Project A: Active Vibration Suppression of Lumped-Parameters Systems using Piezoelectric Inertial Actuators *

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Beverley Carr
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Project A: Active Vibrtion Suppression of Luped-Preters Systes using Piezoelectric Inertil Actutors * A dynic vibrtion bsorber referred to s ctive resontor bsorber (ARA) is considered here, while

1 Project A: Active Vibrtion Suppression of Luped-Preters Systes using Piezoelectric Inertil Actutors * A dynic vibrtion bsorber referred to s ctive resontor bsorber (ARA) is considered here, while exploring its prcticl ipleenttion using piezoelectric ceric (PZT) inertil ctutors. The ARA is pssive bsorber with n dditionl dynic feedbc copenstor within the PZT ctutor (see Figure 1). Without ny controller, it becoes pssive vibrtion bsorber due to internl dping nd elsticity properties of the piezoelectric terils, hence, it is inherently fil-sfe. For ctive opertion, the copenstor preters re designed such tht resonnce condition is intentionlly creted within the bsorber subsection to iic the vibrtory energy fro the syste of concern to which it is ttched. The resonnce condition cn be creted through the pproprite design of the copenstor nd ipleented through djusting the externl electricl voltge pplied to the PZT ctutor. Although PZT terils inherently possess nonliner chrcteristics (e.g., hysteresis), n equivlent linerized odel could be developed for the ctutor subsyste. Inertil ss PCB Series 71 Actutor c u(t) Figure 1 PCB series 71 PZT inertil ctutor (left), schetic of opertion (iddle), nd siple SDOF theticl odel (right). A very iportnt coponent of ny ctive vibrtion bsorber is the ctutor unit. Recent dvnces in srt terils hve led to the developent of dvnced ctutors using piezoelectric cerics, shpe eory lloys, nd gnetostrictive terils. Over the pst couple of decdes, the piezoelectric cerics hve been utilized s potentil replceents for conventionl trnsducers. These terils re copounds of led zirconte-titnte (PZT). The PZT properties cn be optiized to suit specific pplictions by pproprite djustent of the zirconte-titnte rtio. Specificlly, piezoelectric inertil ctutor is n efficient nd inexpensive solution for ctive structurl vibrtion control. As shown in Figure 1, it pplies point force to the structure to which it is ttched. The underlying proposition in ll dynic vibrtion bsorbers is to properly sensitize the bsorber subsection such tht it becoes bsorbent of the vibrtory energy (see Figure ). In this project, n ipleenttion of the tuned vibrtion bsorbers referred to s ctive resontor bsorber (ARA) is discussed. Using siple position (or velocity or ccelertion) feedbc * Jlili, N. nd Knowles, D. W., Structurl Vibrtion Control using n Active Resontor Absorber: Modeling nd Control Ipleenttion, Srt Mterils nd s, 13 (5), (004). Active Vibrtion Control Instruenttion, A Division of PCB Piezotronics, Inc., 3 Spring 007 Dr. Jlili

2 control within the bsorber subsyste, the ARA enforces the doinnt chrcteristic roots of the bsorber subsection to be on the iginry xis, nd hence leding to resonnce (Figure - botto). Once the ARA becoes resonnt, it cretes perfect vibrtion bsorption t this frequency. x Absorber c x 1 Point of ttchent Priry Sensor (Accelertion, velocity, or displceent esureent) x c u ( t) copenstor Absorber x 1 Point of ttchent Priry Figure A generl priry structure with pssive (top) nd ctive (botto) bsorber settings. Overview of PZT Inertil Actutors: The PZT inertil ctutors re ost coonly de out of two prllel piezoelectric pltes (see Figure 1). The resonnce of such ctutor cn be djusted by the size of the inertil ss (see Figure 1). Incresing the size of the inertil ss will lower the resonnce frequency nd decresing the ss will increse it. The resonnce frequency, f r, cn be expressed s f r = 1 π, (1) where is the effective stiffness of the ctutor, e is defined s = + + () PZT nd e PZT is the PZT effective ss, inertil is the inertil ss, nd cc is the cceleroeter ss. Using siple SDOF syste (see Figure 1), the preters of the PZT inertil ctutors cn be experientlly deterined. The odel preters consist of the effective ss, dping nd stiffness of the bsorber s well s the polynoil for of the non-linerity. inertil cc 4 Spring 007 Dr. Jlili

3 Active Resontor Absorber Concept: The concept of ARA is to design copenstor using siple feedbc (position or velocity, or ccelertion) control within the bsorber subsection for the entioned sensitiztion. Such rrngeent plces the doinnt chrcteristic roots of the bsorber subsection to be on the iginry xis, nd hence, creting resonnce (see Figure 3). S-plne Ig Rel Poles of pssive bsorber Poles of ARA Figure 3 Schetic of the ctive resontor bsorber concept through plcing the poles of the chrcteristic eqution on the iginry xis. The ARA requires only one signl fro the bsorber ss, bsolute or reltive to the point of ttchent (see Figure -botto). After the signl is processed through copenstor, n dditionl force is produced, for instnce, by PZT inertil ctutor. By properly setting the copenstor preters, the bsorber should behve s n idel resontor t one or even ore frequencies. As result, the resontor will bsorb vibrtory energy fro the priry ss t given frequencies. The frequency to be bsorbed cn be tuned in rel-tie. Moreover if the controller or the ctutor fils, the ARA will still function s pssive bsorber, thus it is inherently fil-sfe. For the cse of liner ssuption for the PZT ctutor, the dynics of the ARA (Figure - botto) cn be expressed s x t) + c x ( t) + x ( t) u( t) = c x ( t) x ( ) (3) ( t where x 1 (t) nd x (t) re the respective priry (t the bsorber point of ttchent) nd bsorber ss displceents. The ss is given by eqution (), nd control u(t) is designed to produce designted resonnce frequencies within the ARA. The objective of the ARA is to eep the bsorber subsyste rginlly stble t prticulr frequencies in the deterined frequency rnge. The objective of the feedbc control, u(t), is to convert the dissiptive structure (Figure -top) into conservtive or rginlly stble one (Figure -botto) with designted resonnce frequency, ω c. In other words, the control is 5 Spring 007 Dr. Jlili

4 the plceent of doinnt poles t ± jω c for the cobined syste, where j = 1. As result, the ARA becoes rginlly stble t prticulr frequencies in the deterined frequency rnge. Using siple position (or velocity, or ccelertion) feedbc within the bsorber subsection (i.e. U = U X ), the corresponding dynics of the ARA given by (3) in the Lplce doin becoe ( s c s + ) X U X = C( s) X = ( c s ) X (4) The copenstor trnsfer function U (s) is then selected such tht the priry syste displceent, t the bsorber point of ttchent, is forced to be zero, i.e., C( s) = ( s + cs + ) U = 0. (5) The preters of the copenstor re deterined through introducing resonnce conditions to the bsorber chrcteristic eqution, C (s). Tht is, equtions Re{ C ( jω i )} = 0 nd I{ C ( jω i )} = 0 re siultneously solved, where i = 1,,..., l nd l is the nuber of frequencies to be bsorbed. Using dditionl copenstor preters, the stble frequency rnge or other properties cn be djusted in rel tie. Consider the cse where U(s) is ten s proportionl copenstor with single tie constnt bsed on the ccelertion of the ARA given by gs U = U X where U = (6) 1+ Ts In the tie doin, the control force, u(t), is given s t g ( t τ ) T u( t) = e x ( τ ) dτ T (7) 0 To chieve idel resontor behvior, two doinnt roots of eqution (4) re plced on the iginry xis t the desired crossing frequency ω c. Substituting s = ±jω c into eqution (4) nd solving for the control preters, g c nd T c, one cn obtin c g + c = 1, ω ω T c = c ω, for ω = ωc (8) The control preters, g c nd T c, re bsed on the physicl properties of the ARA (i.e. c,, ) s well s the frequency of the disturbnce ω, illustrting the ARA does not require ny infortion fro the priry syste to which it is ttched. However, when the physicl properties of the ARA re not nown within high degree of certinty, ethod to uto-tune the control preters ust be considered. The stbility ssurnce of such uto-tuning proposition will bring priry syste preters into the derivtions, nd hence, the priry syste cn not be totlly de-coupled. 6 Spring 007 Dr. Jlili

Assignents: In order to deonstrte the effectiveness of the proposed ARA, siple single-degree-offreedo (SDOF) priry syste subjected to tonl force excittions is considered.

5 Assignents: In order to deonstrte the effectiveness of the proposed ARA, siple single-degree-offreedo (SDOF) priry syste subjected to tonl force excittions is considered. As shown in Figure 4, two PZT inertil ctutors re used for both the priry (odel 71-A01) nd the bsorber (odel 71-A0) subsections. Ech syste consists of pssive eleents (spring nd dper of the PZT terils) nd ctive coprtent with the physicl preters listed in Tble 1. The top ctutor cts s the ARA with the controlled force u(t), while the botto one represents the priry syste subjected to the force excittion f(t). x c u ( t ) x 1 1 c 1 1 f(t) Figure 4 Ipleenttion of ARA concept using two PZT ctutors (left) nd its theticl odel (right). Tble 1 Experientlly deterined preters of PCB Series 71 PZT inertil ctutors. PZT Syste Preters PCB Model 71-A01 PCB Model 71- A0 Effective Mss [gr], epzt Inertil Mss [gr], inertil Stiffness [N/], Dping [Ns/], c Derive the governing dynics for the cobined syste. For the PZT ctutor odeling, use the constitutive equtions of piezoelectric terils. For this, the one-diensionl liner electroechnicl constitutive reltions between the strin, S 3, the stress, T 3, nd the electric field, E 3, y be expressed s 1 S 3 = d33 E3 + T 3 (9) Y33 where d 33 is the liner electroechnicl coupling coefficient nd Y 33 is the PZT teril s elstic odulus esured t constnt field. In ost cses, PZT ctutors produce displceent. If used in restrinted configurtion, they y generte forces. Force genertion is lwys coupled with reduction in displceent. To siplify the nlysis, liner constitutive equtions re often ssued nd the PZT ctutor is odeled s spring/ss syste. A syste level ctutor description with forces, in the presence of n externl echnicl lod F 0, is shown in Figure 5. The eqution of otion cn be written s 7 Spring 007 Dr. Jlili

33 33 33 33 s v 0 (11) F 0 F 0 x v M x F v V dc Drive Aplifier Figure 5 PZT ctutor-structure. Figure 6 Liner (stced design) piezoelectric ctutor.

6 ( M + /3) x + F + F = 0 (10) where M is the externl lod ss, is the ctutor ss, nd F is the force exerted by the ctutor. Assue tht the ctutor hs n-lyered stc configurtion (see Figure 6) with ech lyer contining n re, A, nd thicness, d. The resultnt ctutor stress (i.e., ctutor force) y be expressed s x d A Y d F = A Y v = ( x ) v = F F nd d d where = AY / 33 nd is the PZT teril equivlent stiffness s v 0 (11) F 0 F 0 x v M x F v V dc Drive Aplifier Figure 5 PZT ctutor-structure. Figure 6 Liner (stced design) piezoelectric ctutor.. Use single tie constnt, bsed on the ccelertion of the ARA, nd estblish the sufficient nd necessry condition for syptotic stbility s functions of copenstor preters, g nd T, using Routh-Hurwitz ethod. If the priry syste is subjected to hronic excittion with unit plitude nd frequency of 800 Hz with the ARA nd priry syste preters ten s those given in 7 Tble 1 nd d 33 = 10 C/N, nuericlly siulte the priry syste nd the bsorber displceents for set of stble copenstor preters. 3. The physicl preters re ssued to be nown exctly. However, these preters re not nown exctly in prctice or vry with tie, so the cse with estited syste preters ust be considered. To deonstrte the fesibility of the vibrtion control, the noinl syste preters (, 1,, 1, c, c 1 ) re fictitiously perturbed by 10% (i.e., representing the ctul vlues) in the siultion. However, the noinl vlues of, 1,, 1, c, nd c 1 re going to be used for clcultion of the copenstor preters, g nd T. Try to tune the syste preters ccordingly such the vibrtion suppression is ttined even for this perturbed syste. Report your nuericl results nd discuss the. 8 Spring 007 Dr. Jlili

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