 0 of the potential to itself. These orbits are given by the following components

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1=2 pﬃﬃﬃﬃﬃﬃﬃﬃ 2a x0 ¼ sec hð atÞ; c 2 1=2 pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2a sec hð atÞ tan hð atÞ; v0 ¼

c

349

ð10aÞ

ð10bÞ

and 1=2 Z t pﬃﬃﬃﬃﬃﬃﬃ 2a expðatÞ expðasÞ sec hð aðs tx ÞÞ ds: z0 ¼ a c 1

ð10cÞ

The Melnikov function is deﬁned by Z þ1 F ½U0 ðtÞ G½U0 ðtÞ; t þ t0 dt; Mðt0 Þ ¼

ð11Þ

1

where t0 is a phase angle. Carrying out the integration, we ﬁnd that 1=2 4kðaÞ3=2 2 px þ pxf0 sec h pﬃﬃﬃﬃﬃﬃﬃ sinðxt0 Þ ckða; tx Þ Mðt0 Þ ¼ 3c c 2 a

ð12Þ

with kða; tx Þ ¼

2acðaÞ3=2 c

Z

þ1

pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ expðatÞ sec hð atÞ tan hð atÞ

1

Z

þ1

pﬃﬃﬃﬃﬃﬃﬃ expð asÞ sec h aðs tx Þ ds dt:

ð13Þ

1

Using the Melnikov criterion for the appearance of the intersection between the perturbed and the unperturbed separatrices [15], it comes that chaos is suppressed when

f0 6 f0cm

4kðaÞ3=2 ckða; tx Þ 3c ¼ 1=2 ; 2 px px sec h pﬃﬃﬃﬃﬃﬃﬃ c 2 a

ð14Þ

where the negative sign corresponds to the case of the left homoclinic orbit at x ¼ xs and the positive sign for the right homoclinic orbit at x ¼ xs . From Eq. (14), we can discuss about the eﬀects of the control parameter and that of the delay-time on f0cm . Let us ﬁrst consider the eﬀects of the control gain parameter c. f0cm is proportional to c with a positive sign for motions inside the right homoclinic orbit and negative sign for the left homoclinic orbit. Thus for motions around x ¼ xs , the fractality of the basin of attraction appears for greater values of f0 when c > 0 as compared to the case where there is no control. Thus here, control is more eﬀective when c > 0 while for c < 0, the control has a negative eﬀect since chaos appears for lower values of f0 . The opposite happens for the motion around x ¼ xs . As concerns the eﬀects of the control speed parameter a and that of the time-delay tx , they are included in the function kða; tx Þ. Thus we need the evaluation of this quantity as a and tx vary. For this aim, some mathematical transformations lead to

pﬃﬃﬃﬃﬃﬃﬃ Z

Z w 8a1 a 1 kða; tx Þ ¼ f1 ðw; aÞ f2 ðp; aÞ dp dw ð15Þ c 0 0 with a=pﬃﬃﬃﬃ a=pﬃﬃﬃﬃ a a w p ð2w 1Þ w1 p1 and f2 ðp; aÞ ¼ ; f1 ðw; aÞ ¼ ð2w2 2w þ 1Þ2 12 p2 þ ðp 1Þ2 pﬃﬃﬃﬃﬃﬃﬃ where 1 ¼ expð atx Þ. Then we can numerically compute the variations of kða; tx Þ. Fig. 3(a) shows the variation of Kða; tx Þ versus a for tx ¼ 0. It increases with a until an asymptotic constant value. This implies that for motion in the well x ¼ xs , f0cm increases with a until a limiting value. Consequently, the control becomes more and more eﬃcient when a increases until a stationary degree which can not change by increasing a further. The opposite happens for the other side of the potential. Fig. 3(b), which reports the variations of kða; tx Þ as tx varies for a ¼ 1; shows that kða; tx Þ decreases

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(a)

(b) Fig. 3. (a) Evolution of kða; tx Þ with tx ¼ 0. (b) Evolution of Kða; tx Þ with a ¼ 1.

when tx increases. Consequently when the motion is around x ¼ xs , the control becomes less and less eﬃcient when the time-delay increases. Again, the opposite happens for motions around x ¼ xs . To complement and validate the analytical predictions, we have simulated numerically the system of Eq. (3) to look for the eﬀects of the control parameters and time-delay on the onset of the fractality in the basins of attraction. Considering ﬁrst the case of the system without control, the fractality begins to appear at f0 ¼ 0:25 and becomes more and more visible as f0 increases. The corresponding value predicted by the Melnikov boundary (see Eq. (14)) is f0 ¼ 0:21. Fig. 4 shows a typical fractal basin boundary for the motion around x ¼ xs . Let us indicate that the plane (x0 , v0 ) is that of initial conditions. Now taking into account the presence of the control, we begin by considering only the eﬀects of c on the critical value f0cm (we set tx ¼ 0). Fig. 5(a) shows that with the same parameters as in Fig. 4, the fractality at f0 ¼ 0:25 disappears when the control is added to the system and this is accompanied by an enlargement of the basin of attraction. By varying f0 , the fractality reappears only when f0 P 0:67. Turning our interest on the eﬀects of the timedelay, we ﬁnd that the fractality appears more earlier. For instance, with tx ¼ 2, the fractality appears for f0 P 0:47 (see Fig. 6).

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351

Fig. 4. A fractal basin boundary diagram for the uncontrolled system with f0 ¼ 0:25.

(a)

(b)

(c) Fig. 5. Evolution of the basin of attraction as f0 increases for the controlled system with a ¼ 1, c ¼ 0:5 and tx ¼ 0 and (a) f0 ¼ 0:25, (b) f0 ¼ 0:64 and (c) f0 ¼ 0:67.

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Fig. 6. Early appearance of the fractal behavior because of time-delay (a ¼ 1, c ¼ 0:5, tx ¼ 2 and f0 ¼ 0:47).

4. Conclusion This paper has dealt with the control of vibration, snap-through instability and horseshoe chaos in a double-well Duﬃng oscillator submitted to an external additive and periodic excitation. The eﬀects of the control parameters as well as that of the time-delay between the motion of the oscillator and the action of the control have been considered. The main conclusion is that the best estimation of the optimal parameters for the eﬃciency of the control should not neglect the eﬀects of time-delay.

Acknowledgements Part of this work was done during the associateship visit of P. Woafo to Abdus Salam International Centre for Theorectical Physics. He would like to thank ICTP for the hospitality and facilities and the Sweedish Agency for Research and Cooperation for ﬁnance support.

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