Frequency Response Optimization Using the Genetic Algorithm in Vibroacoustic Systems

Transcription

1 Frequency Response Optmzaton Usng the Genetc Algorthm n Vbroacoustc Systems Walter J. Paucar Casas, Régs E. Antch, Emanuel M. Cesconeto, Rcardo F. Leuck Flho Departmento de Engenhara Mecânca Unversdade Federal do Ro Grande do Sul walter.paucar.casas@urgs.br, Regs.Antch@gkndrvelne.com, emanuelemc@hotmal.com, leuck.r@gmal.com ABSRAC Although there are many researches carred out n the area o analyss and optmzaton o vbroacoustc systems, ew publcatons show results obtaned by Genetc Algorthms GA and check ts perormance or requency response optmzaton. hs work ntends to contrbute n vbroacoustc analyss and optmzaton, through the development and mplementaton o one knd o algorthm to optmze the sound pressure o coupled and smpled vbroacoustc systems modeled n three dmensons. he response optmzaton s mplemented n our own academc program o nte elements MEFLAB developed nto the commercal sotware MALAB. he optmzaton o the response s also mplemented through an own code developed or genetc algorthms. For the coupled case n study the sound pressure response s reduced adequately n the requency nterval o nterest and allowed us to dsplace one o the two resonance responses to the boundary o the requency nterval showng ts adequacy to optmze the response o coupled systems. Keywords: Global optmzaton, coupled system, lud-structure nteracton 1 INRODUCION A revew o structural and acoustcal analyss technques usng numercal methods lke nte element FEM and boundary element BEM methods, ollowed by a survey o technques or structural-acoustc couplng and a wde dscusson over objectve unctons or passve nose control n structural-acoustc optmzaton s realzed by [1]. Also, [2] demonstrates the potental o the applcaton o normal modes n external acoustcs or optmzaton o radatng structures. hen, although there are many researches carred out n the area o vbroacoustc analyss and optmzaton, ew publcatons present results obtaned usng global optmzaton algorthms. he genetc algorthm method s used or mnmzaton o sound pressure n vbroacoustc systems n [3], where the system s modeled usng BEM. Some works propose a method or nose reducton usng a topology optmzaton [4]. he sound eld s modeled wth the Helmholtz equaton and the topology optmzaton s based on the nterpolaton unctons o contnuous materal, densty and modulus o compressblty. he response optmzaton n coupled systems was explored by [5] and [6] wth the classc method called Sold Isotropc Materal wth Penalzaton SIMP. It was constructed a physcal model o a box composed o thn plates o derent thcknesses and bars o rectangular secton [7]. Insde the lud s ar. he box was nstrumented to allow data collecton rom both the acoustc doman as the doman structure. Ater constructng the physcal 1

2 IV Internatonal Symposum on Sold Mechancs - MecSol 2013 model, a mathematcal model o the box was desgned or smulaton; box s bars were modeled wth Bernoull beam nte elements, plates were modeled wth thn lat plate nte element and lud was modeled wth boundary elements. hs work demonstrates the applcablty and ecency o the developed mathematcal models, when used to analyze and optmze real vbroacoustc systems. hs work s developed n order to mplement algorthms or optmzng the sound pressure o coupled vbroacoustc systems n three dmensons, usng the technque o Genetc Algorthms (GA) to optmze the response. 2 VIBROACOUSIC COUPLED SYSEMS In order to analyze the couplng o a lud-structure system, we consder the eect o lud pressure on the nterace I o the structure surace as beng s I, that added wth the body orces consttute the terms o exctement n the dynamc equaton: sb k, (1) ss u m ss u sγ + I sb where, Ns q da. I A (2) he pressure that the lud exerts on the nterace wth the structure generates dstrbuted orces q normal to the structural surace. Ater substtutng q by p ~, t s obtaned the equlbrum condton at the nterace as: N s N da p. I A (3) where N s are structural shape unctons and N are lud shape unctons. Addng Eq. (3) n Eq. (1): k ss u m ss u + k p (4) s where A s k N N da. (5) Couplng o the structural doman wth the lud doman s orced n the normal drecton nˆ o the nterace surace, through an dentty whch ensures compatblty knematcs. hs can be represented by a slppng condton o the lud n the tangental drecton o the nterace: v (6) nˆ u n ˆ he lud-structure couplng s descrbed n terms o the pressure change or the lud doman near the structural regon, through the ollowng boundary condton: 2

3 p nˆ IV Internatonal Symposum on Sold Mechancs - MecSol 2013 u ˆn, em I (7) Substtutng Eq. (6) n Eq. (7), whch means to replace the normal component v n ˆ by u n ˆ, and adoptng u ~ Ns u to approxmate the value o u n ˆ by u ~, or n dscretzed orm by n u ˆ N s : where orm: m p k p m u (8) m s the matrx wth the terms o nterace, whch can be rewrtten n a sem-dscretzed m N N s di (9) I From Eqs. (4) e (8) we can organze them n a compact matrx, showng the ormulaton u-p: m m ss 0 u kss m p 0 k k u p s (10) As can be seen rom Eq. (10) ths matrx ormulaton s not symmetrc, whch s one o ts man dsadvantages, as preclude the use o several ecent algorthms developed or symmetrc matrces. However, the man benet o ths ormulaton s the small number o degrees o reedom used n the modelng o the lud doman. In the case o ree vbratons, the orce vector o Eq. (10) s zero. he stness matrx o lud-structure nteracton s gven by ntegraton o Eq. (5). he mass matrx o lud-structure nteracton s gven by ntegraton o Eq. (9) and corresponds to: k (11) m 3 MODAL FREQUENCY RESPONSE he modal requency response U s expanded n terms o rght egenvectors Φ o the coupled system accordng to: U ΦQ (12) 1 where the matrx Q consders the let egenvectors Φ and the term Ω relates the egenvalues wth the exctaton requency through: 1 1 Ω dag 2 (13) 1 Q Ω Φ F (14) he soluton u s named as the requency response: 3

4 n 1 IV Internatonal Symposum on Sold Mechancs - MecSol 2013 u (15) q and substtutng ths expresson n Eq. (10), pre-multplyng t by and utlzng the orthogonalty condton o the coupled system s obtaned: where, m q k q ( = 1,2..., n) (16) m M (17) k K (18) F (19) I the exctaton orce actng n the structure s harmonc type, then the response u wll also be harmonc, Fe jt (20) u Ue jt (21) q jt Qe (22) whch ater substtuton n Eqs. (15) and (16) generates the expresson o the requency response as beng a superposton o coupled modes: U n Q n F 2 (23) where Q 1 F 2 (24) he great advantage o the modal superposton nvolves ncludng a lmted number o modes n the calculaton, usually lower than the number o degrees o reedom o the system, ths way the computatonal cost s drastcally reduced. However, a number lower o modes causes errors n the values o natural requences, whle a number hgher o modes can ncrease the processng tme and not compensatng the replacement o the drect method 4

5 4 GENEIC ALGORIHM PROGRAMMING IV Internatonal Symposum on Sold Mechancs - MecSol 2013 he problem ams to nd the mnmum value o a constraned nonlnear multvarable uncton, (x), stated as: x x c 0 ceq 0 Mn ( x ) under A x b (25) Aeq x beq lb x ub where x, b, beq, lb e ub are vector, A and Aeq are matrces, c(x) and ceq(x) are unctons that return vectors, and (x) s a uncton that return an scalar, (x), c(x) and ceq(x) may nvolve nonlnear Start Selects allowed thcknesses, max. and mn. Selects the number o bts o the thckness Selects the number o chromosomes Converson rom bnary to decmal Random selecton o ntal chromosomes (CR) Selects the maxmum repetton Selects the maxmum number o generatons Calculaton o the values o thckness Calculatng the volume o each CR Volaton o volume max/mn Yes Calculaton o penaltes Choose best CR wth penalty No Calculatng the ralng Calculaton o the probablty o CR, objectve uncton or penalty consdered each speces Stop reached? condton Yes hcknesses equal to the decmal values o the best CR selected End No Rale CR or recombnaton Probablty recombnaton acheved? o Yes Recombnes chosen chromosomes No Mantans ntal CR Probablty o mutaton reached? Yes Mutaton chosen chromosomes 0 No 1 2 Keeps CR wthout mutaton Change one CR by the best calculated CR 3 4 Fgure 1: Flowchart o the optmzaton process wth Genetc Algorthms 5

6 IV Internatonal Symposum on Sold Mechancs - MecSol 2013 unctons o the type used n ths work. he uncton to be mnmzed s contnuous. he method o Genetc Algorthms was mplemented nto MEFLAB through a developed program usng Matlab commands to have more reedom, and not just beng used the Genetc Algorthms toolbox avalable n Matlab. Optmzaton by Genetc Algorthm was chosen n ths work because t s a global search algorthm, n ths case over the lowest requency response n structural systems and lud-structure systems. Fgure 1 shows the programmed optmzaton lowchart o Genetc Algorthms. 5 OPIMIZAION OF FREQUENCY RESPONSE here are many ways to reduce the response to vbraton. We could add actve or passve dampng devces, or mody the physcal propertes o materals such as stness. Another way s by changng the geometrcal characterstcs o systems through the shape or dmensons. In ths work, we chose the dmensonal change o the structural doman o the coupled system by varyng the thckness o plates. One o the adopted boundary condtons s to mantan the orgnal structural mass wthn a small varaton. he requency response U j, that n ths work s the sound pressure, n a pont o measurement, resultng rom an exctaton orce appled on one pont o the structure, called the degree o reedom o nput (GDLn), havng an exctaton requency ω j, can be obtaned through the method o modal superposton or n coupled modes. Wth ths method t s possble to calculate the system response usng Equaton (26): GDLn n U 1 j l kl F 2 k k1 l1 l (26) j where l s the component o the rght egenvector l, l s the egenvalue l, kl s the component k o the let egenvalue l. All these components are unctons o the desgn varable, n ths case the thckness varable. he mnmzaton o response s made through a number o dscrete exctaton requences called req. In ths work, the exctaton s appled n the structural doman and the response measurement s realzed n the lud doman; ths conguraton was chosen because t represents the eects ound n a cabn o a vehcle, where the exctaton s transmtted rom the structure to the cabn n the orm o sound pressure (nose), so the pont desred to reduce the sound pressure would be the drver's ear, or example. he readng pont o the response s called the degree o reedom o output (GDLout). A general descrpton o the optmzaton problem to reduce the sound pressure, consderng the structural thckness e as desgn varable, s exposed as ollows:: Mnmze e r req GDLout j1 1 U j e, j (27) subject to constrants: 1 e r mn e A e e 0 max (28) 6

7 IV Internatonal Symposum on Sold Mechancs - MecSol 2013 where e and A are the thckness and area o the nte element, respectvely, s the constant volume o the structure, e mín and e máx are the maxmum and mnmum thcknesses o the nte element, respectvely. 6 RESPONSE OPIMIZAION IN AN ACOUSIC CAVIY OVER A SUPPORED SQUARE PLAE AN EXCIAION FORCE IN A POIN, MULIPLE EXCIAION FREQUENCIES AND A MEASURING POIN For the study s consdered a square plate o sde equal to m (20 n) and thckness equal to m (0.2 n), smply supported at ts our edges. he plate materal s alumnum havng the ollowng propertes: Modulus o elastcty E = x10 9 N/m 2 (1.0x10 7 ps) Posson's rato ν = 0,3 Densty ρ = 2700 kg/m 3 (2.54 x10-4 lb-s 2 /pol 4 ) For the analyss, the plate s dvded n 64 elements wth 4 nodes per element, whch means a total o 81 nodes as can be seen n Fgure 2. he non-conorm plate element s chosen n MEFLAB. Fgure 2: Plate 8x8 64 elements Over the alumnum plate there s an ar cavty o hexaedrcal shape wth m (20 n) on each sde o the base and heght o 2.54 m (100 n). he cavty s lled wth ar havng the ollowng propertes: Densty = 1,29 kg/m 3 (1,21x10-7 lb-s 2 /pol 4 ). Speed o sound n ar c = 330,2 m/s (13000 pol/s). For the analyss the hexahedron o Fgure 3 s dvded n 512 elements wth 8 nodes each one, totalzng 729 nodes. At MEFLAB natural requences are calculated usng ormulatons o hexaedrcal sold elements. he sdewalls o the cavty are consdered rgd. 7

8 IV Internatonal Symposum on Sold Mechancs - MecSol 2013 Fgure 3: Cavty o ar 8x8x8 512 elements We consder the system o an acoustc cavty on a square plate and the correspondng requency response analyss; proposng n ths secton the optmzaton o the sound pressure at a pont o the lud under multple structural exctaton requences through genetc method. For ths optmzaton, the plate thckness s consdered as the desgn varable and volume o the plate as constrant uncton. he requences 60, 62, 64, 66, 68, 70, 72, 74, 76, 78 and 80 Hz are used as exctaton requences or the appled orce, each requency s appled separately and the total response s the sum o the sound pressure o each exctaton requency at the measurng pont. hs way we ntend to obtan a plate wth a dstrbuton o derent thcknesses or each nte element,.e. the plate has a varable thckness, but ts mass should not be changed sgncantly, outsde the speced range gven as constrant. he ollowng data are consdered durng the optmzaton: Force applcaton node: 30, DOF 86 at structural doman. Response node or optmzaton: 405, DOF 563 at lud. Exctaton requences: 60, 62, 64, 66, 68, 70, 72, 74, 76, 78 and 80 Hz, these requences are chosen because there are two derent resonant requences o the coupled system n ths nterval, one nluenced by the lud doman (65.4 Hz) and another (75.7 Hz and repeated because o the symmetry) nluenced by the structural doman. Number o modes or modal analyss: 10 (prevous smulatons showed that ths amount was sucent or the analyss o evaluated requences). Mnmum thckness allowed: 0,1 pol. (0,00254 m). Maxmum thckness allowed: 0,4 pol. (0,01016 m). Mnmum volume allowed: 90% o the orgnal plate volume. Maxmum volume allowed: 110% o the orgnal plate volume. Number o chromosomes: 20. Number o bts per element: 5. Maxmum number o generatons:

9 IV Internatonal Symposum on Sold Mechancs - MecSol 2013 Maxmum number o consecutve repettons o the objectve uncton value o the best ndvdual: 15. Results o 10 smulatons or the Genetc Algorthm can be seen n able 1. able 1 Smulatons o optmzaton or the acoustc cavty over one plate. Smulaton Generatons Decrease o orgnal response (%) Runtme reerence (s) Méda 78, Standar devaton 29, Mn. value Máx. value Fgure 4 shows the obtaned responses or the genetc process, shtng the curve o sound pressure to lower values than the ntal response and repostonng one pont o resonance closer to 79 Hz. Fgure 4: Sound pressure ntal and optmzed or Hz nterval usng genetc algorthm he conguraton o thcknesses or the plate nte elements ound or one o the genetc smulatons can be seen n able 2. 9

10 IV Internatonal Symposum on Sold Mechancs - MecSol 2013 able 2 Dstrbuton o thcknesses ater optmzaton o the acoustc cavty over one plate. 7 CONCLUSIONS Element number hckness (pol) Element number hckness (pol) 1 0, ,32 2 0, ,35 3 0, ,15 4 0, ,27 5 0, ,23 6 0, ,22 7 0, ,10 8 0, ,10 9 0, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,19 Concernng the response optmzaton o the coupled lud-structure system; and observng the dstrbuton o thcknesses, t was detected a conguraton qute derent rom the orgnal. For the studed case, the genetc method reduces adequately the ntal sound pressure but wth an mportant processng tme. 10

11 IV Internatonal Symposum on Sold Mechancs - MecSol ACKNOWLEDGEMENS o programs o scholarshps or undergraduate research; PROBIC-FAPERGS-UFRGS (E.M.C.) and PROBII-FAPERGS-UFRGS (R.F.L.F.). REFERENCES [1] Marburg, S., 2002, Developments n Structural-Acoustc Optmzaton or Passve Nose Control, Archves o Computatonal Methods n Engneerng, Vol. 9, 4, pp [2] Marburg, S., Denerowtz, F., Frtze, D. and Hardtke, H.J., 2006, Case Studes on Structural- Acoustc Optmzaton o a Fnte Beam, Acta Acustca Unted wth Acustca, Vol. 92, pp [3] Lee, J., Wang, S. and Dkec, A., 2004, opology Optmzaton or the Radaton and Scatterng o Sound rom hn-body Usng Genetc Algorthms, Journal o Sound and Vbraton, Vol. 276, n 3-5, pp [4] Duhrng, M.B., Jensen, J.S. and Sgmund, O., 2008, Acoustc Desgn by opology Optmzaton, Journal o Sound and Vbraton, Vol. 317, pp [5] Yoon, G.H., Structural opology Optmzaton or Frequency Response Problem Usng Model Reducton Schemes, Computer Methods n Appled Mechancs and Engneerng, Vol. 199, , [6] Akl, W., El-Sabbagh, A., Al-Mtan, K. And Baz, A., opology Optmzaton o a Plate Coupled wth Acoustc Cavty, Internatonal Journal o Solds and Structures, Vol. 46, n. 10, [7] Marburg, S., Beer H.-J., Ger, J. and Hardtke, H.-J., 2002, Expermental Vercaton o Structural-Acoustc Modelng and Desgn Optmzaton, Journal o Sound and Vbraton, Vol. 252, pp