vaz_IEEE_IM_April_1998

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 47, NO. 2, APRIL 1998 513

Composite Modeling of Flexible Structureswith Bonded Piezoelectric Film

Actuators and SensorsAnthony Faria Vaz,Member, IEEE

Abstract—In this paper, the interaction equations for a flexiblebeam with multiple-bonded piezoelectric films are derived. Anyflexible structure composed of flexible beam members can beanalyzed with these equations. The interaction equations accountfor the effects of shaped piezoelectric films, that is, the film’selectrodes may be etched with geometrical shapes and the polingdirection may be altered by splicing. The derivation is basedon a generalization of the interaction equations for a singlefilm. The interaction equations make use of the vibrationalmodes of a structure. The vibrational modes can be derivedeither analytically or numerically using finite-element techniques.Experimental validation has been done on a cantilever beam withbonded polyvinylidene fluoride and lead zirconium titanate films.These equations are useful for developing techniques for shapecontrol and active vibration damping in flexible structures.

Index Terms—Dynamic modeling, piezoceramic, piezoelectric,polyvinylidene fluoride, smart structures, spatial filtering.

I. INTRODUCTION

I N THE NEXT decade, it is envisaged that large flexiblespace structures (LFSS’s) will be used in a variety of

applications. Some potential applications include the space-based radar, the mobile servicing subsystem, and the spacestation. Due to the light weight and expansive sizes of LFSS’s,these structures will have significant structural flexibility.The elastic body modes will be very lightly damped andrange from very low to very high frequencies. High-frequencymodes can be attenuated by using materials with significantviscoelasticity, whereas low-frequency modes cannot. It is thelow-frequency modes that will limit the performance of acontrol system for an LFSS.

Structural vibrations can be actively damped using piezo-electric films that are bonded to structural members of aflexible structure. The dual nature of piezoelectric films allowsthem to be used as either actuators or sensors. Shaping ofpiezoelectric films, by splicing and electrode etching, can beused to sense and control individual modes in a structure.This enables a feedback controller to be realized in terms

Manuscript received May 18, 1998; revised November 30, 1998. This workwas supported in part by the Canadian Space Agency under Contracts 9F009-0-4140, 9F011-0-0924, and 9F028-5-5106 and in part by the Natural Sciencesand Engineering Research Council of Canada.

The author is with Applied Computing Enterprises, Inc., Mississauga, Ont.,L5V 1E5 Canada and is also with the Department of Electrical and ComputerEngineering, McMaster University, Hamilton, Ont., L8S 4L7 Canada (e-mail:[email protected]).

Publisher Item Identifier S 0018-9456(98)09851-9.

of a suitably shaped film. In principle, the feedback canbe infinite order, as it is only limited by the fidelity withwhich the electrode can be etched [1]. Traditionally, controllerdesign has been done using a temporal paradigm wherecontroller complexity is measured in terms of dynamic order.Piezoelectric films allow control design to be done usinga spatial paradigm [2], [3], where controller complexity ismeasured in terms of film size and electrode geometry. Thisnew control paradigm will enable the development of newcontrol design methodologies. A precursor to this work isthe development of interaction equations for modeling thedynamics of flexible structures with piezoelectric films.

Some researchers have developed a finite-element model(FEM) to represent the composite dynamics of a flexiblesubstructure with bonded piezoelectric films. The disadvantageof this method is that each different placement and shaping ofthe piezoelectric films requires that the FEM be recomputed.This makes it awkward to use film shape and placement as adesign parameter. A complex computational algorithm must beused to deal with the resultant coupling of controller designwith structural modeling [4].

Crawley and de Luis [5] developed a useful two-stepprocedure for analyzing the coupling of piezoelectric filmactuators and a flexible substructure. First, a dynamic modelof a structure without piezoelectric films is developed usinga conventional technique, analytical solution of a partial dif-ferential equation, or numerical solution of a finite-elementsproblem. Second, the influence of piezoelectric film upon thesubstructure is derived in terms of generalized force froma static stress–strain model of the piezoelectric film. Thisquasi-static approach works well under the assumptions thatactuator mass is small compared to total mass of the structure,and the actuator resonance frequencies are much higher thanfrequencies of interest. This method has the advantage thatdifferent film placements and sizes can be studied withoutrecomputation of the substructure dynamics.

Lee and Moon [6] studied the interactions between a piezo-electric film and a flexible plate. The interactions are expressedin terms of eigensolutions to the partial differential equa-tion of plate vibration. Accordingly, a piezoelectric film ischaracterized as a series of modal forces, when used as anactuator, and as a series of modal charges, when used as asensor. The orthogonal nature of the mode shapes can beused to construct modal actuators/sensors which excite/observeparticular vibrational modes.

0018–9456/98$10.00 1998 IEEE

514 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 47, NO. 2, APRIL 1998

Fig. 1. Transverse effects for a piezoelectric film.

In this paper, the modal techniques of Lee and Moon [6]are combined with the quasi-static technique of Crawley andde Luis [5] to yield a novel set of interaction equations. Theseequations are used to characterize the operation of shapedpiezoelectric films that can be used as either actuators orsensors. These equations are used to derive a state-space modelof a flexible structure with multiple bonded piezoelectric films.This model is useful for the design of control techniques forimplementing active vibration damping (see [7]).

This paper is organized as follows. In Section II, electro-mechanical models of piezoelectric films are described. Theoperation of shaped piezoelectric films are discussed inSection III. The static stress and strain equilibrium conditionsfor a flexible beam with bonded piezoelectric films are derivedin Section IV. The piezoelectric film actuator equations arederived in Section V. The piezoelectric film sensor equationsare derived in Section VI. Experimental results are discussedin Section VII. Concluding remarks are made in Section VIII.

II. M ODELING PIEZOELECTRIC FILMS

Consider the rectangular piezoelectric film illustrated inFig. 1. The 1 axis and 2 axis define a plane which is parallelto the film’s surface. The 3 axis is perpendicular to the film’ssurface. The 3 axis points in the opposite direction to theelectric field used to pole the piezoelectric film during itsmanufacture. Some piezoelectric films, such as polyvinylidenefluoride (PVDF), are uniaxially stretched during the polingprocess. In such a case, the 1 axis is oriented in the directionof the uniaxial stretching. Piezoelectric films that are uniaxiallystretched exhibit properties that differ in the directions of the1 and 2 axes (e.g., ). Films that are not uniaxiallystretched, such as lead zirconium titanates (PZT’s), do notexhibit these differences.

The transverse effects in piezoelectric films are illustratedin Fig. 1. The film is assumed to have a length along the1 axis, a width along the 2 axis, and a thicknessalongthe 3 axis. A voltage is measured across the upper andlower electrodes. The upper and lower electrodes are assumedto have free charges of and , respectively. A tensileforce and an elongation displacement are measured

(a)

(b)

(c)

Fig. 2. Electromechanical models based on forceF1 and voltageV3.

along the 1 axis. If the film acts as an actuator, a voltageis applied across the film; the induced polarization in the

piezoelectric generates a tensile force and an elongation. If the film is used as a sensor, a tensile forceis applied

which stretches the film by ; the induced polarization in thepiezoelectric causes an increase of free charges on the contacts,which generates a voltage through a capacitive effect.

A piezoelectric film’s behavior can be modeled in terms offorce and voltage [8]

(1)

where is interpreted asthe film compliance at constantelectric field intensity; the film charge to force ratio; and

, the film capacitance at constant stress. The above filmparameters are defined as follows:

and

where is the Young’s modulus of the piezoelectric at con-stant field intensity, is the permittivity of the piezoelectricat a constant stress,and is the transverse piezoelectriccharge to stress ratio. Equation (1) can be represented in termsof the electromechanical models shown in Fig. 2. Due to theinverted voltage source in the circuit of Fig. 2(b), an externalapplied force causes a negative piezoelectric displacementin the mechanical model in Fig. 2(c); thus, the mechanicalstiffness of the film is augmented by the piezoelectric effect.

Piezoelectric films can also be modeled in terms of voltageand displacement [8]

(2)

VAZ: COMPOSITE MODELING OF FLEXIBLE STRUCTURES 515

(a)

(b)

(c)

Fig. 3. Electromechanical models based on displacementY1 and voltageV3.

where is interpreted as thefilm capacitance at constantstrain, and is the film charge to displacement ratio. Theabove parameters are defined as follows:

and

Equation (2) can be represented in terms of the electromechan-ical models shown in Fig. 3.

III. SHAPED PIEZOELECTRIC FILMS

A shaped filmis idealized as a rectangular piezoelectric filmthat has a poling direction that varies throughout the film,and electrodes that are etched into geometrical shapes. At anypoint in a shaped film, the poling direction may either pointup or down. The electrode shapes are such that a point inthe piezoelectric film may or may not be covered by bothelectrodes. The properties of a shaped film are determined bythe pattern of the poling direction variation and the electrodegeometry. A shaped film can be constructed from severalelectrically connected contiguous sections of piezoelectricfilms. The orientation of the polling direction varies betweensections. This construction is illustrated in Fig. 4.

Shaped films are analyzed in terms of variations of thepoling direction and domain of the outer electrodes. Thefunctions used to represent these quantities are expressed interms of the coordinate system of the beam to which they are

Fig. 4. Shaped piezoelectric film.

Fig. 5. Top view of piezoelectric film bonded to a flat beam.

attached. The , , coordinate frame has its origin on theneutral surface of the beam. The beam has a lengthalongthe axis, a width along the axis, and a thicknessalong the axis. The upper and lower surfaces of the beamhave coordinates of and , respectively.

The shaped film is bonded to the beam as shown in Fig. 5.The film is situated between and inthe axis and and . The coordinate of thebeam surface, to which the film is bonded, is denoted by.If the film is bonded to the top surface, ; otherwise,

for the lower surface. The film is assumed to berectangular, but the electrode is assumed to have a generalshape.

The bending strain of the flat beam varies only in theand directions, and time . Hence, beam strain can

be represented as a function . Note the strainshave the following antisymmetric property:

. Accordingly, the strain in a cross section of thepiezoelectric film is represented by a function . Thestrain variations of in the direction are negligible dueto the thinness of the piezoelectric film. The piezoelectricfilm is assumed to beperfectly bondedto the beam; hence,

. In practice, the errors incurred bythis simplifying assumption are typically negligible.

The properties of a shaped film are characterized in termsof domain function andpoling function . Thedomain function defines the region of a shaped film’setched electrode: , if point is coveredby both electrodes; , otherwise. The polingdirection is specified with respect to the electric field intensityassociated with a particular electrode polarity. Thepolingfunction specifies the variations of the poling directionthroughout a shaped film, , if poling directionat points toward negative electrode; , both


film electrodes do not cover point ; ,if poling direction at points toward positive electrode.These two functions are fundamental to the understanding ofinteractions between piezoelectric films and flexible structures.

Theshape function of a shaped film is the average ofthe poling function over a cross section with a fixed

coordinate. It is defined by

succinctly expresses the key property that determinesthe operation of piezoelectric film sensors and actuators.

The area covered by the film’s electrodes andcapacitanceof the film are given by

and

where is the permittivity of the piezoelectric,and is thefilm thickness. If a voltage is impressed across the film’selectrodes, thepiezoelectric strain at a pointis given by

This follows upon applying (1) and the mechanical model ofFig. 2(c) to an infinitesimal piezoelectric film element. Theaverage strainover the cross section with a fixedcoordinateis . It is defined by

This expression is used in the next section to determine theforces generated by a piezoelectric actuator.

When a shaped film is strained, an electric flux densityis created due to the

polarization in the piezoelectric. This follows upon applying(1) and the circuit in Fig. 3(a) to an infinitesimal element ofa piezoelectric film. The charge corresponding to thecharge source in the circuit of Fig. 3(a) is given by

For a given strain profile , the corresponding open-circuit voltage and the short-circuit current are,respectively, given by

(3)

(4)

Fig. 6. Infinitesimal slice of beam and bonded piezoelectric film.

IV. STATIC STRESS AND STRAIN EQUILIBRIUM CONDITIONS

In this section, the static relationships between stress andstrain of a flexible beam and bonded shaped piezoelectricfilms are derived [9], [10]. Consider a slice of the beam and abonded piezoelectric film, as shown in Fig. 6. The bonding isassumed to be perfect so that the surface strain of the beam isassumed to equal the strain of the bonded piezoelectric film.The corresponding strain profile is shown in Fig. 6. The strainprofile corresponds to the application of a positive moment thatcauses the top of the beam and piezoelectric film to expandunder tension, and the bottom of the beam to contract undercompression.

The relationship between piezoelectric film strain to inducedbeam strain can be derived using elastic strain energy withthe principle of virtual work. Strain energy density in apiezoelectric film is given by the integral [11]

where is the stress as a functionof strain in the piezoelectric film. The strain energy in aninfinitesimal slice of piezoelectric film of length is givenby

The strain energy in a slice of length of a bent beam [11]with a surface strain is

The principle of virtual work [11] can be used to determinethe static interaction between the piezoelectric film and thebeam. Since the beam and the piezoelectric film are assumedto be in equilibrium,


Using the perfect bonding assumption,and , yields

(5)

A flat beam has a cross-sectional moment of inertia. Substituting into (5) yields

(6)

where and . Equation (6) relatespiezoelectric strain to induced beam strain. This relationshipcan be used to derive the effective distributed moment andeffective distributed force that is induced by the piezoelectricfilm. From beam theory [11], a surface strain ofcorresponds to a distributed moment given by

(7)

A distributed moment is equivalent to a distributedupward force given by

(8)

The stress–strain relationships derived by Crawley anddeLuis [5] can be derived by using the above derivation with auniform shape function, that is, . Equations (6)–(8)have the advantage that they have a simple form that doesnot involve the boundary conditions derived from the strainsat the ends of the piezoelectric film. Accordingly, (6)–(8) aresimpler to use than those of Crawley and deLuis [5].

V. DYNAMIC ACTUATOR EQUATIONS

In this section, the equations that determine the dynamicinfluence of a piezoelectric film actuator upon a flexiblesubstructure are derived. The interaction equations are derivedusing theassumed modes method(AMM) [12]. The use ofthis analytical technique is predicated on the assumption thatthe actuator does not significantly affect the mode shape ofthe flexible substructure. This is a reasonable assumption forpiezoelectric films bonded to flexible structural elements.

The response of the flexible structure can be modeled interms of a series , whereis displacement, is the th mode shape, and is theth modal amplitude. This series can be derived analytically

for simple structures, and by FEM for complex structures. Thekinetic energy of the system can be written as

where the symmetric mass coefficients depend on thecontinuous system mass properties and the functions .The potential energy can be written as

where the symmetric stiffness coefficients depend on thecontinuous system stiffness properties and the derivativesof . External forces are nonconservative, so that theLagrange’s equations of motion have the form

for

where is a generalized nonconservative force. Insertingand into Lagrange’s equation of motion yields

for

The dynamic interactions between a shaped piezoelectricfilm actuator and a beam can be most conveniently derivedby adding the strain energy of the piezoelectric film tothe potential energy function for a flexible beam. Thisresults in a composite potential energy function . Thistechnique is adapted from de Luiset al. [13]. This compositeenergy function is used in conjunction with the AMM to derivethe dynamic equations.

Consider the beam with the bonded piezoelectric film shownin Fig. 5. The strain of the piezoelectric film isderived from the surface strain of the beam asfollows:

Piezoelectric strain energy is obtained by integratingalong the length of the film

Using the composite potential energy functionin Lagrange’s equations of motion yields the following


set of differential equations:

(9)

where for , for , isYoung’s modulus of elasticity of the beam material,is themoment of inertia of the beam cross section,is the beammass per unit length, and is the eigenvalue of mode

(10)

(11)

Equation (9) differs from the conventional uncoupled harmonicoscillator equations of a beam by the terms and .The term represents the passive stiffness contributedby the piezoelectric film. It has the effect of coupling the dy-namic modes. The term is the generalized force associatedwith mode . Note that the generalized forces are zero if theelectrodes of the piezoelectric film are short circuited. Equation(8) shows the effect of the piezoelectric parameters, the shapefunction, the mode shape, and the voltage upon the generatedmodal force.

If the beam were bonded with more than one piezoelectricfilm, (10) and (11) would be evaluated for each film. Thecombined passive stiffness terms are determined as follows:

for for

where is obtained by evaluat-ing (10) with the parameters of piezoelectric film PIEZO.Thecombined modal force terms are determined as follows:

for

where is obtained byevaluating (11) with the parameters of piezoelectric filmPIEZO and the voltage across its electrodes. Thecombined terms are used to formulate the following equationfor mode amplitudes:

VI. DYNAMIC SENSORSEQUATIONS

The effect of vibrational modes upon the open-circuit volt-age and short-circuit current can be inferred from a modalexpansion of the strain profile

(12)

Substituting (12) into the expression (3) for open-circuit volt-age , and (4) for short-circuit current , yields

and

where the th modal open circuit voltage is

and the th modal short-circuit current is

The above equations show the fundamental roles played bythe shape function in determining the operation of astrain sensor that measures open-circuit voltage, and a strainrate sensor that measures short-circuit current. The sensorequations have a form that is consistent with the actuatorequations.

VII. EXPERIMENTAL RESULTS

Several experiments were performed to verify the interac-tion equations discussed in Sections V and VI. PVDF filmswere used to construct three sensor types: a rectangular sensor,a first-mode sensor, and a second-mode sensor. A computerprogram is used to generate a postscript file that is laser printedto produce a mask for the PVDF film geometry. The PVDFfilms were bonded to identical aluminum cantilever beamsusing a Kapton substrate. PZT films were used to constructactuators. PZT films were bonded to aluminum beams usingconductive epoxy. Best results were obtained when the PZTwas bonded directly to the beam without the use of a Kaptonsubstrate. The experiments are described in detail in [14].

A tip displacement and mid-beam impact test were per-formed for each shape sensor. During the sensor experiments,the open-circuit voltage and short-circuit currents were mea-sured. The voltages and currents were then compared toMATLAB simulations of both the time response and powerspectral densities. In the tip deflection experiment, the beamtip was pushed against an end stop and then released. Thisgave a repeatable result; however, the vibration is primarilyfirst mode. The mid-beam impact test primarily excites highermodes.

The first-mode and second-mode voltage responses for thetip displacement test are shown in Figs. 7 and 8, respectively.The modal amplitudes corresponding to a 0.02tip displace-ment are andm. This initial condition predominately excites the first mode,since is approximately 5000 times larger than the secondmode . The modal voltages and are a functionof a film’s shape. Using a MATLAB program yields

V/m and V/m for thefirst-mode sensor, and V/m and

V/m for a second-mode sensor. Ideally,


Fig. 7. First-mode sensor response due to tip displacement.

Fig. 8. Second-mode sensor response due to tip displacement.

for a first-mode shape film, and for a second modeshape film; the discrepancies are due to numerical errors. Forthe tip test, the expected amplitude is V forthe first-mode sensor, and V for thesecond-mode sensor. This accounts for the attenuation of thefirst-mode content by the second mode. The first-mode contentstill leaks through the second-mode sensor due to the difficultyof exactly constructing the required shape.

The actuator response was tested by applying a voltage toa PZT film. An acceleratormeter was mounted at the tip ofthe beam. The tip acceleration was compared to the valuespredicted by simulation with a MATLAB program.

The experiments typically showed that the responses of boththe PVDF and the PZT sensors gave voltages and currents thatresembled the theoretical values. The discrepancies were dueto variations in amplitude. The PVDF voltages were within93% for the rectangular sensor, 60% for the first mode, and

50% for the second-mode sensor. When used as a sensor,the PZT film gave voltages that were 20% of the expectedvalues; when used as an actuator, the accelerations achievedwere also 20% of the expected value. These effects can beattributed to deviations , inaccuracies in cutting and etchingthe PVDF films, and imperfections in bonding both PVDFand PZT films. Fortunately, an empirical calibration of a scalefactor for adjusting the actuator and sensor equations can bedone using a sensor test.

VIII. C ONCLUSIONS

The interaction equations in this paper model the dynamicsof flexible substructures with bonded piezoelectric films. Theinteraction equations allow the required placement and shapingof piezoelectric films to be analyzed separately from thesubstructure dynamics. The operation of a shaped piezoelectricfilm is determined by both its placement and its shape function

. The actuator equations have a form that deals with thegeneral case of shaped films, but are substantially simpler thanthose derived by Crawley and deLuis [5]. The sensor equationshave a form that allows signal processing functions to beperformed by the appropriate selection of the shape function.The equations have been empirically verified by comparisonof computer simulations to the response of a cantilever beamwith bonded PVDF and PZT films. An empirical scaling factorcan be used to account for variations in and imperfectbonding. Future research will use these interaction equationsto evaluate potential control techniques for the active dampingof vibrations and shape control of flexible structures.

ACKNOWLEDGMENT

The author would like to thank Dr. S. Yeung, H. Reynaud,and Dr. G. Vukovitch at the Canadian Space Agency for theirassistance.

REFERENCES

[1] D. W. Miller and M. C. van Schoor, “Formulation of full state feedbackfor infinite order structural systems,” inProc. 1st Joint US/Japan Conf.Adaptive Structures,Maui, HI, 1990, pp. 304–331.

[2] D. W. Miller, S. A. Collins, and S. P. Peltzman, “Development ofspatially convolving sensors for structural control applications,” inProc.31st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics andMaterials Conf.,Long Beach, CA, 1990, pp. 2283–2297.

[3] S. A. Collins, D. W. Miller, and A. H. von Flotow, “Sensors forstructural applications using piezoelectric polymer film,” MassachusettsInst. Technol. Space Engineering Research Center, Cambridge, MA,Rep. 12-90, 1990.

[4] T. Iwatsubo, S. Kawamura, K. Adachi, and M. Ikeda, “Simultaneousoptimum design of structural and control system,” inProc. 1st JointUS/Japan Conf. on Adaptive Structures,Maui, HI, 1990, pp. 117–140.

[5] E. F. Crawley and J. de Luis, “Use of piezoelectric actuators as elementsof intelligent structures,”AIAA J.,vol. 25, no. 10, pp. 1373–1385, 1987.

[6] C. K. Lee and F. C. Moon, “Modal sensors/actuators,”J. Appl. Mech.,vol. 57, no. 2, pp. 434–441, 1990.

[7] A. F. Vaz, “Literature survey of active damping using piezoelectricdevices,” Canadian Space Agency, St. Hubert, P.Q., Canada, 1991.

[8] , “Modeling piezoelectric behavior for actuator and sensor appli-cations,” Canadian Space Agency, St. Hubert, P.Q., Canada, 1991.

[9] , “Theoretical development for an active vibration dampingexperiment,” Canadian Space Agency, St. Hubert, P.Q., Canada, 1991.

[10] , “Empirical verification of interaction equations with bondedpiezoelectric films,” Canadian Space Agency, St. Hubert, P.Q., Canada,1991.


[11] E. P. Popov,Mechanics of Materials,2nd ed. Englewood Cliffs, NJ:Prentice-Hall, 1976.

[12] L. Mierovitch, Elements of Vibration Analysis.New York: McGraw-Hill, 1975.

[13] J. de Luis, E. F. Crawley, and S. R. Hall, “Design and implementationof optimal controllers for intelligent structures using infinite order struc-tural models,” Massachusetts Inst. Technol. Space Systems Laboratory,Cambridge, MA, Rep. 3-89, 1989.

[14] T. J. Farr and A. F. Vaz,Validation Tests for Piezolectric InteractionEquations,Canadian Space Agency, St. Hubert, P.Q., Canada, 1997.

Anthony Faria Vaz (S’80–M’81) received theB.A.Sc., M.A.Sc., and Ph.D. degrees in electricalengineering from the University of Toronto,Toronto, Ont., Canada.

He is President of Applied Computing Enter-prises, Inc., Mississauga, Ont., Canada, whichspecializes in system theoretic analysis and softwaredevelopment. He has done consulting work onsmart structures, signal processing, control systems,and electrooptics for both the space and defenseindustries. He is also an Assistant Professor in the

Department of Electrical and Computer Engineering, McMaster University,Hamilton, Ont., Canada. His research interests include the development ofsignal processing and control algorithms for a variety of applications.

Dr. Vaz is a Licensed Professional Engineer in the Province of Ontario.

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