MASS, STIFFNESS, AND DAMPING MATRICES FROM MEASURED MODAL PARAMETERS

Transcription

1 IS 74 Inernionl Insrmenion-omion Conference & Exhibi Ocober, 974 MSS, STIFFNESS, ND DMPING MTRICES FROM MESURED MODL PRMETERS Ron Poer nd Mr Richrdson Digil Signl nlysis HEWLETT-PCKRD COMPNY Sn Clr, Cliforni BSTRCT The heory of complex mode shpes for dmped oscillory mechnicl sysems is explined, sing he mrix of rnsfer fncions in he Lplce domin. These mode shpes re defined o be he solions o he homogeneos sysem eqion. I is shown h complee rnsfer mrix cn be consrced once one row or colmn of i hs been mesred, nd hence h mss, siffness, nd dmping mrices corresponding o lmped eqivlen model of he esed srcre cn lso be obined from he mesred d. INTRODUCTION In recen yers, here hs been considerble civiy in he sdy of elsic srcre dynmics,. in n emp o design srcres h will fncion properly in hosile vibrion environmen. lhogh mch of he erly wor cenered rond fige nd life esing, he les effors hve been direced owrds nlyicl modeling nd simlion of mechnicl srcres. Disribed srcres re generlly modeled s newors of lmped mechnicl elemens, in n effor o predic filres more relibly nd fser hn is fforded by convenionl life esing procedres. Wih he dven of he inexpensive mini-comper, nd comping echniqes sch s he Fs Forier Trnsform lgorihm, i is now relively esy o obin fs, ccre, nd complee mesremens of he behvior of mechnicl srcres in vrios vibrion environmens. Modl responses of mny modes cn be mesred simlneosly nd complex mode shpes cn be direcly idenified insed of relying pon nd being consrined by he so clled "norml mode" concep. Frhermore, he enire sysem response mrix, which comprises he mss, siffness, nd dmping mrices of he lmped eqivlen model, cn be mesred. The following meril covers he heoreicl bcgrond h is needed o ndersnd hese new mesremen echniqes. COMPLEX MODES ND THE TRNSFER MTRIX Le's ssme h he moion of liner physicl sysem cn be described by se of n simlneos second order liner differenil eqions in he ime domin, given by Mx && + Cx& + Kx = f ( where he dos denoe differeniion wih respec o ime. f = f( is he pplied force vecor, nd x = x( is he resling displcemen vecor, while M, C, nd K re he (n by n mss, dmping, nd siffness mrices respecively. In his discssion, or enion will be limied o symmeric mrices, nd o rel elemen vles in M, C, nd K. Ting he Lplce rnsform of he sysem eqions gives B( s X( s = F( s, where ( B( s = Ms + Cs + K Here, s is he Lplce vrible, nd now F(s is he pplied force vecor nd X(s is he resling displcemen vecor in he Lplce domin. B(s is clled he sysem mrix, nd he rnsfer mrix H(s is defined s H( s = B( s which implies h (3 (4 H( s F( s = X( s (5 Ech elemen of he rnsfer mrix is rnsfer fncion. The elemens of B re qdric fncions of s, nd since B, i follows h he elemens of H re rionl frcions in s, wih de(b s he denominor. Ths, H(s cn lwys be represened in pril frcion form. If i is ssmed h he poles of H, i.e. he roos of de(b =, re of ni mlipliciy, hen H cn be expressed s = s s (n by n (6 Pge -

2 IS 74 Inernionl Insrmenion-omion Conference & Exhibi Ocober, 974 The poles occr s = s (zeros of de B, nd ech pole hs n (n x n reside mrix describing is spil behvior. For n n h order oscillory sysem, here will lwys be poles, b hey will pper in complex conjge pirs. The poles re complex nmbers expressed s s = σ + i ω (7 where σ is he dmping coefficien ( negive nmber for sble sysems, nd ω is he nrl freqency of oscillion. The resonn freqency is given by ω = σ + ω r nd he dmping fcor is σ ς = ω r rd/sec (8 In reference [], modl vecors were derived in erms of he eigenvecors of he sysem mrix B. However, hese eigenvecors were only inrodced s n inermediry in he deerminion of modl vecors. Here, he modl vecors re described in erms of he B mrix. Pre-mliplying B imes he expression for H, mliplying by he sclr ( s s, nd leing s = s gives (9 B =, where B = B( s ( Similrly, pos-mliplicion of H by B gives B = ( Ths, ll rows nd colmns of ms comprise liner combinion of homogeneos solion vecors, given by B = ( These homogeneos solion vecors re defined s modl (or mode shpe vecors ssocied wih he pole s =. s Resricing or enion o he specil cse where only one modl vecor exiss for ech pole, i is cler h ll rows nd colmns of ms be some sclr mliple of. Ths, cn be represened by = (n by n (3 where is sclr. In hese erms, H cn be rewrien s = s s nd his is esily wrien in mrix form s (4 θλ θ (n by n (5 where he colmns of θ comprise he modl vecors: θ = (n by (6 nd λ is digonl mrix conining ll s dependence: s s λ = s s,( x (7 Pre-mliplying H by θ, eqion (5 cn be wrien s ( θ θλ ( θ = ( θ so h θ F X (8 rnsforms he spil vecors (F nd X o vecors θ F nd θ X in modl coordines. Similrly ( θ θλ ' is he modl represenion of H. Since B( s = i follows h B( s =, so he modl vecor ssocied wih he conjge pole ( s is (, (he conjge of. Ths, he bove θ mrix lwys conins conjge pirs of modl vecors nd he λ mrix lwys conins elemens corresponding o conjge pole pirs long is digonl. If poles re prely imginry (zero dmping hen ll modl vecors re rel, so only hlf of θ is needed. Oherwise θ is recnglr (n x nd conseqenly, even hogh H cn be wrien in digonl form sing he θ mrix, is inverse B cnno be digonlized sing θ excep in he specil cse when dmping is zero. Pge -

3 IS 74 Inernionl Insrmenion-omion Conference & Exhibi Ocober, 974 IDENTIFICTION OF MODL PRMETERS Becse of he form of he mrix, only one row or colmn of he rnsfer mrix need be mesred nd nlyzed, since ll oher rows nd colmns conin redndn informion. In he process of mesring he rnsfer mrix, nnown prmeers in eqion (4, i.e. he complex vles of s nd he complex vles of he elemens of one row or colmn of re idenified. For exmple, he q h colmn of is given by q = = where q is he q h q q (9 elemen of. Ths he modl vecor (whose normlizion is rbirry cn be recovered once he q h colmn of is idenified. In ddiion he complex sclr cn be recovered sing he forml = q q q ( I is cler h he nmericl vle of depends pon he normlizion of he modl vecor. If we choose =, hen = q q q ( Since complex modl vecors pper in conjge pirs, H cn lwys be wrien in wo prs s n = s s + s s For he h pir of conjge poles, H = + s s s s ( (3 Ech pole of he rnsfer mrix hs corresponding mode shpe vecor nd, frhermore, ech complex conjge pir of poles hs corresponding complex conjge pir of mode shpes. The pq h elemen of H is H = + s s s s pq p q p q (4 The ime domin displcemen poin p de o n implsive force poin q is given by he inverse Lplce rnsform of H pq which is s h pq ( = e + e for. σ = e p q [ ( p q Re cos( ω p q ( p q sin( ω ] Im s = e σ cos( ω + α (5 p q Noe h he pe mplide of he implse response is p q complex reside nd he phse ngle ( α is he ngle of he p q wih respec cosine. The mgnide nd phse of his complex reside re differen, in generl, for ech spil poin on he srcre. MSS, STIFFNESS, ND DMPING MTRICES Recll h B( s = Ms + Cs + K, so i is ppren h K = B( = H( (6 Now, H( is obined by seing s = in λ. modl complince κ cn be defined s So κ = λ ( = s K = H( = θκ ( θκ θ s ( x (7 (8 Pge - 3

4 IS 74 Inernionl Insrmenion-omion Conference & Exhibi Ocober, 974 Ths, he siffness mrix is redily obined from he mesred modl vecors (colmns of θ, nd he idenified nd s complex sclrs. Since HB i follows h nd = I (9 HB' + H' B = (3 HB'' + H' B' + H'' B = (3 where he prime denoes differeniion wih respec o s. The dmping mrix cn be comped wih he following relionship C = B'( = B( H'( B( = KH'( K (3 Or lernively by defining he modl dmping mrix δ s δ = hen nd n ( H'( = θ λ '( θ = θκ ( x (33 δκ θ (34 ( θκ δ( θκ C = K K In similr mnner, we cn wrie (35 M = B''( = B( H'( B'( B( H''( B( = KH'( C KH''( K (36 B H"( cn be wrien s H''( = θκ δκ δκ θ (37 Finlly, he mss mrix cn be obined from ( θκ ( δκ δ( θκ + ( Kθκ δκ θ C ( θκ ( δκ δ( θκ M = K K = K K + CK C Now, from he expression for K, we obin Kθκ ( θκ (38 = θκ θ θκ = θ (39 where θ is he lef-hnded inverse of θ defined by θ θ = I (4 To smmrize, he siffness, dmping, nd mss mrices re obined from he expressions ( θκ K = θκ θ = θ κθ C = θ δθ M = θ µθ + CK C where µ = δκ δ (4 (4 (43 (digonl (44 ll of hese qniies re redily obined from he mesred poles s, sclrs, nd modl vecors. COMMENTRY The fndmenl nre of complex modes in he rel world of mesremen cnno be over-emphsized. Dmping is lwys presen in srcre, nd i cn lwys be observed in mesred rnsfer fncion d. The fc h complex mode implies complex ime wveform shold be of lile concern, becse he conjge wveform is lwys presen o me he observed signl rel vled. complex mode hs he chrcer of "rveling wve" cross he srcre (s opposed o he sl "snding wve" prodced by norml mode s indiced by he chnging phse ngle of he displcemen from poin o poin. Frhermore, i is imporn o noe h mode shpe sill globl (s opposed o locl propery of srcre, even hogh dmping my be hevy. The fc h locl moion ner he poin of exciion in hevily dmped srcre Pge - 4

5 IS 74 Inernionl Insrmenion-omion Conference & Exhibi Ocober, 974 ends o domine is generlly csed by mny closely spced modes h re excied in phse his poin, b end o cncel ech oher elsewhere. Hisoriclly, considerble emphsis hs been plced on simlneosly digonlizing he mss nd siffness mrices, so h he displcemen of priclr poin cold be esily clcled from n rbirry exciion force. Unfornely, i is no possible o simlneosly digonlize more hn wo symmeric mrices, so his echniqe cnno be sed when dmping is presen. However, when sysem is represened by he pril frcion form of is rnsfer mrix, closed form solion for he displcemen ny poin, for ny combinion of modes, is redily obinble sing simple mrix-vecor mliplicion. This is priclrly helpfl when he response o only few modes is of ineres. The definiion of modl vecor s solion o he homogeneos sysem eqion is lso very fndmenl, nd removes mch of he mbigiy bo wh modl vecors relly re, nd when hey exis. This definiion lso mes hem relively esy o clcle or mesre. I shold be ppren from he eqions for obining mss, dmping, nd siffness from mesred d, h he order of hese mrices is eql o he nmber of complex modl pirs h re mesred. This mens h if n modes re idenified, only n spil poins re needed o represen he lmped eqivlen model of he physicl sysem. This resl cn be sed s he following heorem. SPTIL SMPLING THEOREM: lmped sysem modeled wih second order elemens hs excly poles n modl vecors. Ths, if only n modl pirs re fond, here is no need for more hn n spil poins in he lmped model. Finlly, he scope of his discssion hs been limied o symmeric mrices hving disinc poles (ni mlipliciy in complex conjge pirs. Even hogh hese ssmpions re sisfied by lrge mjoriy of he liner sysems esed, modes of vibrion cn sill be defined when ll of hese ssmpions re relxed, nd we hve wored o he heory for his generl cse. The rnsfer mrix (inverse of sysem mrix ws inrodced, nd i ws wrien in pril frcion form, comprising one erm for ech pole. We fond h he rows nd colmns of he reside mrix ssocied wih ech pole re mliples of he corresponding modl vecor. The modl mrix θ, whose colmns re he modl vecors, ws defined, nd i ws shown h θ will rnsform vecor from spil coordines o modl coordines. Finlly, we derived expressions for mss siffness, nd dmping in erms of he poles nd modl vecors, nd indiced how mesred prmeers cn be sed o clcle hese mrices. REFERENCES. Richrdson, M. nd Poer, R., "Idenificion of he Modl Properies,of n Elsic Srcre from Mesred Trnsfer Fncion D", h Inernionl Insrmenion Symposim, lbqerqe, New Mexico, My -3, Thoren,.R., "Derivion of Mss nd Siffness Mrices from Dynmic Tes D", I4 pper 7-346, 3'Ln Srcres, Srcrl Dynmics, nd Merils Conference, Sn nonio, Texs, pril, Rbinsein, M.F., "Srcrl Sysems -Sics; Dynmics nd Sbiliy", Prenice-Hll, Inc., Englewood Cliffs, New Jersey, 97. SUMMRY We hve defined he complex modl (mode shpe vecor ssocied wih ech pole (zero of he sysem deerminn of n elsic mechnicl sysem s he homogeneos solion o he sysem eqions. We hve lso emphsized h hese poles occr in complex conjge pirs, nd h he wo ssocied modl vecors re lso conjges of one noher. Pge - 5