Vibration Control Using Heat Actuators

Transcription

1 World Journal of Meanis, 06, 6, 3-37 Publised Online August 06 in SiRes. ttp:// ttp://dx.doi.org/0.436/wjm Vibration Control sing eat Atuators Ilan uzu Department of Meanial Engineering, California State niversity, Saramento, CA, SA Reeived 3 July 06; aepted August 06; publised 4 August 06 Copyrigt 06 by autor and Sientifi Resear Publising In. is work is liensed under te Creative Commons Attribution International iense (CC BY). ttp://reativeommons.org/lienses/by/4.0/ Abstrat is paper fouses on teoretial investigation of ative vibration ontrol of a antilever beam using eat atuation. e atuator is a tin metal bar rigidly bonded to te beam on one fae and subjet to eat input on te opposite fae. e atuator ten works like a piezoeletri atuator, and expands and ontrats in response to applied eat. We assume tat te atuator is insulated so tat no eat is transferred to te beam, ensuring tat te eat does not alter te beam s termal state. o avoid neessity of ooling, we onsider two atuators working togeter at te same spanwise loation, one on te upper and one on te lower fae of te beam. en, te beam an be bent up and down by applying eat to te lower and upper atuators, respetively. e governing equations are partial differential equations for one-dimensional eat ondution of te atuators and te bending vibration of te beam wit attaed atuators. For an approximate solution, Rayleig-Ritz metod replaes te partial differential equations wit a system of ordinary differential equations. A ontrol model is obtained from a low-dimensional representation of te system, and used to design feedbak ontrol and observer by means of QR and Kalman-Buy filtering teniques. e ontrol signal obtained is introdued into te plant model, a ig-dimensional representation of te system, to mimi te true system as losely as possible. In a numerial appliation, te response of te beam to an initial exitation is simulated, wi demonstrates tat te eat atuators are in fat effetive in ative vibration ontrol of te beam. Keywords Vibration Control, eat Atuator, emperature Control, Euler-Bernoulli Beam, Output Control. Introdution Suppression of vibration in flexible strutures remains to be a allenging problem in te related tenial ommunity. e urrent trend is to use ative means to suppress te vibration. e ative ontrol of vibration in strutures is implemented by means of sensors, atuators, an aurate matematial model of te wole system (flexible struture + sensors + atuators), and a omputer to arry out all required omputations in real time. ow to ite tis paper: uzu, I. (06) Vibration Control sing eat Atuators. World Journal of Meanis, 6, ttp://dx.doi.org/0.436/wjm

2 I. uzu Sensors do all measurements, but usually te states of te system annot be measured diretly. ene, te matematial model of te system is used to estimate all of te states based on te available measurements, provided tat te system is observable. e matematial model is also used to ompute te ontrol signal to be inputed to te atuator based on tese estimates of te states. e atuator produes wat is needed to affet te true system to suppress te vibration. is an be fore, moment, strain, et. e atuators widely used in vibration ontrol are ydrauli, pneumati, and piezo-eletri, wit ea aving its own benefits and disadvantages depending on te speifi appliations in wi tey are used []. A speial attention as been given to piezoeletri atuators in te reent years. ese atuators are bonded on surfaes of strutures at some disrete loations to alter te loal strain terein by te appliation of proper voltage signals to atuators []. Anoter type of atuation tat as started reeiving some attention in vibration ontrol is simple eat. Considering a struture initially at room temperature, an applied load auses temperature to vary over te body of te struture nonuniformly. Areas of te struture under tension are now ooler and areas under ompression are otter tan te room temperature, reating a temperature gradient in te struture. is in turn auses a eat flow by ondution from otter areas to ooler areas. ene, a portion of te meanial energy introdued to te struture is onverted to eat troug tis proess. Sine te eat flow reated is irreversible, te portion of te meanial energy onverted to eat is dissipated. is penomenon is known as termoelasti damping [3]. Wit tis fat, question remains as to weter vibration in a struture an be suppressed by artifiially introduing a termal gradient in te body by means of some external eat atuation. Some earlier studies ad sown tat, wen applied properly, external eat atuation ould in fat introdue signifiant amount of termoelasti damping in strutures [4]-[9]. Pioneering resear by Edberg [4] onsiders ontrol of vibration in a beam using a termoeletri atuator, wi as an ability to eat or ool te beam to produe a termal gradient aross te tikness of te beam. Experimental results reported in te paper sow tat wit a simple lead ompensation, damping ratio for te first vibrational mode of a antilever beam is inreased by more tan nine times. Anoter work by Murozono and Sumi [5] also investigates ative vibration ontrol of a antilever beam, but tis time by eat input into te beam troug te upper and lower surfaes, solely for reating a temperature gradient aross te beam setion. Foil strain gauges bonded on te surfaes are used as eat atuators. e eat input is troug te upper or lower surfae, depending on te diretion of te gradient needed. Experiments reveal tat te damping ratio for te first vibration mode is inreased by about ten times. A omputational study by uzu et al. [6] onsiders a uniform rod in longitudinal vibration and sows tat ontrol of bot vibration and temperature an be aieved by eat input troug termoeletri atuators bonded on te lower and upper surfaes of te rod. uzu and Gonzalez-Roa [7] extended te work of uzu et al. [6] to a beam in bending. Bot vibration and temperature are ontrolled simultaneously using termoeletri atuators. Zang et al. [8] proposes a vibration ontrol of termally indued vibration in spae strutures. Atuators are eaters bounded on te spaeraft struture at some optimal positions. uzu et al. [9] explores te idea of using eat pates as atuators working in pairs on te upper and lower surfaes at some disrete spanwise loations of a uniform beam to simultaneously ontrol vibration and temperature in te beam. A numerial example sows te presene of termoelasti damping and demonstrates simultaneous ontrol of elasti displaement and temperature gradient using a set of displaement measurements. Referenes [4]-[9] ave a major disadvantage: all use eat atuation tat requires eating te flexible struture to be ontrolled, wi as a potential to alter te termal state of te struture signifiantly, wi in turn an prevent te struture from funtioning properly. o overome tis problem, tis paper proposes a simple design of an atuator wi inludes a tin metal bar bonded to te struture to be ontrolled on one fae and subjet to eat input on te opposite fae, as seen in Figure. en, te atuators will work somewat like piezoeletri atuators, expanding and ontrating in response to applied eat, instead of eletri voltage in te ase of piezoeletri atuators. For more effetive ontrol of te struture, multiple of tese atuators an be rigidly bonded to te upper and lower surfaes of te beam in pairs at a desired number of spanwise loations. It is assumed tat te atuators are insulated so tat no eat is transfered to te flexible struture. Sine ea atuator and te struture are rigidly attaed to ea oter, extension in an atuator due to te appliation of eat is transferred diretly to te ost struture. e beam will ten bend down if eat is applied to one or more upper atuators, and it will bend up if eat is applied to one or more lower atuators. ene, te beam's defletion an be ontrolled by oosing proper eat inputs to te atuators. 4

3 I. uzu. Modeling Figure. Cantilever beam wit eat atuators. We will demonstrate our modeling and ontrol approa on a antilever beam, wi is a fundamental element ommonly enountered in strutures. o make te presentation of our approa more lear and fous, we limit ourselves to a simple model by assuming te beam is uniform and tere is only one pair of atuators, as sown in Figure. e beam is slender so tat Euler-Bernoulli beam teory is valid. e upper and lower atuators are assumed to be rigidly bonded to te beam, wi means te displaements of te beam and te atuators are equal at every point on te bonding surfaes. We define te displaement field in te beam as w u ( x, z, t) = z = zwx, v = 0, w = w( x, t) () x were uvw,, are te displaement omponents in xyz-diretions,,, respetively, and subsript x denotes a partial derivative wit respet to x. Displaement omponents of a typial point P on te beam are illustrated in Figure. e beam s left-end is fixed and its rigt-end is free so tat w is subjet to te boundary onditions: w 0, t = 0, w 0, t = 0, w, t = 0, w, t = 0 () x xx xxx were ea subsript x indiates a partial derivative wit respet to x. en, te strain omponents of te beam an be expressed as ε u w xx = = z = zw, 0 xx yy zz xy xz yz x x ε = ε = ε = ε = ε = (3) Sine te ends of te atuators are free, te normal strains of te atuators will be onstant at beam s strains at its interfae wit te atuators [0], wi means ( ε ) w, ( ε ) = = w (4) xx xx xx xx Assuming a linear stress-strain relation, we an express te only nonzero stress omponent in beam as σ = Eε = Ezw (5) xx xx xx were E is te Young s modulus of te beam. On te oter and, te normal stresses in te atuators are ( σxx ) = E a εxx α = Ea wxx α ( σ ) = E ( ε ) α = E ( w α ) (6) xx a xx a xx were E a is te Young s modulus and α te oeffiient of termal expansion for te atuator material, and and are te temperature deviations from te ambient temperature 0 in te lower and upper atuators, 5

4 I. uzu respetively. We assume te atuators are insulated on all faes exept te ones tat are subjet to te eat inputs Q ( t ) and Q ( t ), as sown in Figure. Beause of tese boundary onditions and assumption tat Q and Q are onstant wit x, we onlude tat and are also onstant wit x, i.e. = ( zt, ) and = ( zt, ). We approximate w,, as well as using te expansions were (, ) =Φ, (, ) = Ψ, (, ) = Ψ( ) w xt x q t zt zθ t zt zθ t (7) ( x) φ ( x) φ ( x) φ ( x), ( z) ψ ( z) ψ ( z) ψ ( z) Φ = Ψ = (8) m n are te matries onsisting of m and n number of sape funtions and q θ n ( t) q ( t) q ( t) q ( t) θ ( t) θ ( t) θ ( t) θ ( t) =, =, m n ( t) θ ( t) θ ( t) θ ( t) = are te vetors of generalized oordinates for elasti displaement, and temperatures of te upper and lower atuators. Note tat we use te same matrix of sape funtions for te upper and lower atuators beause we assume te atuators are idential and plaed symmetrially about te x axis. We oose te sape funtions φ i ( x), i =,,, m to be te eigenfuntions of a uniform antilever beam, wi are given as sin βi+ sin βi φi( x) = sin βix sin βix ( os βix os βix), i =,,, m (0) os β + os β i were β i is te i-t β solution of te arateristi equation os β os β =. For example, te first four solutions are β =.875, β = 4.694, β 3 = and β 4 = et us now express te total potential energy, wi is te summation of potential energies of te beam and te lower and upper atuators: = d d d σ ε V + V V V σ ε + σ ε V V xx xx xx xx xx xx i (9) r α ( ) l + r + xr x = EI d d d d w 0 xx x+ Ea Aa xx a xx w x E b w x z xl x xr x + Ea Aa xxd a xxd d w x+ E bα w x z xl x l were Aa = b is te ross-setional area of te ea atuator, V is te volume of te beam, V and V are 3 te volumes of te atuators, and I = z da = B is te area moment of inertia of te beam. Substituting 3 Equation (7) into Equation (), we rewrite te potential energy expression as were ( t) K ( t) ( t) C ( t) () = q q + q aθ () We note tat K is te stiffness matrix. Kineti energy of te beam is 0 xr xx xx a a x xx xx l K = EI Φ Φ dx + E A Φ Φ d, x xr + ( d )( d ), θ x θ θ Ca = Eabα Φxx x Ψ z t = t t l (3) 6

5 I. uzu were te mass matrix is = d d d ρ u + w V + V ρa u + w V + V ρa u + w V V = ( x ) d a( x ) d a( x ) d ρ z w + w V + V ρ w + w V + V ρ w + w V V xr = m w d d d 3 0 x x+ m w x + ρ 0 aaa w x x + w x l = q ( t) Mq ( t) xr M = d d d 0 m ΦΦ x x + mφφ x + ρaaa ΦΦ x x x +ΦΦx 3 x l (5) were ρ is te density of te beam, m mass per unit lengt of te beam, and material. Generalized fore is obtained from te virtual work: (4) ρ a density of te atuator,, d q, d qq (6) 0 0 δw= f xt δw xt x= δ f xt Φ x= δ were f( xt, ) is te applied lateral fore per unit lengt, w( xt, ) = Φ 0 δ virtual defletion of te beam, and Q t f xt, dx (7) is te generalized fore vetor. e equations of motion of te beam an be derived by means of agrange s equations of motion, wi an be written in te vetor as d = Q dt q q were = is te agrangian. Substituting Equations () and (4) into te agrangian, and te result into Equation (8), we get (8) Mq + Kq+ C a θ = Q (9) Equation (9) governs te elasti defletion of te beam. Sine tis equation is oupled wit te temperature differene θ, it must be solved simultaneously wit te eat equations governing te temperatures in te upper and lower atuators. ene, we next derive tese eat equations. e eat ondution in te upper atuator is desribed by te boundary-value problem defined by te following partial differential equation and boundary onditions: κ t z = k = 0 at z = z Q k = at z + z S were κ is te termal diffusivity, k te termal ondutivity, S = bl is te surfae area of te upper fae. We oose te sape funtions ψ i ( z), i =,,, n as te eigenfuntions of te system defined by Equations (0) wit Q = 0, wi are [] (0) z ψ i ( z) = os ( i ) π, i =,,, n () Substituting te seond of Equation (7), premultiplying bot sides by κψ, and integrating te result over 7

6 I. uzu te domain of te upper atuator, we get + + ( Ψ Ψ dz ) = κ ( Ψ Ψ dz ) θ θ () zz Similarly, substituting te seond of Equations (7) into te boundary onditions, we get Q Ψ z ( ) θ = 0, Ψ z ( + ) θ = (3) ks sing te integration by parts, te seond integral in Equation () an be written as ( Ψ Ψ dz) = ( Ψ Ψ ) ( d Ψ Ψ z ) Q + = Ψ ( + ) ( Ψ Ψ dz ) θ θ θ zz z z z ks θ z z Substituting Equation (4), Equation () an be rewritten in te ompat form were (4) θ + θ = Q (5) = + Ψ Ψ dz = κ π = κ d z z z Ψ Ψ = κ κ = Ψ ( + ) = ks ks n ( ) ( n ) Equation (5) ompletely desribes te temperature in te upper atuator for presribed eat input Q ( t ). Note tat by negleting te termoelastiity of te atuator, we assume te temperature is not affeted by te elasti deformation of te beam. Equation (5) an be written in te state-spae form (6) θ = θ + Q (7) wi ten leads to te following salar equations for te individual generalized oordinates: κ θ = Q ks κπ κ θ ks 4κπ κ θ ks θ = θ Q 3 3 = θ + Q ( n ) n κ π n n κ = θ + ( ) Q ks (8) 8

7 I. uzu ene, te eigenvalues of te eat ondution are i κ π for i =,,, n. We note tat te eigenvalue orresponding to i = is zero and all te remaining ones are negative real so tat approaes to a onstant, instead of zero, as Q 0. e eat ondution in te lower atuator is similarly desribed by te boundary-value problem = z κ t k = 0 at z = z Q k = at z = + z S Following te same steps, Equation () troug Equation (4), we get (9) θ + θ = Q (30) t. e matries, and are as given in Equations (6). e salar equations for te individual generalized oordinates for te lower atuator ave te same forms as Equation (8). ey an in fat be obtained by replaing te subsript by in Equation (8). e eigenvalues are exatly same as te eigenvalues of te upper atuators. wi governs te temperature in te lower atuator for presribed eat input Q.. Plant Model Equations (9), (5) and (30) ompletely desribe te motion of te uniform antilever beam wit te upper and lower atuators for presribed inputs f( xt, ), Q ( t ) and Q ( t ). We ast te system in te new first-order form as θ = θ + Q θ = θ + Q q = s s = M Kq M C + M Q ( θ θ ) a wi an be rewritten in te state-spae form x = Ax+ Bu (3) were θ Q θ x =, u= Q (33) q Q s are te state and input vetors, and A, B = = I M Ca M Ca M K M are te oeffiient matries. Note tat te dimensions of x, u, A, and B are ( m+ n), ( m+ n) ( m+ n), and ( m+ n) ( m+ ), respetively. (3) (34) m +, We will all te state-spae model of Equation (3) te plant model, wi, in absene of an experimental model, will be used to simulate te beavior of te atual system and to test te performane of our ontrol design. In order for tis model to approximate te beavior of te atual system better, we will oose m and n (te number of sape funtions used in te expansions of Equations (7)) as ig as possible. Control design, on 9

8 I. uzu te oter and, will be based on a ontrol model tat uses a less number of sape funtions, as explained next. Our objetive in tis paper is to ontrol te defletion of te beam using Q ( t ) and Q ( t ) as te ontrol inputs. e allenge is tat te ontrol as to be done for positive values of Q ( t ) and Q ( t ) so tat ooling of te atuators is never neessary. is paper offers a novel approa to overome tis allenge. e beam s defletion is governed by Equation (9) from wi we note te defletion is atually affeted by te differene θ = θ θ. Subtrating Equation (30) from Equation (5), we get θ + θ = Q (35) were Q = Q Q. ene, for te ontrol objetive, Equation (35) is oupled wit Equation (9) so tat te differene Q = Q Q an be used as te ontrol input instead of te individual inputs Q and Q. e ontrol of te defletion will ertainly require bot negative and positive values of te input Q, wi an be aieved by properly oosing te individual inputs Q and Q. In fat, to avoid unneessary eat inputs into te beam, we oose Q and Q as follows: If negative Q is required, we aieve it by oosing Q = 0 and Q = Q. Similarly, if positive Q is required, we oose Q = Q and Q = 0... Control Model Control design must be as simple as possible for easy implementation and fast real-time omputations. is in turn requires a low-order ontrol model. o tat end, we first note tat te seond integral of te expression for C defined in Equation (3) as a matrix value a [ ] + Ψ dz = 0 0 (36) wi learly means te first olumn is te only non-zero olumn in C a, wi in turn means te only tem- perature omponent appearing in te equation governing elasti defletion, Equation (9), is θ = θ θ, and n n not any of θ = θ θ,, θn = θ θ. ene, te ontrol of te elasti defletion by applying eat input an be aieved only troug θ. For tis reason, in te ontrol model, we will oose n = and keep te governing equation for θ only. en, te oupled system an be expressed in te state-spae model as x = Ax + Bu (37) were θ x = q, u = Q s are te ontrol state vetor and te ontrol input, and κ ks A = 0 0 I, B = 0 (39) M Ca M K 0 0 e dimensions of x, u, A and B in tis ase are ( + m),, ( + m) ( + m), and ( + m), respetively. o keep te dimension even smaller, we will use an m for te ontrol model several times smaller tan for te plant model. Note tat tis model allows us to ontrol not only te defletion in te beam, but also θ, wi is te most dominant temperature omponent among all θ,, θn. 3. Output Feedbak Control Design e system to be ontrolled is inerently underatuated beause antilever beam is a distributed-parameter system wit infinite degrees of freedom wile te number of ontrol inputs tat an be used must always be finite. is inerent property of te system remains a fat even toug we use a finite-order approximation of te atual system, namely te ontrol model, beause te order of te ontrol model is + m wile te (38) 30

9 I. uzu number of ontrol input available is only one. e order of te ontrol model an get very ig depending on te osen m. inear Quadrati Regulator (QR) seems to be one of te best design metods available for tis type of systems. e QR design metod determines an optimal ontrol gain matrix G so tat te ontrol input u t = Gx t minimizes te quadrati performane measure t f J= ( t) Q ( t) u ( t) Ru ( t) dt x x + (40) t0 were Q is a real symmetri positive semidefinite matrix, R a real symmetri positive definite matrix, and t and t are te initial and te final times []. e QR yields te ontrol gain matrix as 0 were f G = R B K (4) K is a real symmetri matrix satisfying te algebrai matrix Riati equation A K + K A + Q KBR B K = 0 (4) x is available for feedbak sine it y of te atual system, wi inludes a number of measurements of defletion and/or its rate on te beam. en, te ontrol input an be obtained from u ˆ t = Gx ( t) were x ˆ ( t) is te estimate of x ( t). sing our ontrol model, te estimate x ˆ ( t) is obtained troug an observer design wose dynamis is desribed by te matrix differential equation xˆ ˆ ˆ t = Ax t + Bu t + y t y( t) (43) e ontrol design desribed above assumes tat te state vetor ( t) uses te input as u ( t) = Gx ( t). owever, in reality, x ( t) must be estimated from te output ( t) were is te observer gain matrix and yˆ ( t) = C x ˆ ( t) is te estimate of te atual output ( t) y []. sing te Kalman-Buy Filter, is obtained from =Σ CW were W is te sensor noise intensity matrix and Σ is te solution of te algebrai matrix Riati equation AΣ+Σ A + V ΣCW CΣ= 0 (44) in wi V is te state exitation noise intensity matrix. We assume defletion measurements at te points P and P sown in Figure are available as output signals. Sine we use te plant model to approximate te atual system, te output of te plant an be written as were y t (, ) (, ) y t w t = = = Cx t y t w t ( ) 0 0 Φ 0 C = 0 0 Φ 0 is te output matrix relating te atual state to te output. e estimate of te output, on te oter and, is written as were yˆ ˆ t w t, yˆ t wˆ, t y t = = = Cx t ( ) 0 Φ 0 C = 0 Φ 0 is te output matrix relating te estimated state to te estimated output. One Q( t) Q ( t) Q( t) u ˆ t Q t G t are osen as follows: = is obtained from = = x, te individual Q ( t ) and Q ( t ) (45) (46) (47) (48) 3

10 I. uzu = = < = = If Q t 0, Q t Q t and Q t 0 If Q t 0, Q t 0 and Q t Q t (49) 4. Numerial Appliation e modeling approa and te ontrol design presented above will be numerially demonstrated for a uniform antilever beam. We will use te same antilever beam and te related parameters used in [9] so tat we an ompare te results from tis paper diretly to te ones from [9]. Pysial parameters of te beam and te atuators are listed in able. For most effetive atuators, we oose x l = 0 and xr = l so tat te atuators are losed to te root were te strain is ig. As revealed in able, te material of te beam is osen to be steel. We oose steel also for te atuators beause it as ig Eaα, wi makes te atuators more effetive. sing te values of te parameters listed in able, we onstrut te matries M, K and C a using te expressions given in Equations (3) and (5). o give te reader an idea about te numerial forms of tese matries, we present tem for some small number of m and n, namely m = and m = 3 : M = kg, K kn m, C a N K = = (50) e matries, and are as given in Equation (6). A and B matries given in (34) an be onstruted by using te numerial forms of te submatries: = K s, = K s W M Ca = m s K, M K s, = (5) M = kg o demonstrate te simulation of te losed-loop system, we oose m = 7 and n = 0 for te plant model and m = 3 and n = for te ontrol model so tat A is 34 34, B 34 9, C 34, A 7 7, B 7, and C 7. For te ontrol and observer designs, we oose te following design parameters: [ ] [ ] R = Q = V = W = I (5) 7 0, diag 0.00,,,,,,, diag 000,,,,,,, were diag means diagonal matrix and I is identity matrix. en, te ontrol and observer gain matries are omputed to be G = 6600, = (53) e orresponding eigenvalues of open-loop plant and ontrol models, losed-loop ontrol model, and observer dynamis are listed in able. e open-loop plant model as 34 eigenvalues wile te ontrol model as only 7. e zero and te negative real eigenvalues of te plant orrespond to te eat ondution of te atuators, as explained earlier, and purely omplex ones orrespond to te vibration of te beam. e ontrol model as a zero eigenvalue and tree pairs of purely omplex onjugate eigenvalues. e tird and te fourt olumns of able sow ow tese eigenvalues are stabilized by te ontrol and observer designs, respetively. owever, te question remains as to weter te ontrol and observer designs stabilize te plant. is question 3

11 I. uzu able. Pysial parameters. Parameter Value nit mm B 4 m m 0.5 mm b 4 m l 40 m ρ = ρ a 7860 kg/m 3 E E a = N/m 6 α 0 /K 6 κ.9 0 m /s k 45 W/(m K) m = ρb kg/m I 3 3 = B Aa S m 4 5 = b 0 m = bl 0.06 m able. System eigenvalues. A A A BG A C ± 4.994i ± 5.4i ± 5.5i.5000 ± 5.049i ± 7.474i ± 7.944i ± 7.944i.80 ± 7.934i ± 0.97i ± 0.73i ± 0.73i ± 0.73i ± i ± i will be answered by te simulation of te losed-loop plant. Blok diagram sown in Figure ompletely desribes ow te ontrol of te plant is implemented. We will test te performane of our ontrol design by simulating te response of te losed-loop plant to te same initial onditions used in [9], namely w x ( x) φ = = (54),0 0 mm, w x,0 0 φ 33

12 I. uzu Figure. Blok diagram of te losed-loop system. φ x wit 0 mm tip displaement. ime istories of openloop, atual, and estimated outputs, namely elasti displaement at x = and x = are plotted in Figure 3. Comparing Figure 3 to te last of Figure of [9], we onlude tat te damping aieved in te present paper is about te same as tat of [9]; in bot, tip displaement redues to 5 mm in approximately 6 yles. Figure 4 sows te eat inputs to te lower and upper atuators needed to aieve tis damping. Considering tat te amplitude of te vibration in tis numerial example is ig, tese eat inputs seem to be reasonable. Even toug [9] uses eat atuators in two spanwise loations, te required eat inputs are quite similar to tose of tis paper. In tis paper, we assume ea atuator is insulated in all faes exept te fae tat is subjet to te eat, inluding te faes normal to x and y axes. is assumption learly means te temperatures everywere in te atuators inrease monotonially until te eat inputs are eased after vibration amplitude redues to a desired value. As an example, temperatures ( t, ) and ( t, ) versus t for 00 s are sown in Figure 5, from wi we read (,00) = 8.3 C and (,00) = 8.3 C. Sine te eat is onfined to a small volume in te atuators, te temperatures get very ig, but still mu less tan te material of te atuators an tolerate. o prevent te temperatures in te atuators to get very ig values, some sort of ooling, su as termoeletri ooling, an be used to ool te atuators. In tat ase, ooling must be introdued into te ontrol design as a disturbane. Contrary to te atuators, no temperature anges our in te antilever beam. In [9], sine te eat is diretly applied to te beam itself, wi as a mu larger volume tan te atuators of tis paper, te temperature inrease in te beam is limited to few degrees. We an also ompare te total eat injeted into te volumes in te two papers. we ompute te total eat input to te atuators witin t seonds as te integral of total eat rate: so tat te beam is initially at rest in te sape of t = s as q ( 00) =.58 kj. On te oter and, in [9], q t Q ξ Q ξ d ξ (55) q t versus t is plotted for 00 seonds in Figure 6, from wi we read te total eat input to te atuators in q 00 =.555 kj, wi is mu larger tan te eat input in te present paper. 5. Conlusions is paper investigates ative vibration ontrol of a struture using a new type of atuator. e struture of interest is a antilever beam. e atuator, on te oter and, is a tin metal bar rigidly bonded to te beam on one fae and subjet to eat input on te opposite fae. e atuator ten an work like a piezoeletri atuator, and expand and ontrat in response to applied eat. We assume tat te atuator is insulated so tat no eat is transferred to te beam. is ensures tat te eat applied to te atuator does not alter te termal state of te beam. One problem tat needs to be addressed is tat expansion of te atuator requires eating wile ontration requires ooling. owever, ooling is not pratial, and to avoid it, we onsider two atuators working togeter at te same spanwise loation, one on te upper and one on te lower fae of te beam. en, te beam an be bent down by applying eat to te upper atuator, and bent up by applying eat to te lower atuator. 34

13 I. uzu Figure 3. Outputs: elasti displaement at x= and x=. Figure 4. eat inputs to te lower and upper atuators. Figure 5. Atuator temperatures. In te first part of te paper, we obtain a ompreensive model of te system tat an be used to design ontrol and simulate te response of te losed-loop system. Euler-Bernoulli beam teory is assumed to be valid to desribe te motion of te beam wit te attaed atuators. en, te agrange s equations of motion are used to derive te governing equations for te vibration of te beam wit te atuators. Sine we assume te elasti motions of te atuators only depend on te elasti motion of te beam, te atuators do not bring any additional elasti degrees of freedom into te governing equations. One-dimensional eat ondution equation is used as 35

14 I. uzu Figure 6. otal eat introdued into te atuators. te equation governing te temperature in ea atuator. We also assume te temperatures of te atuators are not affeted by te elasti motion. e governing equations are a set of tree partial differential equations, ea of wi as an infinite dimension. For an approximate solution, we use Rayleig-Ritz metod to replae te partial differential equations wit a desired number of ordinary differential equations, obtaining a finite, but oterwise ig-dimensional system. e iger te dimension, te more aurate te model is. We obtain te plant model by using a ig-dimensional representation of te system so tat it an be represented as aurately as possible. On te oter and, we obtain te ontrol model by using a low-dimensional representation of te system so tat estimates of te states as well as ontrol inputs an easily be omputed in real time. Sine we do not ave an experimental model, we use te plant model to mimi te true system as losely as possible, wi is te reason to oose a ig dimensional model. e ontrol model is used to design feedbak ontrol and observer by means of QR and Kalman-Buy filtering teniques. e observer design is based on only two displaement measurements, one in te middle and one at te free end of te beam. In a numerial appliation, ontrol signal is omputed using te ontrol and observer gains based on te ontrol model, and inputted into te plant. en, te response of te beam to an initial exitation is simulated. e numerial simulation demonstrates tat te atuators are in fat effetive in ative vibration ontrol of te beam. e results obtained are omparable to some of te results of [9]. For omparison reason, te feedbak ontrol is designed so tat approximately te same damping fator as tat of [9] is aieved. owever, we sow tat te total eat input needed to aieve tis damping is mu smaller tan tat of [9], from wi we onlude tat te atuators of tis paper are more effetive tan tose of [9]. For simpliity of te models, we assume tat ea atuator is insulated in all faes exept te fae it is subjet to eat. Wit tis assumption, te eat ondution is one-dimensional (only in z-diretion), and also we do not ave to make assumptions about te onvetion and/or radiation eat transfer from te faes normal to x and y axes to outside. owever, in real-life appliations, tis assumption is ard to enfore and it must be relaxed. Our future work will aim to make design alterations to relax tis assumption to make te work more appliable to real-life situations. Aknowledgements e autor tanks te editor and te referee for teir omments. Dediation is paper is dediated to te memory of Prof. eonard Meirovit. Referenes [] uber, J.E., Flek, N.A. and Asby, M.F. (997) e Seletion of Meanial Atuators Based on Performane Indies. Proeedings of te Royal Soiety A, ttp://dx.doi.org/0.098/rspa [] Jalili, N. (00) Piezoeletri-Based Vibration Control: From Maro to Miro/Nano Sale Systems. Springer, New York. [3] Nowinski, J.. (978) eory of ermoelastiity wit Appliations. Sijtoff & Noordoff International Publisers. ttp://dx.doi.org/0.007/

15 I. uzu [4] Edberg, D.. (987) Control of Flexible Strutures by Applied ermal Gradients. AIAA Journal, 5, ttp://dx.doi.org/0.54/3.975 [5] Murozono, M. and Sumi, S. (994) Ative Vibration Control of a Flexible Cantilever Beam by Applying ermal Bending Moment. Journal of Intelligent Material Systems and Strutures, 5, -9. ttp://dx.doi.org/0.77/045389x [6] uzu, I., Asmatulu, R. and asanyan, D. (003) Control of te Vibration and emperature in a ermoelasti Rod. Proeedings of 5t International Congress on ermal Stresses and Related opis, Blaksburg, 8- June 003. [7] uzu, I. and Gonzalez-Roa, J. (03) Modeling and Control of ermoelasti Beam. Proeedings of te ASME Dynami Systems and Control Conferene, Palo Alto, -3 Otober 03, V0006A004. ttp://dx.doi.org/0.5/ds [8] Zang, J., Xiang, Z. and iu, Y. (04) Control of te ermally Indued Vibration of Spae Strutures by sing eaters. Journal of Spaeraft and Rokets, 5, ttp://dx.doi.org/0.54/.a360 [9] uzu, I., Moua, J.K. and Olivares, J.G. (06) Control of a ermoelasti Beam sing eat Atuation. Journal of Vibration and Control, in press. ttp://dx.doi.org/0.77/ [0] Crawley, E.F. and de uis, J. (987) se of Piezoeletri Atuators as Elements of Intelligent Strutures. AIAA Journal, 5, ttp://dx.doi.org/0.54/3.979 [] Özişik, M.N. (980) eat Condution. Jon Wiley & Sons, In., New York. [] Meirovit,. (990) Dynamis and Control of Strutures. Jon Wiley & Sons, New York. Submit or reommend next manusript to SCIRP and we will provide best servie for you: Aepting pre-submission inquiries troug , Faebook, inkedin, witter, et. A wide seletion of journals (inlusive of 9 subjets, more tan 00 journals) Providing 4-our ig-quality servie ser-friendly online submission system Fair and swift peer-review system Effiient typesetting and proofreading proedure Display of te result of downloads and visits, as well as te number of ited artiles Maximum dissemination of your resear work Submit your manusript at: ttp://papersubmission.sirp.org/ 37