Journal of Sound and Vibration (1992) 153(2), 335-349
A N E W DISTR_IBUTED SENSOR. A N D A C T U A T O R T H E O R Y FOlK " I N T E L L I G E N T " SHELLS H. S. T z o u
Department of Mechanical Enghleerfltg, Centerfor Robotics and Manufacturfl~g Systems, University of Kentucky, Lexington, Kentucky 40506-0046, U.S.A. (Received 4 September 1989, and accepted fir revisedform 8 Februa0' 1991) An "hltelligent" shell is a shell structure with a self-sensation and action/reaction capabilities contributed by built-in sensors, actuators and control electronic systems. In this study, a new intelligent shell structure composed of a conventional elastic shell, a distributed piezoelectric sensor and a distributed piezoelectric actuator is proposed. The distributed sensor provides a self-monitoring capability to sense shell oscillation and the distributed actuator contributes a self-correction (control) capability to counteract the oscillation. A new generic theory for the intelligent shell system is developed. System equations of motion coupling sensing and control effects are derived. From the theory derived, it is concluded that the distributed sensor is (theoretically) capable of sensing all shell vibration modes and the distributed actuator controlling all shell modes. However, the sensing or control effort of each mode could be different. Two feedback control algorithms, namely direct feedback control and Lyapunov control, are proposed in the paper. The generic theory can be simplified to account for other general geometries, such as cylinder, plates, beams, etc. Applications of the theow are also demonstrated.
1. INTRODUCTION
9
Strong and high-performance structures are always in demand in various engineering designs and applications. In general, conventional elastic structures are "passive" in nature and usually do not possess any "sensation" and "action/reaction" capabilities. Structural response is usually measured by external sensors and transducers---e.g., strain gages, accelerometers, etc.--and structural dynamics control is achieved by external viscoelastic, hydraulic, pneumatic, electromagnetic, etc., actuators [1]. In this study, a new "intelligent" structure concept is proposed. An intelligent structure, by definition, is a structure possessing self-sensation and action/ reaction capabilities. This intelligence can be achieved by integrating sensor/transducer and actuator with the conventional elastic structures. The built-in sensors/transducers can monitor a variety of physical a n d / o r environmental changes and the actuator, through a control system, can respond or correct these changes to maintain a desirable performance. In this study, the sensation and reaction capabilities of the structures are limited to dynamic measurement and control of structures. (Other capabilities, e.g., temperature, pressure, etc., can certainly be added to the, structure to enhance its "intelligence".) Dynamics and control of flexible space structures are o f importance to their highperformance requirement. The inherent natural damping of such structures is usually not sufficient to suppress effectively and quickly the system oscillation. Thus, effective vibration control techniques are usually used to control the structure oscillation. Vibration control techniques are generally classified into two major categories: (1) the "passive" method and (2) the "active" method. Usually, the passive control device is an energy absorption 335 0022--460x/92/050335+ 15 $03.00/0 9 1992Academic Press Limited
336
li. s. "rzou
or dissipation device, e.g., viscoelastic damper, dynamic absorber, shock absorber, etc. [2]. The active device--the actuator/controller--is a counteracting mechanism which generates an "opposing" force or moment actively to counteract the undesirable oscillation. One of the major advantages of active devices over passive devices is a "self-adaptivity", which offers variable control efforts for a variety of operational environments. However, because of this adaptivity, external power supplies (e.g., electrical, hydraulic, pneumatic, etc.) and a decision-making "brain" (such as a computer) are usually required [3, 4]. In addition, in order to monitor the present dynamic state, sensors or transducers are also required to provide feedback signals. Note that conventional transducers/sensors and actuators are discrete in nature, and usually response/activate spatially discrete locations. In this study, a distributed sensor and a distributed actuator integrated with a conventional elastic shell structure are proposed. The proposed new intelligent structure is composed of an elastic structure coupled with a distributed sensor for dynamic monitoring and a distributed actuator for dynamic control. The distributed sensor and actuator are both made of piezoelectric materials~ (The electromechanical behavior'of the piezoelectricity will be briefly discussed later.) Many synthetic and natural piezoelectric materials are available. In general, most of them are crystalline solids, which are dense, brittle and difficult to fabricate into complex shapes. In the sensing and control of"flexible" shells and plates, a "flexible" piezoelectric material is desirable for two reasons. First, the piezoelectric material needs to be closely coupled with the flexible structures, but not to change the dynamic characteristics; e.g., the natural frequencies and modes. Second, the material must possess a certain flexibility, and not be too brittle, so that it will not break during structural vibrations. Thus, a tough and pliant polyvinylidene fluoride (PVDF) is used in this study. (However, it should be noted that the theory developed is general, and not limited to the polymeric piezoelectric PVDF.) Applications of piezoelectric material in vibration control of beams were experimentally and theoretically investigated recently [5-8]. Tzou and Gadre derived a multi-layered shell actuator theory for distributed vibration control of flexible shell structures [9]. Tzou and Tseng also developed a piezoelectric finite element for distributed sensing and control of shells and plates [10]. In this paper, the piezoelectric theory is first briefly reviewed. A generic distributed sensing and control theory of an intelligent shell structure is then developed. Application of the theory to other geometries is also demonstrated.
2. PIEZOELECTRICITY TtlEORY Piezoelectricity is an electromechanical phenomenon in which an electric field is coupled with an elastic field. In general, a piezoelectric material responds to mechanical forces and/or pressures and generates an electrical charge. This is called the direct piezoelectric effect. Conversely, application of an electric field to the material can produce mechanical stress or strain which is called the reciprocal, or converse piezoelectric, effect. In this paper, the direct effect is used for distributed sensing and the converse effect is for the active distributed control of flexible structures. There are two fundamental equations representing the direct and converse piezoelectric effects, respectively [1 1],
{T}=[c~
{E}=[flSl{D}-[h]{S},
(1,2)
where {T} is the stress vector (i.e., {T} = {T,, T22 T33 T23 Tj, Ti2}t), [c O] is the elasticity matrix evaluated at constant dielectric displacement, {S} is the strain vector (i.e., {S} = {SI~ $22 S.~3S2j $3~ SI2}t), [h] is the piezoelectric constant matrix, {D} is the electric displacement vector, [-it indicatesthe matrix transposed, {E} is the electric field vector;
DISTRIBUTED SENSOR A N D ACTUATOR T H E O R Y FOR SIIELLS
337
and [fls] is the dielectric impermeability matrix evaluated at constant strain (a list of nomenclature is given in the Appendix). Alternative expressions can be written as
{S}=tsEl{T}+tal'{e},
{o}=teq{E}+tall{r};
(3,4)
where Is El is the elastic compliance matrix measured at constant electric field, [e r] is the dielectric matrix evaluated at constant strain, and [d] is the piezoelectric constant matrix. The constitutive equations (1)-(4) are assumed to be instantaneously both mechanically and electrically balanced and the two effects can be decoupled. In this research, the piezoelectric material used is a piezoelectric biaxially polarized polyvinylidene fluoride (PVDF) polymer. The piezoelectric matrix [d] of a P V D F polymer can be expressed as
Id0] =
0
00o0 il 0
0
Ld3, d32 d3~
d24
0
0
0
.
(5)
Note that the piezoelectric coefficient d24 is equal to dis for a PVDF that is electrically polarized and not mechanically stretched.
3. SHELL DEFINITION An intelligent structure in this study is modeled as an elastic shell sandwiched between two flexible piezoelectric shell layers (see Figure 1). The top piezoelectric layer serves as a
/ / 9
~z
"N~--/~ Distributed v~ "piezoelectric "~ actuator cl2
Figure I. An intelligent shell with a distributed piezoelectric sensor/actuator.
distributed sensor and the bottom as a distributed actuator. It is assumed that the piezoelectric layer is much thinner than the shell thickness, and that the layers fully cover the shell model. Localization and discretization of the sensor/actuator can be achieved by step functions and/or Dirac delta functions. Note that multi-layers of piezoelectric sheets could also be coupled with the flexible shell. One group can serve as a distributed sensor, while the other serves as a distributed actuator for active vibration suppression and control. The infinitesimal distance ds (see Figure 1) of a shell element in a curvilinear co-ordinate system (al, a2 and a3) is defined by the fundamental form [9, 12] (ds) 2 - A~(da,) 2 +
A~(da2) 2,
(6)
338
H.s. TZOU
where A, and A2 are Lam6's parameters. Following Kirchhoff-Love thin shell theory, one can derive the equations of motion of the flexible shell, by using Hamilton's principle [9; 12], as
O(N,,A2) Oct,
O(N2,A,) Oa 2
OA, OA2 1 FO(M,,A2) N , 2 Oct---~2+ N 22 Oct---~l IRt L
4- O(Mi2A,) "1-M,2 OA, Oa2
a(N,2A2)
~2
.
0,42] +A,A2phff', =A,A2F,
3t"22O a l J
O(N22A,) N,2OA~
OA2
aA,
I [.O(Mi2A2)
0,4,]
4- d(M22A0 ~ - M ~ 2 - - +A,A2ph)i'2=A,A2F2, da2 " 8a, 3/,, d a 2 J
-
a I~ [a(~,_A,) ~(M,,A,) 0tt---~tA, L aa, ") 0a2 ~,,~ T, L ~ FN,, +A,A,
aet2
~,,,
M,,
(7)
(8)
Oal JJ da---T-Mi, oa, JJ
N2:,-]
t ~ + ~ j + A,A, ph,,-,,= A,A~r,:
(9)
where )ih is the acceleration in the ith direction, FI is the externally applied mechanical load, p is the mass density, h is the shell thickness, and I~ and R2 are the radii of curvature of the a, and a2 axes, respectively. NU and M v are the resultant forces and moments,
a3
N~,. Mzl I M21 ~.-*.*"*"C~ ---''''''''~ aIQi5
9
Oj= ~
"~N==. GI z
Figure 2. Definition of force and moment resultants of a shell.
D I S T R I B U T E D SENSOR A N D A C T U A T O R T i t E O R Y FOR SllELLS
339
illustrated in Figure 2 and defined as follows: (1) membrane forces,
4[' ~'~ w~ ,,,,,,,~.~, ~w~ w,-,~
~ ] * " L Z oa, A,A20~
.(~)}
~,,.,~,,-,,, {.,~~_(w~l..,,o 2
~ 3a, \ r i g
~,
,42 3a2
,JJ
,
'
,,o, (il)
(12)
where p is the Poisson ratio, and K is the membrane stiffness, K= (Yh/(l _p2)) where Y is Young's modulus of the shell; (2) bending moments, 1 M,,=D {[;,,•
1
ow;~+,(w~
l
o..~io.,, I
3al \~1 A, Oad d,d2 \R2 ,,423a2] Oa2J +P "~2-~a2\R"-~2 A2 3ag
A,A2 \R,
A, aaJ 3a,JJ'
(13)
o{[, o (,,,~,ow,~+, (,,,,,o~,~io,,~i
[, o (w, l o,,q+, (,.,~, o,..~o.,,-p.
~,4,
.,~._o,,-,,~ {,,,o r, (w~ , o,.,,.,ll +,,, o r ,_(,,,, , ow,llI, ,42 3a2 L,41\~l
,41 aaddJ
~,s~
where D is the bending stiffness, D = {( Y(h)3)/(12(I _p2))}. Note that the resultant foivr and moments could also include piezoelectricity-induced feedback forces and moments, which are to be derived later. The response of a shell structure can be determined by the modal expansion method in which the total dynamic response is a summation of all participating modes Wit with individual modal participation factor r/k [12],
wi(al, a2, t)= ~ 11k(t)Wik(a,, a2),
(16)
k-I
where i= 1, 2, 3 denote the three principal directions. Since, for a distributed system, the number of modes are infinite, k goes from l to infinity. This modal expression will be used to estimate the distributed piezoelectric sensor output for feedback controls. Simplification of the theory to commonly occuring geometries will also be demonstrated in case studies.
4. DISTRIBUTED SENSATION OF SHELLS Recall that the top piezoelectric layer in the intelligent shell structure serves as a distributed sensor which theoretically response to all vibration modes of the structure. In this section, a distributed piezoelectric sensing theory is developed. It is assumed that the
340
II. s.
TZOU
distributed piezoelectric sensor layer is much thinner than that of the shell structure. The piezoelectric sensor strains are assumed to be constant and equal to the outer surface strains of the shell. (Note that in the distributed sensing application, only the direct piezoelectric effect is considered.) Assuming the electric charge is balanced in the piezoelectric sensor layer (an insulator), one can derive an electrical charge equation by using the Gauss theorem [10, 13],
v. {o,} :0,
(17)
where V is the gradient operator. From the system configuration discussed earlier, only the transverse electric field Ej is considered, so that the strains and dielectric displacement D3 are independent ofa~. According to Maxweli's equation, the electric field can be related to the electric potential by {E,} = -V4,.
(18)
The voltage across the electrodes can be obtained by integrating the electric field over the thickness of the piezoelectric sensor layer; i.e.,
gp= -
E3 dct3= S'(h31SI +h32S~-fla3D3),
(19)
where 6* denotes the piezoelectric sensor layer thickness, S~ and S~ are the normal strains in the a, and a2 directions, respectively, and the superscript s denotes the distributed sensor layer. Since the piezoelectric sensor layer is coupled on the outer surface of the shell, the normal strains in the sensor layer can be estimated by [12, 13] [1
a (w,
[, _~ (,,,
s;=a~ ~aa,\m2
OAt(w2
1 aw3/_l.
1
, o,,.q~
, o~(,,,,
1 aw31]'
, o,~.)I '
A2 aad A,A----2act, \m, A, act,I]
~,)
where A~ and A~ denote the distances measured from the neutral surface to the mid-plane of the sensor layer, and wj, )v2 and w3 are the displacements in the three principal directions. Note that it is assumed that bending oscillation dominates the shell motion and, therefore, equations (20) and (21) are the bending strains in two principal directions. Rearranging equation (19), one can write the electric displacement D~ as
D] = (h31S~+ h32S~- ~P/&')/fl~3.
(22)
Since D] is defined as charge per unit area, one can integrate equation (22) over the electroded surface A* to estimate a total surface charge. An open-circuit voltage ~b"condition can be obtained by setting the charge to zero; i.e.,
(h~,S; +h3~$9 d A ' .
=
(23)
I
Substituting the strains into equation (23) yields the distributed sensor output expression in terms of displacements and o.ther system parameters:
-"L {,,,A,[' ~ ('"' l [, o (,,,,
, o,,,.)~
o-, I , o~.(,,,,
+hj2A~ -~2 0ct---~2kR2 A2 aa2] A,A2 aal \RI
, o,,,.)ii~..
A, 0 a J ] }
(24)
DISTRIBUTED
SENSOR AND ACTUATOR
TtIEORY
341
FOR SIIELLS
To examine the modal contribution to the sensor output, one can further substitute the modal expression into equation (24):
q,,(t) IV,k(a,, a,) -~ -
-
--
AIA2 tga2
+h32A~I-~2 9J r
-
-
-
-
AtA2 Oat
k-I
r/~(t)
IV:~(at,
a2)
t3 [ l oo
I 0 ~. tlk(t)wskCa,,a2)) At aal k - ~
A20a2 k- I l
~ ~ rlk(t)tV3~(at, a2))
k-I tlk(t)W,k(a,, a2) At aal k=l
Equation (25) shows that the total output of the distributed sensor is a summation of contributions from all participating modes, and the specific contribution of each mode is determined by the modal participation factor. The modal participation factors ilk are zero for those modes not participating in the shell oscillation. If the surface average and integration is removed from equations (24) and (25), a local voltage amplitude can be estimated by specifying the location (a*, a~) of interest.
5. DISTRIBUTED VIBRATION CONTROL OF StlELLS It is assumed that the piezoelectric actuator layer is not constrained and is free from external in-plane normal forces. Thus, the stress effects can be neglected in the derivation: i.e., this is a stress-free condition. (This stress-free condition implies that the boundaries of the actuator layer cannot be fully constrained.) The induced strains due to imposed control voltages, the converse piezoelectric effect, are used to counteract the shell oscillation. It is also assumed that the applied control voltage ~b" is much more significant than the self-generated voltage r the direct effect, in the distributed actuator. Thus, the selfgenerated voltage ,;b is neglected in the active vibration control system. Note that the distributed sensor output r can also be directly used in a feedback control loop. In this section, this distributed vibration control mechanism is analyzed and two feedback algorithms, namely direct feedback control and Lyapunov control, are proposed. System dynamic equations, including all control effects, are also derived. In the shell definition, the bottom piezoelectric layer is used as a distributed actuator for active vibration suppression and control. It is assumed that the distributed piezoelectric layer is made of a biaxially polarized piezoelectric material. Thus, a voltage r applied to the distributed actuator layer introduces two in-plane normal strains (at and a2 directions) in the layer due to the converse piezoelectric effect:
Sal = (d3t~a)/t5a,
S'~ = (da2~)a)/~a.
(26, 27)
Here 6 ~ is the actuator thickness an~i the superscript a denotes tile distributed piezoelectric actuator. (The induced strains are illustrated in Figure 3.) Note that these strains are generated in the distributed actuator layer which is located some distance away from the shell neutral surface. Thus, these strains introduce counteracting control moments for the shell structure. The sign of the feedback voltage should be carefully controlled so that the induced moments counteract tile shell oscillation.
342
It. S. TZOU
piezoelectric ~ ~ " ~
//"-~...~.. ~"
-
~
~
~"
piezoelecti.ic strains: S~ 9
~,~
.
a= Figure 3. Distributed vibration control by distributed actuator. 5.1. CONTROL FORCES/MOMENTS AND SYSTEM EQUATIONS
The feedback voltage induced in-plane effective forces and moments can be expressed as
N#, =ds, Yr~ba,
M~, =A~ds, Yr~a,
N~2 = ds2 Yp~b",
M~2=A~ds2Ypr", (28-31)
where Yp is the modulus of piezoelectric elasticity and A~' is the moment arm (distance measured from the neutral surface to the mid-plane of the piezoelectric actuator). Note that the feedback voltage ~a is determined by control algorithms which will be discussed later. Substituting these induced normal forces and counteracting moments into the equation of motions yields a(/V,,A2)
O(~[:,A,)_~ll2OA,
aai § Oa~
aa,
~ OA2
aa,
§ a(g122A,) ~_~I,2 OA2_ I~I,,
aa2
Oal
' aa2 ~2L
aA'l
L 9 aa,
at/2
aa2
(33)
Oa,JJ
aa,-M,, a-~a2JJ
F., + A ,A2 llV,, + ,V2,l + A ,A,pln;', = A ,A~Fs, iT
(32)
-~al
-~a2J+ AIA2Ph~'2=A'A2F2'
act2
L •,
aa,
~it, aA, ^ aA~l --_---I+ A,A~ph)~; = A)A~F, Oaz M22 Oa,J
aa2
aaz
1 [O(;I,,A2)
aa--~2+Nz2 aa--~l I~, L
aa2
"l
IR2J
(34)
where the superscript " ^ " terms include the feedback control effects induced by the converse piezoelectric effect. These resultant forces and moments are modified to include the
DISTRIBUTED SENSOR AND ACTUATOR TtIEORY FOR SHELLS
343
induced normal forces and counteracting moments: i.e., ~[ij=Nij-dlj6sYpqb " and f4~j=M~j-M~. N e and 3Lj are defined in equations (10)-(15). Note that the in-plane twisting effect is assumed to be negligible: i.e., M,z= M,2 and Ni2 and Nt2. The feedback voltage q~" is determined by control algorithms, direct proportional feedback and Lyapunov feedback, which will be discussed next. 5.2. DIRECT PROPORTIONAL FEEDBACK CONTROL In the direct proportional feedback control, the feedback voltage ~b~ is generated by directly amplifying the sensor output ~b', i.e., ~b"=~$ ",
(35)
where ff denotes the voltage amplified ratio--the feedback control gain--which can be adjusted depending on the performance requirement of the system, qV is defined in equations (24) and (25). Thus, the feedback voltage induced effective forces and moments are
N~t=ffd3tYpq5", M~, =frAUd3, Y~d?',
N]z=ffd3,Ypr",
(36, 37)
M~=ffAO2d~2YpqS".
(38, 39)
These forces and moments can be substituted into the system equations, and the resulting closed loop system equations are derived. Note that this control algorithm results in a change of system frequency. The feedback signal can be differentiated so that velocity information is obtained. The velocity feedback can enhance the system damping and therefore the oscillation amplitude is controlled. Note that for velocity feedback, the sensor signal ,#' used in equations (36)-(39) needs to be replaced by (~s. In addition, the system feedback gain ff can be extended to a modal feedback gain (~k, from which control of each individual mode can be achieved. 5.3. LYAPUNOV FEEDBACK CONTROL
In Lyapunov feedback control, the feedback voltage amplitude is constant and the sign is opposite to that of the velocity [I 3]. The sensor information ~b'(a,, a2, a3, t) is basically contributed by shell oscillations in all three directions. Thus, the amplitude of the feedback signal can be expressed as ~b"=(# sgn ((~/dt)qS"(a,, az, a3, t)),
(40)
where ff is the feedback gain and "sgn" denotes the singum function, i.e., sgn [z] = - 1 if z < 0, 0 if z= 0, and +1 if z > 0. The forces and moments can be written as
N~l=(Td31Yesgn[dp'(al,a2,a~,t)],
N'j2=ffd3zYpsgntt~'(ctl,a2, a~,t)],
(41,42)
M'~, = C~A~'d31Ypsgn [q~'(a,, a2, as, t)],
(43)
M~z=f#A'Jd~2rr sgn [6"(a,, a2, aj, t)].
(44)
Note that when ~b" is used in the fee~iback control, an averaged dynamic state of the shell is considered. In practical applications, a single-point transverse velocity ~i'3(a~, a*, t) (discrete location) can also be used in the feedback control. Thus, the feedback amplitude becomes q~"=$ sgn 0i'3(a*, a*, t)),
(45)
H.S. TZOU
344
where (a~*, a~) denotes the specific location. Thus, the effective forces and moments are defined as
N~l = f~dji Yv sgn [ffs(t, a?, a*)], M~I =C~A~d3t Yv sgn [~i,a(t, a~, a~)],
N~2 = (~ds2Yp sgn [~a(t, a*, a*)],
(46, 47)
M~2 = f~A~d3,Yt, sgn [r~'3(t, a~, a~)].
(48, 49)
Similarly, a new set of system dynamic equations can be derived accordingly. It can be observed that all three actuator parameters, i.e., the piezoelectric constant d/j, the modulus of elasticity Yp and the moment arm A~, are of importance to the overall control effects. In general, a higher piezoelectric constant, a stiffer piezoelectric actuator, and a longer moment arm contribute better control effects. 6. EXAMPLES In this section, two examples, a hemisphere and a flat plate with distributed piezoelectric sensor and actuator layers, are presented to illustrate the use of the shell sensor/actuator theory. A direct reduction procedure is proposed in order to apply the theory. 6.1. CASE I: A lXEMISPIIERIC SIIELL A hemispheric shell with fully covered distributed piezoelectric sensor and actuator layers is shown in Figure 4. Distributed sensation and control of the hemispheric shell are discussed.
Figure 4. A hemispheric shell with distributed piezoelectric sensor/actuator layers.
The fundamental form defining the hemispheric shell is (ds)2 = R2(d ~t)2 + ~2 sin 2 v(d 0 )2,
(50)
where I~ is the radius of the shell, and V and 0 are defined in Figure 4. Thus, Lam~'s parameters are A,--[~ and ,42 = R sin V; the two neutral surface co-ordinates are a~ =~' and a2 = 0; and the radii are RI = ~2 = R. Substituting these parameters into equation (24) yields
-0'''11,. o, ,.o,,, §
o.(
, o,,I+oo,,~ '\l W v - -O~)J[ sin dIy dO. V
sin v 00 w0 sin V 00]
(51)
Note that A~, = A~ for a piezoelectric sensor layer with uniform thickness. All terms inside braces { } represent a local vo!tage amplitude if the location (V*, 0") is specified. Using equations (36)-(39) and (41)-(44), one can derive the effective control forces and moments
D I S T R I B U T E D SENSOR A N D A C T U A T O R T t l E O R Y FOR SIIELLS
345
for each control algorithm accordingly. The closed loop system equation can also be derived by using the geometric parameters defined above. For demonstration purposes, only the system equation in transverse oscillation is considered:
FOM~,o cos
0 [.O(~fv_z~sin ~')
-a-~L
a~
-a--o ~ +Rsin
ao -
L
a~,
]
~,~foo aO
v/[~lvv+~loo]+RZsinvphff'~= Rz sin vF3.
(52)
NIj and ~l U are the resultant forces and moments, i.e.,/Vi; = Ni;-NZ and ~ t . = M . - M ~ . Ni; and M;; can be derived by substituting the geometric parameters into the general force and moment equations in equations (10)-(15). The derived force and moment expressions are
Nvv =
/(
, o,,,o
+iv3 + p I-wv cot V+ll'3 t~ LkOV / sin g O0
N~176
s i n g -O0 bwvc~
Nvo-K(!-12) [ sin V o 2-----if--
o,,;,--.,r:
~
+12\ag
(~/ -t
o,, -,,, {-=, u~o-
-~@
w3
sin~,
,
sin ~, ~ /
(54)
(55)
sin ~ dO J'
<,.
o(,,, ~ , /
(53)
l o,,,,,.-1
\sin v//
t'v~'-a-VT~,/J*12LsinwO0
)]
-~-]jj,
00/
(56)
o,,,,~l r o (,,, o,,ql ~
o(,,,o
~
, o,q sin v ~J.J
, 0 Ir o,,,a "I/' s~n ,,o.,~0L'i'"-a,,,.n
(58)
-.l---
6.2. CASE 2: A R E C T A N G U L A R PLATE The second case is a zero-curvature shell (a plate) with infinite radii of curvatures for the two neutral surface co-ordinate axes x and ),: i.e., I~ = 1~2= o0 (see Figure 5). It is assumed that the plate has a width b and length a. The fundamental form is defined as (ds) 2 = (dx) 2+ (d),) 2. (59) Thus, LamUs parameters are A~ = 1 and A2 = 1. It is assumed that the biaxially oriented piezoelectric sensor and actuator layers are aligned with the co-ordinate axes x and y. (Otherwise, an orientation matrix needs to be defined.) Distributed sensation and control, direct feedback control and Lyapunov control of the rectangular plate are derived by using the generic shell equations. The distributed sensor output can be derived by using equations (24) and (25), as
+ , : - ~ f f l,,,,z.~ o,,,,, + ,,,,,,;. o~,~.a~.~~., A J.,,J>.L"
" -ax -T
--~/J "
j'
(60)
346
ti. S. T Z O U
Distributed ~ piezoelectric
~-,
-,.. -",,,~.~
~..~"--~-.. -~....~
~
-" . ~ j ~ I ~ ' ~
>,J
D!~,r~.ted \ ~'--.<.,4.........~-~? \
actuator
\ m
b ".-,~y
Figure 5. A rectangular plate with distributed piezoelectric sensor/actuator layers.
and, in modal expansion form,
11kll"3k +ha2Aj.
"
~. tl~iV3k dxdy,
(61)
where A~= Aj. for a uniform thickness piezoelectric layer. For a fully covered distributed sensor, the integration covers the whole plate area, i.e., A'=ab; thus, the integration is J'o J'0 ['" "] dx dy. For a distributed patch sensor covering an area defined by al-a2 in the x f,,2 Jh, f~ [. 9 .] dx dy. direction and b,-b2 in the y direction, the integration is defined as ~,, Using equations (36)-(38) and (41)-(43), one can derive the control forces and moments for the direct feedback control and Lyapunov control accordingly. However, the in-plane control forces N.~:~and Nj"..,.are canceled out because of the infinite radii of curvature: i.e., ~, = 1~2= oo. Substituting the moments and geometric parameters into the generic shell equation, one can also derive the closed loop system equation in the transverse direction, a
/'~
ph ~ +
D \~.r
~
-~-/ ~
8), = F3,
(62)
where the effective control moments are determined by the control algorithms. Note that the plate case can also be directly reduced to an Euler-Bernoulli beam case by considering only one effective axis: e.g., the x direction. Thus, the sensing equation (60) can be reduced to r
f
(hs,A: _'-:-cladx,=Ws'~
A" Jx ',
(63)
where b is the beam width and x defines the beam length direction. The integration depends on the boundary conditions and the effective area of the distributed sensing layer. (For example, the sensing signal is a function of the free end slope for a cantilever beam with a fitlly covered sensing layer:) The above sensing equation can also be extended to modal co-ordinates by assuming that w~.. of the beam case can also be derived - ~ =~'~, qkiVsk. M'~'~ on the basis of the control algorithms. The plate closed loop system equation (62) can be simplified to account for the beam case, as
PAdZws/dl 2 + Yhg'~ws/d.v 4 = - t92~d'~.,,/63x 2 + ~'.a, where A =bh, I is the area moment of inertia, and
F's=bF3.
(64)
D I S T R I B U T E D SENSOR A N D A C T U A T O R T l t E O R Y F O R SltELLS
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7. CONCLUDING REMARKS In this study, an "intelligent" shell concept was introduced. The shell structure is composed o f an elastic shell structure and built-in distributed sensors, actuators and control electronics. These sensors and actuators offer an integrated self-monitoring and control capability. Unlike the conventional "discrete" sensor and actuator, a new "distributed" sensor and actuator design was proposed in this paper. The distributed sensor and actuator layers are both made of piezoelectric materials. On the basis of the direct piezoelectric effect, the distributed sensor can monitor the shell oscillation; and on the basis of the converse effect, the distributed actuator can suppress and control the structural vibration. (Note that control electronics is not discussed here). Anew distributed sensor and actuator theory for distributed dynamic measurement and vibration control of shell structures was developed. Closed loop system equations of the intelligent shell were also derived. When distributed sensation and control are not considered, the system equations can be reduced to the thin shell vibration equations. (It was assumed that bending vibration dominates the shell oscillation when the sensation and control theory was derived.) From the theory, it is observed that the distributed sensor output is contributed by all vibration modes with different modal participation factors: that is, the distributed sensor can theoretically measure all vibration modes of the shell. A voltage distribution contour (modal voltage) can be constructed by graphically connecting all local voltages, calculated by local strains, on a shell provided that the sensor has finite electroded spots. Similarly, the distributed actuator can (theoretically) control all vibration modes by using either the direct feedback control algorithm or the Lyapunov control algorithm. The derived theory is very general, and can be directly simplified to account for other commonly occurring geometries, e.g., plates, rings, cylinders, spheres, beams, cylinder shells, etc. Note that there is a potential problem when calculating the averaged distributed voltage. Zero output could occur in anlisymmetrical modes. In this case, finitearea voltage output may be used in a feedback control system, as discussed in Lyapunov control. The effectiveness of piezoelectric actuators depends on the Young's modulus, the moment ann and the piezoelectric constant of the actuator material, in addition to the feedback voltage and control gains. Note that the induced control forces and moments act on the boundaries of the distributed actuator. These forces and moments could be zero if they are not spatially distributed. In this case, boundary control should be considered. Two cases, a hemispheric shell and a rectangular plate, were used to show the utilities of the generic distributed sensor and actuator theory. On the basis of four geometric parameters, i.e., two LamUs parameters and two radii of curvatures, the original sensation, control and closed loop system equations can be easily simplified to account for these two cases. A similar procedure can be applied to other commonly occurring geometries, such as circular plates, spheres, cylinders, etc. From the closed form system equations derived for the hemispheric shell and plate cases, it is observed that the in-plane control forces are canceled out due to the infinite radii of curvature in the plate case. However, both control forces and moments are preserved due to non-zero radii in the case of the hemispheric shell. Note that the derived iheory is limited to stress-free conditions: i.e., boundaries of the sensor/actuator layers cannot be fully constrained (fixed). In the feedback controls, both a distributed and a finite-area voltage (discrete) can be used in both direct proportional feedback control and Lyapundv control. Since the modal voltage of the distributed piezoelectric sensor can be calculated in the time domain, the vibration control can be extended to a multi-input and multi-output (MIMO) control with a variable gain matrix. This distributed MIMO control algorithm needs to be studied further. In addition, extending the bending vibration assumption to a bending/membrane vibration assumption would certainly advance the theory.
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H s. TZOU ACKNOWLEDGMENT
This research was supported by a grant from the National Science Foundation (No. RII-8610671) and the Kentucky EPSCoR program, and a grant on Flexible Assembly and Intelligent Machines (1988-1989) from the Center for Robotics and Manufacturing Systems ( C R M S ) at the University of Kentucky.
REFERENCES 1. A. M. RE[~HORN and G. D. MANOLlS 1985 Soundand Vibration Digest 17, 7-16. Current state of knowledge on structural control. 2. H. S. Tzou 1988 Journal of Finite Elements in Analysis and Design 4(3), 232-238. Dynamic analysis and passive control of viscoelastically damped nonlinear dynamic contacts. 3. H. S. T z o u and M. GADRE 1988 American Institute of Aeronautics and Astronautics 26(8), 1014-1017. Active vibration isolation by piezoelectric polymer with variable feedback gain. 4. H. S. Tzou and M. GAORE 1990 Journal of Sound and Vibration 136(3), 477-490. Active vibration isolation and excitation by piezoelectric slab with constant feedback gains. 5. J. M. PLUMP,J. E. HUBBARD and T. BAILY 1987 Journal to Dynamic Systems, Measurement, and Control 8, 133-139. Nonlinear control of a distributed system: simulation and experimental results. 6. E. F. CRAWLEYand J. DE LOIS 1987 American Institute of Aeronautics and Astronautics 25(10), 1373-1385. Use of piezoelectric actuators as elements of intelligent structures. 7. H. S. Tzou 1987 Developments in Mechanics 14-C, 1201-1206. Active vibration control of flexible structures via converse piezoelectricity. 8. A. BAZ and S. POH 1988 Journal of Soundand Vibration 126, 327-3~43. Performance of an active control system with piezoelectric actuators. 9. H. S. Tzou and M. GADRE 1989 Journal of Sound and Vibration, 132, 433-450. Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. 10. H. S. Tzou and C. I. TSENG 1990 Journal of Sound and Vibration 138, 17-34. Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a finite dement approach. I I. H. F. TIERSTEN 1969 Linear Piezoelectric Plate Vibrations. New York: Plenum Press. 12. W. SOEDEL 1981 Vibrations of Shells and Plates, New York: Dckker. 13. H. S. T z o u 1988 Recent Developnwnts hz Control of Nonlinear and Distributed Parameter Systans, ASME-DSC-VoL(IO), 51-58. Integrated sensing and adaptive vibration suppression of distributed systems.
APPENDIX: NOMENCLATURE
{" }
vector
[" ] [. ]t
matrix matrix transpose
a, b A,A" AI,A2
plate length and width (m) effective area (m 2) LamUs parameter elasticity matrix evaluated at constant dielectric displacement (N/m 2) feedback gains piezoelectric constant matrix (C/N) or (m/V) bending stiffness (Nm) dielectric displacement'vector (C/m 2) infinitesimal distance (m) electric field vector (V/m) external mechanical force in the a~ direction (N/m) shell thickness (m) piezoelectric constant matrix (N/C) area moment of inertia (m')
[c~ ~,~ [a,:l
D {O,} ds {E,}
F, h
[bu] I
D I S T R I B U T E D SENSOR A N D A C T U A T O R T t t E O R Y F O R SHELLS
membrane stiffness (N/m) resultant membrane forces (N/m) resultant moments (Nm/m) my radii of curvature of the at and a2 axes, respectively (m) Rt, R2 sgn singum function elastic compliance matrix measured at constant electric field (m2/N) IsEl strain vector {s,} stress vector (N/m 2) {7",} displacement in the as direction (m) wt velocity in the al direction (m/s) ~,, acceleration in the as direction (m/s 2) kth modal function in the ith direction Wa Young's modulus (N/m 2) Y Young's modulus of piezoelectric material P (7.1, a 2 , el3 three principal axes in a curvilinear co-ordinate system dielectric impermeability matrix evaluated at constant strain (Vm/C) actuator thickness (m) sensor thickness (m) 8" A", A" distance measured from the neutral surface to the mid-plane of the piezoelectric layer (m) r electric potential (V) sensor output (V) feedback to actuator (V) dielectric matrix evaluated at constant strain (C/Vm) [d-I kth modal participation factor r/k Poisson ratio P density (kg/m 3) P gradient operator V K
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