Synchronized switch damping on inductor and negative capacitance

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1 Article Synchronized switch damping on inductor and negative capacitance Journal of Intelligent Material Systems and Structures 23(18) Ó The Author(s) 2011 Reprints and permissions: sagepub.co.uk/journalspermissions.nav DOI: / X jim.sagepub.com Bilal Mokrani 1, Goncxalo Rodrigues 1, Burda Ioan 2, Renaud Bastaits 1 and Andre Preumont 1 Abstract This article presents a strategy for enhancing the performance of the synchronized switch damping on inductor technique used for the semiactive control of structural vibrations. This enhancement is achieved by adding a negative capacitance to the resonant circuit that dissipates the energy converted by a piezoelectric transducer embedded in the structure. A unidimensional spring-mass system shunted synchronously to a resonant circuit is studied analytically, and the main parameters governing the performances of the system are highlighted. Experimental results obtained with a synthetic negative capacitance demonstrate the enhancement of the performance of synchronized switch damping on inductor and confirm the parametric dependencies identified analytically. Keywords piezoelectric transducers, semiactive damping, synchronized switch damping, negative capacitance Introduction The search for higher comfort, enhanced performance, and longer fatigue life have driven research on vibration control of aerospace and automotive structures, with the privileged solutions being lightweight, compact, and autonomous. One effective way that has been pursued during the last two decades consists of integrating piezoelectric transducers on the vibrating structures for shunting the vibration energy into electrical current and dissipating it through electrical circuits (Davis and Lesieutre, 1995; Forward, 1979b; Hagood and Von Flotow, 1991). The simplest dissipative circuit consisted of a single resistor. However, resonant circuits formed of a resistor and of an inductor (RL shunt) can be tuned to the resonant frequency of the target modes to be damped, enhancing the extraction and dissipation of energy from the transducer (Dell Isola, 2004; Hollkamp, 1994; Moheimani, 2003). Despite offering a high damping, implementations of RL shunt require massive inductors when applied to low-frequency modes and show a sharp decline in effectiveness when the frequency of resonance of the structure detunes from that of the circuit, which can be due to aging, environmental changes, or different operation modes. The loss of performance due to this mistuning motivated the quest for more robust passive techniques. Clark (2000) proposed state switch damping. Based on the fact that a piezoelectric transducer is stiffer in open circuit than it is in closed circuit, damping can be induced by closing the electrodes when the structure starts moving away from its equilibrium position (x = 0) and opening them when the structure starts moving in the opposite sense (_x = 0), in an effect similar to R shunt. Alternatively, Richard et al. (1998) proposed to close the electrodes during a very short period of time, when velocity changes sign, in what became known as synchronized switch damping (SSD). The principle of SSD consists of keeping the sign of the electric charge in the piezoelectric transducer opposed to the sign of velocity, producing an effect equivalent to dry friction. Despite a higher robustness, SSD showed a lower performance than RL shunt. This led the same authors to propose to close the circuit on an inductor, which led to a new technique known as synchronized switch damping on inductor (SSDI; Richard et al., 2000). The presence of the inductor amplifies the electric charge in the piezoelectric transducer leading the way to an 1 Active Structures Laboratory, Université Libre de Bruxelles, Brussels, Belgium 2 Physics Department, Babes-Bolyai University, Cluj-Napoca, Romania Corresponding author: Bilal Mokrani, Université Libre de Bruxelles 50, Av F. D. Roosevelt Brussels 1050, Belgium. bmokrani@ulb.ac.be

2 2066 Journal of Intelligent Material Systems and Structures 23(18) ever-increasing charge being stored in the transducer and therefore higher forces opposing the motion. SSDI requires inductors that can be orders of magnitude smaller than those of shunt and is essentially insensitive to the changes of the resonant frequency of the structure. The challenge of synchronization is tackled by compact and autonomous logic circuits based on solidstate switches, that is, metal oxide semiconductor field effect transistors (MOSFETs), in such a way that it remains globally a passive technique, and some energy harvesting is performed to power the switch (e.g. see the study by Niederberger (2005) on SSD). The principle that the higher the electrical charge that is stored in the piezoelectric transducer, the higher is the resulting damping, led to several semiactive techniques being proposed. Switching the electrodes on to a voltage source (Lefeuvre et al., 2006) allows the piezoelectric transducer to be charged with a sign opposed to velocity and with an amplitude depending on the intensity of the source. The technique became known as synchronized switch damping on voltage (SSDV) source, and when the mechanical energy is dissipated, the transducer continues to recharge and inject energy into the system. An improvement to SSDV was proposed by Badel et al. (2006), which consisted of adapting the voltage source to the amplitude of vibration. However, this technique is rather complex because of the algorithm used to adapt the voltage source. More recent works on this technique were proposed by Ji et al. (2009a, 2009b). Ji et al. (2011) proposed to replace the voltage source by a negative capacitor, which was called synchronized switch damping on negative capacitance (SSD-NC). In this technique, when the circuit is closed, the negative capacitance supplies an electric charge to the transducer and increases the quantity that is stored. However, in the absence of an inductor, the high current resulting from closing the switch produces saturations of the synthetic negative capacitor. In this article, we propose to add a negative capacitor to the resonant circuit being switched, in a new technique called synchronized switch damping on inductor with negative capacitance (SSDI-NC). The resonant characteristic of the dissipative circuit allows the limitation of the current that is supplied by the negative capacitance, providing an improvement with relation to SSD-NC. Negative capacitance increases artificially the electromechanical coupling factor (De Marneffe and Preumont, 2008; Forward, 1979b). This article starts by an analytical description of the SSDI technique, which is based on a 1-degree-of-freedom system, and emphasizes the importance of the electromechanical coupling factor k 2. Despite the fact that the damping mechanism of SSDI consists of a dry friction, an equivalent damping ratio j is then derived, and its very compact dependence on the electromechanical coupling factor and the electrical damping of the shunt circuit is pointed out. This relation can be used as a design chart for predicting the damping ratio achieved by SSDI in terms of the electromechanical coupling factor of the piezoelectric transducer and the connected electrical network parameters. Its extrapolation to multimode structures can be implemented through the modal electromechanical coupling factor K 2 i. The analytical formulation followed makes explicit the increase of damping that can be achieved by canceling the inherent capacitance of the piezoelectric transducer through a synthesized negative capacitor. Since negative capacitors do not exist as passive components, they must be implemented with active circuits employing operational amplifiers. Resorting to energy harvesting will allow the recovery of an autonomous semiactive system. Finally, the article presents the setup and the respective experimental results that demonstrate the increase of damping performance offered by SSDI-NC with relation to SSDI and the consistency observed with the predicted parametric dependence. Electromechanical coupling factor Let us consider the unidimensional spring-mass system of Figure 1(a), where the mass M is supported by a linear piezoelectric actuator consisting of a stacking of n identical piezoelectric elements polarized through their thickness. The constitutive equations of the piezoelectric transducer of Figure 1(b) are V f K a = C(1 k 2 ) 1=K a nd 33 Q nd 33 C x ð1þ where V is the electrical tension between its electrodes, Q is the electrical charge stored, x is its elongation, f is the force applied at its tips, K a is the stiffness of the transducer in short circuit (i.e. V = 0), C is the electrical capacitance when no forces are applied (i.e. f = 0), and d 33 is the piezoelectric constant. Figure 1. (a) Unidimensional spring-mass system. (b) Piezoelectric linear transducer made of n identical elements.

3 Mokrani et al The electromechanical coupling coefficient k measures the capability of the transducer for converting mechanical energy into electrical energy and vice versa, and it can be expressed as k 2 = n2 d 2 33 K a C ð2þ k is also a measure of how the stiffness and the electrical capacitance of the transducer are affected by the changes in the mechanical and electrical boundary conditions: The stiffness of the piezoelectric transducer in open circuit (Q = 0) is related to that in short circuit (V = 0) by x f j Q = 0 = K a (1 k 2 ) ð3þ The capacitance of a blocked piezoelectric transducer (x = 0) is related to that of a free transducer (f = 0) by C S = Q V j x = 0 = CS = C(1 k 2 ) ð4þ Finally, the governing equations of the system are obtained by substituting f = M x in the constitutive equations of the transducer V K a 1=K = a nd 33 Q M x C(1 k 2 ) nd 33 C x ð5þ The dynamic capacitance of the piezoelectric transducer is given by (Figure 2) Q V = C(1 k2 ) Ms2 + K a (1 k 2 ) Ms 2 + K a From the resonance frequency v and the antiresonance frequency O of Q=V, one can compute the electromechanical coupling factor as k 2 = O2 v 2 1 ð6þ Note that O is the natural frequency of the mechanical system with open electrodes transducer, and v is that with short-circuited electrodes, this is true only for 1-degree-of-freedom system; for multidegree-offreedom system, the modal electromechanical coupling factor K i should be used instead of k, suchthat K 2 i = O2 i v 2 i 1 ð7þ In this case, O i is the natural frequency of mode i with open electrodes transducer, and v i corresponds to short-circuited electrodes. In practice, the electromechanical coupling factor can be estimated from the measurement of the piezoelectric transducer impedance (V =I) or from the measurement of the mechanical compliance X =F when the electrodes are open and when short-circuited. SSDI Let us now consider the piezoelectric transducer connected to a resonant circuit RL equipped with a switch (Figure 3). When the switch is closed, the voltage V and the charge Q are related through the impedance of the RL circuit, V = (Ls + R)I = (Ls + R)sQ, leading to Q + RC(1 k 2 ) _Q + LC(1 k 2 ) Q = nd 33 K a x ð8þ and, using the definitions of electrical frequency and damping, such that one gets 1 v e = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and 2j LC(1 k 2 e v e = R ) L Q + 2j e v e _Q + 1 v 2 e Q = nd 33 K a x ð9þ Figure 2. Dynamic capacitance Q=V. It is assumed that the electrical circuit is such that the electrical resonance frequency is significantly larger than the mechanical resonance, 1 typically v e.20v n (Neubauer and Wallaschek, 2008), in such a way that the displacement may be regarded as constant over one half period p=v e, when the switch is closed, and the charge Q essentially evolves as the step response of a second-order system (Figure 4). The overshoot a is related to pffiffiffiffiffiffiffi the electrical damping according to: a = e j ep= 1 j 2 e. The control strategy is the following: When the velocity is zero: _x = 0, that is, the deformation is maximal,

4 2068 Journal of Intelligent Material Systems and Structures 23(18) Figure 3. Piezoelectric linear transducer with a switched RL shunt. Figure 5. Impulse response of the system of Figure 3. (a) The displacement x and electric charge Q and (b) the voltage V (k = 0:03). For low k, x i + 1 x i. Figure 4. povershoot ffiffiffiffiffiffiffi after closing the RL shunt. a = e j ep= 1 j 2 e. The switch remains closed exactly one-half of the electrical period and then opens again. the switch is closed, and remains closed exactly one-half of the electrical period, t = p=v e, and then opens again. Assuming that the piezoelectric transducer is initially not charged and that it starts from nonzero initial conditions, the electric charge after the first switch is (Figure 5) Q 0 =(1 + a)nd 33 K a x 0 When the switch is open, the charge remains the same until the next extremum, at x 1 x 0. Repeating the switching sequence Q 1 = nd 33 K a x 1 + a(nd 33 K a x 1 Q 0 ) The first contribution in the right-hand side is the forcing term of the charge equation (9), and the second one is the overshoot associated with the difference with respect to the previous value. Combining with the previous equation and taking into account that x 1 x 0, Similarly Q 1 nd 33 K a (1 + a)(1 + a)x 1 Q 2 = nd 33 K a x 2 + a(nd 33 K a x 2 Q 1 ) Q 2 nd 33 K a (1 + a)x 2 nd 33 K a a(1 + a)(1 + a)x 1 and so on Q 2 nd 33 K a (1 + a)x 2 ½1 + a(1 + a)š Q n nd 33 K a (1 + a)x n (1 + a + a a n ) Since 0\a\1, the asymptotic value is Q n 1 + a nd 33 K a x n 1 a ð10þ Thus, the ratio between the electric charge and the vibration amplitude tends to stabilize to a constant

5 Mokrani et al where the force in the right-hand side is opposing the velocity and has a constant value during every halfcycle separating two extrema (where the switch occurs); Q is given by equation (10). The equivalent damping ratio may be evaluated by comparing the energy loss in one cycle to that of an equivalent linear viscous damper. The free response of a linear viscous damper is x = x 0 e jv nt. In one cycle, the amplitude is reduced by x 1 = x 0 e 2pj and the strain energy loss 2 is DV V(x 0 ) = V(x 0) V(x 1 ) = 1 e 4pj 4pj ð12þ V(x 0 ) Figure 6. Impulse response of the piezoelectric transducer for two values of the electrical damping j e (k = 0:1). For small values of j e, beat is observed in the damped response. Similarly, the energy loss in one cycle associated with the friction damping is the work of the (constant) friction force F 0 ; it is obtained from equations (10) and (11) DV = 4x 0 F 0 = 4 (nd 33K a ) a C(1 k 2 x 2 0 ) 1 a = 4k 2 K a 1 + a 1 k 2 x a after using equation (2) DV V(x 0 ) = 1 + a 8k2 1 a ð13þ Comparing with equation (12), one gets the equivalent viscous damping j SSDI = a k 2 ð14þ p 1 a Figure 7. Limit value of the electrical damping j e under which the beat occurs (log log scale). value that corresponds to the static response of equation (9) amplified by (1 + a)=(1 a), where a is the overshoot in the electric charge following the closure of the switch. Note that Q n has always a sign opposed to that of the velocity in the following half-cycle, which means that it works as a dry friction. According to equation (10), the friction force is maximized when a is close to 1; however, small values of electrical damping j e lead to beat (Figure 6). Figure 7 shows the lower limit of the electrical damping as a function of the electromechanical coupling factor. Equivalent damping ratio The dynamics of the mass M is governed by the second equation (5) M x + K a 1 k 2 x = nd 33K a C(1 k 2 ) Q ð11þ This result must be compared with j R = k 2 =4 for a purely resistive shunting and j RL = k=2 for a tuned RL shunt (Figure 8; Preumont, 2006). One observes that the performance of the SSDI depends on the electrical damping j e ; smaller values of j e lead to larger equivalent mechanical damping, but a lower limit exists, corresponding to the appearance of beat, which is indicated by a triangle in Figure 8. Enhancement of k 2 We have just seen that the damping ratio j added by SSDI increases linearly with k 2 (equation (14)) and, as a result of equation (2), it therefore increases for lower values of the capacitance C. In this way, the damping performance of a SSDI circuit can be enhanced by reducing the equivalent capacitance of the piezoelectric transducer, which can be achieved by adding a negative capacitor in series or in parallel with the transducer as is illustrated in Figure 9 (Forward, 1979a, 1979b). 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 10 (Philbrick Researches, Inc., 1965).

6 2070 Journal of Intelligent Material Systems and Structures 23(18) Figure 8. Equivalent damping ratio j as a function of the electromechanical coupling factor k 2. Comparison of the resistive shunting (R), inductive shunting (RL), and synchronized switch shunting SSDI, for various values of the electrical damping j e. The triangle indicates the limit value before beat. SSDI: synchronized switch damping on inductor. Figure 10. Parallel negative capacitance (from Philbrick Researches, Inc., 1965). Figure 9. (a) Negative capacitance in parallel and (b) negative capacitance in series. C9 is obtained using the circuit shown in Figure 10. Figure 11. (a) Piezoelectric transducer connected to a parallel negative capacitance C9 and (b) equivalent transducer. Parallel negative capacitance The equivalent piezoelectric transducer obtained by adding a capacitor in parallel as illustrated in Figure 11 preserves the piezoelectric coefficient of the original transducer d 33 as well as the short-circuited stiffness K a (De Marneffe and Preumont, 2008). However, the governing equations of the new system must now be written in terms of the total capacitance C I = C C9 and the total charge Q I V K a 1=K = a nd 33 Q I M x C I (1 k I2 ) nd 33 C I x ð15þ The equivalent electromechanical coupling factor k I can now be expressed in terms of that of the original system as k I2 = C C C9 k2 ð16þ This shows that by making the value of the negative capacitance C9 approach that of the free piezoelectric transducer, the electromechanical coupling factor can be increased and with it the damping provided by the SSDI circuit. However, there is a maximum limit to which C9 can be increased. For k I2.1, the open-circuit stiffness of the equivalent transducer K a =(1 k I2 ) becomes negative, and the system becomes unstable. This occurs when the absolute value of the negative capacitance reaches the blocked capacitance of the original piezoelectric transducer such that C9 = C S = C(1 k 2 ) defines the threshold of stability. Serial negative capacitance Alternatively, the negative capacitance can be integrated in series with the piezoelectric transducer as is illustrated in Figure 12. The governing equations can now be written as (De Marneffe and Preumont, 2008)

7 Mokrani et al Table 1. Equivalent properties of the piezoelectric transducer connected to a parallel or series negative capacitance. Equivalent parameter Original system Parallel NC ( C9) Series NC ( C9) Figure 12. (a) Piezoelectric transducer connected to a series negative capacitance C9 and (b) equivalent transducer. V I K I a 1=K = I M x C I (1 k I2 a nd33 I Q ) nd33 I C I x ð17þ In this configuration, the capacitance of the equivalent piezoelectric transducer is reduced to C I = as is the piezoelectric coefficient and the short-circuited stiffness 1 1 C=C9 C ð18þ d I 33 = 1 1 C=C9 d 33 ð19þ K I a = C9 C C9 C S K a ð20þ The threshold of stability is now attained when the value of negative capacitance equals that of the original piezoelectric transducer C9 = C. The equivalent parameters for the negative capacitance in parallel and in series are shown in Table 1. SSDI-NC As discussed previously, the integration of a negative capacitor in the shunt circuit allows the enhancement of the damping performance of the classical SSDI with a new technique herein called SSDI-NC. Expressing the absolute value of the negative capacitor in terms of that of the stand-alone piezoelectric transducer as C9 = bc S when in parallel, and C9 = C=b when in series, with b\1 (to keep the system stable), leads to an equivalent electromechanical coupling factor in both configurations as k I2 = k 2 1 b(1 k 2 ) which can be approximated for small k 2 as k I2 = 1 1 b k2 ð21þ C I C C C9 C 1 C=C9 d I 33 d 33 d 33 d 33 1 C=C9 k I2 k 2 k 2 1 C9=C k 2 1 C S =C9 OC stiffness K a K a K a SC stiffness 1 k 2 1 k I2 1 k 2 K a K a 1 C=C9 1 C S =C9 K a Stability condition C9\C S C9.C C S = C(1 k 2 ) is the blocked capacitance. NC: negative capacitance; OC: open circuit; SC: short circuit. The equivalent damping coefficient that can be achieved with SSDI-NC in both configurations is given by j SSDI NC = a 1 k 2 ð22þ p 1 a 1 b This result is compared with the classical R and RL shunt and the classical SSDI in Figure 13. It is clear that the curve of the damping coefficient when a serial negative capacitance is used j SSDI NC consists of the curve of j SSDI shifted up by 1=(1 b). Note that the same result is obtained when a parallel negative capacitance is connected such that C9 = 0:91C S. It must be noted that when setting the resonance frequency and the quality factor of the shunt circuit, the values of the inductance and resistance must be tuned according to the value of the equivalent piezoelectric transducer C I that are shown in Table 1. Experimental validation Experimental setup The proposed damping technique SSDI-NC was validated experimentally, with a negative capacitor in parallel with the piezoelectric transducer. The experimental setup is schematized in Figure 14. The structure consisted of a lightly damped cantilever aluminum beam, with the piezoelectric transducer integrated close to clamping for maximizing its authority over the first flexural mode. The excitation consisted of a force applied by a voice coil at the beam tip, being collocated with the velocity measurement with a laser vibrometer. The excitation was applied below 200 Hz, with a constant spectrum, which allowed the response to be dominated by the first flexural mode (v 1 = 109 Hz).

8 2072 Journal of Intelligent Material Systems and Structures 23(18) Figure 13. Equivalent damping ratio j as a function of the electromechanical coupling factor k 2. Comparison of the resistive shunting (R), inductive shunting (RL), SSDI shunting, and SSDI-NC in series (C9 = C=0:91, i.e. b = 0:91), for various values of the electrical damping j e. The same curve is applied to a negative capacitance C9 = 0:91 C S in parallel. SSDI-NC: synchronized switch damping on inductor with negative capacitance. Figure 15. Voltage and current of the SSDI in response to a sinusoidal excitation force: (a) electrical voltage in the transducer and velocity measured at the beam tip and (b) electrical current in the circuit. SSDI: synchronized switch damping on inductor. value of the inductance L must be selected accordingly. This is facilitated by the use of a variable inductor. The voltage of the piezoelectric transducer (without negative capacitance) and the current in the circuit are shown in Figure 15 when SSDI technique is used. The velocity is first filtered by a low-pass filter to remove the high-frequency modes contribution and then a phase delay is introduced by the DSP to synchronize the switch exactly at _x = 0. The negative capacitor was implemented following the schematic in Figure 10 using an operational amplifier OPA445 and variable capacitor and resistor for easily changing the value of the negative capacitance C9. Figure 14. Experimental setup. The switch algorithm was implemented with a digital signal processor (DSP) and consisted of closing the circuit when _x = 0 and keeping it closed during half of the electrical period t = p=v e. In the synchronized switch implementation, the period of the electrical circuit, t = 2p=v e,mustbean integer multiple of the sampling frequency period of the digital controller (DSP), in this case 10 ms. The Estimation of k 2 The effect of the negative capacitance on the electromechanical coupling factor of the equivalent transducer can be observed on the frequency response function (FRF) between the beam tip displacement and the applied force X =F. For this purpose, measurements were made with different boundary conditions of the electrodes: Short circuit Open circuit Open circuit with a negative capacitor in parallel

9 Mokrani et al Figure 16. Frequency response function X=F with open circuit, closed circuit, and negative parallel shunt. Figure 18. Electrical voltage with and without parallel negative capacitance (NC). Table 2. Measured parameters. Parameter Stand-alone transducer With parallel NC v 108:125 Hz 108:125 Hz O 108:64 Hz 109:094 Hz C I 51 nf 26:5nF Ki 2 0:94% 1:8% NC: negative capacitance. Figure 19. Frequency response function X=F with open electrodes, SSDI shunt, and SSDI + negative capacitance shunt. The negative capacitance is connected in parallel. SSDI: synchronized switch damping on inductor. Figure 17. Measured dynamic capacitance of the piezoelectric transducer. The FRFs are shown in Figure 16, and the respective resonance frequencies are shown in Table 2, v for short-circuit electrodes and O for open circuit. The electromechanical coupling factor for mode 1 for the stand-alone transducer and for the equivalent transducer obtained by adding the negative capacitor in parallel can then be calculated from equation (7). The capacitance of C = 51 nf of the stand-alone transducer corresponds to the static value of the measure of Q=V, Figure 17, and the capacitance of the equivalent transducer with the negative capacitance in parallel was calculated at C I = 26:5 nf. Reducing the equivalent capacitance to half resulted in doubling the equivalent k 2 consistently with equation (2). The effect of the negative capacitance on the voltage of the piezoelectric transducer is shown in Figure 18. The voltage increases when the negative capacitor is connected in parallel, this is because the apparent capacitance is reduced consistently with equation (15) when the circuit is opened (Q I = C ste ). One can observe also that higher frequency modes are more excited when the negative capacitance is used; this is because these modes are the first to be destabilized when b approaches 1.

10 2074 Journal of Intelligent Material Systems and Structures 23(18) Experimental results The FRF X=F are compared for open circuit, SSDI, and SSDI with parallel negative capacitance enhancement as shown in Figure 19. The definition of the quality factor Q factor = 1=2j allows the determination of the corresponding damping ratios as j 1 = 0:13% j SSDI = 2:37% j SSDI NC = 4:61% The doubling of the damping coefficient offered by SSDI-NC is consistent with equation (22) with a capacitance of the equivalent transducer that is half of that of the stand-alone transducer. Conclusions The SSDI technique provides structural damping with a high robustness with respect to a change of the natural frequency of the system. Adding a synthetic negative capacitor that cancels the capacitance of the piezoelectric transducer improves its damping performance. The development of the proposed technique follows an analytical formulation that compactly describes the parametric dependence of the performance of SSDIbased techniques, that is, it predicts the damping coefficient provided in terms of the electrical damping of the shunt circuit and the ratio between the synthesized negative capacitance and the capacitance of the standalone piezoelectric transducer. An experimental implementation of the SSDI-NC technique has been demonstrated. The main parameters governing the performance of the system have been highlighted. The performance enhancement offered compared to the classical SSDI confirms the theoretical predictions. The proposed technique is a candidate to be used for the damping of lightly damped aerospace structures. Notes 1. v 2 n = K a=m(1 k 2 ) 2. The strain energy is proportional to the square of the displacement. Funding This research is supported by the Walloon Region of Belgium through the Skywin Project: Health Monitoring. References Badel A, Sebald G, Guyomar D, et al. (2006) Piezoelectric vibration control by synchronized switching on adaptive voltage sources: towards wideband semi-active damping. Journal of the Acoustical Society of America 119(5): Clark WW (2000) Vibration control with state-switching piezoelectric materials. Journal of Intelligent Material Systems and Structures 11(4): Davis CL and Lesieutre GA (1995) A modal strain energy approach to the prediction of resistivity shunted piezoceramic damping. Journal of Sound and Vibration 184(1): Dell Isola F, Maurini C and Porfiri M (2004) Passive damping of beam vibrations through distributed electric networks and piezoelectric transducers: prototype design and experimental validation. Smart Materials and Structures 13: De Marneffe B and Preumont A (2008) Vibration damping with negative capacitance shunts: theory and experiment. Smart Materials and Structures 17(3), doi: / /17/3/ Forward RL (1979a) Electromechanical transducer-coupled mechanical structure with negative capacitance compensation circuit. US Patent No Forward RL (1979b) Electronic damping of vibration in optical structures. Journal of Applied Optics 18(5): Hagood NW and Von Flotow A (1991) Damping of structural vibrations with piezoelectric materials and passive electrical networks. Journal of Sound and Vibration 146(2): Hollkamp JJ (1994) Multimodal passive vibration suppression with piezoelectric materials and resonant shunts. Journal of Intelligent Material Systems and Structures 5(1): Ji H, Qiu J, Badel A, et al. (2009a) Semi-active vibration control of a composite beam by adaptive synchronized switching on voltage sources based on LMS algorithm. Journal of Intelligent Material Systems and Structures 20(8): Ji H, Qiu J, Badel A, et al. (2009b) Semi-active vibration control of a composite beam using adaptive SSDV approach. Journal of Intelligent Material Systems and Structures 20(4): Ji H, Qiu J, Cheng J, et al. (2011) Application of a negative capacitance circuit in synchronized switch damping techniques for vibration suppression. Journal of Vibration and Acoustics 133(4): Lefeuvre E, Badel A, Petit L, et al. (2006) Semi-passive piezoelectric structural damping by synchronized switching on voltage sources. Journal of Intelligent Material Systems and Structures 17(8 9): Moheimani SOR (2003) A survey of recent innovations in vibration damping and control using shunted piezoelectric transducers. IEEE Transactions on Control Systems Technology 11(4): Neubauer M and Wallaschek J (2008) Analytical and experimental investigation of the frequency ratio and switching law for piezoelectric switching techniques. Smart Materials and Structures 17(3), doi: / /17/3/ Niederberger D (2005) Smart damping materials using shunt control. PhD Thesis, Swiss Federal Institute of Technology (ETHZ), Zurich. Philbrick Researches, Inc. (1965) Application Manual for Computing Amplifiers for Modeling Measuring, Manipulating and Much Else. Boston, MA: Nimrod Press.

11 Mokrani et al Preumont A (2006) Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems. Dordrecht: Springer. Richard C, Guyomar D, Audigier D, and H. Bassaler (1999) Semi passive damping using continuous switching of a piezoelectric device. In: Proceeding of the SPIE international symposium on smart structures and materials: passive damping and isolation, Vol. 3672, 01 March 1999, San Diego, CA, pp Richard C, Guyomar D, Audigier D, and H. Bassaler (2000) Enhanced semi passive damping using continuous switching of a piezoelectric device on an inductor. In: Proceeding of the SPIE international symposium on smart structures and materials: passive damping and isolation, Vol. 3989, Newport Beach, CA, 27 April 2000, pp