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Piezoelectric bimorph bending sensor for shear-stress measurement in fluid flow
ELSEVI ER
Sensorsand ActuatorsA 55 (1996) 157-162
Piezoelectric bimorph bending sensor for shear-stress measurement in fluid flow D . R o c h e ~, C . R i c h a r d ~, L . E y r a u d ~, C . A u d o l y b ' Laboratoire de Gdnie Electrique et Fdrro~lectricitd. lnstitut Nationaldes Sciences Appliqude~ de Lyon. B~t. 504. 20. avenue Albert Einstein. 69621 Villeudmnne Cedex. France b DCN Inglnierie Sud/LSM, Le Brnsc, 83800 Toulon Nayal, France
Received 13 June 1995;revised2 February 1996;accept~l3 May 1996
Abstract The measurement of shear stresses developed by a boundary layer is a fundamental problem of fluid dynamics. Knowing the mean shear stress and its fluctuations leads to a better description of the boundary layer and a better understanding of flow noise coupling. The shear stresses to be measured in a laminar or a turbulent flow amount to only a few pascal. The usual types of piezoelectric force sensors cannot perform this measurement with a satisfactory accuracy. The proposed sensor performs a direct measurement of shear stresses, it is composed of several piezoelectric bimorphs, which convert a very low force into a large amount of electrical charge. The typical device sensitivity is 200 nC N- *. Integrating the shear stresses on a surface of a few square centimetres allows shear s ~ to be measured down to ! Pa. ff,ey~,ords: Fluid flow; Piezoelectricbimorphs,Shearstress
1. lntroductio~,
Fluid flow past a boundary produces shear stresses due to the viscosity of the fluid. Near the wall, the velocity of the fluic; increases from zero at the wall to its fu!! value at the distance 8 from the wall which corresponds to the boundarylayer thickness 8 (Fig. 1). Considering a flat plate, whose extremity lies at 0 and in a Ox-oriented flow, the boundary layer can be divided into three regions [ 1 ]: Q the laminar zone, near the leading edge, where the flow is steady ~b the transition zone, where, the flow changes intermittently between laminar and turbulent states o the turbulent zone, in which an irregular fluctuation is superimposed on the main stream.
Y 8
v
~
.........
tamim~
The wall sheafing st~sses in the boundary layer can be considerable because of the large velocity gradient across the flow. Various measurement techniques have already been used to determine the shear stress at the wall surface. For example, devices such as the Preston tube or the Stanton tube and hotfilm anemometers allow the estimation of aerodynamic shear stresses with i r~.¢Erect techniques, through the measurement of a pressure gradient and a heat transfer, respectively. However, they strongly depend on empirical laws, with no guarantee of their accuracy in the considered flow [2]. The floating-element sensor, developed for aerodynamic flows [ 2,3], is another device used for shear-stress measurements (Fig. 2). One model of such a sensor allows ~he measurement of 1 Pa shear stress by measuring the capacitance variation induced by the motion of a sliding armature facing a fixed one [4]. "gX
~
turbal~ zone Fig. I. Boundary-layerdescription.
1~924-4247/96/$15.00© 1996ElsevierScienceS.A. All rightsreserved ~iqlS092,*-4247 (96) 01307-6
m Fig. 2. Moating-element sensorprinciple [2].
158
D. Roche et al. I Sensors and Actuators A 55 (19'96) 157-162
Fig. 3. Bimorphstress.sensor. Surface acoustic wave (SAW) devices have also been used for this kind of measurement. As these surfece waves are very sensitive to external perturbations, the shear stresses are responsible for a shift of the SAW oscillator resonance frequency [5,6]. The proposed device is a floating-element-type sensor (Fig. 3). It is composed of a sensing armature in contact with the flow, integrating the shear stress applied on its surface. This floating armature is at~;::;hed to a fixed one by at least two piezoelectric bimorphs and allows the measurement of the drag force resulting from the shear effort [7,8]. As a first step in the study, onty the bimorph transducer behaviour in air is considered in this paper. The device description and operating principles are first proposed. Next, the calibration procedure and the sensitivity of variot:s prototypes are given. A discussion on the transducer sensitivity optimization is finally proposed with the help of an analytical model and finite-element modelling (FEM). 2. Device description Bimorph bending allows a very low shear-induced force to be converted into a very high flexion deformation. This strain results in high traction and compression stresses in the constitutive piezoelectric plates. Piezoelectric bimorph bending consequently leads to the generation of a large electrical charge through the lateral piezoelectric effect quantified by the coefficient d3, of the piezoceramic plates. Fig. 4 is a schematic view of the double-bimorph prototype structure used in this study. Its global dimensions are 29 m m × 24 mm × 13 ram. This device is composed of: • two symmetricalarmatures (1 and2) madeofastfffmaterial (aluminium, bra~s or epoxy) • two bimorphs (3 and 5), each made up of two piezeelectric plates (20 m m × 12 mm×0.3 mm) (3a, 3b and 5a, 5b). These plates are made of P194 PZT ceramics (Quartz et Silice, Saint-Gobain) chosen for their large piezoelectric coeffici.,mt d3,. They are bonded with an epoxy resin (Epotek E505). This adhesive f,lm ensures good mechanical and ele¢:~xicalcontacts between the electrodes • four err.,~eddings (4a. 4b and 6a, 6b) realized with a rigid epoxy v~sin (Cib~ '~z,gy At aldit D and Hardener HY956 ). The polymer mouldings permit linear clamping with a good reproducibility. In the adopted configuration, the armatu: es ! and 2 present an L shape. For test purposes, it is then possible to apply
Fig. 4. Piezoelectricdouble-bimolphtransducer.
Fig. 5. Straindistributionintoa bimorph. normal forces on the lateral faces. This is similar to a mechanical loading that would lesuit from shear stresses applied to the top and bottom faces. The use of at least two parallel bimorphs allows the sliding armature to move parallel to the fixed one, preventing any rotation of the sensing armature that could perturb the flow. Under a shear-stress loading, the piezoelectric bimorphs bend as shown in Fig. 5: a low stress is converted into a high double-flexion deformation of the bimorphs. The traction and compression stresses are distributed in a piezoelectric bimorph as shown. The centre of the bimorph corresponds to an inflection point where the stresses vanish. Considering the double-flexion strain induced by the clamping, appropriate metallization and poling are necessary to obtain the optimal generated electrical charge. Fig. 6 shows an exploded view of the bimorph. Each piezoelectric plate is
g
b
h
Fig. 6. Explodedviewof a bimorph.
D. Roche et al. /Sensors and Actuators A 55 (1996) 157-162
divided into two parts where the poling vectors PI and P2 have opposite senses. The main electrodes (c end d) are linked together to the first terminal while the secondary electrodes (e, f, g and h), located on each side of the central electrodes (a and b), are connected to the second terminal.
70OO .~e~staaee .........
eooo
159
(Ohm)
B6
B7 ........
5000 ................
....................... i......................... 4000
3. Experimental results
[....
2000 .................
3. !. Frequency response under a shear-stress loading 3.1.1. Experimental set-up Fig. 7 shows the experimental set-up used for the frequency response measurements of the transducer. Dynamic stress is applied with a P-810-10 piezoelectric translator (Physical Instruments). A load cell at the bottom of the device performs the measurement of both dynamic and static stresses applied to the sensor. Two charge amplifiers (Bruel & Kjaer 2635 and Kistler 5011 ) allow respectively the measurement of the dynamic force F and the elecurical charge Q. The transducer frequency response s = Q/Fis then computed with a dual-way FFT spectrum analyser (HP 35665A). 3.1.2. Frequency responses Four different transducers, denoted as B6, B7, B8 and B9, have been test,'d+ The influences of, on the one hand, the number of bimorphs [ 2 or 4) and, on the other hand, the plate thickness (0.3 mm or 0.4 mm) have been observed. Table 1 gives the description of the various prototypes and their experimental sensitivities. The experimental sensitivity is an average over the frequency range 10-200 Hz where the frequency response is quasi-constant. The resistance of each transducer has first been measured with an HP 4194A impedance analyser. The maximum of the resistance corresponds to the free resonance frequency oftbe
lit
ii
+iI
Iooo
1~
Frequency (Hz) Fig. 8. Sensorresistance~ 500
.
.
.
.
.
.
.
.
.
.
.
.
+-+,, :+I........ !..... +; + +++!+........ i+++!+++: ........ .... + .l ........ +:/"=:?+ C++r+:" ~
i
i i i iiii
+ .... + i
o ........ "----_-k-=:-~:-'-:-" .,o ........ - ..... "'"t"'.'"'T"~ ~ i ('~0 ® ........
-.:.....
~...~..::..::..'.+..u:
i
:: ........
~ ....
-~---
........
~ ....
:...
i
.,o 10
100
Fig. 9. B7 frequencyresponseundershear-ser~sloading. sensor. All the transducers have a resonance frequency near 1 kHz (Fig. 8). The frequency response of the B7 sensor is plotted in Fig. 9. Note that the valid measuring range with the set-up shown in Fig. 7 is only 10-200 Hz due to a parasitic resonance of the whole testing structure. Consequently, this device does not allow the resonant part of the frequency response, which is above 200 Hz, to be measured, but allows a quick and reliable measurement of the quasistatic sensitivity.
3.2. Sensor resolution and linearity
I
",
+'
Fig. 7. Experimentalset-up.
Table i Experimentalresults Sensor
B6
B8
B7
B9
Number of bimorphs Plate thickness (ram) Sensitivity ( nC N - I ) Resonance frequency (Hz)
2 0.4 130 1302
4 0.4 140 1444
2 0.3 220 846
4 0.3 205 1055
If the sensor sensitivity must be high in order to measure low stresses, its resolution must also be appropriate. At a 100 Hz driving frequency and for various translator excitations, the dynamic stress and the geue~ated electrical charge have been measured with the experimental set-op of Fig. 7 (Fig. 10). The sensor has a linear behaviour in the range of force used (the sensitivity is d O / d F u 140 nC N - i). With such a sensor it is possible to measure a dynamic force as small as 0.3 mN rms, which theoretically corresponds to a dynamic stress approximately equal to ! Pa rms (the floating armature surface is 3 cm2).
D. Roche et al. /Sensors and Actuators A 55 (1996) 157-162
16o
F
,
~ i
o
~
j.,
Fig. I I. Bimorph bending. !
10
Shear force(mN nns) Fig. I0. B8 sensorresolution.
100
3.3. Normal-stress influence
The proposed sen~ar must detect shear slresses without being perturbed by other quantities. In hydrodynamic flows, the measurement must not be affected by normal stresses generated by the vortex lift up. The influence of f,ormal stresses is measured with the device shown in Fig. 7, rotating the sensor through 90°. Table 2 gives the normal sensitivities S, of sensors B7 anti B9 and a comparison with the shear sensitivitiies S,. Due to the electromechanical symmetry of the device, the normal sensitivity is theoretically equal to zero, but some imperfections in the sensor manufacturing process can cause an electrical charge generation under normal-stress loading. With no special precautions, the ratio of the normal sensitivity to the shear sensitivity is close to 2%. This ratio can be notably decreased by using an enhanced fabrication process (batch sorting or symmetrical poling, for example). The planned testing of the device in a hydrodynamic flow will finally show if this ratio is sufficient to characterize the skin friction accurately. 4.
Theoretical analysis and simulations
4.1. Analytical model
Let us consider the bimorph shown in Fig. 11 subjected to both a bending force F and a voltage V. Its length is 2L, its thickness is 2h and its width is w. Written in matrix form, we have the following relationships [8,9]: [ =
$~IIL3
3d3tL------~ 2
where 28 is the total deflection of'.he bimot ph, Q the electrical charge on the electrode.s, F the shear force and Vthe applied voltage. ~3 and ~t are respectively the permittivit~, and the elastic coefficient of the material according to the IEEE Standard on Piezoelectricity [ 10]. The sensor sensitivity in a static mode is then defined by: s (2/
~2(5/
3L2
=t~)~=o=W)~oo= -- ~h2d3,
(2)
The device is reversible. The transducer can be used as a force sensor by measuring the charge as a function of the applied force or as a micro-displacement actuator with displacement as a result of an applied voltage. The sensitivity depends on the material lateral piezoelectric coefficient d3t. Consequently, 'soft' PZT ceramics, such as P188 (d3t---208 pC N - I ) or P194 (d31 =266 pC N - I ) ale preferred. The sensitivity increases with the aspect ratio L / h , while the global strength of the device decreases with it. 4.2. F E M simulation
This study has been done using the ATILA FEM software [tll. First the influence of the plate thickness has been studied. The family of frequency re~;ponses plotted in Fig. 12 corresponds to a two-bimorph uansducer with various PZT plate thicknesses. A 2D plane s'~rain harmonic analysis has been performed. One armature of the sensor is clamped and the mechanical, dielectric and piezoelectric losses oftbe material are taken into account. sensitivity (~/N)
1 (l)
3d31L2 2h 2
t0' 0.3 mm
Table2 Nnrmalsensitivity $:nsot
B7
B9
Normal sensitivitySn (nC N- t )
4.7 2
4.3
:;./St(%)
2.1
tO tO
!00 tO00 Frequency(az) Fig. 12.Shear-sensorfrequencyresponsefor variousPZTplatethicknesses.
D Roche et al./Sensors and Actuators A 55 (1996) 157-162
1o 4
,~n,~itivity (nC/hO --
: 2 bim~ : 3 bimocfas : 4 bimoq~
161
constant, equal to ! .7 in the three cases (experimental results, FEM, analytical model). The resonance frequencies are also in good agreement. Consequently, an overal~ agreement is observed between the experimental and theoretical results. Moreover, the sensor frequency response can be predicted for any type of structure using finite-element modelling.
":i ~ :., ..,~ !'
5. Conclusions l0 100 1000 Frequency (Hz) Fig. 13. Shear-sensor frequency response for various numbers of PZT bimorphs. The thinner the piezoelectric plates are, the better the sensitivity is. In contrast, the sensor resonance frequency and its pass-band decrease as the plates get thinner. The influence of the number of bimorphs is then analysed. Fig. 13 shows the family of frequency responsescorresponding to various n u m b e ~ of bimorphs ( h - - 0.4 ram). They have been obtained using a harmonic analysis with the same hypotheses as in the previous case. The increase in the number of bimorphs does not affect the transducer sensitivity but its resonance frequency and its pass-band are considerably improved. Table 3 presents the theoretical predictions of the analytical model and FEM simulations and gives a comparison with the experimental results for the four different transducers. The discrepancies observed between experimental and theoretical results can be explained as follows. ( 1 ) The errors on the piezoelectric d31, mechanical ~ l and dielectric 43 coefficients. "['he special polarization of the plates cioes not allow coefficients as high as those determined from the" resonance method to be obtained on a homogeneously pole~d plate. (2) FEM simulations suppose ideal conditions: the adhesive joint, as well as the clampiag conditions, are not taken into account. However, the ratio between the sensitivity of a 0.3 mm plate-thickness sensor and a 0.4 m m plate-thickness sensor is Table 3 Sensor characteristics. EXP. experimental results: ~M. analytical model; FEM, finite-element modelling Sensor
B6
B8
B7
B9
Number of ~imorphs Plate thickness (ram) Sensitivity (nC N- a)
2 0.4 130 176 226 1302 1313 430 380
4 0.4 140 176 226 1444 1604 500 490
2 0.3 220 313 384 846 905 275 270
4 0.3 205 313 384 1055 1128 380 350
Free resonance frequency (Hz) Pas,~:-band(Hz)
EXP AM FEM EXP FEM EXP FEM
A new sensor principle is proposed with a sensitivity appropriate to the measurement of the shear stresses developed by an aerodynamic or hydrodynamic boundary layer. Using the bending of at least two parallel piezoelectric bimorphs, this sensor can perform direct quasistatic and dynamic shear-stress measurements. The device is also reversible: the transducer can be used as a micro-displacement actuator. The transducer has a very high sensitivity: the measured sensitivity is 130 nC N - I for a 0.4 m m plate thickness and 220 nC N - i for ~ 0 3 mm one, It can also be- used for frequencies up to 500 Hz. The sensor has a linear behaviour in the range of forces used and allows shear stresses down to ! Pa to be measured. Moreover, the measurement is only weakly affected by normal stresses. This low parasitic sensitivity should normally ensure a proper selection between normal and shear stresses, although this poim has to be confirmed by a test in hydrodynamic flow. Further work will address other developments, such as the influence of embedding the device, for its waterproofing, and of mounting in a wall-type sensor that will finally be tested in real flow.
Acknowledgements The authors are grateful to R. Vignat for his assistance in the sample preparation. This research was supported by the CERDSM-DCN of Toulon.
References
[ I ] H. Schlichting, Boundary Layer Tlieory, McGraw-Hill, Saint Louis, 1968. [2] J.H. Haritonidis. The measurement of wall shear stress, Advances in Fluid Mechanics Measurements, Sininger. New York, 1989,pp. 229261. | 3 ] K.G. Winter, An outline of techniques available for the measurement of skin friction in turbulent boundary layers. Progr. Aerospace Sci.. 18 (1977) !-57. 14] MA. Schmidt, R.T. Howe, S.D. Senturia and J.H. Halito~dis, Design and calibration of a mictofubricated floating element shear stress sensor, IEEE Trans. Electron Devices. ED-15 (1988) 750-757.
162
D. Roche et a l . / Sensors and ActuattJrs A 55 (1996) 157-162
[5] Y.R. Roh, Development of Ioc~/global SAW sensors for measurement of wall shear stress in laminar and turbu!ent flows, Thesis, Pennsylvania State University ( 1991 ). [6] D. Roche. C. Richard, L. Eyr, ud and C. Audoly, Shear stress sensor using a shear horizontal wave SAW device type on a PZT substrate, Ann. Chimie, to be published. [7] D. Roche, C. Richard, L. Eyraud and C. Audoly, Transductear h bilames pi6zo~lectriques fonctionnant cn double flexion, French Patent No. 9505491 (10 May, 1995).
[8] D. Roche, Conception et r~.alisation d'un capteur pi,~zo~lectriqoe de cc,~traintes de cisaillement pour structure adaptative, Thesis, INSA Lyon, 95 ISAL 0062 (1995). [9] J.G. Stairs, S.I. Dalke and TK. Cooney, The constituent equations of piezoelectric bimorphs, Sensors and Actuators A, 28 ( 1991 ) 41-6 I. [ I O] IEEE Standard on Piezoelectricity, ANS1 / IEEE Std 176, USA, 1987, 54 pp. [11] B. Dubus, ATILA finite element code for piezoelectric and magnetostfictive transducer modelling, Version 5.02, User's Manual, Institat Sup~rieur d'Electroniqoe du Nord, Lille, France, 1991.
Sensorsand ActuatorsA 55 (1996) 157-162
Piezoelectric bimorph bending sensor for shear-stress measurement in fluid flow D . R o c h e ~, C . R i c h a r d ~, L . E y r a u d ~, C . A u d o l y b ' Laboratoire de Gdnie Electrique et Fdrro~lectricitd. lnstitut Nationaldes Sciences Appliqude~ de Lyon. B~t. 504. 20. avenue Albert Einstein. 69621 Villeudmnne Cedex. France b DCN Inglnierie Sud/LSM, Le Brnsc, 83800 Toulon Nayal, France
Received 13 June 1995;revised2 February 1996;accept~l3 May 1996
Abstract The measurement of shear stresses developed by a boundary layer is a fundamental problem of fluid dynamics. Knowing the mean shear stress and its fluctuations leads to a better description of the boundary layer and a better understanding of flow noise coupling. The shear stresses to be measured in a laminar or a turbulent flow amount to only a few pascal. The usual types of piezoelectric force sensors cannot perform this measurement with a satisfactory accuracy. The proposed sensor performs a direct measurement of shear stresses, it is composed of several piezoelectric bimorphs, which convert a very low force into a large amount of electrical charge. The typical device sensitivity is 200 nC N- *. Integrating the shear stresses on a surface of a few square centimetres allows shear s ~ to be measured down to ! Pa. ff,ey~,ords: Fluid flow; Piezoelectricbimorphs,Shearstress
1. lntroductio~,
Fluid flow past a boundary produces shear stresses due to the viscosity of the fluid. Near the wall, the velocity of the fluic; increases from zero at the wall to its fu!! value at the distance 8 from the wall which corresponds to the boundarylayer thickness 8 (Fig. 1). Considering a flat plate, whose extremity lies at 0 and in a Ox-oriented flow, the boundary layer can be divided into three regions [ 1 ]: Q the laminar zone, near the leading edge, where the flow is steady ~b the transition zone, where, the flow changes intermittently between laminar and turbulent states o the turbulent zone, in which an irregular fluctuation is superimposed on the main stream.
Y 8
v
~
.........
tamim~
The wall sheafing st~sses in the boundary layer can be considerable because of the large velocity gradient across the flow. Various measurement techniques have already been used to determine the shear stress at the wall surface. For example, devices such as the Preston tube or the Stanton tube and hotfilm anemometers allow the estimation of aerodynamic shear stresses with i r~.¢Erect techniques, through the measurement of a pressure gradient and a heat transfer, respectively. However, they strongly depend on empirical laws, with no guarantee of their accuracy in the considered flow [2]. The floating-element sensor, developed for aerodynamic flows [ 2,3], is another device used for shear-stress measurements (Fig. 2). One model of such a sensor allows ~he measurement of 1 Pa shear stress by measuring the capacitance variation induced by the motion of a sliding armature facing a fixed one [4]. "gX
~
turbal~ zone Fig. I. Boundary-layerdescription.
1~924-4247/96/$15.00© 1996ElsevierScienceS.A. All rightsreserved ~iqlS092,*-4247 (96) 01307-6
m Fig. 2. Moating-element sensorprinciple [2].
158
D. Roche et al. I Sensors and Actuators A 55 (19'96) 157-162
Fig. 3. Bimorphstress.sensor. Surface acoustic wave (SAW) devices have also been used for this kind of measurement. As these surfece waves are very sensitive to external perturbations, the shear stresses are responsible for a shift of the SAW oscillator resonance frequency [5,6]. The proposed device is a floating-element-type sensor (Fig. 3). It is composed of a sensing armature in contact with the flow, integrating the shear stress applied on its surface. This floating armature is at~;::;hed to a fixed one by at least two piezoelectric bimorphs and allows the measurement of the drag force resulting from the shear effort [7,8]. As a first step in the study, onty the bimorph transducer behaviour in air is considered in this paper. The device description and operating principles are first proposed. Next, the calibration procedure and the sensitivity of variot:s prototypes are given. A discussion on the transducer sensitivity optimization is finally proposed with the help of an analytical model and finite-element modelling (FEM). 2. Device description Bimorph bending allows a very low shear-induced force to be converted into a very high flexion deformation. This strain results in high traction and compression stresses in the constitutive piezoelectric plates. Piezoelectric bimorph bending consequently leads to the generation of a large electrical charge through the lateral piezoelectric effect quantified by the coefficient d3, of the piezoceramic plates. Fig. 4 is a schematic view of the double-bimorph prototype structure used in this study. Its global dimensions are 29 m m × 24 mm × 13 ram. This device is composed of: • two symmetricalarmatures (1 and2) madeofastfffmaterial (aluminium, bra~s or epoxy) • two bimorphs (3 and 5), each made up of two piezeelectric plates (20 m m × 12 mm×0.3 mm) (3a, 3b and 5a, 5b). These plates are made of P194 PZT ceramics (Quartz et Silice, Saint-Gobain) chosen for their large piezoelectric coeffici.,mt d3,. They are bonded with an epoxy resin (Epotek E505). This adhesive f,lm ensures good mechanical and ele¢:~xicalcontacts between the electrodes • four err.,~eddings (4a. 4b and 6a, 6b) realized with a rigid epoxy v~sin (Cib~ '~z,gy At aldit D and Hardener HY956 ). The polymer mouldings permit linear clamping with a good reproducibility. In the adopted configuration, the armatu: es ! and 2 present an L shape. For test purposes, it is then possible to apply
Fig. 4. Piezoelectricdouble-bimolphtransducer.
Fig. 5. Straindistributionintoa bimorph. normal forces on the lateral faces. This is similar to a mechanical loading that would lesuit from shear stresses applied to the top and bottom faces. The use of at least two parallel bimorphs allows the sliding armature to move parallel to the fixed one, preventing any rotation of the sensing armature that could perturb the flow. Under a shear-stress loading, the piezoelectric bimorphs bend as shown in Fig. 5: a low stress is converted into a high double-flexion deformation of the bimorphs. The traction and compression stresses are distributed in a piezoelectric bimorph as shown. The centre of the bimorph corresponds to an inflection point where the stresses vanish. Considering the double-flexion strain induced by the clamping, appropriate metallization and poling are necessary to obtain the optimal generated electrical charge. Fig. 6 shows an exploded view of the bimorph. Each piezoelectric plate is
g
b
h
Fig. 6. Explodedviewof a bimorph.
D. Roche et al. /Sensors and Actuators A 55 (1996) 157-162
divided into two parts where the poling vectors PI and P2 have opposite senses. The main electrodes (c end d) are linked together to the first terminal while the secondary electrodes (e, f, g and h), located on each side of the central electrodes (a and b), are connected to the second terminal.
70OO .~e~staaee .........
eooo
159
(Ohm)
B6
B7 ........
5000 ................
....................... i......................... 4000
3. Experimental results
[....
2000 .................
3. !. Frequency response under a shear-stress loading 3.1.1. Experimental set-up Fig. 7 shows the experimental set-up used for the frequency response measurements of the transducer. Dynamic stress is applied with a P-810-10 piezoelectric translator (Physical Instruments). A load cell at the bottom of the device performs the measurement of both dynamic and static stresses applied to the sensor. Two charge amplifiers (Bruel & Kjaer 2635 and Kistler 5011 ) allow respectively the measurement of the dynamic force F and the elecurical charge Q. The transducer frequency response s = Q/Fis then computed with a dual-way FFT spectrum analyser (HP 35665A). 3.1.2. Frequency responses Four different transducers, denoted as B6, B7, B8 and B9, have been test,'d+ The influences of, on the one hand, the number of bimorphs [ 2 or 4) and, on the other hand, the plate thickness (0.3 mm or 0.4 mm) have been observed. Table 1 gives the description of the various prototypes and their experimental sensitivities. The experimental sensitivity is an average over the frequency range 10-200 Hz where the frequency response is quasi-constant. The resistance of each transducer has first been measured with an HP 4194A impedance analyser. The maximum of the resistance corresponds to the free resonance frequency oftbe
lit
ii
+iI
Iooo
1~
Frequency (Hz) Fig. 8. Sensorresistance~ 500
.
.
.
.
.
.
.
.
.
.
.
.
+-+,, :+I........ !..... +; + +++!+........ i+++!+++: ........ .... + .l ........ +:/"=:?+ C++r+:" ~
i
i i i iiii
+ .... + i
o ........ "----_-k-=:-~:-'-:-" .,o ........ - ..... "'"t"'.'"'T"~ ~ i ('~0 ® ........
-.:.....
~...~..::..::..'.+..u:
i
:: ........
~ ....
-~---
........
~ ....
:...
i
.,o 10
100
Fig. 9. B7 frequencyresponseundershear-ser~sloading. sensor. All the transducers have a resonance frequency near 1 kHz (Fig. 8). The frequency response of the B7 sensor is plotted in Fig. 9. Note that the valid measuring range with the set-up shown in Fig. 7 is only 10-200 Hz due to a parasitic resonance of the whole testing structure. Consequently, this device does not allow the resonant part of the frequency response, which is above 200 Hz, to be measured, but allows a quick and reliable measurement of the quasistatic sensitivity.
3.2. Sensor resolution and linearity
I
",
+'
Fig. 7. Experimentalset-up.
Table i Experimentalresults Sensor
B6
B8
B7
B9
Number of bimorphs Plate thickness (ram) Sensitivity ( nC N - I ) Resonance frequency (Hz)
2 0.4 130 1302
4 0.4 140 1444
2 0.3 220 846
4 0.3 205 1055
If the sensor sensitivity must be high in order to measure low stresses, its resolution must also be appropriate. At a 100 Hz driving frequency and for various translator excitations, the dynamic stress and the geue~ated electrical charge have been measured with the experimental set-op of Fig. 7 (Fig. 10). The sensor has a linear behaviour in the range of force used (the sensitivity is d O / d F u 140 nC N - i). With such a sensor it is possible to measure a dynamic force as small as 0.3 mN rms, which theoretically corresponds to a dynamic stress approximately equal to ! Pa rms (the floating armature surface is 3 cm2).
D. Roche et al. /Sensors and Actuators A 55 (1996) 157-162
16o
F
,
~ i
o
~
j.,
Fig. I I. Bimorph bending. !
10
Shear force(mN nns) Fig. I0. B8 sensorresolution.
100
3.3. Normal-stress influence
The proposed sen~ar must detect shear slresses without being perturbed by other quantities. In hydrodynamic flows, the measurement must not be affected by normal stresses generated by the vortex lift up. The influence of f,ormal stresses is measured with the device shown in Fig. 7, rotating the sensor through 90°. Table 2 gives the normal sensitivities S, of sensors B7 anti B9 and a comparison with the shear sensitivitiies S,. Due to the electromechanical symmetry of the device, the normal sensitivity is theoretically equal to zero, but some imperfections in the sensor manufacturing process can cause an electrical charge generation under normal-stress loading. With no special precautions, the ratio of the normal sensitivity to the shear sensitivity is close to 2%. This ratio can be notably decreased by using an enhanced fabrication process (batch sorting or symmetrical poling, for example). The planned testing of the device in a hydrodynamic flow will finally show if this ratio is sufficient to characterize the skin friction accurately. 4.
Theoretical analysis and simulations
4.1. Analytical model
Let us consider the bimorph shown in Fig. 11 subjected to both a bending force F and a voltage V. Its length is 2L, its thickness is 2h and its width is w. Written in matrix form, we have the following relationships [8,9]: [ =
$~IIL3
3d3tL------~ 2
where 28 is the total deflection of'.he bimot ph, Q the electrical charge on the electrode.s, F the shear force and Vthe applied voltage. ~3 and ~t are respectively the permittivit~, and the elastic coefficient of the material according to the IEEE Standard on Piezoelectricity [ 10]. The sensor sensitivity in a static mode is then defined by: s (2/
~2(5/
3L2
=t~)~=o=W)~oo= -- ~h2d3,
(2)
The device is reversible. The transducer can be used as a force sensor by measuring the charge as a function of the applied force or as a micro-displacement actuator with displacement as a result of an applied voltage. The sensitivity depends on the material lateral piezoelectric coefficient d3t. Consequently, 'soft' PZT ceramics, such as P188 (d3t---208 pC N - I ) or P194 (d31 =266 pC N - I ) ale preferred. The sensitivity increases with the aspect ratio L / h , while the global strength of the device decreases with it. 4.2. F E M simulation
This study has been done using the ATILA FEM software [tll. First the influence of the plate thickness has been studied. The family of frequency re~;ponses plotted in Fig. 12 corresponds to a two-bimorph uansducer with various PZT plate thicknesses. A 2D plane s'~rain harmonic analysis has been performed. One armature of the sensor is clamped and the mechanical, dielectric and piezoelectric losses oftbe material are taken into account. sensitivity (~/N)
1 (l)
3d31L2 2h 2
t0' 0.3 mm
Table2 Nnrmalsensitivity $:nsot
B7
B9
Normal sensitivitySn (nC N- t )
4.7 2
4.3
:;./St(%)
2.1
tO tO
!00 tO00 Frequency(az) Fig. 12.Shear-sensorfrequencyresponsefor variousPZTplatethicknesses.
D Roche et al./Sensors and Actuators A 55 (1996) 157-162
1o 4
,~n,~itivity (nC/hO --
: 2 bim~ : 3 bimocfas : 4 bimoq~
161
constant, equal to ! .7 in the three cases (experimental results, FEM, analytical model). The resonance frequencies are also in good agreement. Consequently, an overal~ agreement is observed between the experimental and theoretical results. Moreover, the sensor frequency response can be predicted for any type of structure using finite-element modelling.
":i ~ :., ..,~ !'
5. Conclusions l0 100 1000 Frequency (Hz) Fig. 13. Shear-sensor frequency response for various numbers of PZT bimorphs. The thinner the piezoelectric plates are, the better the sensitivity is. In contrast, the sensor resonance frequency and its pass-band decrease as the plates get thinner. The influence of the number of bimorphs is then analysed. Fig. 13 shows the family of frequency responsescorresponding to various n u m b e ~ of bimorphs ( h - - 0.4 ram). They have been obtained using a harmonic analysis with the same hypotheses as in the previous case. The increase in the number of bimorphs does not affect the transducer sensitivity but its resonance frequency and its pass-band are considerably improved. Table 3 presents the theoretical predictions of the analytical model and FEM simulations and gives a comparison with the experimental results for the four different transducers. The discrepancies observed between experimental and theoretical results can be explained as follows. ( 1 ) The errors on the piezoelectric d31, mechanical ~ l and dielectric 43 coefficients. "['he special polarization of the plates cioes not allow coefficients as high as those determined from the" resonance method to be obtained on a homogeneously pole~d plate. (2) FEM simulations suppose ideal conditions: the adhesive joint, as well as the clampiag conditions, are not taken into account. However, the ratio between the sensitivity of a 0.3 mm plate-thickness sensor and a 0.4 m m plate-thickness sensor is Table 3 Sensor characteristics. EXP. experimental results: ~M. analytical model; FEM, finite-element modelling Sensor
B6
B8
B7
B9
Number of ~imorphs Plate thickness (ram) Sensitivity (nC N- a)
2 0.4 130 176 226 1302 1313 430 380
4 0.4 140 176 226 1444 1604 500 490
2 0.3 220 313 384 846 905 275 270
4 0.3 205 313 384 1055 1128 380 350
Free resonance frequency (Hz) Pas,~:-band(Hz)
EXP AM FEM EXP FEM EXP FEM
A new sensor principle is proposed with a sensitivity appropriate to the measurement of the shear stresses developed by an aerodynamic or hydrodynamic boundary layer. Using the bending of at least two parallel piezoelectric bimorphs, this sensor can perform direct quasistatic and dynamic shear-stress measurements. The device is also reversible: the transducer can be used as a micro-displacement actuator. The transducer has a very high sensitivity: the measured sensitivity is 130 nC N - I for a 0.4 m m plate thickness and 220 nC N - i for ~ 0 3 mm one, It can also be- used for frequencies up to 500 Hz. The sensor has a linear behaviour in the range of forces used and allows shear stresses down to ! Pa to be measured. Moreover, the measurement is only weakly affected by normal stresses. This low parasitic sensitivity should normally ensure a proper selection between normal and shear stresses, although this poim has to be confirmed by a test in hydrodynamic flow. Further work will address other developments, such as the influence of embedding the device, for its waterproofing, and of mounting in a wall-type sensor that will finally be tested in real flow.
Acknowledgements The authors are grateful to R. Vignat for his assistance in the sample preparation. This research was supported by the CERDSM-DCN of Toulon.
References
[ I ] H. Schlichting, Boundary Layer Tlieory, McGraw-Hill, Saint Louis, 1968. [2] J.H. Haritonidis. The measurement of wall shear stress, Advances in Fluid Mechanics Measurements, Sininger. New York, 1989,pp. 229261. | 3 ] K.G. Winter, An outline of techniques available for the measurement of skin friction in turbulent boundary layers. Progr. Aerospace Sci.. 18 (1977) !-57. 14] MA. Schmidt, R.T. Howe, S.D. Senturia and J.H. Halito~dis, Design and calibration of a mictofubricated floating element shear stress sensor, IEEE Trans. Electron Devices. ED-15 (1988) 750-757.
162
D. Roche et a l . / Sensors and ActuattJrs A 55 (1996) 157-162
[5] Y.R. Roh, Development of Ioc~/global SAW sensors for measurement of wall shear stress in laminar and turbu!ent flows, Thesis, Pennsylvania State University ( 1991 ). [6] D. Roche. C. Richard, L. Eyr, ud and C. Audoly, Shear stress sensor using a shear horizontal wave SAW device type on a PZT substrate, Ann. Chimie, to be published. [7] D. Roche, C. Richard, L. Eyraud and C. Audoly, Transductear h bilames pi6zo~lectriques fonctionnant cn double flexion, French Patent No. 9505491 (10 May, 1995).
[8] D. Roche, Conception et r~.alisation d'un capteur pi,~zo~lectriqoe de cc,~traintes de cisaillement pour structure adaptative, Thesis, INSA Lyon, 95 ISAL 0062 (1995). [9] J.G. Stairs, S.I. Dalke and TK. Cooney, The constituent equations of piezoelectric bimorphs, Sensors and Actuators A, 28 ( 1991 ) 41-6 I. [ I O] IEEE Standard on Piezoelectricity, ANS1 / IEEE Std 176, USA, 1987, 54 pp. [11] B. Dubus, ATILA finite element code for piezoelectric and magnetostfictive transducer modelling, Version 5.02, User's Manual, Institat Sup~rieur d'Electroniqoe du Nord, Lille, France, 1991.