Challenges and Experiences in Model Reduction for Mechanical Systems Illustrated for the Reduction of a Crankshaft

Transcription

1 ODRED 010 Challng and Exrnc n odl Rducton for chancal Sytm Illutratd for th Rducton of a Crankhaft Chrtn Gchwndr, Jörg Fhr und tr Ebrhard Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

2 robotc Alcaton EBS owr ngnrng automotv and drv ngnrng EBS av xrmnt and rototy du to mulaton lghtwght contructon rur condraton of latc ffct Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard rarch modl rducton modl gt mor dtald and largr no ffcnt mulaton obl wthout OR for u of modal rducton t ncary to hav a lot of knowhow and no ml rror-control obl

3 rgd body C multbody ytm rncl of Elatc ultbody Sytm latc multbody ytm barng and coulng lmnt bod f dgr of frdom racton forc rducton of th latc dgr of frdom contnuum latc body dcrtaton fnt lmnt, fnt dffrnc,... Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard latc multbody ytm ar an mortant tool for calculaton of comlx mchancal ytm modl ar gttng largr and mor dtald FE-modl ha to b rducd cau of a larg numbr of dgr of frdom

4 combuton ngn mortant comonnt: barng of a crankhaft rod barng man barng chncal and Indutral Alcaton rgd crankhaft calculaton of hydrodynamc barng forc for dcrt crankhaft oton dformaton of th crankhaft ha grat nflunc for raltc rult: combnaton of lato-hydrodynamc barng and latc multbody ytm latc crankhaft olvng xn lato-hydrodynamc uaton uaton of moton of th latc bod Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

5 Ovrvw motvaton modl rducton of latc bod hort ovrvw of ndutral tat of art and currnt tat of dvlomnt modl rducton bad on th fruncy wghtd Graman matrc wth OD rducton of th crankhaft concluon and outlook Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

6 Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard odl Rducton of Elatc Bod bac uaton lnar modl rducton aroach rojcton of th latc coordnat to a ubac wth to gt a oluton wth vanhng rduum rducd ytm V u B V K V D V W u B W V K W V D W V W u B K D ) ( ) ( dm dm D K 0 u B h a W V r r r r C y D K u B h a r r r r

7 modl rducton tchnu ud for latc bod Ovrvw of odl Rducton chnu ndutral tat of art tradtonal modal rducton (lcton of mod by ur) Crag-Bamton/ Comonnt od Synth (CS) currnt tat of dvlomnt rducton wth momnt matchng and Krylov-ubac mthod bad on Graman matrc combnatd rducton tchnu Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

8 Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard oton and vlocty Graman matrc Graman matrc lay an mortant rol n balancd truncaton modl rducton for cond ordr ytm: Graman matrc of th uvalnt frt ordr ytm arttonng th Graman matrc altrnatv charactrzaton of Graman matrc conncton wth -Norm altrnatv obrvablty Graman matrc for cond ordr ytm wth tructural rort wth am n- and outut Balancd runcaton of nd Ordr Sytm, o o v v S S S S o o v v R R R R S S R R H 1 v H tr tr B B B B H tr tr C C C C 1 v v : B C v

9 motvaton n ngnrng roblm, oftn knowldg about th ntrtng fruncy rang avalabl Fruncy Wghtd nd Ordr Graman atrx fruncy wghtd Graman matrx fruncy rang ar mhazd by alyng an utabl fruncy fltr W( ) 1 H G H jb Wj Wj B G jd 1 wth Gj ( jd K ) band a fruncy wghtd Graman matrx for mchancal ytm th fltr matrc ar oftn not avalabl drctly but th ntrtng fruncy rang known band a Graman matrx for an arbtrary fruncy rang mn, max max mn 1 H 1 H mn, max G j B B G j d G j B B G j mn advantag a ror rror Inttut of Engnrng boundary and Comutatonal chanc wghtng of a cal Unvrty of Stuttgart, Grmany fruncy rang rof. Dr.-Ing. rof. E.h. tr Ebrhard max d dadvantag calculaton of th matrx ntgral only ffcntly obl for mall modl

10 aroxmaton calculaton of th ntgral by numrcal ntgral at dcrt fruncy amlng ont 1 max H, GjB B G j GjB B G j mn max Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard mn nod dlacmnt nahot nahot matrx numrcal ntgraton ummarz of all nahot matrc to on nahot matrx ˆ U ~ U R, mn l Calculaton of th nd Ordr Graman atrx j Gj B Uˆ ImUˆ l max ~ ~, ~ max mn max U U U mn l 1 1 l 1 l Uˆ 1 Uˆ U ~ u u u l 1 ~ max mn mn, max U U c d

11 bac da rducng larg numbr of dndnt varabl to a mall numbr of uncorrlatd varabl nahot fruncy ron of a harmonc xctaton u 1 nahot matrx U u u u wth ror Orthogonal Dcomoton (OD) X( j ) l aroxmaton aroach arch of an orthogonal ba Φ 1 m, whch mnmz lat uar dffrnc btwn nahot and th aroxmatd ubac mnu Φ j1 wth Langrang-functon for contrant gnvalu roblm 1 ror Orthogonal Valu (OV) Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard U U R j m 1 λ u j R max Φ j1 m 1 u j ror Orthogonal od (O) OD - krnl domnant gnubac of OD roblm = domnant gnubac of Graman matrx

12 nahot mthod bad on data vctor u and th ror Orthogonal od k an th am lnar ac OD gnvalu roblm j1 k1 1 vry nahot affct on th O wth Grdy Sarch adatv arch of th nahot rocdng Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard k1 OD Snahot thod wth Grdy Sarch archng for th fruncy at th maxmum rror of an rror mur calculatd for th ntrtng fruncy rang mn, max th fruncy ud for th nxt nahot k a u j j k k u u a u λ a 1 j k u u u a λ a k a k k1 k k k1 wght factor of nahot k nahot mthod rduc th ODroblm to an dmnonal gnvalu roblm k

13 Ovrvw motvaton modl rducton of latc bod hort ovrvw of ndutral tat of art and currnt tat of dvlomnt modl rducton bad on th fruncy wghtd Graman matrc wth OD rducton of th crankhaft concluon and outlook Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

14 bac nformaton h Crankhaft numbr of nod numbr of lmnt numbr of ndndnt DoF numbr of n- and outut 35 modlng of nut and outut rducng th numbr of nut and outut by ntrfac nod (RBE3) wch cau contrant uaton ntrfac nod Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

15 rror n H -norm H [10-10 ] n na f max =00Hz , , n rd , , ,8 65,79 8, , ,39 0, H [10-10 ] f max =000Hz 8 0,69 0,50 8 0,37 0,59 Fruncy Ron rducton to n rd = 8 Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

16 rgd crankhaft m Smulaton rfrnc latc crankhaft Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

17 Ovrvw motvaton modl rducton of latc bod hort ovrvw of ndutral tat of art and currnt tat of dvlomnt modl rducton bad on th fruncy wghtd Graman matrc wth OD rducton of th crankhaft concluon and outlook Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard

18 Concluon and Outlook ummary an ndutral alcaton whr modl ordr rducton mortd ntroducd calculaton of hydrodynamc barng wth th hl of an latc crankhaft ovrvw of ndutral and currnt tat of art of modl rducton tchnu gvn altrnatv modl rducton bad on fruncy wghtd Graman matrc wth th hl of OD xland rult of rducton of th crankhaft hown for th fruncy rang and th tm doman outlook alcaton of OD for larg-cal ndutral alcaton alcaton and mrovmnt of rror tmator for OD rducton Inttut of Engnrng and Comutatonal chanc Unvrty of Stuttgart, Grmany rof. Dr.-Ing. rof. E.h. tr Ebrhard thank you for your attnton