Structural Optimization of Permanent Magnet Motor for Resonance Avoidance

Transcription

1 0 th Wold Congess on Stuctual and Multdscplnay Optmzaton May 9-4, 03, Olando, Floda, USA Stuctual Optmzaton of Pemanent Magnet Moto fo Resonance Avodance Sunghoon Lm, Seungjae Mn, Jung-Pyo Hong 3, Shnj Nshwak 4 Hanyang Unvesty, Seoul, Koea, sleeplm8@hanmal.net Hanyang Unvesty, Seoul, Koea, seungjae@hanyang.ac.k, coespondng autho 3 Hanyang Unvesty, Seoul, Koea, hongjp@hanyang.ac.k 4 Kyoto Unvesty, Kyoto, Japan, shnj@pec.kyoto-u.ac.jp. Abstact hs pape poposes a new desgn method fo avodng esonance of the pemanent magnet (PM) moto by consdeng both the electomagnetc souces and mechancal behavo of the moto stuctue. o avod the esonance band between the electomagnetc souces and the moto stuctue, the optmzaton poblem s fomulated wth the mult-objectve functon to maxmze the lowest egenfequency of moto stuctue and mnmze the hamonc component of magnetc foce. Snce the magnetc and mechancal behavos of PM moto ae hghly senstve to the stuctual boundaes of PM and feomagnetc mateal (FM), the level set method whch can handle a detaled change of moto confguaton s employed. he optmzaton s pefomed by usng the nomalzed desgn senstvtes whch ae calculated fom both the magnetc and stuctual analyss. A desgn example of SPM moto s povded to nvestgate the usefulness of the poposed method and acheve a novel moto desgn that pomses enhanced pefomance.. Keywods: pemanent magnet moto, esonance avodance, stuctual optmzaton, mult-objectve functon 3. Intoducton Snce the esonance of a moto system educes the machne s duablty and causes the dvng nose, the avodance of mechancal esonance s the basc desgn taget n the development of pemanent magnet (PM) moto whch s wdely used n many ndustal applcatons due to hgh powe densty []. he esonance of moto s geneated by the nteacton between the hamonc component of electomagnetc foce and the natual fequency of the moto stuctue [-3]. Hence, many studes have pefomed desgn optmzaton wth two objectves: one s the elmnaton of cetan hamoncs of the exctng magnetc foce [4] and the othe s the adjustment of stuctual fequences to avod the esonance band []. Unfotunately, by pevous desgn methods usng the expemental data and paametc study, the optmal soluton fo esonance avodance can bng the geat change of the moto stuctue such as the pole-slot combnaton and an ncease of the moto sze. hs pape poposes a new desgn method whch can guaantee localzed geometcal change fo avodng esonance of PM moto. Seveal level set functons ae employed as desgn vaables to expess boundaes of PM, FM, coppe col and a n PM moto and obtan an nnovatve optmal desgn [6]. o calculate the magnetc foce and the egenfequency whch s vaed due to the moto confguaton, both magnetostatc and modal analyss ae pefomed wth mateal popetes such as magnetc eluctvty and elastcty, whch ae defned by level set functons. he optmzaton poblem s fomulated to mnmze the hamonc component of magnetc foce whch ceates the esonance band accodng to the otatng speed, and maxmze the natual fequency of moto stuctue. he optmal moto shape s obtaned by mplct boundaes of level set functon whch s moved by the desgn senstvty. he poposed method s appled to the stuctual desgn of PM moto wth consdeaton of ts speed ange and esonance band. 4. Poblem Fomulaton 4.. Bounday Repesentaton of Moto Stuctue o desgn the whole moto stuctue that satsfes ou desgn goal n the conceptual step, thee dffeent level set functons ae employed as desgn vaables fo epesentng the dstbuton of PM, col, feomagnetc mateal (FM) and a n the moto. Fgue depcts how the stuctual bounday of each mateal n the moto can be defned by level set functons. he one level set functon ( ) n the stato s used to dvde col and FM egon, and addtonal two level set functons (, 3 ) ae dstbuted n the oto to expess the confguaton of PM, FM and a.

2 Fgue : Bounday epesentaton of PM moto Each level set functon has the sgn fo dstngushng the dffeent mateal doman ( ) and the zeo level of them epesents the stuctual boundaes ( ) as follows: x 0 fo x () x0 fo x,,3 x0 fo x he mateal popetes fo mechancal and magnetc analyss can be defned by the chaactestc functon ( ) of whch value s 0 o accodng to the sgn of each level set functon. In the stato, magnetc and mechancal popetes of col ( p ) and FM ( col p ) such as the elatve magnetc eluctvty, the cuent densty, Young s FM modulus and densty ae defned as follows: p pcol pfm () o epesent the magnetc popetes of PM ( p ), FM ( PM p ) and a ( FM p ) n the oto of PM moto, the a combnatons of two chaactestc functons ae used as n the followng equaton: p, 3 ppm3 pfm3 pa (3) In ths case, the elatve eluctvty and the emanent flux densty ae accepted as p to expess a chaactestc of PM. 4.. Optmzaton Poblem Fomulaton he objectve functon ( F ) s composed wth two values: one s the fst egenvalue ( ) wth a mnus sgn and the othe s the dffeence between the adal foce ( f ) on the teeth of the stato and the taget value ( f ) fom the fundamental wavefom that hamoncs of adal foce up to m th ode ae elmnated as shown n Fgue. Fgue : aget adal foce

3 By mnmzng F, we can optmze the moto stuctue wth consdeaton of the egenfequency maxmzaton and the hamonc elmnaton of the adal foce whch ae the mpotant ssues fo esonance avodance. he optmzaton poblem s fomulated wth the volume constants of each level set functon ( VF ) as follows: n j j f,, 3 f mnmze F,, 3 j j f stato o oto stato o oto subject to G d d VF 0,,3 stato o oto stato o oto G4,, 3 avg,taget avg,, 3 0 G,,,, 0 3 pple 3 pple,taget whee n s the total numbe of otatng poston, and stato mean the desgn doman n the oto and stato oto of the moto. It s noted that two desgn constants ae added fo satsfyng the taget output toque ( avg,taget ) and mnmzng the toque pple ( pple ) whch also ceates the esonance band. Each level set functon s updated by the followng equaton untl the convegence condtons ae satsfed. F G4 G 4,, 3 () t whee and 4 ae the Lagange multples to satsfy the desgn constants fo toque pefomance and s fo the volume facton of each level set functon.. Desgn Example.. Intal Desgn of SPM Moto he poposed method s appled to the desgn optmzaton of 8-pole -slot SPM moto whch was developed fo an electc powe steeng (EPS) system of electc vehcle. Fgue 3 shows the ntal desgn of SPM moto and the dvng fequency of the adal foce ( ) and the toque ( ) whch s calculated by the followng equatons. N[ pm] pole pa h [ Hz] (6) 60 N[ pm] pole pa 3 h [ Hz] (7) 60 whee N s the otatng speed of the moto and h s the ode of the hamonc component. (4) Fgue 3: Intal desgn of SPM moto: (a) confguaton (b) st egen mode (c) dvng fequency of hamoncs of adal foce (d) dvng fequency of hamoncs of toque 3

4 Unfotunately, of all hamonc ode and of moe than 3 d ode pass though the esonance band whch s geneated by the lowest egenfequency of the stato of the ntal desgn unde the maxmum ated speed (00 pm), as llustated n Fgue 3... Desgn Optmzaton fo Resonance avodance o avod the esonance band between the electomagnetc souce and the moto stuctue, the taget wave of the adal foce s geneated wthout hamonc components up to 3 d ode and the taget toque pple ( pple,taget ) s set to 0.% fo educng the fluctuaton of the toque by moe than 9% compaed wth the ntal desgn. he shape of the stato s optmzed to maxmze the st egenfequency wth satsfacton of the taget output toque, 3. Nm. he desgn doman s the ente aea of the moto except fo the teeth of the stato whee the adal magnetc foce s calculated and the lowe pat of the oto whch s bodeed wth the otatng shaft, as shown n Fgue 4(a). Fgue 4: Confguaton of SPM moto: (a) desgn doman (b) optmal desgn Fgue 4(b) shows the optmal dstbuton of each mateal n compason wth the ntal desgn notced by the sold lnes. he boundaes of the PM n the oto become a ounded shape to elmnate the dstoton of magnetc flux wavefom n a gap and educe the magnetc satuaton effect n the teeth of the stato fo elmnatng the hamonc components of the adal foce. It s noted that the cental PM thckness nceases slghtly to povde the magnetc flux to the stato suffcently and FM aound the edge of PM s elmnated to educe the leakage flux. In the stato, FM s dstbuted to elmnate the shap cone and concentated n the uppe sde of the yoke to maxmze the stffness of the stato wthn the same volume facton as the ntal desgn. Fgue : Optmal desgn of SPM moto: (a) confguaton (b) st egen mode (c) ampltudes of hamoncs of adal foce (d) toque pofles 4

5 Fgue depcts the magnetc and mechancal behavo of the optmal moto desgn. It s confmed that the st egenfequency of the modfed stato shape nceases by % and the ampltudes of hamoncs of the adal foce whch ae ovelapped on the esonance band decease geatly. he fluctuaton of the toque, the othe magnetc souce of esonance, nealy dsappeaed whle mantanng the taget aveage toque as llustated n Fgue (d). able summazes that the optmal moto desgn povdes a lage decease n the hgh hamoncs geneated by the adal foce and the desgn constants fo the toque pefomance ae satsfed. able : Compason between ntal and optmal desgn Ampltude of adal foce [N] [Hz] avg [Nm] 3 d 4 th th Intal desgn Optmal desgn 0. (74.0% ).3 (88.8% ).6 (8.0% ) 03.9 (.0% ) 3. (8.6% ) 0.3 (98.% ) 6. Concluson hs pape poposes a new desgn optmzaton method to avod a esonance of the PM moto. he desgn poblem s fomulated to maxmze the egenfequency of the moto stuctue and mnmze the electomagnetc souce such as the hamoncs of adal foce and the toque fluctuaton whch pass though the esonance band. he desgn example povdes the way to avod the mechancal esonance of an SPM moto and t s expected that the poposed method can povde the novel stuctual desgn fo avodng the moto s esonance n the conceptual desgn stage. 7. Acknowledgements hs wok was suppoted by the Natonal Reseach Foundaton of Koea (NRF) Gant ( D00006). pple [%] 8. Refeences [] A. Cassat, C. Espanet, R. Coleman, L. Budet, E. Leleu, D. oegossa, J. M Boua and A. Maou, A Pactcal Soluton to Mtgate Vbatons n Industal PM Motos Havng Concentc Wndngs, IEEE ansactons on Magnetcs, 48 (), 6-38, 0. [] H.-S. Ko and K.-J. Km, Chaactezaton of Nose and Vbaton Souces n Inteo Pemanent-Magnet Bushless DC Motos, IEEE ansactons on Magnetcs, 40 (6), , 004. [3] S. Yu and R. ang, Electomagnetc and Mechancal Chaactezatons of Nose and Vbaton n Pemanent Magnet Synchonous Machnes, IEEE ansactons on Magnetcs, 4 (4), , 006. [4] J. Hu, J.-W. Reu, B.-W. Km and G.-H. Kang, Vbaton Reducton of IPM-ype BLDC Moto Usng Negatve hd Hamonc Elmnaton Method of A-Gap Flux Densty, IEEE ansactons on Magnetcs, 47 (3), , 0. []. Sun, J.-M. Km, G.-H. Lee, J.-P. Hong and M.-R. Cho, Effect of Pole and Slot Combnaton on Nose and Vbaton n Pemanent Magnet Synchonous Moto, IEEE ansactons on Magnetcs, 47 (), , 0. [6] S. Lm and S. Mn, Desgn Optmzaton of Pemanent Magnet Actuato Usng Mult-Phase Level-Set Model, IEEE ansactons on Magnetcs, 48 (4), , 0.