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A material rigidity effect of a bimorph piezoelectric actuator

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2007 Smart Mater. Struct. 16 1043

(http://iopscience.iop.org/0964-1726/16/4/012)

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IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 16 (2007) 1043–1049 doi:10.1088/0964-1726/16/4/012

A material rigidity effect of a bimorphpiezoelectric actuatorDong-chan Lee1,2,3, Deok-won Yun1 and Chang-soo Han1,3

1 Department of Mechanical Engineering, Hanyang University, Sa-1 dong, Sangnok-gu,Ansan, Kyeonggi-do 425-791, Korea2 CAE Consulting Team, SPACE Solution, Korea

E-mail: dclee@engspace.com and cshan@hanyang.ac.kr

Received 4 August 2006, in final form 25 April 2007Published 5 June 2007Online at stacks.iop.org/SMS/16/1043

AbstractThis paper presents the numerical modeling of a piezoelectric actuator used toavoid the limitation of the lock-in region in a ring resonator. Ring resonatorshave proved suitable for use as inertial sensors for navigation, guidance andattitude controls. However, their accuracy has been limited by the lock-inregion due to frequency coupling between two counter-propagating waves atlow rotation rates. This coupling introduces no phase difference, and noangular increment is detected. The problem can be overcome bymechanically dithering the ring resonator. Mechanically, the vibro-elasticbimorph piezoelectric actuator has been used as a control but is influenced bypiezoelectric rigidity effects. To more accurately predict the ditheringfrequency of a mechanical dither, numerical modeling of a piezoelectricactuator includes the piezoelectric rigidity effect of the actuator, whichoccurs in both the static and dynamic behaviors of composite plates havingpiezoelectric layers symmetrically bonded to the top and bottom surfaces.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Ring lasers have been investigated in order to ascertain theirsuitability for use as rotation rate sensors. When there isan applied rotation about the axis normal to the plane ofrotation, the operation of a gyroscope depends on the phasedifference for beams traveling in opposite directions within aclosed path. However, the accuracy of gyroscopes has beenlimited by the lock-in region due to the fact that at low rotationrates of gyroscope the frequency coupling mechanism arisesfrom backscattering of the mirrors. This effect gives no phasedifference and hence no detect of angular increment. In otherwords, this reduces the angular sensitivities to the electricaland mechanical disturbances, such as low frequency noisecomponents, in the optical measurement. It is important tominimize the transition period of the lock-in region in the beamtraveling path. A good laser gyro consists of mirrors withlow backscatter in conjunction with a stable gas discharge, asuitable dither drive and a stabilized resonator cavity lengthin order to avoid temperature effects. Under the temperature

3 Authors to whom any correspondence should be addressed.

effects, a change of the beam path geometry increases the lock-in threshold. It is very important to stabilize the beam pathgeometry during the operations. By using the piezoelectricmirror, it was possible to change the beam path geometryinside the laser cavity with extreme precision [1, 2]. Afteravoiding of the lock-in effect [3–5], the interests involved inthe development design of ring resonator began to concentrateon the practical solution of stabilizing the beam path [6–10].The purpose of the mechanical dithering is to suppress thedead band, oscillate the block about the rotation axis, and addan external rotation rate. The following describes the designconsiderations.

The systematic considerations for the gyroscope;

• Structural resonant frequency• Peak angular amplitude and peak dither rate• Mechanical properties of piezoelectric actuator material

(choice of material)• Driving technique• Inertia of dithered components.

0964-1726/07/041043+07$30.00 © 2007 IOP Publishing Ltd Printed in the UK 1043

D-C Lee et al

Figure 1. Schematic model of inertial sensor (adapted from [2]).

The considerations for the mechanical performance;

• Torsional stiffness• Bending profile• Points of inflection• Stress induced by bending of the piezoelectric actuator.

The detailed considerations for the piezoelectric actuator;

• Geometry of piezoelectric actuators• Number of piezoelectric actuators• Attachment of dither mechanism to solid block and inertial

reference frame.

Figure 1 shows the typical model of triangular ringresonator with mechanical dither drive. Most of the mechanicaldithering models have the influence of piezoelectric rigidityeffects. Figure 2 depicts a perspective scale-factor stabilityfor the inertial sensors. The existing mechanical gyroscopesoffer the angular rates about two rotation axes. Because ofthe unintentional interferences of rigidity effects, the inertialsensor can decide on how well the sensor reproduces the sensedrate and how much the high-speed angular rate is delayed.The geometrical approach of the aforementioned piezoelectricactuator [5, 11–13] was presented for the loading condition andangular characteristics due to the piezoelectric deformation inprevious research. In this paper, the mechanical performancesof actuators are theoretically presented on the operationcharacteristics due to the piezoelectric rigidity effect andpiezoelectric parameters. Through the presented approach, itis possible to design the dimensions of piezoelectric materialsand elastic metal for the feasible operation of a piezoelectric-driven actuator.

2. Operation of piezoelectric actuator withoutbimorph rigidity

To deform the bimorph element [5, 11–16], a voltageof appropriate polarity should be applied to the externalelectrodes of the piezoelectric elements shown in figure 3. Asa result of the interaction of the polarization and piezoelectricmotion due to tension or compression, the elements will bedeformed. Because of a converse piezoelectric effect in theelement, a stretching force arises where both vectors coincide

in one direction. At the same time in the second element,where the electrical motions are oriented in opposite directions,a contracting force arises, shown in figure 3.

The tension and compression forces created by thepiezoelectric layers symmetrically bonded to the top andbottom surfaces of the non-piezoelectric layer made of ametal material shown in figure 3 depends on its geometricdimensions, piezoelectric properties and the electrical voltageV supplied to the piezoelectric elements [9]:

Fc = εε0ScE

d31= εε0Sc

d31

V

hc(1)

where ε is the relative dielectric permeability of thepiezoelectric, ε0 is the dielectric constant, Sc is thetransverse sectional area of the piezoelectric elements, E isthe piezoelectric field (V/hc), hc is the thickness of thepiezoelectric element and d31 is the piezoelectric constant.

In the piezoelectric actuator, only the external surfacesof the piezoelectric layer can be freely deformed and beinternally hard-mounted on an elastic metal (non-piezoelectriclayer). Considering two bending moments located along anon-piezoelectric layer shown in figure 5 as a set of loadingconditions, we can see that the effect of the piezoelectriclayer mounted on the non-piezoelectric layer is equivalentto the operation of a couple of bending moments applied atdistances (a1 ± lc/4) from a fixed edge of the deflector, wherea1 is a coordinate of the center of the piezoelectric actuator.The transverse deflection of the non-piezoelectric layer, y(z),can be described as a function of the coordinate z withoutconsideration of the rigidity increases due to the piezoelectriclayers and connective seam layers:

EM J AM y = EM J A

M y0 + EM J AM y ′

0z

+mn∑

i=1

Mi(z − ami )

2

1! +pn∑

j=1

Pi(z − ap j)

3

3! (2a)

where EM is the elasticity of metal, J AM = bMh3

M12 is the

transverse area moment of inertia of metal for the neutral, mn isthe number of bending moments, pn the number of transverseforces and

y (z) = y0 + y ′0z + 1

EM J AM

∑Mi (z − ami )

2

+ 1

EM J AM

∑Pj

(z − ap j )3

6. (2b)

The initial conditions for rigid mounting are y(0) =0, y ′(0) = 0. For two moments of opposite signs to each other,the transverse deflection according to the coordinate systemshown in figure 5 will be

y (z) = 1

EM J AM

[−M1 (z − am1)2 + M2 (z − am2)

2] . (3)

Inserting M1 = M2 = MB , am1 = a1 − lC4 and am2 = a1 + lC

4to equation (3), the transverse deflection of the center of thenon-piezoelectric layer can be rewritten by

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A material rigidity effect of a bimorph piezoelectric actuator

Figure 2. Current gyro technology applications (adapted from [10]).

Piezoelectric tension

Piezoelectric compression

(a)

(b)

Figure 3. Tension and compression behaviors of piezoelectric layersdue to the bimorph piezoelectric effect. (a) Piezoelectric tension andcompression actuations, (b) deformed shape of the piezoelectricactuator.

y (z) = 1

EM J AM

[2MBz (am1 − am2) + MB

(a2

m2 − a2m1

)]

= MB

EM J AM

(am2 − am1) [2z − (am2 + am1)]

= MB

EM J AM

lc (z − a1) (4)

where a1 is the position of the center of the piezoelectric layerand lc is the length of the piezoelectric layer.

On the end of the non-piezoelectric layer, z = l , thedeflection is given by

y (l) = MB

EM J AM

lC (l − a1) . (5)

If there are a couple of piezoelectric layers for both sides of theelastic metal, the deflection at z = l will be doubled becauseof the symmetric attachments of piezoelectric layers [11, 13]

y (l) = 2MB

EM J AM

lC (l − a1) . (6)

The equivalent bending force P exerted at the end can bedetermined from a condition of deflection equality given by

Pl3

3EM J AM

= y (l) = 2MB

EM J AM

lC(l − a1). (7)

That is,

P = 6MBlC

l3(l − a1) . (8)

The bending moments shown in figure 5 can be expressed bythe moment due to the piezoelectric force FC, provided by thepiezoelectric:

MB = FCh� = FC (hM + hC + 2hS) /2 (9)

where hM is the thickness of the non-piezoelectric layer usingelastic metal, hC is the thickness of the piezoelectric layer andhS is the thickness of the connective seam.

Using equations (1), (8) and (9), the equivalent force canbe given by

P = 3εε0SCV

hCd31l3lC (l − a1) (hM + hC + 2hS) . (10)

By equating the work of the equivalent force in equation (10)to the magnitude of the kinetic energy at the transition due to anon-piezoelectric layer in a position of equilibrium,

P · y (l) = 1

2

J msys

Nω2

max (11)

where J msys is the polar mass moment of inertia of the non-

piezoelectric layer and resonator without the piezoelectriclayers and connective seams:

J msys = J m

resonator + N

[1

12mM

(h2

M + l2) + mM

(l

2

)2]

N is the number of piezoelectric actuators.

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D-C Lee et al

Figure 4. Schematic configuration of the piezoelectric actuator.

The maximum angular frequency can be given by

ωmax =√

2N Py(l)

J msys

= εε0SCV

d31

(hM + hC + 2hS)

hC

× lC (l − a1)

l

√6N

EM J AM J m

sysl. (12)

Equation (12) is a good approximation for the quasi-staticmode, which is when the frequency of the electrical voltagesupplied to the piezoelectric is sufficiently lower than theresonant frequency of an oscillatory system ‘vibrodrive-monoblock’, or at least is beyond a pass-band of frequenciesof this system. However, this numerical equation does not havethe rigidity effects and geometric dimensions of piezoelectricelements. So, the next section presents the numerical equationsof resonant frequency with the rigidity effects and geometricdimensions of piezoelectric elements.

3. Operation of the piezoelectric-actuator withbimorph rigidity

Under the microdynamic operation condition of an elasticmetal actuator, the piezoelectric attached to the elastic metalprevents the bending motion of the elastic metal actuator dueto the increased equivalent rigidity. Therefore, the optimalthickness dimension ratio between the piezoelectric and elasticmetal actuator has to be studied. The pressure-sensitiveelement is a cantilever type, which has the rotational angleand linear transverse deflection. The bending rigidity of thebimorph element shown in figure 4 can be written as

(E Jx)1 =(

EM + ESJ A

S

J AM

+ ECJ A

C

J AM

)· J A

M

= Ee J AM,

(a1 − lC

4

)< z <

(a1 + lC

4

)

(E Jx)2 = EM J AM,

(a1 + lC

2

)< z < l.

(13)

The equivalent elastic modulus of the bimorph element is givenby

Ee = EM + ESJ A

S

J AM

+ ECJ A

C

J AM

, (14)

Figure 5. Bending behavior of the non-piezoelectric layer.

where EC is the elastic modulus of the piezoelectric, EM is theelastic modulus of the metal, ES is the elastic modulus of theconnective seam,

J AS = 2

[bCh3

S

12+ bChS

(hM

2+ hS

2

)2]

is the transverse inertial moment of the sectional area of theconnective seams for the neutral axis,

J AC = 2

[bCh3

C

12+ bChC

(hM

2+ hS + hC

2

)2]

is the transverse inertial moment of the sectional area of thepiezoelectric element for the neutral axis and

J AM = bMh3

M12 is the transverse inertial moment of the

sectional area of the metal for the neutral.Without the consideration of inertial force, the transverse

deflection according to the length of the elastic actuator can bewritten by

y = MB(E J Ax

)1

·(z − a1 + lC

4

)2

2,

(a1 − lC

4

)< z <

(a1 + lC

4

)(15)

y = MB(E J Ax

)1

(z − a1 + lC

4

)2

2− MB(

E J Ax)

1

(z − a1 − lC

4

)2

2,

(a1 + lC

4

)< z <

(a1 + lC

2

)(16)

y = ya1+ lC2

z

a1 + lC2

,

(a1 + lC

2

)< z < l. (17)

Without the consideration of the rigidity increase due to thepiezoelectric layers and connective seams, the total transversedeflection is given from section 2

y (l) = A1hM + hC + 2hS

2hC· lC

l

(1 − a1

l

)l

= A1

(hM

2hC+ 1

2+ hS

hC

)· lC

l

(1 − lC

2l

)l (18)

where

A1 = εε0ScV(E J Ax

)2

d31, a1 = lC

2. (19)

Assuming χ = hChM

and λ = lCl , equation (18) is rewritten as

y (l) = A2

[1

2 (hC/hM)+ 1

2+ hS

hM · (hC/hM)

]

= A2

2

[1

χ

(1 + 2

hS

hM

)+ 1

]. (20)

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A material rigidity effect of a bimorph piezoelectric actuator

In the range of 0 < λ < 1, the smaller the ratio of hChM

,the more the transverse deflection. If there are two pairs ofpiezoelectric elements for both sides of an elastic metal, thetransverse deflection will be doubled. With the considerationof rigidity increase, the transverse deflections are given by

y (z) = εε0SCV

d31 Ee J AM

· h M + hC + 2hS

hClC

(z − lC

2

),

Ee = EM + ESJ A

S

J AM

+ ECJ A

C

J AM

, 0 < z � lC

(21)

y (z) = y (lC) + εε0SCV

d31 E J AM

· hM + hC + 2hS

hC· lC

(z − lC

2

)

= εε0SCV

d31· hM + hC + 2hS

hC

×[

lC

(lC − lC

2

)

Ee J AM

+ lC

(z − lC

2

)

E J AM

],

E = EM, lC < z � l. (22)

Therefore, the transverse deflection considering the rigidityincrease at z = l is given by

y (l) = εε0SCV

d31· hM + hC + 2hS

hC

[l2C

2Ee J AM

+ lC(l − lC

2

)

EM J AM

].

(23)In the comparison with the total deflection without the rigidityincrease, equation (23) has the effect of an equivalent elasticmodulus which depends on the elasticity modulus of metal,piezoelectric and connective seam. In the comparison withequations (6), (21) and (22) it has the effect of the equivalentelastic modulus. The equivalent bending force P exerted at theend can be determined from a condition of deflection given by

y (l) = P

3

[l3 − (l − lC)3

Ee J AM

+ (l − lC)3

EM J AM

]. (24)

If there are two pairs (couples) of piezoelectric elements forboth sides of an elastic metal, the transverse deflection willbe doubled. By equating the work of the equivalent force inequation (23) to the magnitude of the kinetic energy at thetransition due to the non-piezoelectric layer in a position ofequilibrium,

P · y (l) = 1

2

J msys

Nω2

max (25)

where J msys is the polar inertial moment of mass of the non-

piezoelectric layer and resonator with the piezoelectric layersand connective seams,

J msys

∼= J mresonator + N

[1

12mM

(h2

M + l2) + mM

(l

2

)2

+ 2(

112 mC

(h2

C + l2C

) + mCr 2C

+ 112 mS

(h2

S + l2S

) + mSr 2S

)]

N is the number of piezoelectric actuators. The maximumangular frequency can be given by

ωmax =√

2N Py(l)

J msys

= εε0SCV

d31· hM + hC + 2hS

hC

[l2C

2Ee J AM

+ lC(l − lC

2

)

EM J AM

]√√√√6N

J msys

[l3−(l−lC)3

Ee J AM

+ (l−lC )3

EM J AM

] . (26)

Table 1. Fundamental design parameters for rigidity effect.

Design parameter Value (mm)

Center of piezoelectricelement (a1)

lC/2 + 1.5

Length of metalpressure-sensitive element(lM = l)

21

Thickness of metalpressure-sensitive element(hM)

2.94, 3.08, 3.22

Width of metalpressure-sensitive element(bM, bC)

5

Thickness ofpiezoelectric element (hC)

0.3–0.7

Length ofpiezoelectric element (lC)

0.25lM–0.9lM

Number ofpiezoelectric actuator (N)

4

Table 2. Elastic modulus of components.

Materials

Elasticmodulus(GPa)

Poisson’sratio

Density(kg m−3)

Metal 144.9 0.29 8110Piezoelectric element 40 0.30 7700Connective seam 15 0.31 1800

4. Numerical simulation

For investigating the rigidity effects of a bimorph piezoelectricactuator, the fundamental design parameters shown in table 1are chosen. The elastic moduli of components are shownin table 2. The important characteristics due to thepiezoelectric rigidity effects have influences on the angularrate of mechanical dither. Using equations (12) and (26),the variations of angular rate are investigated through thenumerical simulations. From the results, the feasible geometricdimensions of the piezoelectric element can be studied andreflected in the fabrication of mechanical dithers. The angularrate trends relative to length and thickness of the piezoelectricelement are shown in figure 6. The geometric dimensions ofmechanical dithers presented in [5] are used for verifying theproposed numerical equations. On the condition of l L

C � lC �l UC and h L

C � hC � h UC , the angular rates of the previous

mechanical dithers are calculated by equation (26). Figure 6shows the frequency ranges on the thickness and length of thepiezoelectric element of the mechanical dither type 1. Throughthese contours of frequency ranges, the resonant frequencyabout the rotation mode can be designed with considerationof the geometric dimensions of components and be separatedwith other structural modes.

In order to verify two numerical equations of resonantfrequency of mechanical dither, the experiment is performedusing the rectangular model shown in [5]. Under the electricalsignals, the angular velocities are investigated through therelative angular velocity sensor. Each mechanical dither isdesigned for being separately operated at intervals of 20 Hz.Using the numerical equations, the resonant frequencies ofmechanical dithers are compared in table 3. In table 3, the

1047

D-C Lee et al

(a)

(b)

(c)

Figure 6. Frequency ranges of the mechanical dither type on thedimensions of the piezoelectric element. (a) Dither type 1, (b) dithertype 2, (c) dither type 3.

Table 3. Resonant frequencies of mechanical dithers of variousthicknesses.

Type

ThicknesshM

(mm)

Testfreq.(Hz)

Simulationsfreq.(Hz)

Numericalequa-tion (12)freq. (Hz)

Numericalequa-tion (26)freq. (Hz)

1 2.94 363.00 363.2 359.77 363.502 3.08 382.00 383.1 380.07 382.953 3.22 404.03 406.1 402.52 405.57

calculated resonant frequencies from numerical equation (12)have the relative errors of 1.7% and the ones from numerical

120

100

80

60

40

20

010 100 1000 2000

deg/

sec

Freq. (Hz)

(a)

(b)

(c)

10 100 1000 2000

deg/

sec

Freq. (Hz)

120

100

80

60

40

20

0

10 100 1000 2000

deg/

sec

Freq. (Hz)

120

100

80

60

40

20

0

Figure 7. Frequency response results of ring resonator throughsimulation. (a) Frequency response of the mechanical dither type 1,(b) frequency response of the mechanical dither type 2, (c) frequencyresponse of the mechanical dither type 3.

equation (26) have the relative errors of 0.3%. Consequently,the numerical equations with consideration of the rigidityeffects of the piezoelectric element can more accurately bepredicted. Each mechanical dither-based resonator modelis simulated with the structural damping 3% predicted fromthe previous experiments and the intervals of 1 Hz and thefrequency response results are simulated as shown in figure 7.These simulation models include the finite element modelsof piezoelectric layers and connective seam layers, and theirresults have the relative error 0.5% to the test results, comparedwith the simulation results in [5]. In the numerical equations,the standard material properties of piezoelectric elements and

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A material rigidity effect of a bimorph piezoelectric actuator

seam elements are used, but in fact, they have the stochasticdistributions of material properties. In future work, thestochastic distributions will be considered.

5. Conclusion

From the above numerical results, the motion of thepiezoelectric actuator can depend on the feasible geometricdimensions of piezoelectric layers, connective seam layer andnon-piezoelectric layer. The motions of mechanical ditherwithout consideration of the rigidity effects of the piezoelectricand connective seam layers can more accurately be predicted.Moreover, the manufactured piezoelectric layer and connectivelayer include many uncertain environmental conditions andtheir characteristics have to be included in the inertial sensordesign. Accordingly, the presented numerical equationswith such piezoelectric parameters as charge, capacitance,dielectric permeability, dielectric constant and transversepiezoelectric constant may consider the stochastic distributioncharacteristics of the manufactured environments.

Acknowledgments

This work was financially supported by the Ministry ofEducation and Human Resources Development (MOE), theMinistry of Commerce, Industry and Energy (MOCIE) and theMinistry of Labor (MOLAB) through the fostering project ofthe Lab of Excellency.

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