Chaos, Solitons and Fractals 32 (2007) 73–79 www.elsevier.com/locate/chaos
Active control with delay of horseshoes chaos using piezoelectric absorber on a buckled beam under parametric excitation B.R. Nana Nbendjo *, P. Woafo Laboratory of Nonlinear Modelling and Simulation in Engineering and Biological Physics, Universite´ de Yaounde´ I, B.P. 812 Yaounde´, Cameroun Accepted 17 October 2005
Communicated by Prof. A. Helal
Abstract We consider the control of an undamped buckled beam, subjected to parametric excitations. Control with delay is applied to suppress chaotic vibrations. Using the Melnikov function approach, we obtain the threshold condition for the inhibition of Smale horseshoes chaos. Emphasis is laid on the effect of time delay. Ó 2005 Elsevier Ltd. All rights reserved.
The predictions and control of disturbances suffered by mechanical systems is fundamental for the design and operation of global mechanical structures. An increasing number of studies has been devoted to such a problem in recent years [1–9]. Among the control strategies, the active control plays a particular role. Another reason why active control has been receiving an increasing amount of attention, has to do with the rapid advances that have taken place in allied technologies. In this paper we are interested by the study of the global bifurcation before and after loss of stability [10,11]. Since these bifurcations can be detected analytically, it is important to obtain the condition for theoretical prevention of chaotic dynamics. In order to derive the condition for the appearance of chaos, we use the Melnikov method. It involves transverse intersections of stable and unstable manifolds that represent the starting point for a subsequent route to a chaotic dynamics. This implies the existence of fractal basin boundaries, thus the so-called horseshoes structures of chaos. The model considered here is an undamped buckled beam with hinged ends (see Fig. 1). The partial differential equation governing such beams, obtained by standard methods, can be written in non-dimensional as Z 1 €¼0 w0000 þ Cw00 k wðeÞ de w00 þ dw_ þ w ð1Þ 0
*
Corresponding author. E-mail addresses:
[email protected] (B.R. Nana Nbendjo),
[email protected] (P. Woafo).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.070
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B.R. Nana Nbendjo, P. Woafo / Chaos, Solitons and Fractals 32 (2007) 73–79
Fig. 1. A buckled beam.
where w = w(z, t) is the lateral deflexion, a prime denote differentiation by z and a dot differentiation by s. d represents the viscous damping, k represents the nonlinear stiffness and C is the axial compressible load applied to the beam. The nonlinear term expresses the fact that the axial force in the beam increases with lateral deflexion, leading to increased restoring forces. We assume simply supported boundaries (w = w00 = 0 at z = 0, 1) and that C > p2, so that the beam takes up a stable buckled state at rest. Carrying out the conventional Galerkin averaging, we obtain a set of n second order ordinary differential equations coupled in the terms only. Assuming n = 1, we obtain the following ordinary differential equation as the non-dimensionless equation of motion for the first mode of the beam (see [12,13]): 1 €q þ kq_ þ p2 ðp2 CÞq þ kp4 q3 ¼ 0 2
ð2Þ
Holmes and Marsden [14] studied the buckled beam subjected to linear damping and periodic transverse forcing . They presented a Melnikov-type technique for a class of infinite-dimensional systems and gave a criterion under which the Smale horseshoes chaos appears. Such chaotic vibrations in related experimental models were also found by Tseng and Dugundji [15] and Moon and Holmes [16]. The forced and damped buckled beam is thus one of the earliest example of infinite-dimensional and mechanical systems exhibiting chaotic dynamics. Assuming that the axial compressive force is given by C ¼ C0 ð1 þ f0 cosðXtÞÞ
ð3Þ
We obtain the following equation: €q þ kq_ þ að1 þ f1 cosðXtÞÞq þ cq3 ¼ 0
ð4Þ
with a ¼ p2 ðp2 CÞ;
f1 ¼
C0 f0 ; C0 p2
1 c ¼ kp4 2
The unperturbed system has three fixed points. An hyperbolic fixed point at q = 0 and two elliptic fixed points at qffiffiffiffiffiffi q ¼ 2a . It also possesses a separatrix solution [5,17] c rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2a sechð atÞ q0 ¼ c ð5Þ rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2a p0 ¼ sechð atÞ tanhð atÞ c known as the separatrix orbit, passing through the hyperbolic fixed points (Fig. 2). In order to determine a criterion for the occurrence of transverse intersections of the stable and unstable manifolds of the corresponding Poincare´ map, we compute the Melnikov function which in this case takes the form pffiffiffiffiffiffiffi 4ka a paf 1 X2 Xp þ cosech pffiffiffiffiffiffiffi sinðXt0 Þ ð6Þ Mðt0 Þ ¼ 3c c 2 a
B.R. Nana Nbendjo, P. Woafo / Chaos, Solitons and Fractals 32 (2007) 73–79
75
1 0.8 0.6 0.4 0.2 P0
0
-0.2 -0.4 -0.6 -0.8 -1 -2
-1.5
-1
-0.5
0 q0
0.5
1
1.5
2
Fig. 2. Homoclinic orbits for a = 1 and c = 0.5.
Knowing that Melnikov distance M(t0) at time t0, and checking if M(t0) changes sign for some t0, one can easily write the criterion for the occurrence of Smale horseshoes chaos as pffiffiffiffiffiffiffi 4k a ð7Þ f1 P fc ¼ 3pX2 cosech 2pXpffiffiffiffi a To control the vibration of such structure one can use the configuration as presented in Fig. 3. This configuration integrates piezoelectric materials with an active voltage source and a passive resistance shunting circuit. Depending on the direction taken by the structure in motion one sensor and one actuator will participate in the movement. On one hand structural vibration energy can be transferred an dissipated in the tuned shunting circuit passively. On the other hand the control voltage will drive the piezo-layer, through the circuit, and actively suppress vibration of the host structure [18,19]. In general sensors provide information to the controller to determine the performance of the control system and actuators are used to apply control signals to the system in order to change the system response in the required manner. The governing equation of the controlled system with forcing regarded as a perturbation from an autonomous system is given in the first mode of vibration by (see [20]) €q þ kq_ þ að1 þ f1 cosðXtÞÞq þ cq3 bz ¼ 0
ð8Þ
where z is the control force. This control force can be obtained by solving the characteristic equation of the control given by _ tx_ Þ z_ þ az ¼ aqðt
ð9Þ
where a, b are the control gain parameters and tx_ the time delay between the detection of the vibration and the restoring action of the controller. The controlled system can also be realized experimentally by means of sandwich model [21]. To evaluate the Melnikov distance in the case of controlled system, we compute the corresponding value of control force z in the case of unperturbed system. For that using Eq. (5) and solving Eq. (9), we obtain
Γ
Sensor
Sensor
Controller
Controller Rigid arm Actuator
Actuator
Voltage source
Rr
Γ Fig. 3. Buckled beam under piezoelectric absorber.
Voltage source
Rr
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rffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2a2 at t e sech aðs tx_ Þ tanh aðs tx_ Þeas ds c 1
z0 ¼ a
ð10Þ
Therefore, in the presence of controller system, the Melnikov function becomes pffiffiffiffiffiffiffi 4ka a paf 0 X2 Xp þ cosech pffiffiffiffiffiffiffi sinðXt0 Þ þ bkða; tx_ Þ Mðt0 Þ ¼ 3c c 2 a with kða; tx_ Þ ¼
2aa2 c
Z
þ1
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi sechð atÞ tanhð atÞeat
Z
1
t
ð11Þ
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi sech aðs tx_ Þ tanh aðs tx_ Þeas ds dt
ð12Þ
1
1 0.8 0.6 0.4 0.2 P0 0 -0.2 -0.4 -0.6 -0.8 -1 -2
(a)
-1.5
-1
-0.5
0
0.5
1
1.5
2
1.5
(b) 1 0.5 P0 0 -0.5 -1 -2.5
-2
-1.5
-1
-0.5
0 q0
0.5
1
1.5
2
Fig. 4. Diagram showing the evolution of the transversality phenomenon for X = 1 and k = 0.2: (a) f1 = 0.20, (b) f1 = 0.21.
0.45 0.4 0.35 0.3 Fe
0.25 0.2 0.15 0.1 0.05 0 -2
-1.8 -1.6 -1.4 -1.2 -1
-0.8 -1.6 -0.4 -0.2
0
Fig. 5. Evolution of critical force as function of the control gain parameters b for the appearance of Melnikov chaos.
B.R. Nana Nbendjo, P. Woafo / Chaos, Solitons and Fractals 32 (2007) 73–79
Thus chaos appears if pffiffiffiffiffiffiffi 4ka a þ 3cbkða; tx_ Þ f1 P 3paX2 cosech 2pXpffiffiffiffi a
77
ð13Þ
0.068 0.0675 0.067 0.0665 Fe 0.066
0.0655 0.065 0.0645 0.064
0
0.1
0.2
0.3
0.4
0.5
0.6
lx
Fig. 6. Evolution of critical forcing as function of time delay for the appearance of Melnikov chaos for b = 0.1.
2.5
P0
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
P0
-0.5
0 -0.5
-1
-1
-1.5
-1.5
-2 -2.5 -2.5 -2 -1.5 -1 -0.5 0 (a) q
-2 0.5 1 1.5 2 2.5
-2.5 -2.5 -2 -1.5 -1 -0.5 0 q (b)
2.5
2.5
2
2
1.5
1.5
1
1
0.5
P0
0.5
0
P0
-0.5
0 -0.5
-1
-1
-1.5
-1.5
-2 -2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 q (c)
0.5 1 1.5 2 2.5
-2 2 2.5
-2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 (d) q
Fig. 7. Effects of b on the evolution of the fractality of the basin of attractions for X = 1: (a) f0 = 0.15, b = 0; (b) f0 = 0.19, b = 0; (c) f0 = 0.22, b = 0; (d) f0 = 0.22, b = 0.25.
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First considering the system without control and the Melnikov frontier obtained from Eq. (7), setting ourself at X = 1 and considering the following value of dimensionless parameters a = 1, k = 0.2 and c = 0.5, the chaotic behavior is present if the characteristic of the external excitation are greater than fc = 0.195. Some singularities appear due to the zeros of the sine function in the Melnikov function. Fig. 4 presents the transversality phenomenon: when f1 = 0.20 (Fig. 4a) steady states oscillations inside the homoclinic loop are observed whereas for f1 = 0.21 (Fig. 4b), the system after transitory period crosses the first homoclinic orbit to the second orbit. The initial conditions for the numerical simulation of the differential equations are q0 = 1 and p0 = 0.25. Considering now the effects of the control without delay (a, b 5 0 and tx_ ¼ 0) Fig. 5 shows the variations of fc as function of the control parameter b for a = 1, as b decreases the control becomes more and more efficient since chaos appears for larger value of f1 as compared to the case without control. Taking into account the delay, we have plotted in Fig. 6 the evolution of fc as the parameter tx_ varies for b = 0.1 and a = 1. For these sets of parameters used, the control becomes less and less efficient as tx_ increases. Fig. 7 presents the effects of the parameters b and tx_ on the evolution of fractality of the basin of attraction. When f1 < fc (Fig. 7a) the basin of attraction presents a regular shape. As f1 increases irregularity progressively appears at the boundaries of the basin (Fig. 7b). The Melnikov prediction from Eq. (7) gives fc = 0.195 while that obtained from the numerical simulation is fc = 0.19. Let us now, consider that the parameters of the external excitation are given by fc = 0.22 and x = 1 (case of a chaotic motion for the uncontrolled system), it appears that as b decreases, the fractality reduces to some irregularities on the basin boundaries, and (Fig. 7c) disappears totally for b = 0.25. Turning our interest on the effects of time delay, Fig. 6 permits us to conclude that the fractality will appear more earlier. This implies that although the system is under control, one should take into account the inevitable time delay which can be the source of chaotic behavior in our system. Finally, by comparing the critical forcing for the appearance of horseshoes chaos one notes that active controlled strategy can be used to increase the robustness of the structures and in the realization of the control design one should take into account the effects of inevitable time delay which can be the source of instability in the control process.
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