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Evaluation of piezoelectric active control

systems for vibrations suppression

G.L. Ghiringhelli & M. Zanardi

Diparimento di Ingegneria Aerospaziale

Politecnico di Milano

via C.Golgi, 40, 20133 Milano, Italy

Email: [email protected]. it

Abstract

The paper describes in progress research activities aimed at verifying thepossibility of designing active vibration control systems based on piezoelectricdevices. In particular a wholly clamped rectangular flat panel is mechanicallyexcited and controlled with piezoceramic patches. Vibrations reduction tests andthen correlations with numerical predictions, based on an integrated piezo-structural model, are presented. In this work only simple co-located direct feed-back laws have been used to point out problems related to measurementconditioning circuits. Self sensing actuators have been also tested, the theoreticalpossibilities offered by this kind of devices being evident, but their actualimplementation entails some practical problems.

Encouraging results have been produced, confirming the possibilities ofpiezoelectric active control of thin panels. The correlation with numericalprediction shows that the adopted approach is capable of supplying a suitabledesign tool.

1 Introduction

Active control of vibrations is an important tool that engineers canexploit to improve their designs. Such an approach have been recognisedto be more flexible than passively modified structures. In fact betterperformances can be usually obtained for a wide range of operatingconditions along with a significant weight saving. Piezoelectrics are moreand more adopted to realise active control systems, direct and converse

Transactions on the Built Environment vol 35, © 1998 WIT Press, www.witpress.com, ISSN 1743-3509

piezoelectric eiiects oemg expioiiaoie 10 sense anu 10 exeri me

action. A wide amount of literature is devoted to this subject and noattempt is here made to compile an exhaustive survey, but the interest

into these devices is continuously growing both for vibration and noise

reduction.In this paper some experiences gathered at the Dept. of Aerospace

Engineering of the Politecnico di Milano in testing piezoelectric controlsystems are summarised. These activities are aimed at the vibrationcontrol of a wholly clamped rectangular flat panel, mechanically excitedand controlled with piezoceramic patches. Correlations with numericalpredictions are also presented. These are based on a integrated finiteelement modeling capable of describing the dynamic behaviour of thesystem composed by a generic structure embedding piezoelectric devices(Ghiringhelli et Al.l). It allows a natural integration with actual signalconditioning circuits and other electronic devices used to realise thecontrol system. A completely analog active control system and simpledirect feedback laws have been used to point out problems related tomeasurement conditioning circuits. Self sensing actuators, the usefulnessof which is well known (Dosch et Al.%, Saunders et A1.3, Brusa et Al.tAkishita et A1.5, Anderson et Al.G), have been also tested usingappropriate bridge and conditioning circuits. The theoretical possibilitiesoffered by this kind of devices are evident but the actual realisationentails some problems. The analog choice is due to the goal of testingand tuning electronic devices to be used in actual implementation of thecontrol system: the testing of a digital active control with multi-inputs

multi-ouputs and adaptive control laws is in progress.

2 Integrated Moderation of piezoelectric

In this section the main features of an integrated modeling techniques arepresented, based on a finite element discretisation.

2.1 Piezoelectric effect

The direct and inverse piezoelectric effects, i.e. the generation of anelectric field due to mechanical strains and its ability to induce a strainwhen subjected to an electric field, are described by the linearconstitutive laws of piezoelectric materials, in IEEE standard notation

(ANSI/IEEE, 1988):

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" , ..... v^iui, to/ me siram vector, (b) the electricheld, {D} the electric displacement, [cE] the elastic matrix at constantelectric field, [e] the piezoelectric characteristic matrix ,d [eS] thedielectric matrix at constant strain.

2.2 Response of a controlled system

Starting from Equation (la,b) the electro-structural model can beobtained using the integrated finite element approach described inGhinnghelli et Al.l, the results of which are briefly recalled here Once

the structural displacement d and the electric flux <J> , i.e. the voltage timeintegral *

<t>=)Vdt or <j)=V

are assumed as primary field unknowns, they are expressed a finite

element approach in terms of the corresponding nodal values d and <ftThe dynamics equations of a linear system of this kind are:

(2)

where M, D, K are the mass, damping and stiffness matrices, d thedisplacements vector, /the loads vector, fc the nodal currents vector 6the electro-elastic coupling matrix and Cp the electric matrix.

The electric unknowns $ are naturally limited to a relatively small

number of voltages related to actuators/sensors electrodes and to pointsconnecting discrete electric components. On the contrary Eq (2) cancontain too many structural unknowns to be directly used in the design ofactive control systems. Thus it is generally necessary to take acondensation of the structural part of Equation (14). Here a modaltransformation of the structural part alone has been adopted, based on theeigenvectors matrix X of the structural problem, i.e. d = X q being q thegeneralised co-ordinate array. The previous modal condensation doesnot change the structure of Equation (2), each term simply assuming ageneralised meaning. In the second row of Equation (2) we canimmediately recognise the matrix of the sensing equation that can bederived from the piezoelectric constitutive law: the array arranging the

currents due to the piezoelectric effect is given by: ip =0^q.

The assumption of the time integral of the voltage difference as electricprimary unknown, instead of the most commonly used voltage, allows anatural formulation of the coupled electric/elastic problem in presence ofexternal circuits. In fact a complete analogy between structural

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displacements ana eiecmc nuxes is esmonsneu: iiciwui^b ui

components can be easily modelled, as properly connected generalised

lumped capacitors (masses), resistors (dampers) and inductors (springs),and a completely coupled linear problem is obtained by simplyappending the electric model to Equation (14); expanding the array of

fluxes with the values at the nodes of the external electric network we

have:

M 0

0 -

D -0(3)

where (%, GL, and R̂ are respectively the lumped symmetric

capacitance, conductivity and reluctance matrices.

2.3 Direct position feedback with co-located control

In the case of a direct position feedback with co-located control the twoelements of a pair of piezoelectrics are used to sense and actuate. If thehost structure is thin enough, the measurement on the top facecorresponds to the bottom one and if a pair of piezoelectric devices isused to sense and actuate a co-located control is realised. Being thepiezoelectric charge proportional to the strain, a charge measurement isproportional to the structural displacement. This job is accomplished by aclassic voltage follower. The circuit is a high pass filter and behaves asan ideal displacement transducer, in the frequency range of interest. The

corresponding equations are :

(4)

A velocity feedback has been attempted using a current to voltageconversion circuit to measure piezoelectric current, i.e. Eq. (2b). But thisconditioning circuit failed: high frequency instabilities occurred as thecontrol gain was increased, possibly due to piezoceramic sensorsnonlinearities (Akishita et A1.5).

2.4 The self-sensing actuator

A self-sensing actuator allows to realise an effectively co-located control.For such a device it is mandatory to remove the effects of the controlvoltage from the measurements. This task can be accomplished using ameasurement bridge circuit (see Dosch et A1.2).

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V2

Figure 1 - The bridge circuit

domain the sensing voltage is:

z±?r 7 P

v=v, -v, =•

By introducing the following hypothesis:

a\ r^ 7 /^v i \> ^P^I ~ ̂ 2 b)

the sensor equation becomes:

01 two orancnes, eacn made with two

series impedances, the first

one is cap.) itive, while thesecond is generic one. The

piezoelectric devices plays

the role of a capacitive

impedance. The differentialvoltage across points 1 and 2

leads to a measure of the

piezoelectric current andcancel the control voltageeffects. In the Laplace

(7)

(8)« 1,

Depending on the kind of the complex impedance Z, a displacement,velocity or acceleration measurement can be obtained, respectively, witha capacitor, a resistance and an inductance. Equation (8a) states thebalance condition of the bridge, from which both stability andmeasurement precision depend. The second one leads to a measurementproportional to the piezoelectric current, that is the strain time rate of thehost structure at the piezoelectric, and the impedance Z. Due todifficulties in balancing a capacitive bridge and to satisfy the (8b)condition in the case of inductances, only resistive bridges have beenexperimented, the measurement equation of which is given by:

Vs =RmIp =&, s[0y ]T{q} (10)

It must be outlined that, due to the presence of a resistance between thepiezoelectric device and the ground, the effective voltage acting on thepiezoelectric itself, is given by :

*eff~Vc~Vl (11)If a direct feedback is used, i.e. Equation 5, the effective voltage is:

%#= -#G+% -<?T/2 /= -Gy, , (12)

It approximates the nominal control voltage only if the gain G is highenough. It is possible to observe that, in absence of active control, i.e.

133


(/=(;, a control action is nonetneless present simply cue to me presence

of a loop. In fact the dynamic equation of this system is given by:

M

0

0

O D

0

0(I+G) -9G/ 0

0 / #2

0

K 0 0

0 0 0

0 0 0

The availability of self-sensing actuator allows one to use a couple of

elements as actuators, so that a bending moment is applied. It can be seenthat the sensitivity is not changed while the authority of the actuator is

increased. If the bridge is perfectly balanced the measurement is idealand no stability problems arise. If this is not the case, an asymptotically

stability is obtained if the following inequality is satisfied (Masarati &):RjCp>R2C. The actual resistors must be chosen just to satisfy thisequaton but with the lowest unbalance as possible, to reduce the effect of

the second term of Eq. (7). This holds if a direct feedback is used.The direct feedback on the bridge output, that is a velocity feedback,failed, possibly due to the same reasons that led to bad results whencurrent to voltage conversion has been applied to a piezoelectric sensor.The technique described by Fanson̂ has been attempted: the sensoroutput is conditioned by a second order low-pass filter with highresonance factor so that the signal is amplified at design frequency whilethe higher harmonic content is filtered. The implementation of the moresophisticated adaptive "sensoriactuator" proposed by Cole and Clark̂ isin progress.

3 Experimental set-up

External Excitation

°

Piezoelectric Patches

Figure 2 - Test structure set-up

The structure under test is aflat panel (600x400x2mm),made of a standard 2027Aluminium Alloy, clampedon each side to a steel frame(Fig. 2). It is excited by anelectro-mechanical shaker(B&K Type 4810) and tolimit the dynamic response ofa 450Hz frame torsional

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. nigni pairs of oppositepiezeceramic patches (PZ21A by Ferroperm 50x50x0.5m) have been

p̂ ced near the maximum curvature area of the modes to be controlledThey have been chosen between low frequency modes with an odd

number of half waves along the short side, due to the previouslymentioned reason. ^ ^

4 Numerical and test results

4.1 Position feedback with co-located sensors and actuators

These results are related tothe direct position feedbackcontrol. Figures 3 and 4

report the comparisonbetween the controlled anduncontrolled frequencyresponse functions of the

panel for the experimentaland numerical casesrespectively. They refer to

the measurement of an

accelerometer. The activelycontrolled response is black,while the uncontrolled one isgrey and dashed. Results foractive control with a singleand three piezoelectricdevices are presented. Fig. 3presents the data provided bythe test system while Figure

4 is related to simulationscarried out MATLAB using30 modes to describe thepanel dynamics; a total of 60equations has been attained

One active device (A)

Three active devices (A,B and C)

Fig. 3 - Experimental FRF comparison:controlled Vs uncontrolled

r, ,,. ^4uauun& ims oeen attainedafter adding sensors and actuators dynamics. Data in Table 2 are insteadreferred the active control effectiveness under harmonic excitations usinghe A devices alone: the excitation frequency is close to the listed modefIt u; possible to appreciate a satisfactory agreement between experimentand simulations.

135


MODE

Piv°

VHP

Experiment

ANfg]18.334.63333

Arfg]1.5

13.88517.1

A%

91.859.948.6

Simulation

ANrtg]12.4

45.122,2

Ark]0.983.0212.10

A%

92.0

91.145.5

Table 2 - Comparison of acceleration reduction

400frequency Hz

Three active devices (A,B,C)

30- - If!

200 400 600 800frequency Hz

One active device (A)

Fig. 4 - Numerical FRF comparison: controlled Vs uncontrolled

4.2 The single self-sensing actuator

The use of a single piezoelectric device as a self-sensing actuator,combined with the modalfilter, gave satisfactoryresults. The tests presentedare related to fixedfrequency excitation. Filtershave been tuned by settingthe working frequency inorder to act the specificmode under control.Simulations have been alsoperformed by including thepower amplifier transfer

70--60--50"-

DTest•GspeQGmax

n,cMode & Piezo #

Figure 5 - Acceleration reduction(Single self-sensing actuator)

function and the bridge electric model. In this case a 5% unbalance hasbeen introduced and using the same gains measured during tests. InFigure 6 experimental end numerical results are compared in terms ofacceleration reduction. It is possible to see that significant reductionshave been obtained. The two first bars refer to performancese measuredduring tests and predicted by the simulation using experimental gains: itis possible to appreciate that, in this case, a lower authority is often

136


8070--60 -5040 -30 -20 -10 -

jK

^i*f ~

^(%

1 1

*̂

-^-;1r" %~ ̂i/:

r-rr

- }•','

€'

I I II1(A) 5(B) 5(B)

pr£-&

s

_K- [>*,

7(C)

-T~%#

i'

II7(C)

-1DExperDSimuli

p\l

<?",-%

V7(IIIC)

T 1imentatioii

<^wT

"̂

V3(

-

[IID)

. . i m p r o v e a i r m e maximumgam is used, that is the maximum gain attainable before an instability ofthe control occur. This result is represented by the last bar.

5.7 Double self-sensing actuator

The use of a pair of self-sensing actuators led mostly to better

performances. At the same time both simulations and experimentsshowed lower maximum gains: the system being intrinsically less stable.It is important to note that the bridge allows to sense a strain rate, i.e. a

measurement proportionalto the transversal velocity,but due to the presence of

the modal filter, that

introduce a phase delay of

90°, the control systemoperates a positionfeedback and is notasymptotically stable.Also in this case results oftests at fixed frequenciesare presented. Figure 6

compares experimentaland simulation results.

Again an agreement has been found when simulation gains have beenscaled to the maximum allowable to avoid instabilities.

5 Concluding remarks

The research activities demonstrated once again the remarkablepossibilities of piezoelectric patches to suppress vibrations even usingvery simple control laws, e.g. a direct position feedback. The badbehaviour of velocity sensors should be further investigated becauseprevious experiences with such a kind of measurement, obtained byintegration of an accelerometric signal, gave positive results.The use of a self-sensing actuator, together with a modal filter of theinput signal leads to results comparable to those obtained using separatesensor and actuator devices. The use of a pair of self-sensing actuatorsimproved, but not dramatically, the effectiveness of the active controlThe formulation adopted to simulate the dynamic behaviour of thecoupled system and to design the active control showed a satisfactoryagreement with experimental evidences.

Mode & Piezo #Figure 6 - Acceleration reduction(Double self-sensing actuator)

137


Ine availability or validated models, conditioning circuits and power

amplifiers allows one to continue with design and implementation ofdigital control system. This activity is at a preliminary testing stage: the

digital active control system being implemented on a PC is running in aReal Time Linux kernel performing 18KHz control frequency on fourchannels. Suboptimal and adaptive control laws are currently under test.

References

[1] Ghiringhelli G.L., M. Lanz, P. Mantegazza, 1993. " Numericalmodelling and experimental testing of distribuited piezoeletric

actuators", Forum Int. Aeroelasticite et Dinamique de Structures,

Strasburg[2] Dosch J.J., D.J. Inman, E. Garcia, 1992 "A self-sensing piezoelectric

actuator for collocated control", Journal of Intelligent Material

System and Structures, Vol 3 Jan.[3] Saunders W.R., D.G.Cole and H.H.Robertshaw "The impact of

piezoelectric sensoriactuators on active structural acoustic control",

Proceedings of the SPffi, Vol.1917, No.l,p.578-586[4] Brusa E., S. Carabelli, A. Tonoli, 1996 " Self-Sensing collocated

structures with distribuited piezoeletric transducers", MechatronicsLaboratory Politecnico di Torino, 1C AST 1996 - Roma, Italy

[5] Akishita S., Y. Mitani and H. Miyaguchi, 1994 "Sound transmissioncontrol through rectangular plate by using piezoelectric ceramics asactuators and sensors", Journal of Intelligent Material System andStructures, Vol 5 May

[6] Cole D.G., Clark R.L., 1994 "Adaptive compensation ofpiezoelectric sensoriactuators", Journal of Intelligent MaterialSystem and Structures, Vol 5, pp.665-672

[7] Anderson E.H., Hagood N.W., 1992 "Self-sensing piezoelectricactuation: analysis and application to controlled structures", AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics andMaterial Conf., Dallas TX, pp.2141-2155

[8] Masarati P., 1995 "Travi piezoelettiche: modellazione ed analisi",

Politecnico di Milano, Thesis[9] Fanson J.L. and T.K.Caughey, 1987 "Positive position feedback

control for large space structures", AIAA Paper 87-0902

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