Modal Testing (Lecture 11) - PDF Free Download

Modal Tesing Lecue D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

Response Funcion Measuemen Tecniques Inoducion Tes Planning Basic Measuemen Sysem Sucue Pepaaion Exciaion of e sucue Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Inoducion Te measuemens ecniques used fo modal esing ae discussed: Response measuemen only Foce and esponse measuemen Te nd ype of measuemen ecniques is of ou concen: Single-poin exciaion SISO/SIMO Muli-poin exciaion MIMO Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Tes Planning Objecive of e es Levels accoding o Dynamic Tesing Agency: Level aual Feq Damping aio Mode Sapes Usabe fo validaion Ou of ange esidues Updaing 0 Only in few poins 3 4 Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Tes Planning Exensive es planning is equied befoe full-scale measuemen: Meod of exciaion Signal pocessing and daa analysis Pope selecion of picup poins Exciaion locaion Suspension meod Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Qualiy of measued daa Signal qualiy Sufficien seng and claiy/noise fee Signal fideliy o coss sensiiviy Measuemen epeaabiliy Measuemen eliabiliy Measuemen daa consisency, including ecipociy Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Basic Measuemen Sysem An exciaion mecanism A ansducion mecanism An Analyze Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Basic Measuemen Sysem Souce of exciaion signal: Sinusoidal Peiodic wi specific feq. conen Random Tansien Powe Amplifie Excie Response Funcion Measuemen Tecniques Tansduces Condiion Amplifies Analyzes IUST,Modal Tesing Lab,D H Amadian

Sucue Pepaaion Fee Suppos Gounded Suppo Loaded Suppo Peubed Suppo Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Fee Suppos Teoeically e sucue will possess 6 igid body modes @ 0 Hz. In pacice is is povided by a sof suppo Rigid body modes ae less en 0% of sain modes Suspending fom nodal poins fo minimum inefeence Te suspension adds significan damping o e ligly damped sucues Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Fee Suppos Suspension wies, sould be nomal o e pimay vibaion diecion Te mass and ineia popeies can be deemined fom e RBMs. Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Fee Suppos Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Gounded Suppo Te sucue is fixed o e gound a seleced poins. Te base mus be sufficienly igid o povide necessay gounding. Usually is employed fo lage sucues Pas of powe geneaion saion Civil engineeing sucues Anoe applicaion is simulaing e opeaional condiion Tubine Blade Saic siffness can be obained fom low fequency mobiliy measuemens. Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Loaded Suppo Te sucue is conneced o a simple componen wi nown mobiliy A specific mass Te effec of added mass can be emoved analyically Moe modes ae excied in a ceain fequency ange compaed o fee suspension Te modes of sucue ae quie diffeen Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Peubed Suppo Te daa base fo e sucue can be exended by epeiion of modal ess fo diffeen bounday condiions Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Peubed Suppo Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Exciaion of e sucue Vaious devices ae available fo exciing e sucue: Conacing Mecanical Ou-of-balance oaing masses Elecomagneic Moving coil in magneic field Elecoydaulic on-conacing Magneic exciaion Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Elecomagneic Excies Supplied inpu o e sae is conveed o an alenaing magneic field acing on a moving coil. Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Elecomagneic Excies Tee is a small diffeence beween e foce geneaed by e sae and e applied foce o e sucue Te foce equied o acceleae e sae moving Te foce equied o excie e sucue saply educes nea e esonance poin, Muc smalle an e geneaed foce in e sae and e ineia of e dive od Vulneable o noise o disoion Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Aacmen o e sucue Pus od o singes: Applying foce in only one diecion Flexible dive od/singe inoduces is own esonance ino e measuemen. Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Suppo of saes Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Hamme o Impaco Exciaion Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Oe exciaion meods Sep Relaxaion/sudden elease Cage/Explosive impaco. Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Moving Suppo Coesponds o gounded model Only esponses ae measued Wen e mass popeies ae nown, e modal popeies can be calculaed fom measued daa Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Moving Suppo Response Funcion Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Modal Tesing Lecue D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

Digial Signal Pocessing Inoducion Basics of Discee Fouie Tansfom DFT Aliasing Leaage Windowing Fileing Impoving Resoluion FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Inoducion Te measued foce o acceleomee signals ae in ime domain. Te signals ae digiized by an A/D convee And ecoded as a se of discee values evenly spaced in e peiod T FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Basics of DFT Te specal popeies of e ecoded signal can be obained using Discee Fouie Tansfom/Seies DFT/DFS: Te DFT assumes e signal x is peiodic In e DFT ee ae jus a discee numbe of iems of daa in eie fom Tee ae jus values x Te Fouie Seies is descibed by jus values FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian FRF Measuemen Tecniques Basics of DFT * 0 0 0 0 sin cos, sin cos n n i T n n i n T n n T n n n n n n n n X X d e x T X e X x o d x T b d x T a T n b a a x T x x n n + + + π + + in n n in n n n n n n X X e x X e X x o n x b n x a n b n a a x, sin cos sin cos 0 / 0 / 0 0 0 π π π π π π

IUST,Modal Tesing Lab,D H Amadian FRF Measuemen Tecniques Basics of DFT 0 0 0 / in n x x x X X X e x X M K K M K K M O L L M π

Basics of DFT Te sampling fequency: π f Te ange of fequency specum: fs s π fmax max T Te esoluion of fequency specum: FRF Measuemen Tecniques s s Δf T T s, Δ π T s π T yquis Fequency IUST,Modal Tesing Lab,D H Amadian

Basics of DFT Tee ae a numbe of feaues of DF analysis wic if no popely eaed, can give ise o eoneous esuls: Aliasing Mis-ineopeaing a ig fequency componen as a low fequency one Leaage Peiodiciy of e signal FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Aliasing Digiizing a low fequency signal poduces exacly e same se of discee values as esul fom e same pocess applied o a ige fequency signal < s s FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Aliasing Compae sinπp p < FRF Measuemen Tecniques : sinπ p πp sinπ πp sin IUST,Modal Tesing Lab,D H Amadian

Aliasing FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Aliasing Te soluion o e poblem is o use an ani-aliasing file Subjecing e oiginal signal o low pass wi sap file Files ave a finie cuoff ae; i is necessay o ejec e specal ange nea ayquis fequency s > 08.0 FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Leaage A diec consequence of aing a finie leng of ime isoy coupled wi assumpion of peiodiciy Enegy is leaed ino a numbe of specal lines close o e ue fequency. FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Leaage FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Leaage To avoid e leaage ee ae numbe of scenaios: Inceasing e ecod ime T Windowing Muliply e ime ecod by a funcion a is zeo a e ends of e ime ecod and lage in e middle, e FFT conen is concenaed on e middle of e ime ecod FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian FRF Measuemen Tecniques Windowing Windowing involves e imposiion of a pescibed pofile on e ime signal pio o pefoming e FT T elsewee T a a a a a a w x w x π 0, 0 cos4 cos3 cos cos cos 0 0 4 0 3 0 0 0 0 < < + + +

Windowing FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Windowing Funcion a 0 a a a 3 a 4 Recangula - - - - Hanning - - - Kase- Bessel.98 0.44 0.003 - Fla op.933.86 0.388 0.03 FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Windowing FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Impoving Resoluion Zoom Tee aises limiaions of inadequae fequency esoluion a e lowe end of e fequency ange Fo ligly-damped sysems A common soluion is o concenae all specal lines ino a naow band Wiin f min -f max Insead of 0-f max FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Zoom Meod : Sifing e fequency oigin of e specum x Asin x Asin cos A [ sin ] min + sin + min Te modified signal is en analysed in e ange of 0-f max- f min min FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Zoom Meod : A conolled aliasing effec Applying a band pass file Because of e aliasing penomenon, e fequency componen beween f and f will appea aliased beween 0-f - f FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Modal Tesing Lecue 3 D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

Use of Diffeen Exciaion Signals Inoducion Sepped-Sine Tesing Slow Sine Sweep Tesing Peiodic Exciaion Random Exciaion Tansien Exciaion FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Inoducion Tee ae ee diffeen classes of exciaion signals used: Peiodic Tansien Random FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Inoducion Peiodic: Sepped sine Slow sine sweep Peiodic Pseudo-andom FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Inoducion Tansien: Bus sine Bus andom Cip Impulse FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Inoducion Random: ue andom Wie noise FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Sepped-Sine Tesing Classical meod of FRF measuemen To encompass a fequency ange of inees, e command signal fequency is sepped fom one fequency o anoe Te exciaion/esponses ae measued ampliudes and pases. I is necessay o ensue a e seady-sae condiion is aained befoe e measuemen. FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Sepped-Sine Tesing FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Sepped-Sine Tesing Te exen of unwaned ansien esponse depends on: Poximiy of exciaion fequency o a naual fequency, Te abupness of e cangeove fom e pevious command signal o e new one, Te ligness of e damping of neaby modes. FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Sepped-Sine Tesing An advanage of sepped-sine esing is e faciliy of aing measuemen wee and as ey ae equied. o. poin beween HPP s 3 5 8 Lages Eo % db 30 3 0 5 0.5 0. 0. FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Slow Sine Sweep Tesing Involves e use of a sweep oscillao Povides a sinusoidal signal Is fequency is vaied slowly bu coninuously If an excessive sweep ae is used en disoions of FRF plo ae inoduced FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Slow Sine Sweep Tesing One way of cecing e suiabiliy of a sweep ae is o mae e measuemen wice: Once sweeping up And e nd ime sweeping down FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Slow Sine Sweep Tesing I is possible o pescibe an opimum sweep ae fo a given sucue aing ino accoun is damping levels FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Slow Sine Sweep Tesing Recommended sweep ae: FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Slow Sine Sweep Tesing ISO pescibes maximum linea and log sweep ae oug a esonance as: Linea S max Log S max FRF Measuemen Tecniques < 6 ζ < 30 ζ Hz / min Ocaves min IUST,Modal Tesing Lab,D H Amadian /

Peiodic Exciaion A naual exension of e sine wave es meods: To use a complex peiodic inpu signal wic conains all e fequencies of inees, Te DFT of bo inpu and oupu signals ae compued and e aio of ese gives e FRF Bo signal ave e same fequency conens FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Peiodic Exciaion Two ypes of peiodic signals ae used: A deeminisic signal squae wave Some fequency componens ae ineviably wea. Pseudo-Random ype of signal Te fequency componens may be adjused o sui a paicula equiemens-suc as equal enegy a eac fequency, Is peiod is exacly equal o e sampling ime esuling zeo leaage. FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian FRF Measuemen Tecniques Random Exciaion γ H H S S H S S H S H S S H S S H S xf xx ff fx xf xx ff fx ff xx

Random Exciaion Tee may be noise on one of e wo signals ea esonance is is liely o influence e foce signal A ani-esonances i is e esponse signal wic will suffe FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Random Exciaion H mig be a bee indicaion nea esonances wile H is a bee indicaion nea ani-esonances: S fx H, H S + S ff nn S xx + S S xf mm Auo-speca of noise on e oupu signal Auo-speca of noise on e inpu signal FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Random Exciaion A close opimum fomula fo e FRF is defined as e geomeic mean of e wo sandad esimaes Pase is idenical o a in e wo basic esimaes H H H v FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Random Exciaion Typical measuemen made using andom exciaion: FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Random Exciaion Deails fom pevious plo aound a esonance: H H FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Random Exciaion Use of zoom specum analysis: Impoving e esoluion emoves e majo souce of low coeence FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Random Exciaion Effec of aveaging: FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Tansien Exciaion Te exciaion and e esponse ae conained wiin e single measuemen FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Tansien Exciaion Bus exciaion signals: A so secion of a coninuous signal sin, andom, followed by a peiod of zeo wave. FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Tansien Exciaion Cip exciaion: Te specum can be sicly conolled o be suc wiin fequency ange of inees FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Tansien Exciaion Impulsive exciaion by Hamme: Diffeen impulsive exciaions Signals and speca fo double i case FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Tansien Exciaion Impulsive exciaion by Sae: FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Modal Tesing Lecue 4 D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

RESPOSE FUCTIO MEASUREMET TECHIQUES 3.9 Calibaion 3.0 Mass Cancellaion 3. Roaional FRF Measuemen 3. Measuemen on onlinea Sucues Effecs of Diffeen Exciaions Level Conol in FRF Measuemen FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Calibaion In all measuemen sysems i is necessay o calibae e equipmen. Tee sould be wo levels of calibaion: Absolue calibaion of individual ansduces Te oveall sensiiviy of insumenaion sysem FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Calibaion Te oveall sysem calibaion Te scale faco sould be ceced agains compued faco using manufacues saed sensiiviy Sould be caied ou befoe & afe eac es FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Mass Cancellaion ea esonance e acual applied foce becomes vey small and is us vey pone o inaccuacy. Some applied mass is used o move addiional ansduce mass X&& αt equied F T ft fm m && x X&& αm measued F FRF Measuemen Tecniques M IUST,Modal Tesing Lab,D H Amadian

Mass Cancellaion Added mass o be cancelled and e ypical analogue cicui A deiving poin a elaion beween measued and equied FRF s can be obained FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Mass Cancellaion Re F T Re F M m Re X&& Im F T Im F M m Im X&& o Re/ α T Re/ α M m Im/ α T Im/ α M FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Roaional FRF Measuemen Measuemen of oaional FRFs using wo o moe ansduces: x o x A + x B θ o θ A + L θ B FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Roaional FRF Measuemen Applicaion of momen exciaion X F, X M, θ, F θ M FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Measuemen on onlinea Sucues Many sucues, especially in viciniy of esonances, beave in a nonlinea way: aual fequency vaies wi posiion and seng of exciaion Disoed fequency esponses nea esonances Unsable o unepeaable daa FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Measuemen on onlinea Sucues Examples of diffeen nonlinea sysem esponse fo diffeen exciaion levels Sofening effec Incease in damping FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Effecs of Diffeen Exciaions FRF measuemen on nonlinea sysem: Sinusoidal Exciaion Compaible wi eoy Random Exciaion Lineaized sysem Tansien Exciaion FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Effecs of Diffeen Exciaions Mos ypes of nonlineaiy ae ampliude dependen: A lineaized beaviou is obseved wen e esponse level is ep consan Te obained linea model is valid fo a paicula vibaion level FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Level Conol in FRF Measuemen Response level conol, Bes linea epesenaion nonlineaiies ae displacemen dependen Foce level conol O no level conol FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Level Conol in FRF Measuemen Invese FRF plos fo a SDOF Real pa is expeced o be line w fequency squaed Imaginay pa sould be linea/consan Any deviaion fom e expeced beaviou can be deeced as nonlineaiy in e sysem FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

Level Conol in FRF Measuemen Use of Hilbe ansfom o deec non-lineaiy Te Hilbe ansfom expess e elaions beween eal and imaginay pas of e Fouie Tansfom FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

oes: Hilbe Tansfom Te Hilbe ansfom expess e elaions beween eal and imaginay pas of e Fouie Tansfom Fouie Tansfom is consideed o map funcions of ime o funcions of fequency and vice vesa Hilbe ansfom map funcions of ime o fequency o e same domain FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

oes: Hilbe Tansfom Fo causal funcions: g g g even odd g even + g odd g /, g /,, > < 0 0 g /, > 0 g /, < 0 FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian

oes: Hilbe Tansfom ReG I ImG I Since I ReG ImG { sign } { g } I{ g sign } even { g } I{ g sign } odd even i based on π i i ImG, π i ReG. π convoluion eom FRF Measuemen Tecniques IUST,Modal Tesing Lab,D H Amadian odd,, :

Modal Tesing Lecue 5 D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

Modal Paamee Exacion Inoducion Peliminay cecs of FRF daa Visual cecs Assessmen of muliple-frf daa se using SVD Mode indicao funcions SDOF modal analysis meods Pea ampliude meod Cicle fi meod Invese o line fi meod Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Inoducion Some of e many available pocedues fo fiing a model o e measued daa ae discussed: Tei vaious advanages and limiaions ae explained, o single meod is bes fo all cases. Tis pase of e modal es pocedue is ofen called modal paamee exacion o modal analysis Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Inoducion Types of modal analysis: Fequency domain of FRFs Time domain of Impulse Response Funcion Te analysis will be pefomed using SDOF meods, and MODF meods. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Inoducion Anoe classificaion of meods elaes o e numbe of FRFs used in e analysis: Single-FRF meods, and Muli-FRF meods: Global meods wic deals wi SIMO daa ses and Polyefeence wic deals wi MIMO daa Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Inoducion Difficuly due o damping: In pacice we ae obliged o mae ceain assumpion abou e damping model, Significan eos can be incued in e modal paamee esimaes as a esul of conflic beween assumed and acual damping effecs. Decision on e issue of eal and complex modes. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Peliminay cecs of FRF daa Low-fequency asympoes, Siffness-lie caaceisics fo gounded sucues Mass-line asympoes fo fee sucues Hig-fequency asympoes, Mass line o siffness line Incidence of aniesonances Fo a poin FRF ee mus be a esonance afe eac aniesonance Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Peliminay cecs of FRF daa Mode Indicao Funcions: Te Pea-Picing Meod Sum of ampliudes of all measued FRFs o locae e esonance poins Te fequency-domain decomposiion meod Defined by e SVD of e FRF maix Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Case Sudy: MODES OF A RAILWAY VEHICLE Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Case Sudy: Tes se-up Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Case Sudy: Senso Locaions Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Case Sudy: Exciaion Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Case Sudy: Measuemens Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Te Pea-Picing Meod Sum of ampliudes of all measued FRFs o locae e esonance poins # Mode 3 4 5 6 7 8 9 Hz Fequency.67 4.67 5.33 8.33.33 3.33 4.00 6.00 4.67 Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Te fequency-domain decomposiion meod A moe advanced meod consiss of compuing e Singula Value Decomposiion of e specum maix. Te meod is based on e fac a e ansfe funcion o specum maix evaluaed a a ceain fequency is only deemined by neigboing modes. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Te fequency-domain decomposiion meod [ ] { } { } { } [ ] [ ] [ ][ ][ ] [ ] [ ] [ ] Σ Σ Σ T T np MIF V U H H H H H K

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods SDOF modal analysis meods Te SDOF assumpion j j j s s s s s j s j j s s s s j s j B i A i A i A i A + + + + + + η α η η α η α

SDOF modal analysis meods SDOF modal analysis meods Pea ampliude meod Cicle fi meod Invese o line fi meod Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

SDOF modal analysis meods: Pea Ampliude Individual esonance peas ae deeced fom e FRF Te fequency of e maximum esponses is aes as e naual fequency of a mode, Te pea ampliude and e alf powe poins ae deemined, Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods SDOF modal analysis meods: Pea Ampliude H A A H en H nowns b a b a b a ˆ, ˆ,,, ˆ η η η ζ η

SDOF modal analysis meods: Pea Ampliude Anoe esimae fo modal esidue: Hˆ A maxre + minre η maxre + minre Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Modal Tesing Lecue 6 D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

Modal Paamee Exacion Cicle-fi meod Popeies of e modal cicle Cicle-fi analysis pocedue Inepeaion of damping plos Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Popeies of e modal cicle Assuming a sysem wi sucual damping e basic funcion o deal wi is: Aj α / + iη Since e effec of modal consan is o scale e size and oae e cicle, we conside: α / + iη Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Popeies of e modal cicle Finding e naual fequency: + / an / an an90, / an d d η η θ θ η η θ γ η γ

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Popeies of e modal cicle Damping d d fequency aual d d d d ae sweep d d + 0.@ / η θ θ η η θ

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Popeies of e modal cicle b a b a b a b a b a b a a a b b wen fo η θ θ θ θ η η θ θ η η θ η θ + + o K 90 an an 3% % an an / an / an

Popeies of e modal cicle Te final popey elaes o e diamee of e cicle D: D j A η j Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Cicle-fi analysis pocedue Selec poins o be used Fi cicle, calculae qualiy of fi Locae naual fequency, Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Cicle-fi analysis pocedue Obain damping esimaes Calculae muliple damping esimae and scae Deemine modal consan module and agumen. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Inepeaion of damping plos oise may conibue o e ougness of e suface. Sysemaic disoions due o: Leaage Eoneous esimaes fo naual fequency onlineaiy Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Cicle-fi Minimizing e algebaic disance: 0. : 0. + + + + + + + + + + C B A y x y x y x y x y x y x Soluion Squaes Leas C By Ax y x R b y a x n n n n M M M M

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Cicle-fi Minimizing e geomeic disance: 3 3 min, U d u u u U Le u u u X d R b x a x n i i i i + + +

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Cicle-fi + + + + Δ + + Δ + min min 0 0 0 0 0 m m m m m m x u x u x u x u x u x u x u x u x u x u x u x u p d p p d d d p p d d d M M M L

Cicle-fi Leas-Squaes Fiing of Cicles and Ellipses By: Wale Gande, Gene H. Golub, and Rolf Sebel You may find i in fpmec.ius.ac.i Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Home wo Deemine e modal popeies of e beam esed in e lab Fequency ange of 0-400Hz aual fequencies Damping cape plos Mode Sapes Due ime 87// Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Impoing e ASCII files Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Modal Tesing Lecue 8 D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

MDOF Modal Analysis in e Fequency Domain SISO In some cases e SDOF appoac o modal analysis is simply inadequae o inappopiae: closely-coupled modes, e naual fequencies ae vey closely spaced, o wic ave elaively eavy damping, ose wi exemely lig damping Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

MDOF Modal Analysis in e Fequency Domain SISO One sep MDOF cuve fiing meods: on-linea Leas Squaes Meod Raional Facion Polynomial Meod A meod paiculaly suied o vey ligly damped sucues Global Modal Analysis in Fequency Domain Global Raional Facion Polynomial Meod Global SVD Meod Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods on-linea Leas Squaes Meod ec A A A q dq de w E H H M K i A H H j j j p l l l l m l l l m m l j l l j,,,,, 0.,, 3 ε ε η L + + + Te diffeence beween measuemen and analyical model

on-linea Leas Squaes Meod Te se of obained equaions ae nonlinea o diec soluion ieaive pocedues on-uniqueness of soluion Huge compuaional load Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Raional Facion Polynomial Meod j i a i a i a a i b i b i b b H i A H 0 0 ζ + + + + + + + + + L L

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Raional Facion Polynomial Meod m m m m m m m m i a i a i a a H i b i b i b b e o H i a i a i a a i b i b i b b e 0 0 0 0 + + + + + + + + + + + + + + + + L L L L Ode of model is seleced

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Raional Facion Polynomial Meod { } { } m m m m m m i H a a a a a i i i H b b b b i i i e 0 0 M L M L

Raional Facion Polynomial Meod A se of linea equaions using eac individual measued FRF is fomed. Te unnowns a i and b i ae obained using a leas squae soluion. Te modal popeies ae exaced fom obained coefficiens a i and b i. Te analysis may epea fo a diffeen model ode. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Ligly Damped Sucues In ese sucues i is easy o locae e naual fequencies, Is accuacy is equal o e fequency esoluion of e analyze Te damping aio is assumed o be zeo. Te modal consans ae obained using cuve fiings. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Ligly Damped Sucues Ω Ω Ω Ω Ω Ω M M M M M M M M L L M M j j j A A H H A H Te naual fequencies ae nown

Global Modal Analysis in Fequency Domain So fa eac measued FRF is cuve fied individually, Muli-esimaes fo global paamees naual fequencies and damping Anoe way is o use measued FRF cuves collecively. Fequency and damping caaceisics appea explicily. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Global Raional Facion Polynomial Meod If we ae seveal FRF s fom e same sucue en e denominao polynomial will be e same in evey case. A naual exension of RFP meod is o fi all n FRFs simulaneously m- values of a i and, and nm- values of b i Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Global SVD Meod { } [ ] [ ] { } { } + Φ n n n R s i H H H H φ M

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Global SVD Meod { } [ ] [ ] { } { } { } [ ] [ ] [ ] { } { } n n R s i s H R s i H + Φ + Φ φ φ & { } [ ] { } { } [ ] { } { } { } [ ] [ ] { } { } n n R g s H R g H s i g + Φ + Φ φ &

Global SVD Meod { ΔH } { H } { H } i i i+ c { ΔH } [ Φ] { Δg } i n i { } ΔH& [ Φ] [ s ]{ Δg } i n i Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Global SVD Meod Conside daa fom seveal diffeen fequencies o obain fequencies and damping: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] { } [] [ ] T T T L n L n L n L n z z H s H g s H g H + Φ Δ Δ Δ Φ Δ Δ Φ Δ 0, & &

Global SVD Meod Te eigen-poblem is solved using e SVD. Te an of e FRF maices and eigenvalues ae obained. Ten e modal consans can be ecoveed fom: Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Modal Tesing Lecue 8- D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

MDOF Modal Analysis in e Time Domain Te basic concep: Any Impulse Response Funcion can be expessed by a seies of Complex Exponenials Te Complex Exponenial Seies conain e eigenvalues and eigenvecos infomaion. Te IRF is obained by aing invese Fouie ansfom of e measued FRF. Modal Paamee Exacion Meods ζ + i s j Aje ; s ζ IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Complex Exponenial Meod + s j j j j j j j e A IRF s i A o s i A s i A FRF : α α

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Complex Exponenial Meod Single FRF Δ q q q q l l s l s A A A V V V V V V V V V A V A e A e 0 M M L L M L L M M K K L L L L M

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Complex Exponenial Meod Single FRF q q q T q q T q A A A V V V V V V V V V 0 0 0 M M L L M L L M M K K L L L L M M M β β β β β β β β j q i i j i j q i i i V A 0 0 β β

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Complex Exponenial Meod Single FRF Te ae seleced o be coefficiens of e polynomial: i i i i i i i i j i j q i i j i j q i i i q q V V A q Se V V V 0 0 0 0 0 0 0. 0, : 0. + + + + β β β β β β β β β L β i

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Complex Exponenial Meod Single FRF + + 4 0 4 3 0 0 i i i M M M M L M L M M M M L M M M L L β β β β

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Complex Exponenial Meod Single FRF Te values and A i ae obained fom: s i e V Δ 0. 0 + + + + V V V β β β β L A A A V V V V V V V V V 0 M M L L M L L M M K K L L L L M

Complex Exponenial Meod Single FRF Implemenaion Pocedue: Ode of modal model is seleced, Modal model is idenified using e defined seps in pevious slides, FRF is egeneaed fom modal infomaion and compaed wi e measued FRF Te pocedue epeaed using anoe ode fo e modal model unil sable esuls ae obained. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Sabilizaion Diagam Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Global Analysis in Time Domain Ibaim Time Domain Meod Te basic concep is o obain a unique se of modal paamees fom a se of vibaion measuemens: Scaled mass nomalized mode sapes wen e foce is nown, Un-scaled mode sapes wen e foce is no measued. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Modal Paamee Exacion Meods Ibaim Time Domain Meod m s i i e x ψ q m m q q s s s s s s m n n n m m q n n n q q e e e e e e x x x x x x x x x,,, L L M L L M L L L L L M L M M L L L M L M M L L ψ ψ ψ ψ ψ ψ ψ ψ ψ [ ] [ ] [ ] Λ Ψ X

Ibaim Time Domain Meod A nd se of eqns: x i l + Δ m m ψ s Δ s l ie e ψ i e s l +Δ [ ˆX ] [ ] Ψ ˆ [ Λ] m ψˆ i e s l Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Ibaim Time Domain Meod [ ] [ ] [ ] A Ψ Ψˆ [ X ] [ Ψ ] [ Λ] [ ] [ ] Xˆ Ψ ˆ [ Λ] [ ] [ ] [ ] A X Xˆ [ ] [ ] A Xˆ [ X ] + Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Ibaim Time Domain Meod [ A]{ ψ } e s { } Δ ψ Eigenvecos of maix [A] ae e mode sapes, Te naual fequencies and damping aios ae obained fom eigenvalues of [A]. Modal Paamee Exacion Meods IUST,Modal Tesing Lab,D H Amadian

Modal Tesing MDOF Modal Analysis in e Time Domain D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

MDOF Modal Analysis in e Time Domain Te basic concep: Any Impulse Response Funcion can be expessed by a seies of Complex Exponenials s j Aje ; s i Te Complex Exponenial Seies conain e eigenvalues and eigenvecos infomaion. Te IRF is obained by aing invese Fouie ansfom of e measued FRF. Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Complex Exponenial Meod CE D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods

Complex Exponenial Meod A A FRF j j i s i s IRF o j : Modal Paamee Exacion Meods j i A j A e j s s j D H Amadian, Modal Tesing Lab, IUST

Complex Exponenial Meod Complex Exponenial Meod Single FRF l l s s A V A e A e l A A V A e A e A A V V V 0 V V V q q q A V V V D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods q A V V V

Complex Exponenial Meod Complex Exponenial Meod Single FRF T T A A A V V V V V V 0 0 0 V V V q q q q q q A V V V q q j q i i j i j q i i i V A 0 0 D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods j i i 0 0

Complex Exponenial Meod Complex Exponenial Meod Single FRF Te ae seleced o be coefficiens of e i polynomial: q qv V V 0. 0 i j i q i q q V 0 0, i i i i j i i j i j i i i V A q Se 0 0 0 0. 0 : i i i 0 D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods i 0

Complex Exponenial Meod Complex Exponenial Meod Single FRF 0 i i i 0 3 0 3 D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods 4 4

Complex Exponenial Meod Complex Exponenial Meod Single FRF Te values and A i ae obained s i e V fom: 0. 0 V V V A A V V V 0 A V V V V V V A V V V D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods

Te Leas Squaes Complex Exponenial Meod LSCE D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods

Te Leas Squaes Complex Exponenial Meod LSCE Te LSCE is e exension of CE o a global pocedue. I pocesses seveal IRF s obained using SIMO meod. Te coefficiens β a povide e soluion of caaceisic polynomial ae global quaniies. Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Te Leas Squaes Complex Te Leas Squaes Complex Exponenial Meod LSCE 0 0 One Typical IRF q q o 3, 4 4 Exending o all measued IRFs G T G G T G G G o, D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods q p

Te PolyRefeence Complex Exponenial Meod PRCE D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods

Te PolyRefeence Complex Exponenial Meod PRCE Consiues e exension of LSCE o MIMO. A geneal and auomaic way of analyzing dynamics of a sucue. MIMO es meod ovecomes e poblem of no exciing some modes as usually appens in SIMO. Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Te PolyRefeence Complex Te PolyRefeence Complex Exponenial Meod PRCE Consideing q inpu efeence poins: s j j e A l l Q A s j j e A s j j e A jl l j l j jl A W A Q A s j j e A W s e A l l W s e A W jq jq e A j q jq e A W D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods Modal Paicipaion faco

Te PolyRefeence Complex Te PolyRefeence Complex Exponenial Meod PRCE s j j e A j j s j j A e W e A W s j q jq e A W 0 0 0 0 j s s j A A e W W W 0 0 j s s j A A e W W W W W W D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods 0 0 j s q q q jq A e W W W

Te PolyRefeence Complex Exponenial Meod PRCE 0 W A j j W V A j j L L W V A j j, V e L W V W V W V 0, L q 0 L Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Te PolyRefeence Complex Te PolyRefeence Complex Exponenial Meod PRCE 0 0 0 j j A W 0 0 j j j j A V W j j A V W j L L j L A V W L L L j j A V W D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods 0 0

Te PolyRefeence Complex Te PolyRefeence Complex Exponenial Meod PRCE 0 0, L L j I 0 0 L j j L 0 j j j 0 j j j L L L L 0 j j j j j j B L L L L L L D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods j j T B

Te PolyRefeence Complex Te PolyRefeence Complex Exponenial Meod PRCE B j j T B Consideing fo eac esponse locaion j,,p: p p T B B g p j,,p T T T T T B B T T T T T T T B Knowing e coefficien maix [B], we mus now deemine [V] D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods

Te PolyRefeence Complex Te PolyRefeence Complex Exponenial Meod PRCE L, 0 0 L L V W V W V W L V W 0 0 0 L L 0 0 0 0 0 0 L s L W e V W L 0 0 0 0 0 L s L W e V W 0 0 L W V 0 0 L s L W e V W D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods 0 0 0

Te PolyRefeence Complex Exponenial Meod PRCE L 0 V W 0 L L V V V W V W 0 L z0w z V W Vz0 z V W V z z L V W V z 0 z0 z z V z L L L z V W V z L L Modal Paamee Exacion Meods L L L D H Amadian, Modal Tesing Lab, IUST

Te PolyRefeence Complex Exponenial Meod PRCE An sandad eigenvalue poblem o obain V z L zl L L 0 z L zl I 0 0 0 V z z 0 0 I 0 z z 0 0 Te eigenveos z 0 coespond o W Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Te PolyRefeence Complex Te PolyRefeence Complex Exponenial Meod PRCE L A V W j j, 0,,, j j j 0 j j V W W jq j V j j L j A W H o A V W L j V W L H W A j V j H W A Te esidue calculaion is epeaed fo all meas ed poins j p D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods all measued poins, j,,,p.

Te PolyRefeence Complex Exponenial Meod PRCE Te meod povide moe accuae modal epesenaion of e sucue. I can deemine muliple oos o closely spaced modes. Socomings: Sensiive o nonlineaiies and any lac of ecipociy in fequency esponses, Some difficulies in analyzing sucues wi moe an 5% viscous damping. Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Global Analysis in Time Domain Ibaim Time Domain Meod D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods

Ibaim Time Domain Meod m s s i i e x q q s s s s m m q q e e e e x x x x x x,, q m m s s m n n n q n n n q e e x x x,, X D H Amadian, Modal Tesing Lab, IUST Modal Paamee Exacion Meods

Ibaim Time Domain Meod A nd se of eqns: x i m l m i e s l m s s l ie e ˆ ˆX ˆ i e s l Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Ibaim Time Domain Meod X A ˆ ˆ ˆ X AX Xˆ A Xˆ X Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Ibaim Time Domain Meod A e s Eigenvecos of maix [A] ae e mode sapes, Te naual fequencies and damping aios ae obained fom eigenvalues of [A]. Modal Paamee Exacion Meods D H Amadian, Modal Tesing Lab, IUST

Modal Tesing Lecue 9 D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

Deivaion of Maemaical Models Spaial Models mass, siffness, damping eeds measuemen of mos of e modes Requies measuemen in many DOFs Response Models FRF eeds measuemen in fequency ange of inees Requies measuemen in seleced DOFs Modal Models naual fequencies and mode sapes eeds measuemen of only one mode Requies measuemen in andful of DOFs Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Deivaion of Maemaical Models Modal Models Requiemens o consuc Modal Models Refinemen of Modal Model Convesion o eal modes Compaibiliy of DOFs Reducion Expansion Response Models FRF Tansmissibiliy Base Exciaion Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Requiemen o consuc Modal Models Minimum equiemens One column in case of fixed exciaion o One ow wen esponse is measued a a fixed poin. Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Deivaion of Maemaical Models Requiemen o consuc Modal Models Poof: ii ni mi i i n i i m n m mn X F F X F X F X α α α α

Requiemen o consuc Modal Models Seveal addiional elemens of FRF o even columns ae measued o: Replace poo daa, To povide cecs Modes ave no been missed Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Refinemen of Modal Models Complex o eal convesion: Taing e modulus of eac elemen and assigning a pase of 0 o 80. Finding a eal mode wi maximum pojecion o e measued one: max Muli poin exciaion Ase s meod φ φ T R R φ C φ C Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Compaibiliy of DOFs Employmen of e measued modes in updaing/modificaion of analyical models equies e compaibiliy of DOFs. Tee ae wo appoaces in compaibiliy excusive: Analyical model educion Expansion of measued modes Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

IUST,Modal Tesing Lab,D H Amadian Deivaion of Maemaical Models Reducion of Analyical Model Guyan Reducion { } [ ]{ } { } { }, 0 f x T K K K K T x T x x x K K I x x x K K x f x x K K K K T T T T T

IUST,Modal Tesing Lab,D H Amadian Deivaion of Maemaical Models Dynamic Model Reducion { } [ ]{ } { }. 0, 0. φ φ φ φ φ φ φ φ φ φ φ T M M M M K K K K T T M K M K I M K M K M M M M K K K K T T T T T T T T T

Expansion of Models In ode o compae analyical model wi e measued modal daa on may expand e measued daa by: Geomeic inepolaion using spline funcions Using analyical model spaial model Using analyical model modal model Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Expansion in Spaial Domain K K T K K M M T M M φ φ 0. K M K T M T φ φ Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Expansion in Modal Domain Φ Φ Sol : 0 R T Assuming Mass Maix is coec min R ~ ΦR ~ Φ 0 s : R T R I. ~ Φ T ~ Φ 0 UΣV T R VU T, Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Response Model Fequency esponse funcions [ ] [ ] H Φ λ Tansmissibiliies [ ][ Φ] T i X e T j, T j i i j X e H H ji i Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Tansmissibiliy Plos Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Response Model Te ampliude s peas of e ansmissibiliies do no coespond wi e esonan fequencies. Tansmissibiliies coss eac oe a e esonan fequencies becomes independen of e locaion of e inpu i T j H H i ji i T j φ φφi j φ φ φ i j Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Base Exciaion An applicaion aea of ansmissibiliy. Inpu is measued as esponse a e dive poin. {} x { } x el + x ef M Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Base Exciaion [ M ]{&& x } + [ K ]{ x } && x [ M ]{}{} g, g [ ]{} ef xef M g { X } x { g} [ ] H { X } x { g} o el x ef ef el ef [ H ][ M ]{}. g,. M Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Base Exciaion { } { } X x g { Q} Q i x ef ef Deivaion of Maemaical Models [ H ][ M ]{} g { } { } X x g ef, {} u [ M ]{} g x ef φ i j u j j φ IUST,Modal Tesing Lab,D H Amadian,,

Spaial Models [ M ] [ Φ] T [ Φ] [ K ] [ Φ] T [ λ ][ Φ] Deivaion of Maemaical Models IUST,Modal Tesing Lab,D H Amadian

Modal Tesing Lecue 0 D. Hamid Amadian Scool of Mecanical Engineeing Ian Univesiy of Science and Tecnology amadian@ius.ac.i

Deivaion of Maemaical Models Inoducion Equaion Eo Meod Sec. 6.3.6 page 456 Idenificaion of Rod FE Model Paamee Idenificaion Soluion of Ove-deemined se of Equaions Soluion of Unde-deemined se of Equaions Eo Analysis Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Inoducion Consucion of Spaial Model fom modal daa: K Φ T ΛΦ, M Φ T Φ, C Modal model mus be complee: All modes mus be pesen Mode sapes ae measued in all DOF s Φ T Measuemen of complee Modal Model is impacical. ΓΦ Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Inoducion Alenaive meods ae equied o consuc e spaial model fom incomplee and noisy measued modes. Te difficuly wi incompleeness is emoved by educing e numbe of unnowns in spaial model. Te noise effecs ae emoved by aveaging. Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Equaion Eo Meod We ave some infomaion egading e spaial model foma: Symmey Paen of zeos We may incopoae ese infomaion ino e idenificaion pocedue and econsuc e spaial model. Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Equaion Eo Meod In is meod e eigen poblem is eaanged o obain e spaial model: KΦ MΦΛ 0. K M [ T T ΛΦ ] 0 Φ Te DOF s of measued modes mus be compaible wi e DOF s of spaial model. Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

D H Amadian, Modal Tesing Lab, IUST Deivaion of Maemaical Models Reaangemen Example: 0. 0 0 0 : 0, 0 0 + m m o m m φ φ φ φ φ φ φ φ φ

D H Amadian, Modal Tesing Lab, IUST Deivaion of Maemaical Models Idenificaion of A Rod FE Model Conside a fixed-fee od wi n elemens. Te mass and siffness maices ae:,,,, 3 3 n n n n n n n n m m m m diag M K + + + L L L L

D H Amadian, Modal Tesing Lab, IUST Deivaion of Maemaical Models Idenificaion of A Rod FE Model Te equilibium sae a modes and s ae:. 0 0. s s M K M K φ λ φ λ Te las ows of equilibium sae equaions ae:. 0 0,,,,, + + n s n n n s n n n n n n m m φ λ φ φ λ φ

D H Amadian, Modal Tesing Lab, IUST Deivaion of Maemaical Models Idenificaion of A Rod FE Model,,,,,,,,,,,,,,, 0 0, 0. n n n n n n n n s n s n s n s n n n n n s s n s n s n n n m m m φ φ λ φ φ φ φ φ φ φ λ λ λ φ φ φ λ φ φ φ

D H Amadian, Modal Tesing Lab, IUST Deivaion of Maemaical Models Idenificaion of A Rod FE Model Fom oe ows one obains: n n n n n n n n m m m m m m m m m,,,,,,, K K Using oal mass infomaion m m is obained: + n l n l n oal m m m m

Idenificaion of A Rod FE Model Only wo modes and one naual fequency ae equied o consuc e mass and siffness maices. Moe deails can be found in: GML Gladwell, YM Ram, Consucing Finie Elemen Model of a Vibaing Rod, Jounal of Sound and Vibaion, 69,9-37,994. Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Paamee Idenificaion In a geneal case e mass and siffness maices ae paameeized and ae obained by eaanging: Equaion of moion in modal domain Oogonaliy equiemens, ec. Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Paamee Idenificaion Paameeizaion : K K,, K, n, M M m, m, K, m n. EOM : KΦ MΦΛ Exas : KΦ R 0, 0, Φ Φ T R T MΦ MΦ R I, Φ m, I T xx KΦ Λ., I xy, K Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Paamee Idenificaion Re aangemen : Ax b A A Φ, Φ R, Λ, b b Λ, moal, I x x,, K,, m, m,, m n K m A full an n m m > m < n xx, K n x A b n Unde de e min ed n Ove de e min ed Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

min x Soluion Soluion of undedeemined case o min x, ST : A x b min T T T T x x x A b λ AA T x : 0, ST : Ax b λ b λ Ax 0 b x A T T AA b x A T AA b T λ 0. Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

D H Amadian, Modal Tesing Lab, IUST Deivaion of Maemaical Models Soluion of Ove-deemined se of Equaions [ ] [ ] b A A A x b A Ax A x b b b A x Ax A x Soluion E E Assume b Ax n m b Ax T T T T T T T T T T T T ij j i min min 0.,, + < ε ε ε ε ε ε ε σ δ εε ε ε

D H Amadian, Modal Tesing Lab, IUST Deivaion of Maemaical Models Eo Analysis [ ] [] [ ] [ ] [ ] [ ] [] [ ] [ ] [ ] [ ] [ ] x E x E A E noisy A If x E x E E m as A A A E x E x E A A A E x E b A A A E Ax b T T T T T T T + + + 0 0, ε ε ε ε ε

Example: Te paamees o be updaed ae e 0 siffness and 6 masses Te measued daa consiss of e s ee naual fequencies and mode sapes added wi unifomly disibued andom noise Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Example: Eigenvalue equaions aeaangmen: 3 equaions 3*6 equaions fo eac eigenveco em, *6 symmeic oogonaliy equaions, and oal mass equaion 6 paamees and Te ems in A and b conain noisy daa. Paamees EOM : KΦ T Exas : φ Mφ R {,, K,, m, m, K, m } : x 0 6 MΦΛ R m. 0, Φ T MΦ I, Φ T, KΦ Λ. Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST

Example 3 4 5 6 7 8 9 0 m m m 3 m 4 m 5 m 6 Exac 000 50 500 0000 000 000 5000 7000 000 000 0. 0. 0. 0. S/ 00 04 66 58 0008 006 008 994-44 035-63 0. 0. 0.99 0. -0.0 S/ 0 954 43 50 9844 60 0-04 35-83 33 0.97 0. 0. 0.96-0.0 0.8 Deivaion of Maemaical Models D H Amadian, Modal Tesing Lab, IUST