Voltage-Controlled Synthetic Inductors for Resonant Piezoelectric Shunt Damping: a Comparative Analysis

applied sciences

Article Voltage-Controlled Synthetic Inductors for Resonant Piezoelectric Shunt Damping: A Comparative Analysis

Marian Vatavu 1, Vasile Nastasescu 1, Flaviu Turcu 2 and Ioan Burda 2,*

1 Weapons Engineering Systems and Mechatronic Department, Military Technical Academy, 050141 Bucharest, Romania; [email protected] (M.V.); [email protected] (V.N.) 2 Physics Department, Babes-Bolyai University, 400084 Cluj-Napoca, Romania; [email protected] * Correspondence: [email protected]

Received: 15 October 2019; Accepted: 5 November 2019; Published: 8 November 2019

Abstract: In this paper, the design, simulations, and experimental results related to new analog circuits for voltage controlled synthetic inductors (VCSI) are presented. The new circuits based on a generalized impedance converter (GIC) are proposed for adaptive resonant piezoelectric shunt damping. The VCSIs are implemented using (1) an analog multiplier and (2) an operational transconductance amplifier (OTA) as voltage-controlled resistor. The simulation and experimental results for the new proposed VCSIs are presented and a comparative analysis follows. The proposed VCSIs work in a stable manner in parallel with negative impedance converters (NIC) to enhance structural damping in resonant piezoelectric resistive-inductive shunt applications. The behavior of the synthetic inductor is identical to a real inductor only in a specific frequency range and this situation can explain the reported spreading performance in the literature for resonant piezoelectric shunt damping. The simulation results are validated by a group of experimental investigations that confirm the improved stability and linearity of the new circuits proposed as VCSIs. Experimental results show that the VCSI based on an analog multiplier have an enhanced linearity in comparison with the OTA version in a limited voltage control range.

Keywords: synthetic inductor; voltage control; resonant piezoelectric damping; analog multiplier; operational transconductance ampliﬁer

1. Introduction In structural vibration control, piezoelectric materials [1,2] have been used with regularity for a few decades. The ceramic perovskite (Pb[ZrxTi1 x]O3, 0 x 1) is a piezoelectric material transducer − ≤ ≤ called a PZT patch, which is bonded onto a mechanical structure to obtain a very eﬃcient passive damping technique using an external resonant resistive-inductive circuit known as an RL shunt. Two versions of the parallel and series RL shunt circuits are presented in Yamada et al. [3]. The series and parallel resonant shunts are equivalent in terms of vibration control performance, but the parallel RL shunt is less sensitive to the mistuning of the optimal resistance value. A synthetic inductor is based on a generalized impedance converter (GIC), which is well-known as a gyrator [4]. The gyrator was used instead of a real inductor in the analog circuits era [5] to implement low frequency RC ﬁlters. The drawbacks of a real inductor are related to the physical dimensions, residual resistance, and electromagnetic interferences. Synthetic inductors are also used for mechatronic applications [6–8], as in the case of resonant piezoelectric RL shunt vibration damping. The most performant and frequently used circuits to implement synthetic inductors in mechatronic applications are Riordan [9] and Antoniou [10] gyrator circuits. The RL shunt damping technique is very sensitive to resonance frequency mistuning, and adaptive circuits [9,10] were proposed based on an additional piezoelectric sensor and a voltage-controlled synthetic inductor (VCSI). The analog VCSIs presented in the literature

Appl. Sci. 2019, 9, 4777; doi:10.3390/app9224777 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 10

Appl.sensor Sci. and2019, 9 a, 4777voltage-controlled synthetic inductor (VCSI). The analog VCSIs presented in2 ofthe 10 literature are based on a Junction Field Effect Transistor (JFET) [11] or photoresistive opto-isolator are[12] based used as on a a voltage Junction-controlled Field Effect resistor Transistor (VCR). (JFET) These [11 circuits] or photoresistive controlled by opto-isolator an external [voltage12] used have as a voltage-controlledthe advantages of a resistor wide range (VCR). of Theseresistance circuits values controlled but unfortunately by an external have voltage a nonlinear have the response. advantages The ofvactrol a wide (photoresistive range of resistance opto-isolator) values has but an unfortunatelyextended response have time, a nonlinear a significant response. memory The effect vactrol, and (photoresistiveretains the advantage opto-isolator) of optical has isolation. an extended Considering response time,digital a significantelectronics, memory we can obtain effect, andequivalen retainst thebehavior advantage with ofprevious optical isolation.analog VCSIs Considering versions digital by using electronics, a digital we potentiometer can obtain equivalent [13] as a behaviorVCR. To withavoid previous constrain analoging the VCSIsabovementioned versions by VCS usingI based a digital on analog potentiometer circuits presented [13] as ain VCR. literature, To avoid two constrainingnew approaches the have abovementioned been used to VCSI implement based onVCR analogs based circuits on (1) presented an analog in multiplier literature, and two (2) new an approachesoperational havetransconductance been used to implementamplifier (OTA). VCRs based The best on (1)achievement an analog multiplier of this paper and is (2) related an operational with the transconductancelinear response of amplifier the resonant (OTA). frequency The best achievementto the voltage of control this paper in isresonant related withpiezoelectric the linear RL response shunt ofvibration the resonant damping. frequency to the voltage control in resonant piezoelectric RL shunt vibration damping. This paper is organized as follows:follows: In Section 22,, thethe classicalclassical AntoniouAntoniou [[10]10] synthetic inductor circuit design design and and simulation simulation results results are presented. are presented. In Section In Section3, two new 3, two circuits new for circuits a voltage-controlled for a voltage- syntheticcontrolled inductor synthetic are inductor simulated. are The simulated. experimental The results, experimental comparative results, analysis, comparative and mistuning analysis, e ffandect ofmi thestuning VCSI effect non-linearity of the VCSI in RL non shunt-linearity vibration in RL damping shunt vibration are presented damping in Sectionare presented4. Finally, in Section 54. concludesFinally, Section the paper. 5 concludes the paper.

2. Synthetic Inductors Circuit Design In 1948, Bernard D. H. Tellegen [[4]4] proposedproposed aa two-porttwo-port passivepassive electricalelectrical network element as a kind of transformer in which the primary voltage is converted to a secondary current current,, and vice versa versa,, as it is shown in Figure1 1.. HeHe calledcalled thisthis devicedevice aa gyratorgyrator becausebecause ofof aa theoreticaltheoretical analogyanalogy withwith aa ﬂywheelflywheel (gyroscope).

Figure 1.1. Gyrator: two two-port-port passive electrical network element.

Currently, the gyrator is mostly known as an active electronic circuit [9,10], which in a limited Currently, the gyrator is mostly known as an active electronic circuit [9,10], which in a limited range of voltage, current, and frequency, works as an ideal gyrator based on a transconductance model range of voltage, current, and frequency, works as an ideal gyrator based on a transconductance given by the following equation: model given by the following equation:

v v v v jωC C 1= =1 = 1 = 1 = = = , (1) Z = = = = = =jω = jωLeq, (1) i1 g2v2 g2i2Zc g2 g1v1Zc g1 g2 g1 g2 where v is the voltage source, g1 and g2 are transconductances and C is the capacitive load. whereFirstv is, thethe voltagesynthesis source, withg active1 and g elements2 are transconductances of the gyrator and to implementC is the capacitive efficient load. RC filters [5] happenedFirst, many the synthesis years later with as activea result elements of the analog of the circuit’s gyrator innovation to implement period. effi Scientignificant RCfilters progress [5] happenedwas added manyby operational years later amplifier as a result (op of-amp) the analogintegrated circuit’s circuits innovation used in Riordan period. [9] Significant or Antoniou progress’s [10] wasconfiguration added by operationalthat ensures amplifier a stable (op-amp)performance integrated and a high circuits-quality used infactor Riordan of the [9 ]synthetic or Antoniou’s inductor. [10] configurationAfter some significant that ensures applications a stable in performance analog processing and a high-qualityof audio signals, factor a recent of the useful synthetic technolog inductor.ical Aftersupport some was significant offered by applications gyrators in resonant in analog piezoelectric processing ofRL audio shunt signals, vibration a recent damping. useful To technological demonstrate supportthe real performance was offered byof gyratorssynthetic in inductor resonant based piezoelectric on an Antoniou RL shunt op vibration-amp circuit damping., the PSPICE To demonstrate® simulator ® the(version real performance 9.1, OrCAD of®, syntheticCadence Design inductor Systems, based on San an AntoniouJose, CA, op-amp95134, USA circuit,) has the been PSPICE used, simulatoras shown ® (versionin Figure 9.1, 2a. OrCADTo satisfy, the Cadence real experimental Design Systems, condition, San Jose, the CA, high 95134, voltage USA) FET has-input been op used,-amps as (OPA445) shown in Figure2a. To satisfy the real experimental condition, the high voltage FET-input op-amps (OPA445) are Appl.Appl. Sci. 2019Sci. 2019, 9, 4777, 9, x FOR PEER REVIEW 3 of 103 of 10

are used in a parametric (R5 = Rval) circuit simulation. The circuit impedance simulation results shown usedin in Figure a parametric 2b confirm (R5 = theR vallimited) circuit bandwidth simulation. characteristic The circuit of impedance the synthetic simulation inductor results with a shown self- in Figureresonance2b conﬁrm peak the in limited perfect bandwidth agreement characteristic with the behavior of the of synthetic the real inductor inductor with without a self-resonance residual peakresistance. in perfect For agreement the low frequency with the range, behavior the synthetic of the real inductor inductor is a withoutvery reasonable residual solutio resistance.n for very For the lowlow frequency mass-volume range, applications the synthetic in dimensional inductor is agreement a very reasonable with a PZT solution patch used for as very smart low materials mass-volume in applicationsvibration indamping. dimensional As is agreementexpected, the with large a PZTfrequency patch range used ascan smart be easily materials controlled in vibration by changing damping. the value of the R5 resistor based on the following equation (symbols are defined in reference to As is expected, the large frequency range can be easily controlled by changing the value of the R5 Figure 2): resistor based on the following equation (symbols are deﬁned in reference to Figure2): = (2) R1R3R5 L = Cg (2) In conclusion, the Antoniou circuit can be easRil2y adapted based on Equation (2) for numerous real applications by changing the value of only two components Cg and R5.

(a)

(b)

FigureFigure 2. 2.Synthetic Synthetic inductor: inductor: (a) ( a Antoniou) Antoniou gyrator gyrator implemented implemented with an with OPA445 an OPA445 Op-Amp Op-Amp.. (b) Impedance simulation results for Rval = {500 kΩ, 200 kΩ, 100 kΩ, 50 kΩ, 20 kΩ, 10 kΩ, 5 kΩ, 2 kΩ, 1 (b) Impedance simulation results for Rval = {500 kΩ, 200 kΩ, 100 kΩ, 50 kΩ, 20 kΩ, 10 kΩ, 5 kΩ, 2 kΩ, 1 kΩkΩ}.}.

In conclusion,In vibration the damping Antoniou experiments circuit can with be easily an RL adapted shunt to based ensure on Equationvery stable (2) results for numerous for the real Antoniou gyrator circuit, the best choice is R1 = R2 = R3 = R in the range of 1 kΩ to 10 kΩ. Considering applications by changing the value of only two components Cg and R5. the previous notation, the Equation (2) can be rewritten in simplified form as L = R∙R5∙Cg. The stability In vibration damping experiments with an RL shunt to ensure very stable results for the Antoniou of the Antoniou gyrator circuit has been experimentally demonstrated [14] by using it in parallel with gyratornegative circuit, impedance the best converters choice is (NIC).R1 = R2 = R3 = R in the range of 1 kΩ to 10 kΩ. Considering the previous notation, the Equation (2) can be rewritten in simpliﬁed form as L = R R Cg. The stability of · 5· the Antoniou3. Voltage Controlled gyrator circuit Synthetic has been Inductors experimentally demonstrated [14] by using it in parallel with negativeThe impedance robustness converters of RL shunt (NIC).s used in resonant piezoelectric vibration damping is improved by using adaptive circuits based on the phase shift between voltages at the electrodes of two PZT 3. Voltage Controlled Synthetic Inductors The robustness of RL shunts used in resonant piezoelectric vibration damping is improved by using adaptive circuits based on the phase shift between voltages at the electrodes of two PZT patches. Appl. Sci. 2019, 9, 4777 4 of 10

Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 10 One PZT patch used an actuator is shunted with an RL circuit and a second PZT patch is used as a referencepatches. strain One sensorPZT patch [12 ].used To a developn actuator a VCSIis shunted with with a linear an RL response circuit and in frequencya second PZT of thepatch RL is shunt circuitused for as a a large reference range strain is challenging, sensor [12]. butTo develop it is an importanta VCSI with step a linear to implementing response in frequency simplified of adaptivethe RL shunt circuit for a large range is challenging, but it is an important step to implementing simplified applications for resonant piezoelectric shunt damping. adaptive applications for resonant piezoelectric shunt damping. 3.1. Voltage-Controlled Syntetic Inductor Based on an Analog Multiplier 3.1. Voltage-Controlled Syntetic Inductor Based on an Analog Multiplier TheThe VCSI VCSI based based on anon AD633an AD633 analog analog multiplier multiplier used used as a as VCR a VCR is shown is shown in Figure in Figure3a. The 3a. behaviorThe of thebehavior analog of multiplier the analog in multiplier the VCR in configuration the VCR configuration [15] is described [15] is described by the equation by the equation (symbols (symbols are defined in referenceare defined to in Figure reference3): to Figure 3): 10Rm RVCR = 10 (3) = 10 VCTRL (3) 10 −− with the constant value “10” expressed in volts. with the constant value “10” expressed in volts.

(a)

(b)

FigureFigure 3. Voltage-controlled3. Voltage-controlledsynthetic synthetic inductor:inductor: (a ()a )based based on on AD633 AD633 analog analog multiplier multiplier,, and ( andb) the (b ) the resonantresonant frequency frequency simulation simulation results results forfor voltage control control (V (VCTRL) from) from −10 V10 to V10 toV 10with V 1 with V step. 1 V step. CTRL −

Considering Equations (2) and (3), for R1 = R2 = R3 = R, we obtained: Considering Equations (2) and (3), for R1 = R2 = R3 = R, we obtained: 10 = (4) 10 10− Rm L = RCg (4) 10 VCTRL where the constant value “10” is expressed in volt−s. where the constant value “10” is expressed in volts. Appl. Sci. 2019, 9, 4777 5 of 10

In Figure3b, the frequency resonance of an RL shunt in a parametric simulation based on frequency Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 10 swapping in the Thevenin equivalent of the cantilever beam with an attached PZT patch is given [11]. The PSPICEIn Figure® simulation 3b, the results frequency for resonance VCSI based of an on RL AD633 shunt in in an a RL parametric shunt application simulation illustrate based on that the resonantfrequency frequency swapping wasin the not Thevenin linear forequivalent the entire of the voltage cantilever control beam range with froman attached10 V PZT to 10 patch V and is the − expectedgivenbehavior [11]. The was PSPICE only® simulation true for negative results forvoltage VCSI values.based on AD633 in an RL shunt application illustrate that the resonant frequency was not linear for the entire voltage control range from −10 V 3.2. Voltage-Controledto 10 V and the expected Syntetic behavior Inductor wa Baseds only on true an for OTA negative voltage values.

3.2.The Voltage OTA- LM13700Controled Syntetic was used Inductor to implement Based on an theOTA VCR stage of the VCSI, as shown in Figure4a. The behavior of the OTA conﬁgured as a VCR was described by the equation: The OTA LM13700 was used to implement the VCR stage of the VCSI, as shown in Figure 4a. The behavior of the OTA configured as a VCR was described by the equation: R8 + R4 RVCR = (5) 19.2 I+ABCR4 = (5) 19.2 1 where IABC is the ampliﬁer bias current [16] and the constant value “19.2” is expressed in−1 A− . where is the amplifier bias current [16] and the constant value “19.2” is expressed in A .

(a)

(b)

FigureFigure 4. Voltage-controlled 4. Voltage-controlled synthetic synthetic inductor: inductor: (a) based (a on) an based LM13700 on an operational LM13700 transconductance operational ampliﬁer,transconductance and (b) the resonanceamplifier, and frequency (b) the resonance simulation frequency results forsimulation the voltage results control for the ( Vvoltage) fromcontrol 10 V CTRL − to 10(V CTRL with) from 1 V − steps.10 V to 10 V with 1 V steps.

Considering Equations (2) and (5), for R1 = R2 = R3 = R, we obtained the equivalent inductance: Appl. Sci. 2019, 9, 4777 6 of 10

Appl. Sci.Considering 2019, 9, x FOR Equations PEER REVIEW (2) and (5), for R1 = R2 = R3 = R, we obtained the equivalent inductance:6 of 10

R8 + R4 L = + RCg (6) =19.2 IABCR4 (6) 19.2 1 wherewhere thethe constantconstant valuevalue “19.2”“19.2” isis expressedexpressed inin AA−−1.. InIn FigureFigure4 4b,b, the the frequency frequency resonance resonance of of the the RL RL shunt shunt in in an an equivalent equivalent simulated simulated situation situation used used in in SectionSection 3.1 isis shown.shown. In contrastcontrast with the wholewhole voltagevoltage control range from −1100 V to ++1100 V V,, thethe linearlinear − behaviorbehavior of the resonant frequency for the RL shunt withwith a VCSI based on an LM1700 waswas observedobserved onlyonly forfor positivepositive voltagevoltage values. values. For For both both versions versions of of VCSIs, VCSIs, the the PSPICE PSPICE® ®simulated simulated results results showed showed a lineara linear response response for for the the resonance resonance frequency frequency of of the the RL RL shunt shunt only only in in a limiteda limited voltage voltage control control range. range.

4.4. ExperimentalExperimental Results and Comparative Analysis ® ForFor this study, study, an an equivalent equivalent experimental experimental setup setup with with PSPICE PSPICE® simulationssimulations was was used. used. The Themethodology methodology was wasbased based on a on network a network analyzer analyzer to measure to measure the theresonan resonantt frequency frequency in equivalent in equivalent RL RLshunt shunt damping damping [12] [12 experiments.] experiments. Based Based on onEquations Equations (4) (4)and and (6) (6),, the the value value of ofthe the VCSI VCSI inductance inductance is is a function of RC , and in practice, can be easily modified using a new value for the capacitor. Some a function of , gand in practice, can be easily modified using a new value for the capacitor. Some R R R R adjustmentsadjustments are also also pos possiblesible using using new new values values for for resistors resistors R1, R1,2, and2, and R3, or3 ,only or only R3 to ensure3 to ensure very verysimple simple tuning tuning facilities. facilities. The static The static capacitance capacitance of the of PZT the PZTpatch patch and andVCSI VCSI value value of inductance of inductance are are responsible for the resonance frequency of the electrical circuit ωe = 1/ √LCPZT and the static responsible for the resonance frequency of the electrical circuit = 1/ and the static capacitancecapacitance of the PZT patch is thethe firstfirst elementelement thatthat hashas toto bebe identifiedidentified beforebefore startingstarting thethe VCSIVCSI design.design. A schematic of the experimental configurationconfiguration isis shownshown inin Figure5 5.. TheThe measurementmeasurement system system ® isis built built around around a a virtual virtual vector vector network network analyzer analyzer (SpectraPLUS-SC (SpectraPLUS-SC®, versionversion 5.3.0.2,5.3.0.2, PioneerPioneer Hill Hill SoftwareSoftware LLC, Poulsbo, WA 98370,98370, USA)USA) toto investigateinvestigate aa VCSIVCSI inin parallelparallel withwith aa capacitorcapacitor withwith anan equivalentequivalent value of the staticstatic capacitancecapacitance of the PZT patch (26(26 nF)nF) usedused inin thethe RLRL shuntshunt vibrationvibration dampingdamping experiments. The virtual vector network analyzer wa wass configured configured to measure a complexcomplex transfertransfer functionfunction inin 2424 bitsbits samplingsampling formatformat withwith aa 0.7320.732 HzHz spectralspectral lineline resolution.resolution. AA whitewhite noisenoise excitationexcitation signalsignal isis generatedgenerated byby softwaresoftware usingusing aa DDSDDS (direct(direct digitaldigital synthesis)synthesis) algorithm.algorithm. TheThe USBUSB interfaceinterface used used for for data data acquisition acquisition contained contain aed performant a performant coder-decoder coder-decoder circuit (CA0187-IAQ) circuit (CA0187 boosted-IAQ) withboosted a high with impedance a high impedance and very and low very noise low amplifiers noise amplifiers for both for input both channels. input channels. The excitation The excitation signal fromsignal the from output the output of the coder-decoder of the coder-decoder circuit was circuit passed was through passed athrough power amplifiera power amplifier with a standard with a 50 Ω output impedance. An external power supply (+/ 24 V, HP-6236B) was used and voltage control standard 50 Ω output impedance. An external power− supply (+/− 24 V, HP-6236B) was used and was derived from it in a 10 V to 10 V range based on a precision voltage reference (AD584) and voltage control was derived− from it in a −10 V to 10 V range based on a precision voltage reference op-amps(AD584) and (OP27) op- inamps a voltage (OP27) follower in a voltage configuration. follower configuration.

Figure 5. Experimental setup based on virtual vector network analyzer.

TheThe simulation and experimental results for both versions of the VCSIs are shown in Figure 66a.a. ToTo ensureensure aa realisticrealistic comparisoncomparison forfor bothboth versionsversions ofof thethe VCSI,VCSI, anan equalequal rangerange forfor thethe voltagevoltage controlcontrol waswas considered.considered. For VCSI implemented with an AD633 analog multipliermultiplier asas a VCR, the linear response to keep the error less than 0.5 Hz was obtained for the voltage control from −10 V to 2 V. In the case of a VCSI implemented with OTA LM13700 as a VCR, in the condition aforementioned, the linear response was obtained for the voltage control between −2 V to 10 V. For particular implementations presented in this paper, the frequency range was 8.4 Hz for the VCSI based on LM13700 and 8.9 Hz Appl. Sci. 2019, 9, 4777 7 of 10 to keep the error less than 0.5 Hz was obtained for the voltage control from 10 V to 2 V. In the case − of a VCSI implemented with OTA LM13700 as a VCR, in the condition aforementioned, the linear response was obtained for the voltage control between 2 V to 10 V. For particular implementations Appl. Sci. 2019, 9, x FOR PEER REVIEW − 7 of 10 presented in this paper, the frequency range was 8.4 Hz for the VCSI based on LM13700 and 8.9 Hz forfor the version with AD633. It It can can be be considered considered that that both both VCSIs VCSIs cover covereded equal equal linear frequency domains for for an an equivalent equivalent volt voltageage control control range range with with a a medium medium resolution resolution of of 0.7 0.722 H Hzz/V./V. With With the experimentalexperimentallyly validated validated linearity linearity and and its its unique unique very very high high resolution, resolution, the the new new VCSIs VCSIs analyzed analyzed in this in paperthis paper outperform outperformeded the previous the previous analog analog circuits circuits [11,12] [ 11presented,12] presented up until up now until in now liter inature. literature. From analysisFrom analysis of the ofresiduals, the residuals, for the for identical the identical voltage voltage control control range range presented presented in Figure in Figure 6b, 6itb, can it can be concludedbe concluded that that the the VCSI VCSI based based on on AD633 AD633 ha hadd a abetter better linearity linearity in in comparison comparison with with the the LM13700 version. Both Both VCSIs VCSIs ha hadd an an equivalent equivalent frequency frequency do domainmain for for identical linear linear voltage control range rangess and met met the the condition condition of of having having a frequency a frequency deviation deviation less than less 0.5 than Hz 0.5 to produce Hz to produce with linear with behavior. linear behavior.From the experimentalFrom the experimental point of view, point the of VCSI view, based the VCSI on an based analog onmultiplier an analog (AD633)multiplier was (AD633) more easily was moreimplemented easily implemented using only a using resistor only mounted a resistor external mounted to the external integrated to the circuit. integrated circuit.

(a)

(b)

FigureFigure 6. 6. Voltage-controlledVoltage-controlled synthetic synthetic inductor: inductor: (a) (simulation,a) simulation, experimental experimental results, results and, linear and ﬁtting; linear fitting;and (b )and comparative (b) comparative residuals residuals analysis analysis for equal for voltage equal volta controlge control ranges. ranges.

Considering a mechanical structure with a PZT patch attached by it and Ω as the modal vibration frequency, the maximum vibration damping with an RL shunt method can be obtained for optimal value of the resistor and equal resonant frequency of the electrical circuit, i.e., Ω = . The resonant frequency mistuning effect in the RL shunt vibration damping has been intensive investigated in References [2,6] and adaptive tuning techniques have been proposed. The VCSI with a linear response for the RL shunt resonant frequency can be considered a big step forward as Appl. Sci. 2019, 9, 4777 8 of 10

Considering a mechanical structure with a PZT patch attached by it and Ωn as the modal vibration frequency, the maximum vibration damping with an RL shunt method can be obtained for optimal value of the resistor and equal resonant frequency of the electrical circuit, i.e., Ωn = ωe. The resonant frequency mistuning effect in the RL shunt vibration damping has been intensive investigated in References [2,6] and adaptive tuning techniques have been proposed. The VCSI with a linear response Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 10 for the RL shunt resonant frequency can be considered a big step forward as technological support towardtechnological implementing support new toward simplified implement adaptiveing new methods. simplified By adaptive increasing methods. the robustness By increasing of the RL shuntrobustness against mistuning, of the RL shunt future against integration mistuning, of the future method integration in more of practical the method applications in more practical is ensured and theseapplications investigations is ensured are and to be these a constant investigations research are activityto be a in constant smart materials research activity and the in intelligent smart systemsmaterials community. and the intelligent systems community. To demonstrateTo demonstrate the the capabilities capabilities of of the the new new proposed proposed VCSIs in in vibration vibration damping damping applications applications basedbased on an on RL an shunt, RL shunt, Figure Figure7 shows 7 shows the simulation the simulation results results of the of mistuning the mistuning e ffect effect due to due the to resonant the frequencyresonant nonlinearity frequency ofnonlinearity the electrical of the circuit. electrical In Figure circuit7.a, In the Figure simulation 7a, the simulation results for results VCSI basedfor VCSI on an based on an analog multiplier is presented in accordance with two reference situations: an open analog multiplier is presented in accordance with two reference situations: an open circuit and perfect circuit and perfect tuning where Ω/ = 1. tuning where Ωn/ωe = 1.

35 Open circuit / = 1 30 n e / at -10V n e / at -4V n e 25 / at 2V n e

0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Frequency ratio (a)

35 Open circuit / = 1 30 n e / at -2V n e / at 4V n e 25 / at 10V n e

0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Frequency ratio (b)

FigureFigure 7. Simulations 7. Simulations of theof the mistuning mistuning e ﬀ effectect of of the the voltagevoltage controlled synthetic synthetic inductors inductors (VCSI) (VCSI) nonlinearitynonlinearity in RL in shunt RL shunt vibration vibration damping: damping: (a) VCSI(a) VCSI with wi anth an AD633 AD633 multiplier multiplier as as a VCRa VCR and and (b ()b VCSI) VCSI with an LM13700 OTA as a VCR. with an LM13700 OTA as a VCR. Three extreme points of residual analysis, as is shown in Figure 6b, were chosen for vibration damping simulation in this case: −10 V, −4 V, and 2 V. The mistuning effect of nonlinear behavior is very low with less than 1.84 dB depreciation of the vibration damping performance of the RL shunt. The simulation results for the VCSI based on an OTA are shown in Figure 7b in accordance with two

reference situations: open circuit and perfect tuning where Ω/ = 1. Based on the residual analysis, Appl. Sci. 2019, 9, 4777 9 of 10

Three extreme points of residual analysis, as is shown in Figure6b, were chosen for vibration damping simulation in this case: 10 V, 4 V, and 2 V. The mistuning effect of nonlinear behavior is − − very low with less than 1.84 dB depreciation of the vibration damping performance of the RL shunt. The simulation results for the VCSI based on an OTA are shown in Figure7b in accordance with two reference situations: open circuit and perfect tuning where Ωn/ωe = 1. Based on the residual analysis, as is shown in Figure6b, three extreme points were chosen for vibration damping simulation in this case: 2 V, 4 V, and 10 V. Nonlinear behavior regarding voltage control induced a mistuning − effect that involved a depreciation of vibration damping performance for the RL shunt of less than 1.15 dB in this case. For both VCSI circuits, the nonlinearity regarding the voltage control induced a mistuning effect in an insignificant range and the circuits can be used with a very good performance for practical applications. In a very large bandwidth for the voltage control from 10 V to 10 V, the linearity of the − VCSI version with an OTA used as a VCR was better in comparison with the VCSI based on an analog multiplier, as shown in Figure6a. The VCSI based on an analog multiplier was more practical from an experimental point of view and considering only limited voltage control range, it had a better linearity, as shown in Figure6b. New adaptive and simplified techniques based on open or closed loop strategies can be imagined in the future to improve RL shunt robustness in the presence of uncertainty under cumulated effects induced by environmental temperature, materials aging, or modifications in the geometry of the mechanical structures. The well-known difficulties in tuning the multi-modal RL shunt [6] for more than two modal vibration frequencies of mechanical structures can be avoided using VCSIs with a linear response of the RL shunt resonant frequency regarding voltage control.

5. Conclusions In this paper, new VCSIs for adaptive resonant piezoelectric shunt damping were presented. The best achievement of this study was related with a linear response to the voltage control of the resonant frequency of the RL shunt in vibration damping with piezoelectric materials. It was demonstrated that the new VCSIs had an improved linearity and perfect performances, and can be used in closed or open loop adaptive control. Validations of the proposed VCSIs were done using experimental measurements and circuits were implemented using only analog components. The experimental investigations confirmed improved stability and the linearity of new circuits proposed as VCSIs. The PSPICE® (version 9.1, OrCAD®, Cadence Design Systems, San Jose, CA, 95134, USA) simulations showed that the behavior of the synthetic inductor was like a real inductor only in a frequency range and that this limitation can affect the performance of the resonant piezoelectric shunt damping. Experimental results showed that the VCSI based on an analog multiplier had a better linearity in comparison with the OTA version. Nonlinear behavior of both VCSI regarding voltage control induced a mistuning effect that involved an expected and insignificant depreciation of vibration damping performance for the RL shunt method.

Author Contributions: Conceptualization, investigations, and writing—original draft, M.V.; supervision, V.N.; validation, F.T.; conceptualization, writing, and editing, I.B. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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