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SSDI-Max Control with Negative CapacitanceBabesse E1, Belkhiat S1, Cherif A2, Meddad M2 and Eddiai A3*

AbstractIn this paper, the performance of a semi-active control of structural vibrations which is the Synchronized Switch Damping on Inductor Max called SSDI-Max technique is improved. This improvement is achieved by adding a negative capacitance to the resonant circuit that dissipates the energy converted from the structure by the piezoelectric transducer. A model of system shunted to a resonant circuit, using the SSDI-MAX control and the negative capacitance, is analyzed analytically using Matlab/SIMULINK environment. The results show that a light modification with the same used energy can highly increase damping performances.

Keywords

Semi-active control; SSDI-Max technique; Piezoelectric transducer

*Corresponding author: Eddiai A, Laboratory of Condensed Matter Physics, Faculty of Science BenM’sik, Hassan II University of Casablanca, Morocco, Tel: +212 663540999; E-mail: aeddiai@gmail.com

Received: July 21, 2016 Accepted: December 02, 2016 Published: February 01, 2017

can be mostly divided into three categories: passive, active, and semi-active or semi-passive. Passive control technique is the oldest and the simplest among the three categories because it uses only the R-L shunting [5,6]. But its control performances are very sensitive to the system parameters variations and usually need large inductance in low frequency domain, which is difficult to realize. Moreover, the passive control strategies must be tuned to a certain frequency (mode of vibration) of the structure, which has to be known in advance and which should not change during operation. Active control systems aim at imposing a force or a displacement in certain points of the system to be controlled, in function in particular of the measured state or the history of this one [7,8]. On one hand the active methods can achieve an excellent efficiency. But on the other hand, they require processors and power amplifiers to drive actuators, which are not suitable for many practical applications.

In order to compromise the limited damping performance of passive strategies and the complexity of active ones, several semi-active approaches have been proposed. Semi-active control is defined by Harari [9] as an innovative alternative to active control. It is a technique which carries out a nonlinear treatment of the voltage generated by the piezoelectric elements, without need for a great quantity of external energy (moreover it can be self-supplied [10].

The first installation of the semi-active methods was by switching the piezo-element from an open circuit to a short circuit state called later “Synchronized Switch Damping in short circuit (SSDS)” [11]. Richard [11] in proposed to short-circuit the piezoelectric element at each end of deformation (or displacement). Next time, this technique has been improved by adding an inductor in the dissipative circuit to increase the electric charge in the piezoelectric element. This strategy called ‘’ Synchronised Switch Damping on Inductor “(SSDI) was proposed by Richard et al. [12]. It is almost identical to SSDS, but instead of forcing the voltage to zero in each maximum of displacement, it is inversed. Another improvement can be achieved by connecting the piezoelectric elements to a DC voltage source (SSDV) [13], this strategy allows improving the damping performance; however, the need for an external power source is rather a drawback of its progress.

Nowadays, improving the SSDI method takes in two main areas: i) the generalization of the method for multimodal cases and ii) improving the method of damping performance such as the work done in Ji et al. [14] and Mokrani et al. [15] where the improvement has been obtained by adding a negative capacitance to the resonant circuit that dissipates the energy converted by the piezoelectric transducer. In one hand, Ji in [12] proposed a new SSD technique, called SSDNC (synchronized switch damping on negative capacitance). This uses a negative capacitance in the shunt circuit, instead of the inductance used in SSDI technique. By choosing an appropriate value of the negative capacitance, a significant transducer voltage can be obtained, thus ensuring good damping performance. In other hand, Mokrani et al. [15] improved the damping performances, by combining SSDI and negative capacitance, in a technique synchronized switch damping on inductance and negative capacitance (SSDI-NC). By adding a negative capacitance, the capacitance of the piezoelectric transducer decrease, thus improves damping.

In the first case, the SSDI method has been enhanced to control

IntroductionVibration in new structures like airplanes, satellites or bridges

can result malfunctions, fatigue damages or radiate unwanted noise. So, it has to be suppressed [1-3]. Since conventional passive damping techniques have reached their limits to damp vibration because it requires more space and weight. Furthermore, it is not very effective at low frequencies, but it can typically be optimized for one eigen form of the structure. For this, the new control designs with novel actuator systems have been proposed. Through smart materials, vibrations can be controlled and suppressed with efficient and intelligent way without causing much great additional weight or cost. Because of their natural electromechanical coupling, piezoelectric materials are promising candidates for sensors and actuators application especially in the field of the structures known as intelligent. This importance can be achieved in order to demonstrate the abilities to convert mechanical energy into electrical energy and vice versa.

In piezoelectric materials, the direct piezoelectric effect is a response of piezoelectric material to a mechanical force by generating an electric charge or voltage. However, mechanical strain is induced when an electric field is applied to the material; this phenomenon is the converse piezoelectric effect [4]. Due to their excellent electromechanical coupling characteristic and excellent frequency response, piezoelectric materials have been widely used in structural vibration control, structural health monitoring, precision positioning, energy harvesting, and so on.

Vibration control methods based on piezoelectric transducers

Citation: Babesse E, Belkhiat S, Cherif A, Meddad M, Eddiai A (2017) SSDI-Max Control with Negative Capacitance. J Comput Eng Inf Technol 6:2.

doi: 10.4172/2324-9307.1000166

• Page 2 of 6 •Volume 6 • Issue 2 • 1000166

systems subject to broadband excitations. The Modal SSDI-Max is among these methods. This modal technique was developed for the first time [9] and has been used in many papers as in Cherif et al. and Babesse et al. [16,17]. In this paper, configuration and performance of the improved SSDI-Max technique with negative capacitance inserted in parallel was investigated. Results of the proposed work was done using numeric simulation, using Matlab/SimulinkTM

environment, in the case of a cantilever beam equipped with four piezoelectric inserts.

Modeling of the intelligent structure

The electromechanical behavior of an intelligent structure (instrumented of piezoelectric elements) is given by the following equations [9,17]

,...

FVqKCM E βαδδ +−=++ (1)

,.

0

.VCI t −= δα (2)

With δ is the vector of displacement, M, C and KE are respectively the matrices of mass, damping and rigidity when the piezoelectric patches are in short-circuit. α is the electromechanical matrix of coupling, V is a voltage vector of the piezoelectric patch, I is the electrical current vector, and C0 is the diagonal matrix of patches capacitances. F is the force applied to the system.

Carrying out the change of variable according to:

,qφδ = (3)

Where ϕ is the mode matrix of the structure limited to n modes and q is the vector of modal displacement of the structure. Equations (1) and (2) become:

,FVqKqCqM E βθ +−=++ (4)

,0VCqI t

−= θ (5)

With ,.αφθ t= is the modal matrix of electromechanical coupling of size [n, i]. M, C, KE are respectively the modal matrices of the mass, damping and rigidity. The equation (2) is standardized in order to have:

( )2; 2 ( ) ( ); , = = ξ =ω ω

D EE

dM C diag diag diagI K

With ξ is the vector of modal damping, ωE is the vector of the frequency in short-circuit and ωD the vector of frequency in open circuit.

While separating the voltages of the transducers and the sensors, Is and Vs respectively, and when the sensor voltage is supervised by a voltage amplifier (whose intensity of the sensor is null). Equations (4) and (5) become:

,FVVqKqCqM ssaaE βθθ +−−=++ (6)

,00 =− ssts VCqθ (7)

By substitution of (7) in (6) one found:

( ) ,01 FVqCKqCqM aa

tsss

E βθθθ +−=

+++

− (8)

The system of linear equations (7) and (8) can be written in the form:

,,

= + = =

x A x Bu qx

y C x q (9)

[ ],aVFU = is the control vector; [ ],qqU = Vs is the output vector, A, B, C are the state matrices:

( ) ,0

110

1

−+−

= −−− CMCKMIA t

sssE

d

θθ

,0

11

−+

= −− θβ aMMB ,

00

0

10

=

− θ tss

d

d

CI

IC

Va is calculated by the following relation:

,10 qCV t

aaa θ−= (10)

C0a and C0s are the capacity matrices of transducers and sensors respectively.

The intelligent structure used in the simulation is a cantilever beam fixed on one side and equipped by four piezoelectric inserts (PZT) two of them are used actuators and the others are used as sensors (Figure 1). Dimensions and physical properties of the beam as well as the PZT transducers are given in Tables 1 and 2 respectively. The structure has been identified according to the model described previously.

Electromechanical coupling coefficient

The electromechanical coupling coefficient (EMCC) is an important parameter for the description of piezoelectric materials; it measures the conversion of electrical energy into mechanical one and vice versa.

Consider the piezoelectric linear transducer of Figure 2: it is made of n identical slices of stacked piezoelectric material, each of them polarized through the thickness. If one assumes that the stress, the strain, the electric field and the electric displacement are one-dimensional and parallel to the direction of polarization. The electromechanical coupling factor k2 can be defined by:

CKdn

Sdk a

TE

233

22332 ==ε

(11)

Figure 1: Configuration of the cantilever beam structure used for the simulation. Sizes of the piezoelectric elements are: 20 mm in width and 430 µm in thick and 0, 05 m in length for patches 1 and 2 and 0,025 m for patches 3 and 4.

Parameter Real valueLength 250 mmWidth 20 mm

Thickness 2 mmYoung modulus 69 GPaPoisson ratio 0.3

Density 2800 Kg/m3

Table 1: Characteristics of the cantilever beam used in simulation.

Citation: Babesse E, Belkhiat S, Cherif A, Meddad M, Eddiai A (2017) SSDI-Max Control with Negative Capacitance. J Comput Eng Inf Technol 6:2.

doi: 10.4172/2324-9307.1000166

• Page 3 of 6 •Volume 6 • Issue 2 • 1000166

The practical way to determine the electromechanical coupling factor, from admittance Frequency Response Function (FRF) measurements, can be given by:

2

222

zpzk −

= (12)

p is the natural frequency with short-circuited electrodes, and z is the natural frequency with open electrodes. According to Eq.11, the value of k2 is strongly limited by the electrical capacitance of the piezoelectric transducer C. In 1979, forward proposed to use an active electronic circuit that decrease artificially the capacitance of the transducer and increase in consequence k2, the so-called `negative capacitance’ shunt (Figure 3). Although Forward took a patent for his invention, he did not publish any experimental results. This concept was re-introduced in practical after more than 20 years [18,19]. Since then, the concept has been used in various experimental set-ups [20,21].

It is clear in eq.11 that the electromechanical coupling factor k2 can be increased by reducing the capacitance C of the transducer. This is the origin of the idea of adding a negative capacitor in parallel with the transducer.

From the Figure 4 the global capacitance of the transducer becomes:

* 'C C C= − (13)

Here: *C is the global capacitance, C is the transducer capacitance

and 'C is the inserted negative capacitance.

By replacing (13) in (11):

2 2 2 22 233 33*

* ' 'a an d K n d K Ck k

C C C C C= = =

− − (14)

In the equation (14) it is clear that the negative capacitance 'C

increases the electromechanical coupling factor k2, and thus, the performances of any connected shunt, including the SSDI-Max. Hence, by increasing the negative capacitance value, one increases the k2 value and as a consequence all the control performances. However, for k2>1 the system becomes unstable. This occurs when the absolute value of the negative capacitance reaches the blocked capacitance of the transducer so equation (15) defines the threshold of stability [15].

2' (1 k )C C= − (15)

Mokrani [15] resumes the equivalent parameters for the negative capacitance in parallel in the following Table 3:

From the SSDI to the SSDI-MAX technique

The SSDI control consists in using an electronic switch controlled during brief moments in a synchronous way with the vibration (Figure 5). When the voltage of the piezoelectric elements is extreme, the switch connects it to an electric circuit composed of an inductance which causes to reverse the voltage. This inversion induces a mechanical force of sign opposed at the speed, thus creating the desired damping [16].

The SSDI-Max technique is an improvement of SSDI technique by the avoidance of commutation in local maxima. It consists in delaying the switching moment to following voltage extreme after the targeted modal coordinate extreme.

Property Symbol Real valuedensity ρ 7700 Kg*m3

Compliances CC 11Es33Es

15.44×10-12 Pa-1

20.09×10-12 Pa-1

permittivityS11εS33ε

12.75 nF.m-1

7.411 nF.m-1

Piezo. coefficient 11d33d

-186 pC.N-1

425 pC.N-1

Table 2: Characteristics of the PZT piezoelectric patches.

Figure 2: Piezoelectric linear transducer.

Figure 3: Negative capacitor equivalent circuit.

Figure 4: The new configuration with negative capacitance.

Citation: Babesse E, Belkhiat S, Cherif A, Meddad M, Eddiai A (2017) SSDI-Max Control with Negative Capacitance. J Comput Eng Inf Technol 6:2.

doi: 10.4172/2324-9307.1000166

• Page 4 of 6 •Volume 6 • Issue 2 • 1000166

Hence, the algorithm of the strategy is summarized as fellow:

The window of time of limitation starts when a maximum of modal displacement appears. Thus the signs of the voltage Va and its derivative are considered during the window:

• If the voltage is positive and the derivative is negative, the switch trigger is immediate.

• If the voltage is positive and the derivative is positive, the system waits for the next maximum voltage. This delay is nevertheless limited by the time window.

• If the voltage is negative, the system waits for one of the above conditions.

• If no switching occurred and the end of the time window is reached, the switching is triggered.

This algorithm is anti-symmetric if a minimum modal displacement is reached. Some examples of inversion times, using this method, are presented in Figure 6. SSDI-Max and SSDI techniques were compared in many studies [9,16,17]. Here, we present the two methods tested with a pulse square of 50 μs for the previously described structure.

It is clearly visible, in Figures 7 and 8 that a strong improvement of damping by using SSDI-Max method is carried out.

SSDI-MAX technique with the negative capacitance

In order to enhance the limited damping performance of passive shunting, active elements have been proposed. The mostly studied element is a negative capacitance, which can be realized by a negative impedance converter circuit [22]. Initially considered by Forward [23], a negative capacitance proves to be able to increase the effective piezoelectric coupling factor.

In this work, we propose to add a negative capacitor to the resonant circuit being switched (Figure 4).

Negative capacitance increases artificially the electromechanical coupling factor. Negative capacitors do not exist as passive components; however, they can be synthesized by an active circuit employing operational amplifiers as the one shown in Figure 3.

Simulation ResultsIn this part, conventional SSDI-Max and the improved one

with negative capacitance were compared with a pulse square of 50 μs. Figure 9 shows clearly that the structure damping is improved by association of negative capacitance and SSDI- Max method. The effect of the negative capacitance on the voltage of the piezoelectric transducer is shown in Figure 10. The voltage increases when the negative capacitor is connected in parallel that is a result of the negative capacitance effect on the electromechanical coupling factor. The time window, which is used to limit the shift as possible before switching, is a very important and critical parameter.

• If it is too small, the voltage will not have the ability to increase and no significant improvement of damping will be seen.

Equivalent parameter Original system Parallel negative capacitance

Capacitance C* C C-C’Piezoelectric constant d*33 d33 d33

Electromechanical coupling factor k*2 k2 2

'C k

C C−

Stiffness on open circuit 21aK

k−2*1

aK

k−

Stiffness on short circuit aK aK

Stability condition 2' (1 k )C C= −

Table 3: Equivalent properties of the piezoelectric transducer connected to a parallel negative capacitance.

Figure 5: (a) The SSDI electrical circuit (b) waveforms for a sinusoidal excitation: aV is the piezoelectric actuator voltage and Ԑ is the voltage corresponding to the strain (c) zoomed view of the voltage inversion.

Figure 6: SSDI-Max strategy illustration: the definition of the switching moment according to the targeted modal coordinate, the piezoelectric actuator voltage in open circuit and the authorized maximum time.

Figure 7: Displacement simulation of the first mode with the impulse excitation.

Citation: Babesse E, Belkhiat S, Cherif A, Meddad M, Eddiai A (2017) SSDI-Max Control with Negative Capacitance. J Comput Eng Inf Technol 6:2.

doi: 10.4172/2324-9307.1000166

• Page 5 of 6 •Volume 6 • Issue 2 • 1000166

• If it is too long, there is a risk of losing synchronization between the actuator voltage and targeted mode.

In order to define optimal time window, simulations were made by varying the value of the window from zero (purely SSDI) then from 1/30 to 1/5 of the targeted mode period. Figure 11 illustrates the change in damping of the target mode according to the time window for the excitation pulse.

As shown in Figure 11, the time window between 1/10 and 1/5 of the mode period is optimal for SSDI-Max control, ensuring an increase in the voltage of the actuator whereas causing a larger phase shift between the transducer voltage and the targeted mode, thus resulting an additional structure damping.

ConclusionThe SSDI-Max control provides very consistent performances in

comparison with the SSDI control however this technique can be also improved using a negative capacitance.

In this work, configuration and performance of the new proposed SSDI-Max technique with negative capacitance in parallel was investigated. The validation of the proposed work was done using numeric simulation, using Matlab/ SimulinkTM environment, in the case of a cantilever beam equipped with four piezoelectric inserts. Future works target at implanting and validating experimentally the proposed work.

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Figure 8: The actuator voltage simulation with an excitation by impulse in the case of mode 1 targeted by control.

Figure 9: Cantilever displacement, using SSDI-Max (Blue) and SSDI-Max + Negative Capacitance (Red), in the pulse case.

Figure 10: Transducer voltage, using SSDI-Max (Blue) and SSDI-Max + Negative Capacitance (Red), in the pulse case.

Figure 11: Influence of the time window on the damping in the pulse case.

Citation: Babesse E, Belkhiat S, Cherif A, Meddad M, Eddiai A (2017) SSDI-Max Control with Negative Capacitance. J Comput Eng Inf Technol 6:2.

doi: 10.4172/2324-9307.1000166

• Page 6 of 6 •Volume 6 • Issue 2 • 1000166

10. Lallart M (2010) Vibration Control Sciyo Croatia.

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Author Affiliation Top

1DAC HR Laboratory, University of Sétif, Algeria 2DAC HR Laboratory Sétif, University Mohamed el Bachir el Ibrahimi BBA, Algeria 3Laboratory of Condensed Matter Physics, Faculty of Science BenM’sik, Hassan II University of Casablanca, Morocco

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