the effect that PVDF is ferroelectric (Kepler and Anderson,. 1978), which means that the piezoelectric strength can be changed or reversed by poling. One thing ...
Reprinted from June 1990, Vol. 112, Journal of Applied Mechanics
I
Modal Sensors/Actuators C.-K. Lee IBM Research Division, Almaden Research Center, San Jose, CA 95120-6099 Assoc. Mem. ASME
F. C. Moon Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853·7501 Mem.ASME
A piezoelectric laminate theory that uses the piezoelectric phenomenon to ejject distributed control and sensing oj structural vibration oj a flexible plate has been used to develop a class oj distributed sensor/actuators, that oj modal sensors! actuators. The one-dimensional modal sensors!actuator equations are jirst derived theoretically and then examined experimentally. These modal equations indicate that distributed piezoelectric sensors!actuators can be adopted to measure!excite specific modes oj one-dimensional plates and beams. Ij constructed correctly, actuator! observer spillover will not be present in systems adopting these types oj sensors/ actuators. A mode 1 and a mode 2 sensor jor a one-dimensional cantilever plate were constructed and tested to examine the applicability oj the modal sensors! actuators. A modal coordinate analyzer which allows us to measure any specific modal coordinate on-line real-time is proposed. Finally, a way to create a special two-dimensional modal sensor is presented.
Introduction Investigating the use of discrete point sensors/actuators to control the vibrations of flexible structures has been a major research subject in the last ten years (Balas, 1978a, 1978b, 1979; Oz and Meirovitch, 1980; Bar-Kana et al., 1983; Meirovitch et al., 1983; Schafer, 1985; Natori et al., 1988). These control strategies often suffer from actuator/observer spillover as shown by Balas (l978b), where actuator and ob server spillover due to the residual (uncontrolled) models leads to instabilities in the closed-loop system. Techniques such as prefiltering (Balas, 1978b) and modal-filtering (Meirovitch and Baruh, 1982) have been developed over the years to overcome the spillover of the independent modal-space control method. It should be noted that the modal filtering is carried out in space and not in time (Meirovitch and Baruh, 1985). Use of PZT (lead, zirconate, titanate) ceramic and PVDF (polyvi nylidene fluoride) thin film as a distributed active damper for beam vibration have been studied (Bailey and Hubbard, 1985; Burke and Hubbard, 1987; Plump et aI., 1987; Hanagud et al., 1985, 1986; Crawley and de Luis, 1985, 1986; Fanson and Caughey, 1987; Fanson and Garba, 1988). In this paper, using a concept similar to modal-filtering, a novel class of distributed sensors/actuators, modal sensors!actuators for certain types of beams and plates will be presented. These modal sensors! actuators will sense or actuate the modal coordinate of a par ticular mode of the beam or plate directly without extensive on-line real-time computational requirements. These types of sensors!actuators can be easily implemented into structural systems using piezoelectric laminae, such as PZT, PVDF, etc. Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the JOURNAL OF APPLIED ME CHANICS. Discussion on this paper should be addressed to the Technical Editor, Leon M. Keer, The Technological Institute. Northwestern University, Evanston. IL 60208, and will be accepted until two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, June 30, 1988; final revision, February 15. 1989.
4341 Vol. 57, JUNE 1990
Theory of One-Dimensional Distributed Sensors/Ac tuators There are two essential ideas in this paper: first, to use the directional properties of a piezoelectric material to design a composite structure, and second, to shape the electrodes to achieve the desired input-output characteristics of the sensor. The basic configuration of the laminates that will be studied in this paper is shown in Fig. I, where the shim metal layer (layer 2) is made from isotropic material such as aluminum, stainless steel, etc., and can be considered here as part of the existing structure. The piezoelectric laminae can be made from any piezoelectric materials and are considered as added layers for sensing, actuating, and even controlling the structure vi bration. An interesting fact about PVDF worth mentioning is that the Young's modulus of PVDF manufactured under cer tain conditions is the same in all directions (Tamura et al., 1977; Kepler, 1978). These types of mechanically isotropic PVDF will be used as the piezoelectric laminae in this paper. Even if mechanically anisotropic PVDF is used as the piezo electric laminae, the mechanical stiffness of PVDF is usually ten times smaller than the existing structure, thus the me chanical anisotropy of the PVDF can generally be neglected. Since all the laminae are made from isotropic material, the laminates constructed will be transverse isotropic in the xy plane as shown in Fig. 1. The skew angle (J in Fig. 1 is defined as an angle rotating from the x-axis to the Xl -axis and obeying the sign convention of the right-hand rule with respect to the z-axis. One thing that should be noted is that the z -axis can either point to the positive or the negative z-axis direction. These facts will be used later to develop the modal sensors! actuators. Extending the Kirchoff hypothesis (Ashton, et al.), which results in a plane stress state for a thin plate, to piezoelectric . laminae will yield the following constitutive equations for pi ezoelectric laminae (Lee, 1987, 1988) written in IEEE compact matrix notation (IEEE std., 1978): I
Transactions of the ASM E
80 0> I
0
40
N E
20
:::::::J
Z 3jl3yer 2 Isotropic
",~~":'-'"~
, f' i : : j '. = 1.
I
x,l
'
I
60
>
.,cD .,N .,
-20
OM
-40
0;;;
-60
--..........
...., ....
-------
/.",........-
.................
>::.... ............------~ >', ......... _............ .,-'
--.
-80 '----I_.....I._.....L._-'-_-'-_......_ . . I . -......... 0,00 0,39 0,79 1.18 1,57 1,96 2,36 2,75 3,14 Z,31
Skew Angle (rad) Fig.3
Fig. 1 Three·layer PVOF·shlm metal composites (x, y. zrepresent lam· Inate axes; x'. y' , z' represent laminae axes)
t :
z,3 (poling direction)
~
Piezoelectric stress/charge constant of PVOF versus skew angle
for PVDF is that experimental data published by Fukada and Sakurai (1971) showed that the polarization profile can have the same variation for different piezoelectric constants. Thus, we can use the same function Po(x,y) to model the variations in this paper. The skew angle effect (Fig. 1) is modeled by performing coordinate transformation for the piezoelectric constants between the laminae and laminate axes:
electrode
[Tm(O)1
[ ~'I'] ~'2'
,
(4)
e3 '6'
poling direction (1 )arrow:surlace subject to negative poling field, (2) tail :suriace subject to positive poling field.
(5)
Fig. 2 Coordinate axes of PVOF film
where the transformation matrix [Tm(O)] has the following form:
-2mn] m2 2mn, -mn m2 _n 2
(1)
D J , = EJ 'J,E3, + ¢l,],poT], +¢l'2,poT z ' + ¢l'6 'PoT6"
(2)
where the vectors T, S, D, E represent the stress, strain, ele(;tric displacement, and electric field, respectively, E is the permi tivity constant of a piezoelectric lamina, and the prime on the subscript is used to designate that the constant is written with respect to the laminae axes. In addition, Ypf(l
[el
[
II~)
IIpYp/(l
IIpYp/(l- II~) Yp/(1
o
0
II~) II~)
0 0 Yp/2(1
] •
(3)
+ lip)
is the stiffness matrix of the piezoelectric lamina, [e~ 'I' e~ '2' ~ '6']/ = [c][df'I' df'2' df'6' j1, where the superscript "t" represents the transposition of a matrix, the constants e and d are the piezoelectric stress/charge and strain/charge con stants of the piezoelectric laminae, the superscript "0" indi cates that the piezoelectric quantity specified is a constant, while the variation of the polarization profile of the laminae is modeled solely by Po(x,y). More specifically, Po(x,y) rep resents the polarization profile which is introduced to model the effect that PVDF is ferroelectric (Kepler and Anderson, 1978), which means that the piezoelectric strength can be changed or reversed by poling. One thing that should be noted
Journal of Applied Mechanics
I·
(6)
m = cosO, n =sinO, and 0 is the skew angle . For the commercially available PVDF thin film (Kynar Film Manual, 1983) used in this paper, when written with respect to the coordinate system shown in Fig. 2, df, 1' 23 x 10- 12m/V (or coullN) and df'2' = 3 X 10- 12 m/V, and df'6' =0. Therefore, the change of the e constants versus the skew angle can be shown in Fig. 3. Equations (1) and (2) allow us to introduce the piezoelectric property into the classical laminated plate theory (Ashton et al.). In fact, equation (1) forms the base of the distributed actuator equation and equation (2) the base of the distributed sensor equation. Upon introducing equation (1) into the classical laminated plate theory and using equations (4) and (5) to model the skew angle effect, the actuator equation for a laminated piezoelectric plate as shown in Fig. 1 can be obtained as follows (Lee, 1987, 1988):
\ a4 w Irw a4 w} a2w DIl lax" + 2aray + af' + phiJi2
where JUNE 1990, Vol. 57/435
(8)
is the flexural rigidity of the plate; w w(x, y, /) is the out of-plane displacement of the plate; Y, P, h represent the Young's modulus, Poisson's ratio, and the thickness of the laminae, respectively; subscripts p and s are used to designate that of the PVDF layer or the shim metal layer; zp is the z-coordinate of the midplane of the third layer, i.e., Zp (h p + hs )/2 in Fig. 1; p represents the equivalent density of a laminate, e.g., for a three-layered PVDF-shim metal laminate as shown in Fig. 1 where p
(EPkhk) /h=(2pphp+pA)/h.
When an electric field is connected to a piezoelectric lamina, only the portion of the lamina that is covered by electrode on both sides of the surfaces, termed effective surface electrode, will be affected by the externally applied electric field. There fore, we can use the surface electrode to control the location of the force induced by the electromechanical interactions of the piezoelectric laminae. This effective surface electrode prop erty is modeled by F(x, y). More specifically, F(x, y) equals one if (x, y) is covered by electrode on both sides of the lamina, otherwise it is zero. A similar effect can be found in the pie zoelectric sensor. As the piezoelectric material is dielectric, the electric charge generated due to the external mechanical dis turbance will be detected only if the charge is collected through the surface electrode to an external measurement device. In other words, this selective charge collecting phenomenon per formed by a surface electrode is equivalent to doing signal processing through the integration of spatial domain. This is also one of the fundamental concepts that is used in this paper to create modal sensors/actuators. If we assume that there is no y-dependence in the displace ment w, i.e., w=(x, t), equation (7) can be reduced to a one dimensional version:
a
a
4 2 w w 0 tf;r; (x) Dlla~ +phiJii= -2hpzpG(t)e31~'
equation, (equation (9», instead of the beam equation should be used to model the dynamic behavior (Pao, 1988). Expanding this discussion and recognizing that the differ ence between a single-layer one-dimensional plate and a one dimensional laminated plate is only in the definition of used, we can use equation (9) to model a laminated piezoelectric beam of unit width simply by setting Pp5 P,50 in the definition of DII presented in equation (8). Substituting equation (2) into Gauss' law and recognizing the fact that the equivalent circuit of a piezoelectric lamina, while it is not resonant in its thickness mode, is a capacitor with a parallel leakage resistor, we can approximate the closed circuit charge signal qk(t) measured through the electrode of the kth lamina (Fig. 1) as follows (Lee, 1987, 1988):
DII
qk(l) =
~I~; +~2~y~ +2~6::~]dXdY, k= 1,3, (11)
where Zk is the z-coordinate of the midplane of the kth layer, e.g., in Fig. I, ZI= Z3 zp= -(hp +hs )/2, S is the area that is covered by the piezoelectric lamina (with and without electrode). For a single-span slender plate with the same configuration as that shown in Fig. I, w=w(x, t) can again be substituted into equation (II) to obtain the one-dimensional sensor equa tion. Furthermore, since there is no moment or force resultants along the y-direction, the summation of F(x, y)Po(x,y) along the y-direction can again be adopted to represent the spatial dependent part of the interactions between the PVDF and the electrical field as that done in the actuator equation. In other words, one can use ;r; (x) in equation (10) to model the effect of the effective electrode F(x, y) and the polarization profile Po (x, y) summed along they-direction. Therefore, the sensor equation of a single span plate becomes
a
2w _ fa z~IJo;r;(x)ardx,
qk(t)=
(12)
where a is the length of the plate (Fig. 1). (9)
Theory of Modal Sensors!Actuators
where
H b
;r; (x)
=
F(x,y)Po(X,y)dy,
(10)
The transverse deformation of a one-dimensional plate can be decomposed into the modal summation
2
is introduced to represent the spatial dependent part of the interactions between PVDF and the electric field summed along the y-direction, b being the width of the beam. This one dimensional plate equation, equation (9), when reduced to a single layer case, i.e., hp-O in D II , has the same form as that of the beam equation except that the flexural rigidity constant differs by a factor of (I p2). That is, the cylindrical bending effect of a slender plate in the y-direction appears as an increase in stiffness since the flexural rigidity EI of a unit width beam is (1 p2) times the flexural rigidity, D lI , of a one-dimensional slender plate. This (1-;) factor can be further understood by using the concepts of plane-strain approximation and plane stress approximation (Love, 1927). Taking a slender plate as an example, the plane-strain approximation is used along the width direction whenever the thin plate theory is used (Ti moshenko and Woinowsky-Krieger, 1959). The plane-stress approximation, on the other hand, is used along the width direction of a beam. These two concepts indeed unearth the difference and similarity between a one-dimensional plate and a beam as a result of the (1 ;) factor between the formulation of the plane-stress and plane-strain approximation (Love, 1927; Chou and Pagano, 1967). Furthermore, these discussions also indicate that if the thickness of a slender plate as shown in Fig. 1 is much smaller than its width, a one-dimensional plate
4361 Vol. 57, JUNE 1990
I'
ZktFPo [
W(x,t)= EAm(t)cPm(X),
W
(13)
m;1
where Am(t) and cPm(x) are the modal coordinate and mode shape of mode m, respectively. Substituting equation (13) into equation (9) yields that qdt) = EAm(l)Bm,
(14)
m~1
where -
0
fa
Bm=-Z~3IJo;r;(X)
[tfcPm (X)] dr dx.
(15)
Since a one-dimensional plate as well as a beam is a self adjoint system as discussed previously, the modes are orthog onal to each other. This property can be used to help us design a modal sensor/actuator. In addition, since the difference be tween a one-dimensional plate and beam (laminated or ho mogeneous) can be attributed to the value of the flexural rigidity used, the following discussions will remain valid for all cases (beam or plate, laminated or homogeneous) as long as the flexural rigidity is changed accordingly. For example, since the mode shapes of a beam and a one-dimensional plate (laminated or homogeneous) depend on the boundary conditions only, they will be identical for all these cases. However, since the Transactions of the ASME
x ~--------a------~~
I tP1 (x)
poling direction of PVDF
I tPa(x)
a
.--===--
/.
~~
a
Fig. 4
x
~
Mode shape of a one·dlmensional cantilever plate
(b)
natural frequencies are proportional to the square root of the flexural rigidity, they will have to be changed accordingly. Consider a one-dimensional cantilever slender plate with length a as shown in Fig. 4, i.e., DlI has the form of equation (8), the mode shapes cf>m (x) for both the one-dimensional cantilever plate and the cantilever beam are (Blevins, 1985) cf>m (x) = [cosh(AmXla) ~
(c)
sin(AmX1a )] ,
(16)
where Al = 1.87510407, A2 = 4.69409113, etc. satisfy the tran scendental equation cos AcoshA + 1 0, and 0"1 = 0.734095514, 0"2= 1.018467319, etc. are obtained from the formula O"rn = (sinhAm - sinAm)/(coshAm + COS Am)· Combining equations (14) and (15) and the fact that
I:
[ifcf>n (x) 1ti.x2][ifcf>m (x) Idxl]dx =
onm(A~/a3),
(17)
where onm is the Kronecker delta, we know that if we choose ff (x) equal to a scaling constant ILn times the second derivative of a particular mode shape, a modal sensor will be created. Physically, ff(x) is proportional to the modal strain distri bution along the length of the plate. In other words, if ff (x) ILn[ifcf>n(x) Idxl] , (18) equation (15) becomes Bm=
PVDF
cos(AmXla)]
O"m[sinh(AmX1a)
(i.e:lIILn A!la 3)onm.
(19)
Therefore, equation (14) reduces to the modal sensor equation as follows: (20) This equation allows us to measure a particular modal coor dinate directly. In other words, observer spillover will not be present in systems adopting this type of sensor. Several concepts need to be cleared up before equation (20) can be turned into a real physical system. First, if we choose Po == 1, then ff (x) can be achieved by varying the surface elec trode pattern F(x, y) according to equation (10). Second, sim ply varying the surface electrode pattern cannot generate a negative value for ff (x) which is required for portions of ifcf>n(x)ldxl. The answer is to vary the polarization profile Po (x, y). Several configurations that create a mode 2 sensor, which measures the mode 2 coordinate directly are shown in Fig. 5. As we can see, either re-poling the PVDF or wiring the output polarity of different portions of PVDF can be used to create negative values for 3' (x). Instead of measuring a charge signal to create a modal sen sor, we can apply voltage to the effective surface electrode (equation (18» to create a modal actuator. Taking the one dimensional modal sensor just discussed as an example, and substituting equation (18) into equation (9) yields that
Journal of Applied Mechanics
poling direction of PVDF
Fig. 5 Configurations for PVDF mode 2 sensors (the dotted area rep· resents the effective surface electrode)
(21) Substituting equation (13) into equation (21) and using the fact that (22) cfcf>m(x)Idx"= (Amla)4cf>m(x), we can obtain that .;... 4 ';"'ifAr(t) Dl1L."Ar(t) (Aria) cf>r(x) +phL.,,~cf>r(x) r= I
1
=
-2[hp iAIILn(Anla)4]G(t)cf>n(x).
(23)
Multiplying both sides of (23) by cf>m (x), integrating over the length of the one-dimensional plate and employing the fact that Sgcf>mcf>,dx = aOmr yields the modal actuator equation: ifAn(t)
ph~+
4
[DlI(Anla) ]An(t) (24)
This equation shows that if ff (x) has the form of equation (18), only the modal coordinate An(t) of mode n is under the infiuence of the PVDF mode n actuator. In other words, we can create a distributed actuator which excites each particular mode independently so that actuator spillover will not be an issue in one-dimensional flexible plates actuated by this type of actuator. The results shown can be easily extended to all other boundary conditions of one-dimensional plates and beams.
Experimental Setup and Procedures An experimental setup as shown in Fig. 6 is used to check how well different modal sensors can distinguish clifferent modes. Mode I and mode 2 sensors were fabricated by first pasting two pieces of I100ILm thick PVDF on both sides of a 203.2-ILm (8 mil) thick stainless steel shim to form a 14.66 cm long, 1.0 cm wide cantilever slender plate. The surface electrode of PVDF is then etched according to equation (18) with ILl = (al Al)212, IL2 = (a/}..0 2/4. From the discussion of the effective sur face electrode concept, we know that only the electrode on one surface of the piezoelectric lamina needs to be etched to JUNE 1990, Vol. 57 1437
70
Impedance Match Circuit
T
60 f «i e
Ol
i:i5 output
:; 0
50
40 I 30
'">
20 I
Ol
10
u..
0
i:i5
PVDF Mode 1 Sensor
I-
.f «i e
I
peak: 64.532 at 7.727 Hz
peak: 12.032 at 47.577 Hz
.A
)\......
0
> a. '0 .2 Fig. 6 Experimental setup for modal sensors (the dotted area repre· sents the effective surface electrode)
til
a::
Experimental Results The transfer function, magnitude ratio of the PVDF sensor signal and the dynamic signal analyzer output driving signal, is shown in Fig. 7. As can be seen here, the natural frequencies of mode 1 and mode 2 of this piezoelectric laminate are 7.270 Hz and 47.577 Hz, respectively. Substituting Ys= 200 x 1O~/ m 2 , Yp=2x 109N/m 2, Ps=7800 Kg/m 3, Pp= 1780 Kg/m3, hs = 203.2 ILm, hp = 1l0ILm, and lis = lip = 1/3, into equation (8) yields that the flexural rigidity, Oil' equals 0.17 N-m. There fore, the natural frequency of the first two modes can be calculated by substituting the corresponding constants into f(Hz) Ar(D Il /ph)1I2/(21rcr). Therefore, the predicted natural 4381 Vol. 57, JUNE 1990
PVDF Mode 2 Sensor
50 I 40
peak: 67.061 at 47.577 Hz
Q)
"e :E
30 l-
co
20
Ol
achieve the selective charge collecting phenomenon predicted by the theory. Therefore, only the electrode on the two outer surfaces of the piezoelectric laminates was etched. The PVDF film used in this experiment was sputtered with a nickel-alu minum electrode (150 A nickel base layer and 400 A aluminum cover layer). The most common etchant for printed circuit board processing-ferric chloride (FeCI) was used to etch the nickel-aluminum electrode despite the fact that nickel is gen erally considered not to react to FeCl3 solution. It is believed that the thin film nature of the electrode and the strong chem ical activity of the aluminum are the main reason that FeCI3 is usable here. This finding makes the physical implementation of the surface electrode effect readily available. A HP 3562 dynamic signal analyzer is used to perform the sweep sine measurement. The sine wave from 5 Hz to 55 Hz with a sweep rate of 6.87 mHz/sec is used to drive the shaker which in turn drives the PVDF modal sensors. The PVDF sensor signal is measured by connecting the PVDF surface electrode such as shown in Figs. 5 and 6 into the impedance match circuit. This in-house impedance match circuit is made from a National Semiconductor LM163 precision instrument amplifier. The twin shield drivers of this 16-pin dual-in-line package eliminates the capacitive loading to PVDF and reduces sensor signal bandwidth loss due to cable capacitance. Also, the high input impedance of LM163, 20 GO for both the com mon and differential mode with a gain of 10, makes the corner frequency of the high pass filter circuit formed by PVDF sensor and the LM163 below 0.1 Hz. Thus, for a frequency higher than 1 Hz, the signal measured will not be subjected to dis t~rti
