A Survey on Model Reduction Methods to Reduce Degrees of Freedom of Linear Damped Vibrating Systems

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1 opdaa Aavakom Mah, pg 003 A uvey o Model Reduco Mehods o Reduce Degees o Feedom o Lea Damped Vbag ysems ABRAC hs epo descbes he deals o he model educo mehods o educe degees o eedom o he dyamc aalyss o geeal lea ad damped vbag sysems. he epo sas wh a oduco o a equao o moo o a wo-soy buldg a eample o a damped vbag sysem. he esposes o he smla equaos o moo o damped sysems ae suded hough egevalue poblems. he oce-depede mode shape, quas-sac mode shape, ad Raylegh Rz mehods ae appled o oba some appomae mode shapes coespodg o he lowes udamped equeces o he lage sysems. hs s because he lowes udamped equecy modes ae a lo moe mpoa o he hgh ad udamped equecy modes sucue egeeg. hese he lowes udamped equecy mode shapes ca he be used o asom he lage sysems o he smalle oes coag he asomed coodaes coespodg o he lowes udamped equecy modes o he lage sysems. hese model educo mehods deeme he appomaos om oly some low equecy modes hus helps educe me ad cos o compuao o he esposes.. IRODUCIO he aalyss o sucues o dyamc ecao s dcaed by he compley o he sucues, ad seveal huded o a ew housad degees o eedom may be ecessay o he accuae evaluao o he oces he comple sucues. he eed modelg ca be used o dyamc aalyss o he sysem, bu may be uecessay ad ewe degees o eedom could be eough. ha s he case because he dyamc espose o may sysems ca be epeseed by he s ew aual vbao modes, hese modes ca be deemed accuaely wh sgcaly ewe degees o eedom ha equed o sac aalyss. hus we ae eesed educg he umbe o degees o eedom as much as possble beoe poceedg wh compuao o aual equeces ad modes. Fs, we eed o sa wh he lea damped equaos o moo o he sucues o be suded. he equaos o moo wll be secod ode deeal equaos ma oms wh he mass, dampg ad sess maces as he coeces. Whou he dampg em he equaos, he espose o hese equaos ca be poduced by solvg he egevalue poblems ad usg he modal ma o decouple he sysems o modal equaos. hs appoach s vey ece o udamped sysems ad sysems wh specal ypes o dampg e.g. popooal dampg whee he dampg ma s a lea combao o he mass ma ad he sess ma. I he case o a damped sysem, he lea asomao s appled o he ogal equao o

2 opdaa Aavakom Mah, pg 003 moo o asom he mass ma o a dagoal ma. he appomae espose ca be deemed by meas o solvg he decoupled equaos o moo by eglecg he o-dagoal elemes o he modal dampg ma ad he modal sess ma. hs cocep sll woks he educed equaos o moo ae eeded. he model educo mehod s appled o he ogal equao o ge he some o he appomae lowes equecy mode shapes whch wll be omed as a aso ma. he he ogal equao s he asomed o a educed equao a he same me as he mass ma s chaged o a dagoal ma. he educed sysem obaed ca he be solved o he esposes wh less wok ha solvg he ogal sysem. he model educo mehods, wheeby he umbe o degees o eedom he sysem s educed, s appled o lage sysem o gve ase compuao o he aual equeces ad mode shapes o a sucue. hee ae may dee eave pocedues o compug he egevecos o so called mode shapes coespodg o a chose se o egevalues o a symmec ma, e.g. subspace eao, he Laczos mehod ad he ace mmzao mehod. A classcal mehod o compug appomae egevecos s he subspace eao, a mehod o hadle moe ha oe veco. Foce-depede mode shape veco mehod [] has log bee used o appomae he dyamc espose o sucues ad as a model educo echque o educe he sze o lage-scale sysems. Povded ha he appomae mode shapes spa he same coguao space, hey ae a aacve aleave o he coveoal egeveco mehod omal mode mehod, sce he mode shape vecos o he educed sysems ca be compued wh sgcaly less compuaoal eo. Foce-depede mode shape vecos ae a pacula goup vecos whch he omao abou he loadg o he sucue s used o geeae vecos. he coveoal mode shape veco mehod employs sac ecuece pocedues o geeae he appomae modes shapes, whch sases he sac compleeess codo. he oce-depede mode shape veco mehod has smla advaages ad dsadvaages as he mode acceleao mehod. Cosequely, hs mehod s bes sued o elavely low-equecy poblems. Fo hghe-equecy, o baded equecy poblems, lage ses o mode shape vecos ae eeded o spa he coguao space assocaed wh he hgh equeces. hs wll decease he oce-depede mode shape veco mehod s ececy. I also esuls he loss o ohogoaly o he mode shape vecos, whch causes umecal eos solvg he educed sysem. A appopae se o mode shape vecos should sasy wo codos. Fs, he bass omed by mode shape vecos should be complee wh espec o ocg loadg paes o he poblem, a leas o a equecy o ees. he oce-depede mode shape vecos sasy hs codo. ecod, he mode shape vecos should spa all desed equecy space. omal modes always mee hs codo sce hey coss o all modes he equecy age. Howeve, he oce-depede mode shape vecos do o sasy hs codo. A ew oce-depede mode shape vecos mehod called quas-sac mode shape veco mehod s oduced [] o sasy boh codos above. hs mehod

3 opdaa Aavakom Mah, pg 003 employs a quas-sac ecuece pocedue based upo a ew modal supeposo echque. Compag wh he coveoal oce-depede mode shape vecos mehod, hs mehod s moe ece ad moe accuae ems o eos. he subspace eao mehod s he oduced o ge some mode shapes smulaeously. o uhe mpove he subspace eao appomao, oe ca use he Raylegh-Rz pocedue [7]. hs s a well kow mehod o cease he accuacy he esmaes, by lle ea wok. he Rz values ad Rz vecos obaed wh hs pocedue ae he opmal appomaos kowg some o he appomae mode shapes aloe. Devg Equaos o Moo o a wo-soy Buldg Model he sucue dyamcs poblem ca be omulaed o sucues dscezed as sysems wh a e umbe o degees o eedom. he equaos o moo ae cosdeed as a mul-degee-o-eedom sysem; e.g. a wo soey ame ha s subeced o eeal oces o eahquake. he equaos o moo ae appled o he buldgs ad he espose wll be aalyzed. A wo-soy buldg s a vey smple mul degee o eedom sysem. he buldg ame s subeced o eeal oces p ad p he gue. he beams ad loos ae assumed o be gd. he aal deomao o he beams ad colums ae egleced. he mass s dealzed as coceaes a he loo level because mos o he buldg mass s a he loo level. cly speakg he mass s acually dsbued houghou he buldg. he umbe o degees o eedom s he umbe o depede coodaes ha s equed o dee he posos o all he masses. he model o a wo-soy buldg s show Fgue wh mass a each loo, he laeal dsplaceme ad o he loos. Wh ewo s Law o Moo, gves he ollowg equao o each mass: m & + + p D whee m ae he masses o he h loo; p, D, ae he eeal oces, he elasc essg oces ad he dampg oces especvely. 3

4 opdaa Aavakom Mah, pg Fgue : a wo-soy ame whou oces ad b wo-soy ame wh oces hee ae wo equaos o moo o he Fgue whe ad, hey ca be we he ma om as: p p m m D D & & o D p M + + & 3 whee &, 0 0 m m M, D D D,, p p p Assumg he elasc essg oce s lea; s elaed o he loo dsplaceme. k 4 whee k s he laeal sess depedg o he soy hegh ad a colum wh modulus ad secod mome o ea. Wh he sess deed ad he ewo s laws o moo appled, he elasc essg oces ad ae elaed o he loo dsplacemes as ollows: p p m m m m c c a b

5 opdaa Aavakom Mah, pg k k k + 5 I he ma om: + k k k k k 6 o K 7 he dampg oces D ad D ae elaed o he loo veloces &ad &. c D & 8 whee c s he dampg coece. I he smla mae as Equao 5, we have + c c c c c D D D & & 9 o D C & 0 he Equaos 7 ad 0 ae subsued o Equao 3 o oba he ollowg equao ha s he geeal equao o moo o a lea vbaoy sysem p K C M + + & & whee he al codos ae 0 0 0, 0 & & ; M s he mass ma, C s he dampg ma, ad K s he sess ma. All o hem ae o ode. he dsplaceme ad he eeal ecao p ae -dmesoal vecos. I he case o passve sysems, whch oly have passve elemes, M, C, ad K maces ae all eal, symmec ad posve dee.. HE EIGEVALUE PROBLEM he equaos o moo he om o Equao ae lea secod ode deeal equaos. I ode o solve hese sysems, s ease o solve he smla

6 opdaa Aavakom Mah, pg 003 equaos bu whou he dampg em o he aual equeces ad aual modes s. Ad he apply hose esuls o he ogal sysems o ge he appomae esposes. I he absece o vscous dampg ad eeal oce, he sysem Equao would be M& + K 0 A mpoa case he sudy o vbaos s ha whch all coodaes have he same moo me. o eame he possbly ha such moos es, we cosde a soluo o Equao he epoeal om e s φ 3 we Ioducg Equao 3 o Equao ad dvdg hough by s e, we ca K φ λmφ whee λ s 4 I ode o d he vbao popees; such as, he udamped aual equeces ad modes o he sysem, he soluo o he ma egevalue poblem 4 s equed. Le φ be he egevecos o he geealzed symmec egevalue poblem: o K φ λ Mφ o... 5 Kφ ω Mφ 6 whee λ ω ae he egevalues assocaed wh he egevecos φ. Physcally he veco φ s he h aual mode whle ω s he h aual equecy o vbao. Le Φ deoe he modal ma assocaed wh sysem. he colums o Φ ae he egevecos φ o he ogal sysem Equao. [ φ φ ] Φ... 7 φ he egevalues λ ae he oos o he chaacesc equao ω p λ de[ K λ M] 0 8 whee p λ s a polyomal o ode, he umbe o degees o eedom o he sysem. oe ha hs mehod s o paccal o he lage sysems lage umbe o degees o eedom because eques much wok o evaluae he coeces ad he umecal oud-o eos mgh be sgca. 6

7 he ohogoaly o he modes opdaa Aavakom Mah, pg 003 he aual modes coespodg o dee aual equeces ca be show o sasy he ollowg ohogoaly codos. Whe λ λ φ φ 0 9 φ φ 0 0 Fuhemoe, ca be show ha Φ s eal ad osgula. he modal ma Φ s usually omalzed accodg o Φ MΦ I whee Φ s he aspose o Φ ad I s he dey ma. Moeove, Φ KΦ Λ dag[ λ,..., λ ] dag[ ω,..., ] ω s he ma amed a specal ma whose dagoal ees ae he squaes o he sysem s aual equeces;.e. ω. Fom Equaos 5 ad 6, we have KΦ MΦΛ 3 By applyg a lea modal asomao o Equao wh Φq, 4 Equao s omalzed o I q& + Dq& + Λq 5 whee Φ p ad q s he veco o omal coodaes o -dmesoal modal dsplaceme veco. D Φ CΦ s called he modal dampg ma ad s symmec. Whe D s dagoal, Equao 5 s a se o decoupled, secod-ode deeal equaos, whch ca be solved depedely o he ohes. hus, we have q& + d q& + ω q 6 whee d s he h dagoal eleme o ma D, s he h compoe o he moded ocg veco. 7

8 opdaa Aavakom Mah, pg 003 Howeve, he modal dampg ma D s usually o dagoal. Equao 6 s he coupled by he o-zeo o-dagoal elemes D. A commo mehod o solve hs damped sysem s o goe all o he o-dagoal elemes o he modal dampg ma. hs mehod s called he decouplg appomao. By applyg he mehod, he sysem s modal dampg ma s dagoalzed o ucouple he sysem s equao o moo ad he Equao 6 s obaed. Raylegh s Quoe We have show ha he sysem possesses eal ad posve egevalues λ ad he assocaed wh he eal egevecos φ sasyg he egevalue poblem, Equao 5, K φ λ Mφ. he egevalues ae aaged ascedg ode o magude, so ha hey sasy he equales λ λ... λ 7 Equao s pemulpled by φ, φ Kφ λφ Mφ 8 he posve deeess o M guaaees ha φ Mφ cao be zeo. heeoe, λ ω φ Kφ φ Mφ 9 he quoe s called Raylegh s quoe ad s a uco o egevecos φ. he behavo o Raylegh s quoe as φ ages ove he ee -dmesoal space s o ees. Accodg o he epaso heoem, ay abay -veco φ ca be epessed as a lea combao o he sysem egevecos φ φ,..., φ,. φ c φ + φ φ c c 30 c φ 3 Φc 3 whee Φ s he ma o egevecos o he sysem ad [ c c... ] veco o coodaes o φ wh espec o he bass φ,..., ad 3, we oba φ Kφ Φc KΦc λ φ Mφ Φc MΦc c s he - c, φ φ. Fom Equaos 9 8

9 opdaa Aavakom Mah, pg 003 λc c Λc 33 c c c As he abay veco φ moves ove he -dmesoal Eucldea space, wll eveually ee a small eghbohood o a gve egeveco, say φ. he coeces c epese he coodaes o φ wh espec o he bases φ, φ,..., φ. Because φ sde he small eghbohood o φ, ollows ha o c >> c 34 c c ε 35 whee ε ae small umbes. Iseg Equao 35 o Equao 33, usg bomal appomao, ad gog hghe-odes ems, we oba λ λ + λ λ ε 36 Bu Equao 35 mples ha φ des om φ by a small quay. Equao 36 saes ha he coespodg Raylegh s quoe λ des om λ by a small quay oo. he esul says ha Raylegh s quoe coespodg o a lea vbaoy sysem has saoay values he eghbohood o he egevecos, whee he saoay values ae equal o he assocaed egevalues. I we le Equao 36, we wll have λ λ + λ λ ε λ 37 whee we ecogze ha he sees s always posve. Iequaly 37 saes ha Raylegh s quoe s eve lowe ha he lowes egevalue λ. I s geeally hghe ha λ, ecep whe φ φ, ha case Raylegh s quoe has a mmum value a φ φ. he equaly above also gves a uppe boud o he lowes egevalue λ. Followg he smla agume, o, Equao 36 yelds λ λ λ λ ε λ 38 9

10 opdaa Aavakom Mah, pg 003 o Raylegh s quoe s eve hghe ha he hghes egevalue λ. I s geeally lowe ha λ, ecep whe φ φ, ha case Raylegh s quoe has a mmum value a φ φ. 3. MODEL REDUCIO MEHOD hese mehods ae mos geeal echques compug some appomaos o he lowe aual equeces ad modes o he udamped lage sysems M & + K p by solvg he symmec egevalue poblem 5. eleco o Mode hape Vecos he mode shape vecos mehod peomace depeds o how well he lea combaos o he mode shape vecos ψ appomae he aual modes o vbao. hee ae may appoaches o selec he appomae mode shape vecos; such as, physcal guess o he shapes o he aual modes ad a sep-by-sep compuaoal pocedue.. Foce-Depede Mode hape Vecos Mode shape vecos ae deemed o aalyss o a sysem wh eeal oces. p sp 39 he spaal dsbuo o oces s does o deped o me; howeve, he me depedece o he oces s gve by he scala uco p. he s mode shape veco ψ wll be he sac dsplaceme due o he appled oces s, whch s Ky s 40 he dsplaceme veco y s omalzed o be mass ohoomal: y ψ 4 y My he secod mode shape veco ψ s compued om he sac dsplaceme veco y due o he appled oces gve by he ea oce dsbuo assocaed wh he s mode shape veco ψ. he veco y s obaed om Ky Mψ 4 0

11 opdaa Aavakom Mah, pg 003 he secod mode shape veco ψ s he omalzed veco o ˆψ whee ˆψ s ceaed o be ohogoal o, ad hece lealy depede o ψ by Gam-chmd ohogoalzao pocedue. he veco ˆψ s gve by ψˆ y a ψ 43 ad ψ My a 44 Fally he veco ˆψ s omalzed so ha s mass ohoomal o oba ψ. ψ ψˆ 45 ˆ ˆ ψ Mψ he pocedue s geealzed so ha he h mode shape veco ψ s compued om he sac dsplacemes y due o appled oces gve by he ea oce dsbuo assocaed wh he - h mode shape veco ψ. he veco y s deemed om Ky Mψ 46 he veco ψˆ s p ψ ˆ y a ψ 47 p p Ad he mode shape veco ψ s ψˆ ψ 48 ˆ ˆ ψ Mψ he sees o mode shape vecos ψ, ψ,..., ψ ae muually mass ohoomal ad hece hey ae lea depede o each ohe. hese popees mee he equeme o he mode shape veco mehod.. Moded Foce-Depede Mode hape Veco Mehod

12 opdaa Aavakom Mah, pg 003 Eve hough, he Gam-chmd ohogoalzao pocedue heoecally gves a ew veco ha s mass ohogoal o he pevous vecos, he acual ew veco ca sue loss o ohogoaly because o he umecal oud-o eos he compue. o oba a moe sable mode shape veco geeao algohm, a addoal se o empoay vecos ad ohogoalzao pocedue ae oduced Chopa []. he pocedue s moded ad summazed as ollows:. Deeme he s mode shape veco ψ a. Deeme y by solvg: Ky s 49 y b. omalze y : ψ 50 y My. Deeme addoal mode shape vecos ψ,..., ψ,..., ψ a. Deeme y by solvg: Ky Mψ whee... 5 b. Ohogoalze y wh espec o ψ,..., ψ by epeag he ollowg seps o p,,..., a p ψ My 5 p p ψˆ y a ψ 53 y c. omalze ψˆ : p ψˆ 54 ψˆ ψ 55 ˆ ˆ ψ Mψ he oce-depede mode shape vecos mehods above use sac ecuece pocedues o geeae mode shape vecos. heeoe, hese vecos ae bes sued oly o some low-equecy poblems. Fo hgh-equecy o baded equecy poblems, he oce-depede mode shape vecos mehods eed lage ses o mode shape vecos o spa he hgh-equecy coguao space ad hus cease compuaoal cos. 3. Quas-ac Mode hape Veco Mehod he quas-sac mode shape veco mehod [] eeds he pevous ocedepede mode shape vecos mehods by employg a quas-sac ecuece pocedue, based o he cocep o quas-sac compleeess o he mode shape vecos bass. he basc dea s o le he mode shape vecos spa he coguao space a desed equeces ad ecely possess all dyamc deomaos o hose equeces. he s quas-sac mode shape veco ψ s chose as a quas-sac mode coespodg o he loadg pae s by solvg he ollowg equao: K ωc M y s 56

13 opdaa Aavakom Mah, pg 003 y K ωc M s 57 whee ω c s he ceeg equecy [3], whch s usually chose a he mdpo o he equecy age. omalzao o y gves he s quas-sac mode shape veco y ψ 58 y My Fo, he quas-sac ecuece pocedue wll gve addoal mode shape veco ψ,...,ψ a. Deeme y by solvg: K ω cm y Mψ 59 b. Ohogoalze y ψ ˆ y ψ My ψ 60 ψˆ c. omalze ψˆ : ψ 6 ˆ ˆ ψ Mψ Physcally, ψ epeses a omalzed equecy espose deomao mode o he udamped sysem ude he loadg pae s a he equecy ω c. By usg a quassac soluo, he dyamc eec o he loadg o he ea em egleced he sac soluo o Equaos 40 ad 49, s cluded. 4. Mode hape Veco emao Pocedues I ode o deeme how may mode shape vecos ae eeded o a poblem, a pacpao aco, ρ, was oduced by Wlso e al. [6] o measue he sgcace o oe pacula mode shape veco, ψ, he espose ρ ψ s 6 he pacpao aco ρ s compued o each mode shape veco, ad s used o emae he veco geeao pocess. he aco ρ Equao 6 does o clude he dyamc eecs,.e. s a sac measue; heeoe, s oly suable o lowequecy poblems. Gu e al. poposed a ew measue o he pacpao aco o he sysem wh hamoc eeal oces: 3

14 opdaa Aavakom Mah, pg 003 whee ψ ρ 63 [ ψ Mψ M ] K ω M s 64 s he equecy espose a a speced equecy ω due o he ocg pae s. Whe he pacpao aco s oe, he mode shape veco peecly maches he equecy espose shape. Whe he aco s zeo, he mode shape veco s ohogoal o he equecy espose shape ad does o aec he educed model espose a all. he equecy ω s chose o be he doma equecy o he ocg pae s. he pacpao aco Equao 63 cludes he dyamc eecs o he espose; hece s a moe ealsc aco ha he oe Equao 6. he pacpao aco s used o emae he ecuece pocedues whe he pacpao veco dops below a speced oleace value. he sequece o quas-sac ecuece pocedues also sops whe he umbe o he mode shape vecos deemed s oo lage. 5. Kylov ubspace he Kylov subspace s a ype o subspace o compug egepas o B. hs kd o subspace s deemed by a ozeo veco k. Kylov maces ae m m K k k,bk,...,b m m k ad Kylov subspace Κ k spa[ K k ]. I pcple all Kylov maces ae saved, hey ca be used he Raylegh Rz appomaos whee he Kylov subspace has bee compued. We ca see ha he model educo mehods,.e. oce-depede mode shape, he moded oce-depede mode shape ad he quas-sac mode shape mehods, ae based o Kylov subspace eao mehod. hs s because oms he Kylov subspace o each mode shape eaed. I ode o show ha s he case, he egevalue poblem wll be moded o a sadad om. Evey eal symmec posve dee ma A ca always decomposed o A LL 65 whee L s a uque osgula agula ma wh posve dagoal elemes. Equao 65 s kow as he Cholesky Decomposo. We ecall ha he mass ma s eal symmec ad posve dee. o we ca we he mass ma as ollows: M LL 66 4

15 Fo ou case whee he mass ma s dagoal, ha opdaa Aavakom Mah, pg 003 Hece; L M 67 M M M 68 he gve egevalue poblem K φ λ Mφ 69 ca be ewe as φ KM M λ M M φ 70 M KM M φ λ M φ 7 he equao above ca be we he sadad om o he egevalue poblem whee ad Av λ v 7 A M KM 73 M φ v 74 he s pocedue oce-depede appomae mode shapes geeaes he appomae mode shape vecos v om he veco sequece,k M,K M,... whch s geeaed he vese eao mehod. I we mulply M o y,,, we wll have heeoe; y M y 75 y M y 76 o o he sac dsplaceme, we ca sa wh solvg o y K y s 77 5

16 he dsplaceme veco y s omalzed o be ohoomal: opdaa Aavakom Mah, pg 003 y v 78 y y he secod mode shape veco v s compued om veco y obaed om y M K M v 79 he secod mode shape veco v s he omalzed veco o ˆv vˆ y a v 80 ad a 8 v My Fally he veco ˆv s omalzed v vˆ 8 ˆ vˆ v he veco y s deemed om M K M v y 83 he veco vˆ s v ˆ y a pv p 84 p Ad veco v s vˆ v 85 ˆ vˆ v m m I s smla o om a Kylov subspace K k k,bk,...,b k show wh he ma B as 6

17 B opdaa Aavakom Mah, pg 003 A M K M 86 Fo he quas-sac veco mehod, he cocep s smla o he above equaos show o oce-depede mode shape mehod bu B s o be chaged o B M K ωc M M ubspace Ieao Mehod he subspace eao s aohe way o d he appomae egevecos o he sysem. I s dee ha he oce-depede mode shape veco, moded ocedepede mode shape veco ad quas-sac mode shape veco mehods whee oe appomae mode shape s ceaed a a me bu he subspace eao mehod caes ou eaos o a gve umbe o modes smulaeously [6, pg. 38]. Wokg wh seveal colums a oce wll mpove he lea covegece o successve subspaces. Whe seveal low egevalues ae cluseed, hs mehod wll covege o he egevecos vey as povdg ha he al guess vecos have some decos he desed egevecos. We popose o make coeco bewee he egevalue poblem 5 ems o wo eal symmec maces M ad K. ad he sadad egevalue poblem Au λ u ems o a sgle eal symmec ma. By ollowg he Equaos 68 hough 74, we have whee ad sasy Av λ v 7 A M KM 73 M φ v 74 he muually ohogoal egevecos ae assumed o be omalzed so as o v v δ. ubspace eao s deed by he elao. V ˆ whee p,, 88 p A Vp whee V p s a ma o muually ohoomal vecos v elaed o he ma V ˆ p o depede vecos vˆ by 7

18 p p p opdaa Aavakom Mah, pg 003 V Vˆ U whee p,, 89 whee U p s a uppe agula ma. Equao 89 epesses he ohoomalzao o depede vecos called he QR acozao. he ohoomalzao ca be caed by meas o he Gam-chmd, ad hs pocess mus be doe a evey eao sep. heoem : QR Facozao Ay m ma B ca be we as BQR whee Q s a m ma sasyg Q Q I, R s a uppe agula ma wh oegave dagoal elemes, akb ad boh Q ad R ae uque. he QR acozao s he ma omulao o he Gam-chmd pocedue o ohoomalzg he colums o B he ode b, b,..., b. he se q, q,... q s oe ohoomal bass o he subspace spaed by b, b,..., b Povdg ha V 0 s o ohogoal o he desed egevecos coveges wh he esul V, he eao pocess lm V V p p 90 lm U Λ p p 9 whee V [ v v... v ] ad Λ [ λ λ... λ ] dag egevalues. s he ma o he lowes ohoomal egevecos s he dagoal ma o he lowes ohoomal he covegece ae o he eao depeds o he gap o he closes egevalue o amog he waed oes. hs mehod s heeoe well sued o cluseed egevalues, whee he Powe mehod has a bad covegece ae due o he small gap, ad may somemes pay o compue a couple o ea vecos o oba a as covegece. 7. he Raylegh-Rz Pocedue he Raylegh-Rz pocedue ad he heoy assocaed wh hs mehod ae cosdeed. he Raylegh-Rz pocedue wll mpove he accuacy o he appomae mode shape vecos obaed om he ubspace Ieao mehod pevously show. 8

19 opdaa Aavakom Mah, pg 003 Assume ha we have a ma V wh ohoomal colums, whch ae appomaos o egevecos o A. he uhe mpoveme o he esmaes ca be obaed by usg he Raylegh-Rz pocedue. Gve: V, V V I a. Fom H : V AV 9 b. Compue he egevalues ad egevecos Hg τg ad he Rz vecos y Vg o,,, c. Compue he esduals s Ay τy o,,, he ma H s hee a ma, so s small compaed o A oly a ew egevecos ae o be compued. I s also symmec ad posve dee, so ep b ca be cheaply compued o he symmec egevalue poblem. Hece, he ea wok equed hs pocedue s maly he wok omg H. he egevalues τ o H ad he vecos y ae used as ew appomaos o he egevalues ad egevecos o A, especvely. I [7 ad 8] Pale demosaes hee ways whch hese Rz values ad Rz vecos ae opmal:. he egevalues o A ca be deed by Coua-Fsche Mma heoem [0, pg. 4] whee A λ A m ma 0 ad,,, 93 F C F F s a dmesoal subspace o C. he Rz values sasy: g Ag λ A m ma g 0 ad,,, 94 G V g G g g whee V spav ad G s a dmesoal subspace o aual deo o he bes appomao o λ A he subspace. Dee he esdual ma V. hs s a V. RΒ ΑV VΒ. 95 he he ma B H. H : V AV mmzes hs esdual,.e. RH < RB 9

20 3. he Rz pas ae he egepas o A's poeco oo spas he closes subspace V o spaa. opdaa Aavakom Mah, pg 003 V,.e. he ma whch he mmum value o he om o he esdual ma RB ca be see as a measue o how a V s om beg a vaa subspace o A. I V s a vaa subspace o A he he poduc A v, whee v s a colum o V, equals a lea combao o he colums o V,.e. A v Vb. Hece, hee s a ma B such ha RB s zeo. I V s ohogoal he B V AV : H ad H s he esco o A o V. I V s o a vaa subspace o A, he hee s o ma B such ha RB 0,bu he ma H V AV sll mmzes RB. w s ay ohoomal bass o V ad D dag d, d,..., d s ay dagoal ma, he AW WD s mmzed whe ad oly whe w y ad d τ o,,,. hs ollows sce, w y ad τ o,,,, he d Moeove, W [ w w... ] 96 AΨ ΨΤ AVG VGΤ AV VGΤG 97 AV VH 98 RH 99 Hee we have used he oaos Y [ y y... ] G dag g, g,... g. y, Τ dag τ, τ,... τ ad Bu whe W Y o Τ, we ca sll epess W he bass V, W V, I sce hey spa he same subspace. We oba, AW WD AV VD 00 AV VD 0 > RH 0 0

21 hs meas ha whe egepas o A. opdaa Aavakom Mah, pg 003 V s a vaa subspace o A he Rz pas ae he ue Whe he egevalues o ma A ae well sepaaed hee ae easly obaed bouds o he Rz values oce he esduals s have bee compued. hese bouds ae gve he heoem. heoem : esdual eo boud Le V C be a ma sasyg V V I ad le τ, y o,,, be he coespodg se o Rz pas o A wh esduals s Ay τ y τ s, τ + s coas a egevalue o A.. he he eval [ ] hs heoem, ogehe wh he ollowg heoem, ca also be oud [7 ad 8, Chap. ]. Poo: Le λˆ be he closes egevalue o A o τ. I λˆ τ he esul s mmedae. I λˆ τ he he ma A τ I s o-sgula. Usg y A τi A τi y gves y A τ I A τ I y 03 m λ A τ s 04 Hece, λˆ sases λˆ τ 05 s whch poves he heoem. I all he evals coespodg o he Rz values ae dso we kow ha sde each o hese evals hee s a egevalue o A. hus, we have appomae egevalues o kow accuacy. Howeve, some o he evals ovelap hee may be wo Rz values appomag he same egevalue. A addoal boud, o he Rz values ovelappg evals as well, s gve heoem below. heoem 3: Le V C be a ma sasyg V V I ad le τ, y o,,, be he coespodg se o Rz pas o A wh esduals s Ay τy. he hee ae egevalues o A, λ,,,, such ha

22 opdaa Aavakom Mah, pg 003 τ λ 06 whee Poo: [ s s... s ] AY YΤ 07 I s always possble o d a uay ma V ~ ~, so he ma P [ V V] squae ad P P I. he mulplyg ma A om le ad gh by especvely, gves s P ad P, V AV P AP ~ V AV ~ V AV H ~ ~ : V AV F F Q 08 Le RRHA V - V H. he P R P AV P VH 09 P APP V P VH 0 H F I I H F Q F so R P R F 3 pl he ma P AP o H 0 0 F ~ ~ P AP + : Q + F 0 Q 4 F 0 he by he Weyl Mooocy heoem [7, pg. 9] he egevalues o P AP sasy ~ ~ λ A λ P AP λ Q + λ F 5

23 opdaa Aavakom Mah, pg 003 ow, sce he egevalues o Q ~ ae he uo o he egevalues o H ad Q, each Rz value τ equals a egevalue o Q ~. Hece, hee ae dces such ha ~ λ Q o,,, 6 τ he secod em Equao 5 s obaed by compug ~ F 0 F F 0 0 F F F F FF 7 ce F F ad FF have he same egevalues we ge ~ λ F λ F F F R 8 Wh R, Equaos 5 ad 6 whe we have λ A τ 9 mlaly, ~ ~ λ Q λ P AP + λ F 0 ad, ecallg he egevalue dsbuo o he ma F ~. We kow ha Hece, by he same agumes as above we have ~ λ F. τ λ A ad he equaly 06 s esablshed. Whe cosdeg he accuacy o he Rz vecos he poblem s o as smple as s o he Rz values. he easo s ha egevecos assocaed wh mulple egevalues ae o uquely deemed. Ay lea combao o egevecos coespodg o he same egevalue s a egeveco. mlaly, Rz vecos coespodg o egevalues whch ae close ed o be vey sesve ad gve bad esmaes o he coespodg egevecos, bu he subspace hese vecos spa may be a good appomae o he subspace assocaed wh he cluse. A boud o how well a Rz veco appomae a egeveco o A does es, bu s oly useul whe he assocaed egevalues ae well sepaaed. 3

24 opdaa Aavakom Mah, pg 003 heoem 4: esdual bouds Le V C be a ma sasyg V V I ad le τ, y o,,, be he coespodg se o Rz pas o A wh esduals s Ay τy. he he egevalues o A, λ,,,, sasy whee τ λ Gap [ s s... s ] AY YΤ 3 ad Gap s he gap bewee he mamum o he egevalues o A ad he mmum o hose o F. Poo: A s smla o P AP Equao 08 V AV P AP ~ V AV ~ V AV H ~ ~ : V AV F F Q 4 ake he deema o λ I A, oe has λi H F de λi A de 5 F λi Q [ λi H F λi Q F] de λi Q de 6 he egevalues o A ea he egevalue o H ae he egevalues o H + F λi QF, heeoe F τ λ 7 Gap 8 Gap ome echques e.g. pevously descbed subspace eao poduce he appomae egevecos ha ae o muually ohogoal whe he egevalues ae close ogehe. hs also happes whe s lage. Le V be a ma wh 4

25 opdaa Aavakom Mah, pg 003 ohoomal colums, whch ae appomaos o egevecos o A, pobably he ma o Rz vecos. Whe he smalles sgula value o V s.e. σm V, he σmaz V oo. hus V s ohoomal. heoem 3 wll be weak σm V deceases ad ges close o 0. heoem 3 would be τ λ σ V whe V s o ohoomal. uppose we have a ma V, a se o Rz pas o A.e. τ, y ad he esduals s Ay τ y. Fo each,,, hee s a egevalue λ o A such ha λ τ s. Bu hee may o be dsc λ s o each. heeoe we eed he smalles η such ha a dsc λ may be oud each eval [ τ η τ + η] η σ V. m Whe V s o que ohoomal, heoem 3 has o be moded as ollows. m, ad heoem 5: Le λ λ, λ,..., λ hold he egevalues o A ad τ τ, τ,..., τ hold he egevalues o H. Ad le V be a ma wh ull colum ak. he hee ae a leas locaos λ, λ,... λ,..., λ λ such ha, o,,, whee τ λ σ V 9 m [ s s... s ] AY YΤ 30 Please see [8, pg. 58] o he poo. 8. Reduced Equaos o Moo Fom Equao, he equao o moo o a sysem wh degees o eedom subeced o a oce p sp s M & + C& + K sp 3 I he mode shape veco mehods, he dsplacemes ae epessed as a lea combao o he seveal shape vecos ψ, ha s 5

26 opdaa Aavakom Mah, pg 003 z ψ Ψz 3 whee z ae geealzed coodaes, ψ ae appomae mode shape vecos compued om he pevously eplaed model educo mehods ad Ψ s he ma whose colums ae ψ. hese appomae mode shape vecos ae lealy depede vecos sasyg he geomec bouday codos. Howeve he subspace eao mehod o he Raylegh Rz mehods ae used, oduce he aso ma ad Ψ M V 33 Ψz 34 Hece, subsug he appomae mode shape vecos Equaos 3 ad 34 o he equao o moo wll esul MΨ & z + CΨz& + KΨz sp 35 Pemulply by whee Ψ gves Ψ MΨz & + Ψ CΨz& + Ψ KΨz Ψ sp 36 ~ ~ ~ I & z + Cz& + Kz Lp 37 ~ C Ψ CΨ ~ K Ψ KΨ ~ L Ψ s 38 Equao 37 s a sysem o deeal equaos he geealzed coodaes z. he coodae asomao o equao 34 ca educe he ogal se o equaos 33 o he odal dsplaceme o a smalle se o equaos he geealzed coodaes z. I s quesoable ha oly a ew mode shape vecos s much smalle ha would be good eough o epese he dsplacemes o he sysem. Also he seleco o he mode shape vecos s ccal. Wh Equao 37, we ca d he egevecos o he educed model z wh less wok because he degees o eedom o hs equao o moo wee educed om o ad s small. 6

27 opdaa Aavakom Mah, pg 003 Appomao I he aalyss o Equao 37, he appomae aual equeces ae om he squae oo o he appomae egevalues obaed om he model educo mehods. ω ~ λ 39 Moeove, hey cao be smalle ha he acual equeces accodg o he Raylegh s saoay codo ω ω ~, ω ~ ω,, ω ω ~ 40 Oe ca see ha he dampg ad he sess maces Equao 37 ae o dagoal. I he o-dagoal elemes o he dampg ma C ~ ad sess ma K ~ ae small compag o he dagoal elemes, hee s a commo mehod o solve such a sysem ha s o goe all o he o-dagoal elemes ad keep oly he dagoal elemes. he oe ca solve he ucoupled deeal equaos. ~ ~ ~ & z + c z& + ω z L p,,, 4 4. COCLUIO hs epo descbes a sucual dyamc aalyss by usg appomae mode shape vecos he lage sysems o oba some mode shapes o he small sysems ad solvg o he esposes based o hose small sysems. A small sysem always eeds less compuao cos ad me o compue he aual equeces, mode shapes ad esposes. Mos o he me a small sysem obaed seves eally well povdg a good aalyss o a sucue. hee ae oce-depede mode shape, moded ocedepede mode shape, quas-sac mode shape, subspace eao ad Raylegh Rz mehods descbed hs epo. he oce-depede mode shape veco mehod compues he s mode shape veco based o he sac espose mode, bu he ea em s egleced. he he ea s appled as a sac load he e sep o geeae a ew mode shape veco. hs pocess s epeaed ll hee ae eough mode shape vecos whch sasy sac compleeess codo. he algohm loses ohogoaly due o he umecal oud-o eos whe he umbe o he mode shape vecos becomes lage. he oce-depede mode shape veco mehod s he moded wh a addoal se o empoay vecos ad a ew Gam-chmd ohogoalzao pocedue o ge moe sable mode shape vecos. Howeve, whe he mode shape veco se om he ocedepede mode shape veco mehod becomes vey lage, he mode shape vecos become ealy lealy depede, causg loss o accuacy. I ode o oba moe accuae esuls, he quas-sac mode shape veco mehod s peeed sce he 7

28 opdaa Aavakom Mah, pg 003 dyamc eec o he loadg o he ea em s cluded he quas-sac soluo. Howeve, wh subspace eao mehod, specc umbe o mode shapes equed ae appomaed oce evey eao. Raylegh Rz show seco 7 eques some ea wok; bu he Rz vecos ae moe accuae ha he appomae mode shapes om he ohe mehods eplaed hs epo. eveheless; hee s aohe good opo called he Laczos algohm bu s o coveed hs epo. Laczos algohm mplemes he Raylegh Rz pocedue o he sequece o Kylov subspace whee he Raylegh Rz pocedue s smpled. 5. REFERECE. A. K. Chopa, Dyamcs o ucues, Pece Hall, ew Jecy, J. Gu, Z. D. Ma, G. M. Hulbe, A ew Load-Depede Rz Veco Mehod o ucual Dyamcs Aalyses: Quas-sac Rz Vecos, Fe Elemes Aalyss ad Desg, 36, P. Lege, E. L. Wlso, R. W. Clough, he Use o Load Depede Vecos Fo Dyamc ad Eahquake Aalyses, Eahquake Egeeg Reseach Cee Repo, Uvesy o Caloa, Bekeley, UCB/EERC-86/04, P. Lege, Load-depede ubspace Reduco Mehods o ucual Dyamc Compuaos, Compues ad ucues, 9, Z. D. Ma, I. Hagwaa, Impoved Mode-supeposo echque o Modal Fequecy Respose Aalyss o Coupled Acousc-sucual sysems, Ameca Isue o Aeoaucs ad Asoaucs, 9, L. Meovch, Pcples ad echques o Vbaos, Pece-Hall, Uppe addle Rve, ew Jecy, B.. Pale, he ymmec Egevalue Poblem, Pece-Hall, Eglewood Cls, ew Jecy, B.. Pale, he ymmec Egevalue Poblem, d ed., ocey o Idusal ad Appled Mahemacs, Phladelpha, E. L. Wlso, M. W. Yua, J. M. Dckes, Dyamc Aalyss by Dec upeposo o Rz Vecos, Eahquake Egeeg ad ucual Dyamcs, 0, G. H. Golub ad C. F. Va Loa, Ma Compuaos, d ed., Johs Hopks Pess, Balmoe, MD.,

29 9 opdaa Aavakom Mah, pg 003