Improved Dynamic Response of Piezoelectric Composite Shells using Multiobjective Optimization and Closed Loop Control

Transcription

1 Improved Dynamc Response of Pezoelectrc Composte Shells usng Multobjectve Optmzaton and Closed Loop Control Wllam Lee * and Adt Chattopadhyay Department of Mechancal and Aerospace Engneerng Arzona State Unversty Tempe, AZ Abstract A multobjectve optmzaton procedure s developed for mprovng the vbratory response of composte shells wth dstrbuted pezoelectrc patches under a varety of loadng condtons. The objectve s to mnmze the overall maxmum deflecton assocated wth the frst fve modes of vbraton and the statc deflecton as the shell s subjected to electrcal, mechancal and combned loadng. Constrants are mposed on the natural frequences and stresses. The stacng sequence s used as a desgn varable to nvestgate the most effcent stffness dstrbuton. A closed loop control system s desgned usng Lnear Quadratc Gaussan (LQG) controller. The multple objectve functon problem s formulated usng the Kresselmeer-Stenhauser (K-S) functon approach. Usng ths approach, the multobjectve constraned optmzaton problem s reduced to the unconstraned optmzaton of a sngle envelope functon (the K-S functon). A Smulated Annealng algorthm s used as the search algorthm. Numercal results are presented for a varety of loadng condtons to show the sgnfcant mprovements n vbratory response from optmzng the stacng sequence. The nfluence of the orentaton of the pezoelectrc patch s also nvestgated. σ j = stress ε j = stran E = electrc feld e = pezoelectrc constants j D = electrc dsplacement c = elastc constants jl b j = delectrc permttvty φ = electrc potental u = deformaton on dsplacement boundary t = tracton on dsplacement boundary q = charge accumulated (sensng sgnal) γ = structural dampng rato f = components of force per unt volume S D = charge boundary S = stress boundary σ F = th objectve functon (φ) Nomenclature * Graduate Research Assstant, Mechancal and Aerospace Engneerng Dept, Arzona State Unversty, Mal Stop 6106, Tempe, AZ, Professor, Mechancal and Aerospace Engneerng Dept Arzona State Unversty, Mal Stop 6106, Tempe, AZ,

2 g max = maxmum constrant value f * = th reduced objectve functon f max = largest constrant value of the reduced objectve functon ρ = pull down factor L(t) = Kalman flter gan matrx K(t) = optmal feedbac gan matrx K = gan matrx J = performance ndex R = nput energy weght matrx Q = ntermedate state weght matrx U = n-plane dsplacements (=, ) denotes the -th layer of the lamnate u, φ = lamnate unnowns for (=, ) w = lamnate unnowns θ, ψ = layerwse unnowns for (=, ) f (z),g (z) = through-lamnate-thcness functons of hgher order odd and even dstrbutons (=, ) ϕ j = rotatonal degrees of freedom for (j=, ) I. Introducton Aerospace vehcles are susceptble to hgh vbratory loads due to the hgh unsteady load and complex aerodynamc envronment n whch arcraft components must operate. Over the last decade, a sgnfcant amount of research has been conducted by usng nduced stran actuaton for mproved vbraton and nose control n arcraft components 1-4. The use of these transducers allows the mposed changes to be talored accordng to the condtons sensed by a partcular structural component. Most mportantly, by usng dstrbuted sensors and actuators, control can be acheved over a much larger bandwdth than by usng conventonal mechansms. Pezoelectrc materals can be used as actve vbraton dampers for structures where they act as both sensors and actuators. When placed n dscrete locatons on a structure, these materals can sense movement and control that moton va localzed strans. Although these materals generate very low stran, they cover a wde range of actuaton frequency and, therefore, have more practcal applcatons. Compostes have an advantage over metals due to ther excellent toughness and strength whle mantanng low strength-to-weght rato. Therefore, research has also been reported n usng embedded/surface bonded actuators and sensors to mprove the vbratory characterstcs of composte structural elements 5-8. A multtude of aerospace components, rangng from fuselage to engne contanment systems, can be modeled as composte shells. Therefore, mproved vbratory performance and dynamc response of ansotropc shell structures s an mportant research area and s the topc of the present wor. A sgnfcant amount of wor has been reported n usng formal optmzaton technques to mprove the performance of composte plates and shells usng passve technques. Yuan 9 nvestgated the effect of stacng sequence on the dstrbuton of thermal stresses. Hwang 10 used the Tsa Wu falure crteron as the objectve functon and nvestgated optmum stacng sequence usng a state-space approach. Waler 11 examned the multobjectve desgn of lamnated composte shells wth respect to buclng and torsonal loads. They formulated the performance ndex as the weghted sum of ndvdual objectves n order to obtan Pareto optmal solutons of the desgn problem. Hafta et al. 12 used Genetc Algorthms (GA) wth bult-n repar strateges for the stffness talorng of compostes aganst buclng. A GA wth eltst selecton for best-ft sngle ndvdual was also used by Hafta et al. 13 n optmzng the stacng sequence of a composte as the objectve functon. Optmzaton of pezoelectrc composte plates for mproved vbratory performance was addressed by Thornburgh and Chattopadhyay 7. They used a hgher order plate theory wth passve control to mprove the vbratory response of lamnated plate structures wth dstrbuted PZTs. More recently, Km et al. addressed the modelng and closed loop control of free vbraton of pezoelectrc shell structures 8. The goal of the present paper s to further extend ths wor by mprovng the dynamc response of composte shells under a varety of loadng condtons. The effcent desgn of smart structural systems wth dstrbuted sensors and actuators requres the ntegraton of hgh fdelty analyss tools, robust controllers, and optmzaton technques. In the analyss of composte lamnates, classcal lamnate theory (CLT) and frst-order shear deformaton theory (FSDT) have tradtonally been used. These theores are lmted by the fact that CLT gnores transverse shear stresses and FSDT uses ad hoc correcton 2

3 factors to compute them. Also, valuable nformaton regardng local stress dstrbutons s neglected n both theores. Although three-dmensonal approaches are more accurate than two-dmensonal theores, ther mplementaton can be very expensve n practcal applcatons. Layerwse approaches are alternatves snce they are capable of modelng lamnated stress dstrbutons. However, computatonal effort ncreases wth ply number, whch maes them computatonally prohbtve. A computatonally effcent new layerwse lamnate theory has been developed by Zhou et al 14 addressng transverse shear stress contnuty and ensurng accuracy at local levels. In ths paper, the refned layerwse theory s used n the analyss of composte shells wth dstrbuted pezoelectrc patches under a varety of mechancal and electrcal loadngs. In the analyss of ntegrated pezoelectrc structures, t s mportant to accurately model the electro-mechancal feld nteractons. A completely coupled theory, ntegratng thermal, mechancal and electrcal felds was developed by Gu et al 15 and further refned by Zhou et al 16. A smplfed two-way electro-mechancal model was later developed by Thornburgh and Chattopadhyay 7. In ths paper, the completely coupled model from Zhou et al 16 s used n conjuncton wth the refned layerwse theory 5 n the dynamc response analyss of composte shells wth dstrbuted PZTs. A multdscplnary procedure s developed, ntegratng the refned analyss wth a multobjectve hybrd optmzaton algorthm and closed loop control to nvestgate the desgn tradeoffs assocated wth stffness talorng and changes n pezoelectrc patch orentaton angle on the vbraton control. II. Problem Statement The objectve s to mnmze the vbratory response of the shell structure under mechancal, electrcal and combned electro-mechancal loadng condtons. Therefore, the maxmum deflectons assocated wth the frst fve modes of vbraton as well as the statc deflecton are used as objectve functons. The stacng sequence s used as desgn varable and constrants are mposed on the natural frequences and stresses. In addton, geometrc constrants are also mposed on the desgn varables to avod unrealstc desgn. Snce the problem nvolves multple objectve functons, the Kresselmeer-Stenhauser multobjectve formulaton s used 17. In the K-S functon approach the orgnal objectve functons are transformed nto reduced objectve functon, whch tae the form of constrants. The problem reduces to the unconstraned mnmzaton of the K-S functon, whch sgnfcantly smplfes the search procedure. A Smulated Annealng algorthm 18 s used n conjuncton wth the K-S functon approach to address the closed loop control of composte shells. The nfluence of the orentaton of the pezoelectrc patches n vbraton reducton s also nvestgated. III. Analyss of Pezoelectrc Composte Shells Composte shells wth dstrbuted pezoelectrc actuators and sensors and smply supported boundary condtons are modeled usng a combnaton of the refned layerwse theory and the two-way coupled electro-mechancal feld formulaton. Consder a shell element wth N ples descrbed n orthogonal coordnates, and, along the shell md-surface as shown n Fg 1. 3

4 AC3 AC6 AC9 AC2 AC5 AC8 AC1 AC4 z AC7 L R φ h Fgure 1. Shell geometry The area of an nfntesmal rectangle s denoted ds and the volume of an nfntesmal paralleleppe s denoted dv. ds = A (1 + z R dv = A (1 + z R ) A (1 + z R ) A (1 + z R ) d d ) d d dz (1) Where A represents the coordnate shell coeffcent and R s the geometrc coordnate radus. The thcness coordnate, z, s measured from the md-surface and R and R are the rad of curvature of the and -curves. The orthogonal coordnate system can express the volume and surface area of the shell n a dfferental form 8. The mplementaton of the ntal curvature s mportant n the development of an accurate thc shell theory. It must be noted that the terms ( 1+ z R ) and ( 1+ z R ), whch address the varaton of surface curvature for dfferent values of thcnesses, are ncluded to tae nto account the ntal curvature effects n thc lamnates. The layerwse zgzag dsplacement feld accounts for the nterlamnar contnuty of transverse stresses at each nterface of lamna as well as the tracton free boundary condtons on the top and bottom surfaces of the lamnates. The dsplacements of a pont wth the coordnates (,, z) are descrbed usng the superposton of frst-order shear deformaton and layerwse functons. The layerwse functons descrbe the zgzag n-plane deformaton through the lamnate thcness. When the PZT patches are actuated, the stresses are located at the nterface between actuator and composte shell. The hgh stress concentraton s descrbed wth one more rotatonal degree of freedom for the dsplacement feld of the PZT lamnate. Then, the equlbrum of the composte shell wth pezoelectrc actuators s descrbed wth the layerwse functons of 8 : 4

5 U U U z (,, z) = (1 + (,, z) = (1 + z R z R (,, z) = w(, ) ) u ) u + g + g (, ) + zφ (, ) + f ψ (, ) + zϕ (, ) δ ψ (, ) + zϕ (, ) δ (, ) + zφ (, ) + f θ (, ) θ (, ) (2) Where: 0 δ = for (3) 1 = In these equatons, a superscrpt denotes the -th layer of the lamnate and superscrpt denotes the -th PZT layer. Lamnate unnowns ( u, u, w, φ and φ ) and layerwse unnowns ( θ, θ, ψ and ψ ) are used to address n-plane deformaton. The rotatonal degrees of freedom ( ϕ and ϕ ) are ntroduced n the PZT layer to descrbe electrcal actuaton and stress concentraton. Next, the Kronecer Delta ( δ ) s ntroduced to descrbe addtonal rotatonal degree of freedom n the PZT layer. The through-lamnate-thcness functons f, f, g, and g descrbe the global deformaton of the composte shell. f = f g = g = snh = cosh (4) Where the functons f, f, g, and that Eq. (2) results n p total number of layers and N p s the number of PZT layers. g are hgher order odd and even dstrbutons. Note 4 N N number of structural unnowns n the dsplacement feld, where N s the Conventonal sensor modelng neglects the nonzero electrc feld nduced by mechancal deformaton. Ths leads to a volaton of the conservatve charge law. It also mspredcts the charge accumulated on electrodes sgnfcantly. In ths paper, a hgher order feld descrpton of electrcal feld s used to accurately descrbe the nonunform dstrbuton of electrc feld through the thcness of pezoelectrc layers. Two-way couplng between the mechancal and the electrcal felds s modeled to extract accurate sensng nformaton generated by pezoelectrc sensors 15-16, 19. The consttutve relatons governng stress, stran, charge and electrc feld can be wrtten as follows: σ D j = c = e jl j ε ε l j e + b j j E E j (5) Where the quanttes ε j and σ j denote the components of stran tensor and stress tensor, respectvely and E and D denote the components of electrc feld and electrc dsplacement, respectvely. The quanttes c jl, e j and b j represent elastc constants, pezoelectrc constants and delectrc permttvty, respectvely. The relatonshp between charge varatons generated by the sensors and delamnaton parameters wll be quantfed usng varatonal prncple as follows 15-16,19 : 5

6 t t0 dt V + γu + ) + δu σ jδε j dv fδudv tδ uds Dδφ dv qδφds = 0 V S V S D σ ( ρuδu, (6) Where the quantty φ denotes electrc potental appled n the pezoelectrc devce. The quanttes u, t, φ and q denote the deformaton on the dsplacement boundary S u, tracton on the stress boundary S σ, voltage on the potental boundary S φ, charge accumulated (sensng sgnal) on the charge boundary S D, respectvely and ρ, γ and f denote mass densty, structural dampng rato and components of the force per unt volume, respectvely. The procedure wll provde means to dentfy structural characterstcs usng the charge accumulaton, q. The above formulaton represents the pezoelectrc and converse pezoelectrc effects smultaneously, as opposed to the conventonal sequental approach. Thus, energy transformaton from the electrcal to the mechancal feld and vce versa s accounted for. Ths theory provdes an accurate descrpton of the pezoelectrc on the composte shell. A pezoelectrc patch may possess ansotropc characterstcs f the mechancal propertes are unque n each drecton. Lewse, the confguraton of the three mutually orthogonal crystal axes affects the polar axs that nduces stran on the host meda out of the plane 20. Dependng on how the cubc crystal axs or element face algns to the polarzaton axs, t can elevate or degrade the mechancal propertes of the pezoelectrc materal. The pezoelectrc patch rotates about the rhombohedral crystal axs, whch can elevate or degrade ts mechancal propertes. Generally, the user orents the materal n the axs of polarzaton (best mechancal ft) drecton perpendcular to the surface of the electrc potental drecton. Because actuator propertes change based on the polarzaton angle, t s necessary to optmze the sensor confguraton and locaton under a specfc bendng load. Addtonally, t s pertnent to search for the best confguraton of the pezoelectrc patch to that provdes the best dampng performance. To nvestgate the effect of physcal orentaton of the PZTs on vbraton control, the optmzaton process s repeated over a range of orentaton angle, Φ ( 90 < Φ < 90 ). IV. Optmzaton Problem Formulaton The maxmum deflectons of the frst fve modes and the statc deflecton are used as objectve functons and the ply stacng sequence s the desgn varable. Three dfferent loadng condtons are studed. These nclude the electrcal load due to appled voltage, the radally appled mechancal load and combned electro-mechancal load. Constrants are mposed such that the stresses n the shell do not exceed the ultmate falure or maxmum fber stress n ether compresson or tenson. That s, σ < σ σ m compresson m < m tensox To avod changes n frequency, wth reference to the host structure, constrants are mposed on the frst fve natural frequences. Geometrc constrants are also mposed on the desgn varables; where Ψ represents the stacng sequence angle n the range of 90 < Ψ < 90. The K-S functon s used to handle the multple objectve functons. Frst, the orgnal objectve functons, F o (φ) (whereφ s the desgn varable vector) are converted to reduced objectve functon, whch tae the form of constrants, as shown n Eq.(5). F ( φ) f* = 1 g max 0 F ( φ) o (7) Where g max s the maxmum possble physcal constrant. The K-S envelope functon s then expressed as follows. F s M ρ ( fm ( φ ) fmax ) ( φ ) = f + ln e (8) max 1 ρ m = 1 6

7 Where Fs (φ ) s a sngle composte functon that combnes all of the objectve functons and f max s the largest constrant correspondng to the new reduced constrant functon f m (φ ). It must be noted that f max s not equal to g max. The quantty ρ s a pull-down factor. A hgh value of ρ ensures that the K-S functon curve les close to the maxmum reduced objectve constrant surface and a low value of ρ yelds an envelope that represents all the constrants. The Smulated Annealng algorthm s used as the search algorthm. The procedure mmcs the probablty of smulated annealng of metals by generatng a random number n Boltzmann s probablty dstrbuton functon, a step sze, a temperature reducton factor, and a step reducton factor. The step sze controls the ncrement for generatng random numbers, and the temperature reducton factor changes accordngly for convergence ssues. Fnally, the step reducton factor controls the magntude of the step sze. The general strategy of ths technque warrants that a hgher value of an objectve functon (K-S composte functon n ths case) s acceptable under a set of condtons because a probablty exsts that an absolute mnmum may stll be present f more runs are generated. Ths operaton leads to a robust method for fndng the global mnmum by reducng the chances of neglectng the absolute mnmum of the system f the ntal condtons are not favorable mentoned by Seeley 21. V. Control System Desgn In the desgn of control system, the use of Lnear Quadratc Regulator (LQR) controller s not always feasble because n most practcal applcatons not all states are measurable. Unle the standard LQR, whch requres full state feedbac, a LQG controller allows the user to specfy the measurable states resultng n a much more practcal approach n terms of autonomous applcatons. Therefore, the LQG controller system s used n ths research. The equatons of moton n state-space can be expressed as follows. x ( t) = Αx( t) + Bu ( t) + w( t) y( t) = Cx ( t) + v( t) u ( t) = Kx ( t) x ( t) = Ax ( t) + Bu ( t) + L[ y( t) Cxˆ( t)] (9) Where x (t) s the state space, y(t) s the output vector, u(t) s the nput vector and L defnes the Kalman flter gan matrx for the LQG controller. The Kalman flter L substantally mproves the autonomous systems by estmatng the past, present, and future states even when the exact condton of the system s unnown. It s an teratve soluton to dscrete lnear data flterng whch averages the lmts of data. At the root of the controller s a recursve algorthm, whch has n dscrete-tme Rccat equatons 22. The goal s to mnmze all the ponts of the steady state deflectons as the states are approachng zero. The performance measure ndex s a matrx provdng nsght on the strength of the controller. T T J = E [x ( t)qx ( t) + u (t)ru (t)]. (10) t= 1 J() s a hybrd sum of everythng whch s desred to be small 22, wth weghtng matrces provded for the tolerances n these dynamcs. R s the nput weghted energy matrx, and P s the pror estmated error covarance. VI. Results Table 1 shows the fxed geometrc parameters of the composte shell for the case studes n ths wor. The composte shell s smply supported wth nne pezoelectrc patch actuators mounted on top and bottom surfaces. The stress lmts used n the optmzaton are 2.1GPa as the ultmate tensle strength and 1.72 GPa as the ultmate compressve strength. Table 2 shows the mechancal and electrcal propertes of the composte structure and pezoelectrc patches. Because the stffness of the composte shell changes as a result from the stacng sequence, the natural frequences may shft durng optmzaton. To avod drastc changes n natural frequences, relatve to the host structure, the frst four modes are constraned to reman wthn 15 percent of the orgnal values. The ffth mode had a hgher frequency fluctuaton, so the constrant was set at 20 percent. For the electrcal loadng case, a potental of 200 V s appled to each of the nne PZT patches. For the mechancal loadng case, a unform load of magntude 1000 Pa s appled radally on the composte shell. A stacng sequence of [90 0 /0 0 /90 0 /90 0 /0 0 /90 0 ] s used 7

8 as the reference desgn. The optmum stacng sequence obtaned for the three loadng cases, electrcal, mechancal, and combned, are presented n Tables 3 and 4 for L/h=100 and 200, respectvely. Note that each stacng sequence s unque for a predefned pezoelectrc orentaton. Fgures 2-8 llustrate the nfluence of the PZT orentaton on the objectve functon. The reducton n modal deflecton, assocated wth the frst fve modes, for the plate wth L/h=100 are shown n Fgs. 2 4 (for all three loadng condtons). Smlar results are presented n Fgs. 5 7 for the plate wth L/h = 200. In these fgures, a negatve percentage ndcates an ncrease n deflecton. Fgure 8 shows an example of the frst fve mode shapes of the optmzed shell structure subject to electrcal loadng. The dfferences n statc deflectons, before and after optmzaton are llustrated n Fgs for L/h=100. Table 1. Shell dmensons. Radus.3m Arc Angle ο 60 Length.6m Length/thcness rato 100,200 Table 2. Mechancal and electrc propertes. Property Composte Pezoelectrc Elastc Modulus 1-dr (GPa) Elastc Modulus 2-dr (GPa) Elastc Modulus 3-dr (GPa) Shear Modulus 1-dr (GPa) Shear Modulus 2-dr (GPa) Shear Modulus 3-dr (GPa) Posson Rato (1-2)-dr.3.28 Tensle Yeld Strength (MPa) (Dyn. Tensle Compressve Yeld Strength (MPa) (Statc loadng) Densty (g/m3) Delectrc perm. 1-dr (nfarad/m) N/A 15.3 Delectrc perm. 2-dr (nfarad/m) N/A 15.3 Delectrc perm. 3-dr (nfarad/m) N/A 15 Pezoelectrc charge constant 1-dr (um/v) N/A 250e-6 Pezoelectrc charge constant 2-dr (um/v) N/A 230e-6 Pezoelectrc charge constant 3-dr (um/v) N/A 230e-6 8

9 PZT Angle Table 3. Optmum Stacng Sequence; L/h=100. Elec. load Ply Ply Ply Ply Ply Ply Mech. load Ply Ply Ply Ply Ply Ply Combned load Ply Ply Ply Ply Ply Ply PZT Angle Table 4. Optmum Stacng Sequence; L/h=200. Elec. load Ply Ply Ply Ply Ply Ply Mech. load Ply Ply Ply Ply Ply Ply Combned load Ply Ply Ply Ply Ply Ply

10 Examnng the frst, second, and ffth modes of vbraton n Fg. 2, for the electrcal loadng s case, the mprovements n reducton range from 2-5 percent for all PZT orentatons consdered. A sgnfcant beneft reducton of nearly 14 percent s acheved for the fourth mode. The optmzaton procedure s neffectve n suppressng the maxmum deflecton assocated wth the thrd mode (12.5 percent ncrease, compared to reference confguraton). Smlar observatons are made on all other loadng cases (Fgs. 2 4, L/h = 100). Ths phenomenon s not due to the bendng-torson couplng, whch can cause the maxmum deflecton of a sngle dynamc mode to ncrease. As seen from Fg. 8, for the case wth PZT orentaton and electrcal loadng, such couplng s observed only n the fourth and ffth mode shapes. Therefore, the ncrease s perhaps due to the fact that whle the K- S functon approach mnmzes the envelope functon, a unform mprovement n all of the ndvdual objectves may not be possble. In such cases, t may be of nterest to apply weght factors to the ndvdual objectve functons 23. In all three loadng cases, the PZT orentaton has a strong effect on the modal response. The parametrc studes conducted llustrate that larger reductons n the modal deflectons are acheved at certan angles Percent reducton Mode1 Mode2 Mode3 Mode4 Mode pezoelectrc angle Fgure 2. Deflecton reducton under electrcal loadng; L/h=

11 15 10 Percentage Reducton Mode1 Mode2 Mode3 Mode4 Mode pezoelectrc angle Fgure 3. Deflecton reducton under mechancal loadng; L/h= Percentage Reducton Mode1 Mode2 Mode3 Mode4 Mode pezoelectrc angle Fgure 4. Deflecton reducton under electro-mechancal combned loadng; L/h=

12 Examnng Fg.5, when the dmensons of the shell s set to L/h=200, the same trends are observed for the frst, second, and fourth mode as Fg. 3. However, the reducton n the deflecton assocated wth the ffth mode s more sgnfcant. Smlar observatons can be made for the other loadng cases (Fgs. 6 and 7). For example, the frst and second modes of deflecton were reduced by about 2-4 percent and the deflecton of the ffth mode undergoes a reducton of 4-10 percent, compared to the reference desgn. Also, comparng the electrcal to mechancal loadng of Fgs. 5 and 6, t s apparent that stffness talorng of a composte depends upon also the type of determnstc load encountered. Once agan, t s observed that the PZT orentaton plays an mportant role n the dynamc response and close loop control Percent reducton 5 0 Mode1 Mode2 Mode3 Mode4 Mode pezoelectrc angle Fgure 5. Deflecton reducton under electrcal loadng; L/h=

13 20 15 Percentage Reducton Mode1 Mode2 Mode3 Mode4 Mode pezoelectrc angle Fgure 6. Deflecton reducton under mechancal loadng; L/h= Percent Reducton 10 0 Mode1 Mode2 Mode3 Mode4 Mode pezoelectrc angle Fgure 7. Deflecton reducton under electro-mechancal combned loadng; L/h=

14 Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Fgure 8. Optmzed mode shapes under electrcal load; 45 0 /45 0 PZT orentaton. The statc deflecton also shows sgnfcant reductons, on the order of percent, as seen from Fgs. 9-11, for the shell wth L/h=100. The results obtaned llustrate the sgnfcance of stacng sequence n mprovng the dynamc and statc response of shell structures wth closed loop control. Ths demonstrates how the stacng sequence s very crucal to the stffness talorng of the composte shell n the statc as well as dynamc loadng cases. The parametrc studes conducted shows that the PZT orentaton angles affect the dynamc response and closed loop control of composte shells. Further studes are necessary to examne the observed trends. 14

15 5.E-05 5.E-05 4.E-05 Dsplacement (m) 4.E-05 3.E-05 3.E-05 2.E-05 2.E-05 Wth Optmzaton Wthout Optmzaton 1.E-05 5.E-06 0.E+00 pezoelectrc angle Fgure 9. Comparson of statc deflecton under electrcal load; L/h= E-04 1.E-04 Dsplacement (m) 8.E-05 6.E-05 4.E-05 Wth Optmzaton Wthout Optmzaton 2.E-05 0.E+00 pezoelectrc angle Fgure 10. Comparson of statc deflecton under mechancal load; L/h=

16 2.E-04 1.E-04 1.E-04 Dsplacement (m) 1.E-04 8.E-05 6.E-05 4.E-05 Wth Optmzaton Wthout Optmzaton 2.E-05 0.E+00 pezoelectrc angle Fgure 11. Comparson of statc deflecton under electro-mechancal load; L/h=100. VII. Concludng Remars A procedure has been developed to mprove the vbratory response of composte shells wth dstrbuted pezoelectrc patches under a varety of loadng condtons. The maxmum deflecton assocated wth the frst fve modes of vbraton and the statc response s mnmzed usng stacng sequence as desgn varable. A refned layerwse theory s used n conjuncton wth a two-way coupled electro-mechancal feld formulaton to determne the response of composte shells subjected to electrcal, mechancal and combned electro-mechancal loadng condtons. Constrants are mposed on the natural frequences and stresses. A closed loop control system s desgned usng Lnear Quadratc Gaussan controller. The Kresselmeer-Stenhauser (K-S) functon approach s used to formulate the multple objectve functon problem. A Smulated Annealng algorthm s used as the search algorthm. Parametrc studes are conducted to nvestgate the nfluence of the orentaton of the pezoelectrc patch on the dynamc response. The followng observatons can be made from the present study. 1) The developed framewor s effcent and applcable for the desgn of composte structures wth mproved vbratory characterstcs. 2) The K-S functon approach n conjuncton wth the LQG controller s effcent n mprovng the overall dynamc modal response under all loadng condtons. 3) Sgnfcant mprovements are observed n the reductons of the ndvdual objectve functons assocated wth the frst, second, fourth, and ffth modes of vbraton. 4) Sgnfcant reductons are also observed n the statc deflecton. 5) The lamnate stacng sequence and the orentaton of the pezoelectrc transducers play an mportant role n mprovng the dynamc response. 16

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