STRUCTURAL DYNAMICS. m P(t)

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1 STRUCTURAL DYNAMICS We ca cosder dyamc aalyss as a exeso of he ools of srucural aalyss o cases where era ad dampg forces ca o be egleced. We oe ha he presece of forces ha vary wh me does ecessarly mply ha oe s dealg wh a dyamc problem. For example, f we cosder a massless sprg subjeced o a force P(), as show he skech, k P() u() he dsplaceme s clearly gve by; u () P() k As we ca see hs s o a dyamc problem bu a successo of sac problems. If we aach a mass a he ed of he sprg ad subjec he sysem o a load P() he suao oe faces s he oe depced he skech. k m P() m u ku m P() u() From eulbrum cosderaos oe ca readly see ha he force he sprg has corbuos from he appled load ad from he era he mass. The sprg elogao, herefore, s gve by; The prevous euao would ypcally be wre as; P() mu u() () k m u k u P() () Needless o say, he problem s a dyamc oe sce eral forces are volved. I hs class we exame echues o formulae he euaos of moo as well as he mehods

2 used o solve hem. The Paral Dffereal Euao Formulao for Dyamc Problems Sce he moveme of ay maeral po s a fuco of me ad he mass of a srucure s dsrbued, he formulao of srucural dyamc problems leads o paral dffereal euaos. The esseal feaures of he couous formulao ca be clarfed by cosderg he case of a rod subjeced o a arbrary dyamc load a x L. Cosderg eulbrum of he dffereal eleme ( he horzoal dreco) we ge; u(x, ) σ ( ρa dx) σ A ( σ dx)a x (3) assumg he maeral behaves learly we ca express he sress as he produc of he sra mes he modulus of elascy of he rod, E, amely; subsug e.4 o e.3 ad smplfyg oe ges; u(x, ) σ E (4) x u(x, ) E ρ u(x, ) x (5) whch eeds o be solved subjec o he boudary codos; u(,) u(l, ) E x P() A (6a) (6b) The boudary codo e.6a s smply a saeme ha he rod s held a x whle ha 6b dcaes ha he sress a he free ed mus eual he appled sress.

3 As he rod example shows, cosderao of he dsrbued aure of he mass a srucure leads o euaos of moo erms of paral dervaves. Cosderable smplfcao s realzed, however, f he problem s formulaed erms of ordary dffereal euaos. The process of passg from he paral dervaves o ordary dffereal euaos s kow as dscrezao. The cocep of degree-of-freedom (DOF) s maely assocaed wh he process of dscrezao. A degree of freedom s ay coordae used o specfy he cofgurao of a physcal sysem. As oed, dscrezao covers he dyamc problem from oe of paral dffereal euaos o a se of ordary dffereal euaos. I hese euaos he ukows are he emporal varao of he DOF. I s worh og ha he dscrezao ca be acheved by eher physcal lumpg of he masses a dscree locaos. x() x() x(3) x(4) x(5) or by usg assumed modes shapes ad reag he DOF as scalg parameers of he shapes. The dea s llusraed he skech, where we show 3 shapes ha may be used o approxmae he dsplaceme feld. φ φ φ 3 I parcular, we would wre; u() φ (7) () φ () φ 3 () 3 3

4 where he 's are he DOF. Calculag he me hsory of hese 's s he objec of he dyamc aalyss for he dscrezed problem. I geeral, he assumed modes approach approxmaes he dsplaceme feld as; where N umber of assumed shapes. N u (x, y,z, ) () φ (x, y,z) (8) Fe Eleme Mehod The FEM s a parcular form of dscrezao where he assumed shapes do o cover he complee srucure bu are resrced o small poros of (he FE). Temporal Dscrezao If me s reaed as a dscree varable he he ordary dffereal euaos ha are obaed as a resul of he spaal dscrezao become dfferece euaos whch ca be solved wh algebrac echues. Dscusso of varous umercal echues where me s reaed as a dscree varable are preseed laer secos. Types of Models: I s worh og ha he problems we face may clude: a) Models where he doma s fe. b) Models where he doma s sem-fe. 4

5 or sol c) Models where he doma s fe (a example s a me wh a exploso or he compuao of vbraos of he earh from a very deep earhuake) Sysem Classfcao I he sudy of vbraos s mpora o deerme wheher he sysem o be examed s lear or olear. The dffereal euao of moo ca be coveely wre operaor form as; Ru ( ) f ( ) () 5

6 where R s a operaor. For example, he case of a sysem wh he euao of moo mu cu ku f() () he operaor R s; R m d c d k (3) d d Say R(u ()) f() ad R(u ()) f () he a sysem s lear f R(áu () âu ()) áf() âf () Example Tes leary for he dffereal euao show y a()y f() (4) we have; y y a()y a()y f f () () (5a,b) summg es (5a) ad (5b) oe ges; y y ) a( )( y y ) f ( ) f ( ) (6) ( he f we defe 6

7 y (7) 3 y y ad we ca wre f (8) 3 f f y (9) 3 a()y3 f3 whch shows he euao s lear. Noe ha he euao sasfes leary eve hough he erm a() s a explc fuco of me. Example Tes leary for he dffereal euao show 3 y a( ) y f ( ) () followg he same approach as he prevous example we wre; ad y y a( ) y a( ) y 3 3 f ( ) f ( ) (a,b) 3 3 y y ) a( )( y y ) f ( ) f ( ) () ( akg y (3) y y3 ad 7

8 oe ges f (4) f f 3 so y ( f 3 3 a )[ y3 somehg] 3 y ( y f (5) 3 a ) ad we coclude he sysem s olear. Oher Termology Assocaed wh Classfcao Whe he coeffces of he dffereal euao deped o me he sysem s me varyg. Whe hey do o deped o me, he sysem s sad o be me vara. Tme vara sysems have he desrable characersc ha a shf he pu produces a eual shf he oupu,.e., If he R ( u( )) f ( ) R( u( τ )) f ( τ ) Whe a sysem s lear he respose ca be bul by reag he excao as he sum of compoes ad vokg he prcple of superposo. Ths possbly smplfes he reame of complex loadg cases. We should oe ha he assumpo of leary s ofe reasoable provded he amplude of he respose s suffcely small. Sources of oleary:. Maeral Noleary Oe source of oleary ha s commoly ecouered s ha due o olear sress-sra relaoshps he maerals. 8

9 . Geomerc Noleary Is ecouered whe he geomery suffers large chages durg he loadg. I hese problems he eulbrum euaos mus be wre wh referece o he deformed cofgurao. Dyamc Eulbrum The euaos of moo ca be vewed as a expresso of eergy balace or as a saeme of eulbrum. I a dyamc seg eulbrum volves o oly appled loads ad eral ressace bu also eral ad dampg forces. The era forces are readly esablshed from Newo s Law as he produc of he masses mes he correspodg acceleraos. I coras wh eral loads, dampg forces derve from a geerally udeermed eral mechasm. For aalyss, he rue dampg mechasm s replaced by a dealzed mahemacal model ha s capable of approxmag he rae of eergy dsspao observed o occur pracce. Cosder he suao where a sysem s subjeced o a cera mposed dsplaceme u(). Assume ha we ca moor he force f() whch s reured o mpose he dsplaceme hsory ad ha he ressace of he sysem from era ad from sffess s kow as a fuco of he dsplaceme ad he accelerao. The ressace ha derves from dampg ca he be compued as f d () f() - era - elasc resorg force. The work doe by he dampg force a dffereal dsplaceme (whch euals he dsspaed eergy) s gve by; dw dw dw f f f D D D ( ). du du ( ) d d ( ) ud (6a,b,c) 9

10 so he amou of eergy dsspaed bewee mes ad s; W f () ud D (7) The mos wdely used dampg model akes he dampg force as proporoal o he velocy, amely; f D ( ) cu ( ) (8) where c s kow as he dampg cosa. To exame he mplcao of he assumpo of vscosy o he dsspaed eergy we subsue e.8 o e.7 ad ge; W c u d (9) Cosder he suao where he mposed dsplaceme s harmoc,.e u( ) u ( ) As Ω AΩcosΩ (a,b) subsug es o e.9 we ge W W π / Ω ca ca Ω Ω (cosω) d π / Ω (cosω) d () akg

11 oe ges dθ Ω Θ, d () Ω W π (cosθ) ca Ω dθ Ω (3) whch gves W π ca Ω (4) We coclude, herefore, ha whe he dampg model s assumed vscous he eergy dsspaed per cycle s proporoal o he freuecy ad proporoal o he suare of he amplude of he mposed dsplaceme. Tes resuls dcae ha he eergy dsspaed per cycle acual srucures s deed closely correlaed o he suare of he amplude of mposed harmoc moo. Proporoaly bewee dsspaed eergy ad he freuecy of he dsplacemes, however, s usually o sasfed. I fac, for a wde class of srucures resuls show ha he dsspaed eergy s o sgfcaly affeced by freuecy (a leas a cera lmed freuecy rage). The vscous model, herefore, s ofe o a realsc approxmao of he rue dsspao mechasm. A model where he dsspaed eergy W s assumed o be depede of Ω ca ofe provde a closer represeao of he rue dsspao mechasm ha he vscous model. Whe dsspao s assumed o be freuecy depede he dampg s kow as maeral, srucural or hyserec. A ueso ha comes o md s - how s he dampg force f d () relaed o he respose u() f he eergy dsspao s depede of Ω? Afer some mahemacal mapulaos ca be show ha he dampg force s relaed o he respose dsplaceme as; f D() u( τ) G dτ τ (5) where G s a cosa. Nog ha he Hlber rasform of a fuco f() s defed by

12 H ( f ( ) ) π f ( τ ) dτ τ (6) we coclude ha he dampg force he hyserec model s proporoal o he Hlber rasform of he dsplaceme. I s worh og ha he Hlber rasform of a fuco s a o-causal operao meag ha he value of he rasform a a gve me depeds o oly o he full pas bu also o he full fuure of he fuco beg rasformed (oe ha he lms of he egral are from - o ). Sce physcal sysems mus be causal, he hyserec assumpo ca o be a exac represeao of he rue dampg mechasm. Neverheless, as oed prevously, he hyserc dealzao has prove as a useful approxmao. The euao of moo of a SDOF wh hyserec dampg he me doma s gve by; m u H ( u( )) k u f ( ) (7) Whch ca oly be solved by eraos because he dampg force s o-causal. I pracce hyserec dampg has bee used fudameally he freuecy doma (where he lack of causaly s reaed a much smpler fasho). Euvale Vscous Dampg Alhough hyserec dampg ofe provdes a closer approxmao ha he vscous model, he vscous assumpo s much more mahemacally covee ad s hus wdely used pracce. Whe a srucure operaes uder seady sae codos a a freuecy Ω he varao of he dampg wh freuecy s umpora ad we ca model he dampg as vscous (eve f s o) usg: (Wcycle ) expermeal (8) ð A Ù c For rase excao wh may freueces he usual approxmao s o selec he dampg

13 cosa c such ha he eergy dsspao he mahemacal model s correc a resoace,.e whe he excao freuecy Ω euals he aural freuecy of he srucure. Ths s approach s based o he premse ha he vbraos wll be domaed by he aural freuecy of he srucure. W cycle Slope of le gves he euvale c real dsspao Naural freuecy of he srucure Types of Excao Deermsc - Excao s kow as a fuco of me Radom (sochasc) excao s descrbed by s sascs. A radom process s he erm used o refer o he uderlyg mechasm ha geeraes he radom resuls ha are perceved. A sample from a radom process s called a realzao. A radom process s kow as Saoary whe he sascs are o a fuco of he wdow used o compue hem. The process s o-saoary whe he sascs deped o me. The compuao of he respose of srucures o radom excaos s a specalzed brach of srucural dyamcs kow as Radom Vbrao Aalyss. 3

14 Paramerc Excao We say a excao s paramerc whe appears as a coeffce he dffereal euao of moo. I he example llusraed he skech he vercal excao P() proves o be paramerc. h P() F() mθ h F Assumg he roao s small ad summg momes abou he base we ge M F()h P() θh m θ h K r θ m θ h (K r P() h) θ F() h Paramerc Excao Types Of Problems I Srucural Dyamcs The varous problems srucural dyamcs ca be coveely dscussed by referrg o he euao of moo operaor form, parcular; R (u()) f() (8) R ad f() are kow ad we wsh o compue u() Aalyss R ad u() are kow we wsh o oba f () Idefcao of he Excao u() ad f() are kow ad we wa R Sysem Idefcao (used for example he desg of acceleromeers) 4

15 I addo o he prevous ypes we may wsh o corol he respose. If he corol s aemped by chagg he sysem properes (whou exeral eergy) we have Passve Corol. If exeral eergy s used o affec he respose he corol s kow as acve. The basc euao acve corol s; R (u() ) f() G u() (9) where G s kow as he ga marx (he objecve s o defe a G o mmze some orm of u()). Soluo of he Dffereal Euao of Moo for a SDOF Sysem The euao of moo of a vscously damped SDOF ca be wre as; The homogeeous euao s:... M U C U KU f () (9) ad he soluo has he form: subsug e.3 o 3 oe ges;... M U CU KU (3) U s h ( ) G e (3) s ( Ms Cs K) Ge (3) herefore, excep for he rval soluo correspodg o G s ecessary ha; Ms Cs K (33) dvdg by he mass ad roducg well kow relaoshps oe ges so he roos are; K s C s (34) s M M K ω (35) M C ωζ (36) M (ωζ ) s ω s (37) s ωζ ± ( ωζ ω (38) 5

16 s ωζ ± ω ζ (39), The homogeeous par of he soluo s, herefore; U s s h() Ge G e (4) The parcular soluo U p () depeds o he fuco, geeral: U s s G e G e U ( ) (4) ( ) p where he cosas G G deped o he al codos. I parcular, a ad herefore, ad U ( ) U (4) U ( ) U (43) G G U p ( ) I marx form he euaos ca be wre as: U (44).. sg sg U p( ) U (45) s ad, solvg for G G oe ges; G G Free Vbrao ( U ) : From e.(47) oe ges; p s s G s G s s U U U. U U U p() p() U. U p() p() (46) (47) G. su s U (48) s s G s U. U (49) s s 6

17 so he soluo s; U s s ( ) Ge Ge (5) Sep Load P () Posulae he parcular soluo The he euao of moo becomes: U () N (5)... M U CU KU P (5) subsug he fuco ad s dervaves o he euao oe ges; K N P (53) herefore; where s s he sac defleco. G G are obaed from e.47 usg; P N s (54) K U p() U p () s ; A MATLAB roue ha mplemes he prevous euaos s preseed ex. %*********************** % FVC.M %*********************** % Program o compue he respose of a SDOF sysem usg complex algebra. % Eher free vbrao or he vbrao for a sep load ca be obaed. % Varables % T Perod 7

18 % z Fraco of Crcal Dampg % u Ial Dsplaceme % v Ial Velocy % ds Sac Deformao due o Sep Load % d me sep /5 of he perod o esure a smooh plo. T ; z.; u ; v ; d T/5; ds ; w (*p)/t; :d:4*t; %Tme rage from o 4T wh cremes of d % Evaluae he roos s -w*zw*sr(z^-); s -w*z-w*sr(z^-); A [s -; -s ]; A /(s-s)*a; G A*[u-ds v]'; u G()*exp(s*) G()*exp(s*) ds; % Whle u wll be real, he compuer wll calculae a magary compoe ha s very small % due o roud-off error. plo(,real(u),,mag(u)); %plos boh real ad magary pars of u grd; xlabel('me'); ylabel('u, dsplaceme'); FIGURE u,

19 The soluo for a sep load where s s llusraed Fgure. I hs case, he dampg s % ad we see ha he peak s approxmaely eual o.78. I ca be show ha f he dampg s zero, he peak would be * (See Fgure ). Therefore dampg s o ha effce s reducg he respose for a sep load. Ths s geerally rue whe he maxmum respose occurs early. FIGURE u, Revsg he Roos of E me E.39 gves he roos of he homogeeous soluo as; s, ωζ ± ω ζ sce he dampg rao s ypcally much less ha oe s covee o wre he roos as; 9

20 s, ωζ ± ω ζ (55) whch hghlghs he fac ha he roos are ypcally complex cojugaes. Desgag oe ca wre; ω D ω ζ (56) s, ωζ ± ω (57) D Whle he roos ad he cosas G adg are geerally complex, he soluo u() s, of course, always real. Ths ca be easly demosraed aalycally by roducg Euler s dey e θ cos( θ ) s( θ ) ad carryg ou he algebra. Numercal cofrmao s also foud he oupu from he program sce he magary par of he resul s ploed ad foud o be o op of he x-axs, showg ha s zero. Compuao of he Respose by Tme Doma Covoluo I he prevous seco we looked a he compuao of he respose as a drec soluo of he dffereal euao of moo. I hs seco we exame a alerave approach where he soluo s oba by covoluo. A advaage of he covoluo approach s ha oe obas a geeral expresso for he respose o ay appled load. We beg by defg he Drac dela or mpulse fuco δ() erms of s basc propery, amely δ(-a) for a (58) From he prevous defo follows ha; δ ( a) d (59) U (a) U() δ( τ a) dτ (6) A graphcal represeao of he Drac Dela fuco s depced he fgure. ε ε d d

21 Cosder a SDOF sysem subjeced o a Drac Dela fuco ad desgae he soluo by h(), amely:... M U C U KU δ () U h() (for a mpulse a ) h() h() s kow as he mpulse respose of he sysem. Now cosder a arbrary load: P τ We ca hk of he respose a me as resulg from he superposo of properly scaled ad me shfed mpulse resposes. U()

22 U P( τ ) h( τ dτ (6) ( ) ) The egral e.6 s kow as he covoluo egral (also refered as Duhamel s egral he srucural dyamcs leraure). Noe ha oce h() s kow, he respose o ay load P s gve by a mahemacal operao ad does o reure ha oe cosder how o oba a parcular soluo. I summary, for a lear sysem ha sars a res he respose s gve by; U ( ) P( τ ) h( τ ) dτ (6) If he sysem s o a res a ( ), we mus add he free vbrao due o he al codos. Freuecy Doma Aalyss I he prevous seco we foud ha he respose of a lear sysem ca be compued he me doma by meas of he covoluo egral. whch s also symbolcally wre as u() h( τ ) p( τ ) dτ u() h()*p() A dsadvaage he prevous approach s ha he umercal cos of covoluo s hgh, as we shall see he developme ha follows, by rasferrg he euao o he freuecy doma he covoluo s redered a mulplcao ad sgfca savgs he umercal effor ca resul. Before eerg o deals of he acual freuecy doma soluo, however, some prelmares o Fourer Aalyss are preseed.

23 Fourer Trasform The Fourer Trasform of a fuco f() s defed as; F( ω) f ()e ω d A( ω) e φ( ω) as ca be see, he FT s geerally a complex fuco of he real varable ω. I s llusrave he o wre as; F(ω) R(ω) Ι(ω) where, by roducg Euler s dey, oe ca see ha; ad R f ()cos( ω) d Ι f ()s( ω) d Noe ha f f() s real R(ω) R(-ω) ad I(ω) - I(-ω). For F(ω) o exs f () d < Properes of he Fourer Trasform ) Leary f() F(ω) f() F(ω) 3

24 he a f() b f() a F(ω) b F(ω) ) Tme Shfg f() F(ω) A(ω) e ( φ( ω)) f(- ) F(ω) exp(-ω ) A(ω) e ( ( φ( ω)-ω o ) ) Illusrao f() - f(- ) f() A - A A A A A 3 A A 4 A 3 A 3 4 f(- ) A

25 Proof: F(ω) f()e *w* d f * () f(- ) F * (ω) f * ()e w* d akg - herefore d d F * (ω) f * *w*( ) *w* ()e d e f * ()e *w* d F * (ω) e * w* f()e *w* d * w* F * (ω) e F(ω) The verse formula s f() π F(w)e *w* dw Proof s o preseed bu s eresg o oe ha volves usg he dey I summary: δ() π e *w* dw 5

26 F(ω) f()e *w* d f() π F(w)e *w* dw The Covoluo Theorem (lks me doma o freuecy doma) u() P( τ ) h( τ ) dτ Provded boh P() ad h() are causal he expresso ca be wre as; u() P ( τ ) h( τ ) dτ say u() u(ω) u(ω) u reversg he order of egrao we ge; w* w* ( ) e d P( ) h( ) d e d τ τ τ or u(ω) P(ô(ô)hô)e w* d d ow akg u(ω) P(ô( h( ô)e w* d dô we have ad subsug oe ges -τ d d τ u(ω) P(ô( h()e *w*( ô) d dô 6

27 u(ω) P(ô(ô *w*ô h()e *w* d dô or u(ω) P(ô(ô * w* ô h(w)dô u(w) h(w) P(ô(ô *w* ô dô u(ω) h(ω) P(ω) Numercal cos: u() P( τ ) h( τ ) dτ say hs reures Z us of cpu me I he freuecy doma he followg operaos are reured; (mulplcao) P() P(w) x us h() h(w) y us P(w) h(w) ( very small) Takg he verse F.T g us Toal cpu he freuecy doma approach xyg. Whle hs oal used o be larger ha Z, he developme of he Fas Fourer Trasform algorhm has chaged hs resul makg he freuecy doma soluo he more effce approach may cases. Propery of dffereao of Fourer Trasformao: To egrae by pars, we defe F(ω) f()e w* d f() u f () d du 7

28 herefore e * w* d dv v e w *w* ad we ge f()e *w* d -* ω f()e - ω * *w* f ()e - d ù *w* f ()e d w w f *w* ()e d or ω f w e * ( ) * d f * w* ( ) e d d We coclude ha he Fourer Trasform of f() s smply (ω)* he Fourer rasformao of d f(), amely; d F f () ω F(ω) d Ths ca be easly exeded o sae ha, d F d f () (ω) F(ω) Cosder ow he euao of moo of a SDOF sysem, m u c u ku P() akg he Fourer rasform of boh sdes oe ges; m (ω) u(ω) c(ω)u(ω) ku(ω) P(ω) 8

29 u(ω) (-ω m k cω) P(ω) ad sce P( w) u(ω) w m k cw u(ω) P(ω) h(ω) we coclude ha h(ω) w m k cw we ca esablsh, herefore, ha î*wd* *w* e m wd s(wd ) e d ( w m k) cw Fourer rasformao of Drac dela fuco F(δ()) ä()e w d We have prevously oed ha he respose o a Drac Dela fuco s he mpulse respose of he sysem, s easy o show ha he Fourer rasform of he mpulse respose euals he freuecy respose fuco. Cosder a SDOF subjeced o a Drac Dela fuco; m h() c h() k h() δ() where he oao h() has bee roduced o emphasze ha he compued respose s he mpulse respose fuco. Takg a Fourer rasform we ge; (ω) [(-ω h m k) cω] 9

30 whch s he expeced resul. Mahemacal Formulao of Freuecy Doma Aalyss Say he euaos of moo for a MDOF sysem are: [ M ]{} u []{} c u [ K]{} u { f() } ad for oaoal coveece we do o use he brackes from here o. Ths euao ca be rasformed o he freuecy doma as or M(ω) u(ω) c(ω) u(ω) K u(ω) f(ω) M K) c ω] u(ω) f(ω) herefore [(-ω u(ω) [(-ω M K) c ω] - f(ω) so or u() u(ω)e u() IFT u(ω) ω dω A specal case ofe foud pracce s ha where f() g z(), where g s a fxed spaal dsrbuo. I hs case oe ges u(ω) [-ω MKcω] - {g} z() Numercal Implemeao of Freuecy Doma Aalyss I pracce we usually ca o oba he FT aalycally ad we resor o a umercal evaluao. 3

31 If f() s dscrezed a spacg, he he hghes freuecy ha ca be compued he umercal rasform s he Nyus freuecy: W NYQ π/ If he durao of f() s " max ", he he freuecy spacg of he dscree FT s dω π / max. If you have he FT aalycally ad you are compug he IFT umercally, he he me fuco you ge, wll repea every p π / dω where dω s he freuecy sep used he evaluao. Reasos for usg a freuecy doma aalyss - I may be more covee ha a soluo he me doma - The mahemacal model has freuecy depede erms. Models wh freuecy depede erms are ofe ecouered whe descrbg he properes of a dyamc sysem erms of a reduced se of coordaes. Cosder for example he case of a rgd block o sol subjeced o a vercal dyamc load P(). P() Rgd block Sol If we assume he sol as lear he he mpulse respose may be obaed (albe approxmaely) by subjecg he area where he block ress o a load ha grows ad decays very uckly (hrough a rgd lghwegh plae). If we ake he Fourer Trasform of he mpulse respose we oba a freuecy respose fuco whch ca be expressed as; h (ω) e m (ω)^ k e e c e ω If he sol medum where ruly a SDOF vscously damped sysem we would fd ha here are values of m e, k e ad c e for whch he prevous euao maches he resuls obaed. However, sce he respose of he medum s much more complex ha ha of a SDOF we fd ha s 3

32 o possble o mach he expermeally obaed fuco wh he hree free parameers m e, k e ad c e. Whle we ca ry o oba a bes f soluo s also possble o rea m e, k e ad c e as freuecy depede parameers. I hs case we ca selec values for hese coeffces o mach he expermeal curve a each freuecy. Sce a each freuecy he acual value of he freuecy respose s a complex umber (wo eres) we oly eed wo free parameers ad s cusomary o ake m e. Noe, herefore, ha we have arrved a a model ha looks lke a SDOF sysem bu wh parameers ha ca oly be specfed he freuecy doma. Aoher way o say he same hg s ha he freuecy doma we separae he load o s harmoc compoes ad we ca calculae he respose o each compoe usg properes ha chage he full respose s, of course, obaed by superposo. % ********** % OFT.M % ********** % Roue o clarfy some ssues assocaed wh he use of he FFT algorhm Fre. Doma % Aalyss. % ************************************ % The program cosders he me fuco % y()u() exp(-a*) whch has he FT y(w) (a - w)/(a^w^) % ad ca be used o exame how he exac rasform ad he FFT compare or o es how he % IFFT of he rue rasform compares wh he real fuco. a; dur4; d.; :d:dur; yexp(-a*); wmaxp/d; dw*p/dur; % The exac rasform s % ywa-*w/(a^w^); % Compare he exac rasform wh he FFT yff(y*d); ymffshf(y); ome-wmax:dw:wmax; yw(a-ome*)./(a^ome.^); plo(ome,real(ym),ome,real(yw),'g'); pause % Now we use he exac rasform ad go back o me. om:dw:wmax; om-wmax:dw:-dw; om[om om]; yw(a-om*)./(a^om.^); yreal(ff(yw)/d); plo(,y,,y,'g'); 3

33 % Noe ha whe you ake he FT of a me fuco you mus mulply by d. % Noe ha whe you ake he IFT of a freuecy fuco he resul mus be dvded by d. % Noe ha he IFT of hs fuco here s "vbrao" aroud he org. Ths s a resul of % Gbbs pheomeo, whch appears whe he fuco y() s dscouous. Noe ha our % fuco s dscouous a he org. % The FFT of a me fuco wll be ear he real rasform f: % ) The fuco s eglgble for mes afer he durao cosdered. % ) The me sep d s such ha he values of he rue rasform for w> wmax are % eglgble - where wmax p/d. % I ay case, he FFT of a dscree me fuco gves coeffces of a perodc expaso of % he fuco. % The IFFT of a freuecy fuco s perodc wh perod T*p/dw, he frs cycle wll be % close o he real me fuco f he par of he fuco beyod he rucao s eglgble (a) (b) (c) (a) Tme fuco; (b) Comparso of real pars of FFT ad Fourer Trasform; (c) Comparso of magary pars of FFT ad Fourer Trasform; (d) Comparso of IFFT ad he fuco self. (d) Observaos: 33

34 . I he case show here he rue rasform ad he FFT are very close because he rue Fourer Trasform s eglgble for freueces > ω Nyus.. The IFFT gves a very good approxmao of he fuco because dω s small eough ad ω Nyus s large eough. You ca use he program provded o es wha happes whe you chage he parameers. % ********** % EFDS.M % ********** clear % Program o llusrae how a MDOF ca be arbrarly codesed whe he aalyss s carred % ou he freuecy doma. I he lmg case, f oe s ersed he respose a a sgle % coordae, he codesed sysem akes he form of a SDOF wh freuecy depede % parameers. % Theory % ******************************************************************* % dof oal umber of degrees of freedom % rdof reduced umber of degrees of freedom o keep % v umber of loads havg a dffere me varao % u(w) (A'*h(w)*g)*z(w); % where: % A marx (rdof, dof). I each row s full of zeros wh a oe a each oe of he DOF of % eres. For example, f we have a 4dof sysem ad we waed o rea ad 4 % A [ ; ]; % h(w) [-w^*mkcw]^- % g spaal dsrbuo marx (dof, v) % z(w) fourer rasform of emporal varao of he loads (v, ) % ************************* % Specal case where rdof * % ************************* % The euao for u(w) ca also be wre as % u(w) he(w)*z(w); % from where s evde ha he(w) plays he role of h(w) he % case of a SDOF sysem. % ************ % Sce he(w)(a'*h(w)*g)ab ad h(w) /((k-mw^)cw) % we ca choose m ad solve for k ad c erms of a ad b % he resuls are k a/(a^b^) ad c -b/(w*(a^b^)) % ************** 34

35 % The frs par of he program compues he euvale parameers for he roof respose of a 3 % sory shear buldg havg he parameers gve ex. % ****** Daa ******************* m eye(3); k[4-4 ;-4 8-4; -4 8]; c.5*k; A[ ]'; g[ ]'; dw.; 5; wmax(-)*dw; % ******************************* sr(-); for j:; w(j-)*dw; hea'*v(k-m*w^c*w*)*g; hee(j)he; areal(he); bmag(he); dhe*coj(he); ke(j)a/d; ce(j)-b/(w*d); ed; omeg:dw:(-)*dw; subplo(,,),plo(omeg,ke,'w'); subplo(,,),plo(omeg,ce,'c'); pause % **************************************************************** % I hs par we ake he o he me doma o oba % he mpulse respose. % The rck used akg he IFFT ca be argued based % o he causaly of he mpulse respose heechee*; heec()[ ]; hee[hee heec]; dp/wmax; h*real(ifft(hee))/d; :d:*p/dw; fgure plo(,h); pause % ****************************************************** % I hs par he program compues he respose of he roof of he buldg o a emporal % dsrbuo z()u()**s(*p*)*exp(-.5*); z*s(*p*).*exp(-.5*) fgure plo(,z,'r'); pause 35

36 % Compue he respose me by covoluo; ucov(h,z)*d; rlegh(); uu(:r); % Calculae freuecy uw*(ff(z*d).*hee); ureal(ff(uw))/d; % Check usg umercal egrao [,dd]caa(m,c,k,d,g,z,g*,g*,(r-)); fgure plo(,u); pause plo(,u,'g'); pause plo(,dd(:,),'m'); pause % Show all ogeher subplo(3,,),plo(,u); subplo(3,,),plo(,u,'g'); subplo(3,,3),plo(,dd(:,),'m'); 36

37 Euvale Parameers for he roof respose (k ad c) Impulse respose for roof respos, h() 37

38 op respose compued wh covoluo ceer respose compued freuecy low respose compued wh umercal egrao (CAA) Numercal egrao So far we have examed he bascs of me doma (covoluo) ad freuecy doma (Fourer Trasform) soluos. Whle acual pracce we may have o mpleme hese echues umercally, he expressos ha we operae wh are exac ad he error derves exclusvely from he umercal mplemeao. A severe lmao of boh covoluo ad freuecy doma aalyss, however, s he fac ha her applcably s resrced o lear sysems. A alerave 38

39 approach ha s o lmed o lear sysems s drec umercal egrao. The basc dea of umercal egrao s ha of dvdg me o segmes ad advacg he soluo by exrapolao. Depedg o how oe ses ou o do he exrapolao, may dffere echues resul. A esseal dsco bewee umercal egrao mehods ad he umercal mplemeao of eher covoluo or freuecy doma echues s ha he case of umercal egrao we do sar wh exac expressos bu wh a scheme o approxmae he soluo over a relavely shor perod of me. Of course, all cases we reure ha he exac soluo be approached as he me sep sze approaches zero. I seg up a advacg sraegy oe ca follow oe of wo basc aleraves. The frs oe s o formulae he problem so he euao of moo s sasfed a a dscree umber of me saos. The secod alerave s o rea he problem erms of weghed resduals so he euaos are sasfed he average over he exrapolao segme. Some Impora Coceps Assocaed wh Numercal Iegrao Techues Accuracy: Accuracy loosely refers o how good are he aswers for a gve me sep. A mehod s sad o be of he order (Ο( ) ) f he error he soluo decreases wh he h power of he me sep. For example, a mehod s sad o be secod order f he error decreases by a facor of (a leas) four whe s cu half. Geerally, we are lookg for mehods whch are a leas Ο. Whle s cusomary o ake as a cosa hroughou he soluo, hs s o ecessary or s always doe. For example f we have o model a problem ha volves he closure of a gap he sffess of he sysem may be much hgher whe he gap s closed ha whe s ope ad may be approprae o adjus he me sep accordgly. Also, o accuraely capure whe he gap closes may be ecessary o subdvde he sep whe he eve s deeced. Sably: We say ha a umercal egrao mehod s ucodoally sable f he soluo obaed for udamped free vbrao does grow faser ha learly wh me, depedely of he me sep sze. Noe ha lear growh s permed wh he defo of sably. O he oher had, we say ha a mehod s codoally sable f he me sep sze has o be less ha a cera lm for he codo of sably o hold. 39

40 Some Classfcao Termology for Numercal Iegrao Mehods Sgle Sep: A mehod s kow as sgle sep f he soluo a s compued from he soluo a me plus he loadg from o. Mul sep: A mehod s mul-sep whe he soluo a depeds o oly o he soluo a me ad he loadg from o bu also o he soluo a seps before me. Comme: As oe would expec, mul-sep mehods are geerally more accurae ha sgle-sep echues. Mul-sep mehods, however, ypcally reure specal reame for sarg ad are dffcul o use whe he me sep sze s o be adjused durg he soluo erval. Explc: A mehod s explc whe exrapolao s doe by eforcg eulbrum a me. I s possble o show ha explc mehods he marx ha eeds o be vered s a fuco of he mass ad he dampg. Needless o say, whe he mass ad he dampg ca be dealzed as dagoal he verso s rval. Implc: A mehod s mplc whe he exrapolao from o s doe usg eulbrum a me. Commes: Explc mehods are smpler bu are less accurae ha mplc oes. I addo, he sably lm of explc mehods s geerally much more resrcve ha ha of mplc mehods of smlar accuracy. There are o ucodoally sable explc mehods. Sably Aalyss Le's exame he sably of a raher geeral class of sgle sep algorhms. Assume a algorhm ca be cas as: {L} spaal dsrbuo of loadg Xˆ vecor of respose uaes [A] egrao approxmaor { Xˆ } [A]{Xˆ } {L}P( ν) 4

41 Look a he seuece Xˆ AXˆ ( say ν ) LP Xˆ AXˆ A(AXˆ LP LP ) LP A ALP Xˆ LP Xˆ AXˆ Xˆ 3 LP3 A(A ALP LP ) LP3 3 A A LP AP Xˆ LP 3 he follows ha geeral Xˆ A Xˆ A k Sably reures ha he maxmum egevalue he marx A have a modulus < ( he scalar case meas ha A<). To explore he sably reureme whe [A] s a marx we frs perform a specral decomposo of [A], k LP k [A] [P][ λ][p] he above s kow as he Jorda form of [A] ad he operao s kow as smlary rasformao. I he prevous expresso [λ] s a dagoal marx lsg he egevalues of [A]. A covee expresso for he expoeal of [A] ca ow be obaed by og ha; A A 3 A A PλP A A Pλ P PλP PλP PλλP Pλ λp Pλ P 3 Pλ P herefore A Pλ P I s evde, herefore, ha o keep A from becomg ubouded s ecessary ha he larges λ <. The larges egevalue s called he specral radus of he marx, amely; ρ max λ 4

42 A umercal egrao mehod s ucodoally sable f ρ < depedely of he me sep. A echue s codoally sable f ρ > for >"SL" ad SL s kow as he sably lm. The Ceral Dfferece Algorhm: u (- ) ( ) (u u ) u () Cosder a Taylor seres for he dsplaceme a u u u u... serg o e. oe ges; ( expaded abou u() ): u [u u u ] () Noe ha he above euao does o reure formao furher ha oe me sep away from me. Eulbrum a me gves: m u cu ku P() (3) Subsug es. ad o e.3 ad solvg for u au bu u P c( ) m, oe ges afer some smple algebra: 4

43 43 where c m k m a c m c m b c m m c I ca be show ha for a SDOF sysem (or he jh mode of a MDOF sysem): ) ( a ωξ ω b ωξ ωξ c ωξ Defe: u u Xˆ u u Xˆ he m P c u u b a u u

44 Le's exame sably: a a λ b aλ λ a b λ (a λ)( λ) b λ ± b ( a / ) b Assume sably lm λ a a a a 4 a b (a / ) a a b 4 b b eer he values for a ad b: ( ω ) ωξ To compue he egevalues we eed o solve: ωξ ωξ ωξ ( ω ) ωξ 4 ( ω ) ω π T he mehod, herefore, s codoally sable ad he sably lm s; T π For he ceral dfferece mehod we eed a specal sarg echue. To derve we cosder he euao ha predcs he dsplacemes forward, amely u au bu P c m 44

45 he, from es. ad oe ges: u u ( u u ) u ( u u ) u u u he u ( u u u ) u Therefore, u u u u whch s he desred resul. Newmark's Mehod A famly of mehods kow as he Newmark-β mehod are based o assumg he form of he accelerao wh he me sep. u u u τ The dea he approach s evde he euaos preseed ex; u u u 45

46 Iegrag he accelerao oe ges u ( τ) u for oaoal coveece we desgae Iegrag aga oe ges ad we ow defe u( τ ) u u α( τ) α ( τ) a τ α ( τ) a τ τ τ u α( τ)dτ u τ u τ u ( τ) α( τ)dτ δ( τ) u τ δ( τ)dτ u τ u Evaluag a τ τ δ( τ)dτ ( τ) u u whch ca be also wre as: u uδ( ) u u u ( ) u u u u u u [( γ)u γ u ] u [( β) u β u ] where γ adβ deped o α (τ) ad are easly relaed o δ( ) ad ( ). Specfc values of γ adβ defe specfc members of he Newmark Famly. Sably of he ewmark algorhm Resuls show ha for he specral radus o be < 46

47 .5 γ β( ϖ ) < ad o esure ha he egevalues are complex β ( γ) 4 ( ω ) Ucodoal sably s realzed whe; γ ad β (.5 4 γ) Members γ β Sably Codo CAA / /4 Ucodoally Sable LAA / /6 3 T π Fox-Goodma / / 3/ T π Parabola /3 49/44 Ucodoally Sable O he Seleco of a Numercal Iegrao Mehod If sably s a ssue (say we have a large F.E.M. model wh may modes), he we wa a ucodoally sable mehod. Aoher ssue s how he mehod behaves regardg perod dsoro ad umercal dampg. Numercal dampg 47 udamped

48 Typcally, umercal dampg depeds o. T Exac umercal egrao mehod for SDOF sysems wh pece-wse lear excao The basc dea hs approach s o oba he exac soluo for a SDOF sysem for a learly varyg load ad o use he soluo o se up a marchg algorhm. Whle oe ca prcple have me saos oly a he locaos where he load chages slope, s cusomary o advace usg eually spaced me seps. P() P P A m u cu ku P A o m u cu ku P Takg he dfferece oe ges We fd he exac soluo o Euao () ad evaluae a he form; Pτ m u( τ) c u( τ) k u( τ) () u AP u A'P τ BP (C )u DU B' P C' U (D' ) U. The resul ca be wre: u A u A' B P C B' P C' D u D' u where he cosas are gve Appedx A. 48

49 STATE FORM OF THE EQUATIONS OF MOTION Ths seco cocludes he par of he course where we look a echues for solvg he euaos of moo. The esseal dea of he sae form s ha of reducg he sysem of euaos from secod order o frs order. As wll be show, hs ca be doe a he expese of creasg he DOF he soluo from N o N. The frs order euaos offer may advaages, cludg he possbly for obag a closed form (albe mplc) expresso for he respose of a MDOF sysem o arbrary loadg. Cosder he euaos of moo of a lear vscously damped MDOF sysem; defe ad ge [ m ]{} u []{} c u []{} k u { P() } {} u {} x [ m ]{} x []{} c x []{} k u { P() } where {} x [ m] ({ P() } []{} c x []{} k u ) he above euaos ca be coveely wre as; u x m k I m u c x [ m] { P() } he vecor ha coas he dsplacemes ad veloces s he sae vecor, whch we desgae as u {y}. We ca wre;{ y} x where [A] s he sysem s marx gve by ad [B] ad {f()} are gve by; {} y [ A]{} y [ B]{ f() } m [ A] k I m c 49

50 [ B] { f() } [ m] P() Now cosder he soluo of y Ay Bf() () Cosder mulplyg e. by a arbrary marx K(), we ge ad sce follows from (3) ha subsug (4) o () d d we ow mpose he codo; ad ge K () y K()Ay K()Bf () () d d K () ( K()y) K() y K() y (3) d y (K()y) K()y (4) d (K()y) K()y K()Ay K()Bf () (5) d d Iegrag boh sdes of e.7 yelds K ()A K() (6) (K()y) K()Bf () (7) K ()y K( τ)bf ( τ)dτ C (8) 5

51 a we have; K y C C K y ad we ca wre mulplyg e.9 by K() K ()y() K( τ )Bf ( τ )d τ K y (9) gves y () K() K( τ)bf ( τ)dτ K() K y () Les exame he mplcaos of e.6. Assumg ha he marces K() ad A commue we ca wre; e. s of he form whch has he soluo AK () K() () u αu u be α herefore, he soluo o e. s; A K() be () a K () b so he soluo s K() K e A (3) subsug e.3 o e. we coclude ha 5

52 A Aτ A e e K Bf ( τ)dτ K e K y y K (4) sce K s arbrary we ake eual o he dey ad wre; A( τ) A y e Bf ( τ)dτ e y (5) Proof of assumpo abou A ad K() beg marces ha commue. K() e A From a Taylor Seres expaso abou he org follows ha 3 3 A A A K() e I A... (6)! 3! from where s evde ha pre ad pos mulplcao by A lead o he same resul. We are jusfed, herefore, swchg K()A o AK(). A Also as oe gahers from he Taylor Seres expaso, e s accurae whe s o large sce as creases we eed more ad more erms of he seres. Neverheless, we ca always use he followg echue A e e Where all he erms are accurae. A(... ) e A e A...e A I addo s cusomary o call e A Φ() Tras Marx. Therefore subsug, reversg, ad foldg (5), follows ha Evaluao of Φ (). Noe ha y Φ( τ)bf ( τ)dτ Φ()y (7) A A Φ( ) e hs seres always coverges. o! A A A Φ () e e (e ) whch ca be used o mprove accuracy. A If he marx e has a specral radus greaer ha, he as creases Φ() creases ad he al codos grow dcag he sysem s usable. 5

53 I summary, Sably of a physcal lear sysem reures ha he specral radus of Φ () < (for ay ). Numercal Implemeao. I follows from (7) ha y( ) Φ( τ)bf ( τ)dτ Φ ( ) y() Le's use he smple approach, whch s explaed by he followg euao y ( ) ( ) ( ) ( ) Φ Bf Φ y() (8) The advaage of (8) over more complcaed forms s he fac ha o verso s reured. We oly eed o compue he marces A Φ e ad ( ) A Φ ( ) e A more refed way. Assume he load wh he erval s approxmaed as a cosa evaluaed a some eror po say I follows ha Ad evaluag he egral we ge where ( < < ) Aτ y ( ) e (Bf ( )dτ Φ ( ) y() (9) Aτ y ( ) A e l Bf ( ) Φ ( ) y() Ad evaluag he lms follows ha 53

54 y ( ) Aτ ( ) A [e I]Bf ( ) Φ y() () Exac Soluo for Pece-Wse Lear Loadg by he Traso Marx Approach Y() e A ( τ ) B f( τ )d ô e A y () assume. fτ f(τ) f () The dervao o be preseed volves he egral, e a d (3) we solve he egral frs o allow a smooher flow he dervao laer; Iegrag by pars : e a d dv a e a u dv d a e d a e a a a e d a - a a e d a e (a ) e (a ) (4) Cosder ow e. wh he lear load varao descrbed by e., oe ca clearly wre A Aτ Y() e e B(f fτ )dτ. e A y (5) Y() e e A A τ Bfdτ A A τ e e Bf τ dτ e A y (6) 54

55 solvg he frs egral oe ges; or e A Aτ A Aτ ( A) e Bf e Bf τdτ Y() [ ] e A e y (7) e A ( A) A Ι Bf Y() [ ] e A Aτ e e Bf A τdτ e y (8) Sce A e ad A commue we ca wre; ( A) Ι A Bf Y() [ ] e A Aτ e e B f A τdτ e y A e A Ι Bf Y() [ ] A A τ e e Bf τ dτ e A y P e A Bad f are cosa Afer akg he cosas ou of he egral he form s ha e.4. I hs case a -A ad τ, we fd; A [ ] Bf e A A A Y() f e ( A) e ( A ) e ( A ) P (9) y A A [ e ] Bf e Y() f ( A) ( A ) ( A ) P y A A Y() P f A ( A ( e Ι ) Bf A Bf e y whch ca be wre as A Y() P f A Pf A Bf e y () E. ca be furher smplfed by recogzg ha ad f f f (a) f f() (b) 55

56 subsug e. o e. oe ges Y() ( f f ) A B( f f ) A P P A f e y A Pf A Pf A Y() P f A Bf A Bf e y A P A P A Y() P A B f A b f e y So he resul ca be wre as; P P A Y() P A B f A B f e y T P P P () A P f P f e y Y() ( ) MATLAB IMPLEMENTATION A MATLAB roue ha mplemes he raso marx approach s preseed hs seco. %**************************************************************** % % ***** TMA ***** % % Prepared by D.Beral - February, 999. % % %**************************************************************** % Program compues he respose of lear MDOF sysems usg he % raso marx approach. The basc assumpo s ha he % load vares learly wh a me sep. Excep for he % approxmao he compuao of he expoeal marx, he % soluo s exac for he assumed load varao. % INPUT 56

57 % m,c,k mass dampg ad sffess marces for he sysem. % d me sep % g vecor ha scales he appled load a each DOF % p load marx. Each row coas he me hsory of he load % appled a each DOF. Sce he case where he loads have he % same hsory a each DOF s commo, a opo o pu p as a % sgle row s provded. I parcular, f p has a sgle row, % he program assumes ha hs row apples o all he DOF. % d,v al codos % seps umber of me seps he soluo. % OUTPUT % u marx of dsplacemes % ud marx of veloces % udd marx of acceleraos (f flag7); % (each colum coas oe DOF); % LIMITATIONS % Mass marx mus be posve defe %**************************************************************** fuco [u,ud,udd] ma(m,c,k,d,g,p,d,v,seps); % To elmae he compuao of acceleraos flag~7; flag7; % Basc marces. [dof,dum]sze(m); A[eye(dof)* eye(dof);-v(m)*k -v(m)*c]; B[eye(dof)* eye(dof)*;eye(dof)* v(m)]; % Place d,v ad g colum form case hey where pu as a % row vecors ad make sure ha he load s ordered rows; [r,c]sze(g); f r; gg'; ed; [r,c]sze(d); f r; dd'; ed; [r,c]sze(v); f r; vv'; ed; [r,c]sze(p); f r~dof; f r~; pp'; ed; ed; 57

58 % Compue he marces used he marchg algorhm. AAA*d; Fexpm(AA); Xv(A); PX*(F-eye(*dof))*B; PX*(B-P/d); % Ialze. yo[d;v]; %Perform he egrao. for :seps; y(:,)(pp)*[g*;p(:,).*g]- P*[g*;p(:,()).*g]F*yo;yoy(:,); ed; uy(:dof,:)'; u[d';u];udy(dof:*dof,:)'; ud[v';ud]; % Calculae he accelerao vecor f reuesed. f flag7; Xv(m); for :seps; uddx*(p(:,).*g-c*ud(,:)'-k*u(,:)'); udd[udd udd]; ed; uddudd'; ed; Coeco Bewee e A ad he Impulse Respose. I he al reame of he mpulse respose we lmed our aeo o a SDOF sysem. By examg he geeral soluo for he raso marx s evde ha he marx e A plays he role of he mpulse respose he case of MDOF sysems. Two resuls ha may be approprae o oe explcly are: ) If a load fuco has a cosa spaal dsrbuo he respose a DOF "j" ca be obaed by he covoluo of a sgle "mpulse respose fuco" wh he me hsory of he load. The mpulse respose fucos for each oe of he DOF are gve by: A { h( )} e (3) m g where m s he mass marx ad g s vecor ha descrbes he spaal dsrbuo of he loadg. I hs case oe ca hk abou he respose a DOF as ha of a SDOF sysem wh freuecy depede parameers subjeced o he load hsory. The freuecy depede parameers are obaed from he Fourer Trasform of he approprae mpulse respose fuco. 58

59 ) I he more geeral case he load ca be expressed as k P( ) g p ( ) (4) where k #DOF. As s evde, he respose a each DOF s ow gve by he superposo of "k" covoluos wh mpulse respose fucos defed by e.3. As a geeralzao of he saeme ha cocluded he prevous ery, we ca ow vsualze he respose a DOF "j" as he sum of he respose of "k" freuecy depede SDOF sysems wh parameers defed by he Fourer rasforms of he correspodg mpulse respose fucos. 3) Euvale o () s he dea ha oe ca defe a mpulse respose marx where he h,j s he respose a DOF due o a mpulse a DOF j. By speco of es. 3 ad 4 oe cocludes ha h,j s he "h" erm e.3 whe he load dsrbuo g j s uy a DOF "j" ad zero elsewhere. 4) I s also of eres o oe ha he mpulse respose marx s he upper dof x dof paro of he marx e A (remember hs marx s dof x dof). Ispecg he Taylor seres expaso of e A oe ca wre a seres expresso for he mpulse respose marx he frs few erms are: [ h] [ I] [ m] 3 [ k] ( [ m] [ c] ([ m] [ k]) ) 6 (5) whch shows ha as he mpulse respose approaches he dey mes oe also ha he off-dagoal erms. PART II I he frs par of he course he focus was he soluo of he euaos of moo bu lle was sad abou how hese euaos are obaed. I hs seco of he course we focus o he formulao of he euaos for elasc MDOF sysems. Iroduco o Aalycal Mechacs Oe approach o ge he euaos of moo for a sysems s by usg Newo s law..e. u P() k 59

60 From a free body dagram of he mass oe ges; m u ku P( Whle oe ca always oba he euaos of moo usg eulbrum cosderaos, he drec applcao of Newo s laws s dffcul complex sysems due o he fac ha he uaes volved are vecors. A alerave approach s o use eergy coceps he eergy approach s reaed he brach of mechacs kow as Aalycal Mechacs. Impora Corbuors o Aalycal Mechacs: Lebz : Noed ha euao of moo could also be expressed erms of eergy. D alember : Frs o apply he prcple of vrual work o he dyamc problem. Euler: Derved (amog may oher hgs) he calculus of varaos. Hamlo: Hamlo s Prcple s he corersoe of Aalycal Dyamcs. Lagrage: Hs applcao of Hamlo s prcple o he case where he dsplaceme feld ca be expressed erms of geeralzed coordaes leads o a covee form of he prcple kow as Lagrage s Euao. Geeralzed Coordaes ( Lagrage ) z z Aleravely *.5 * 6

61 We ca descrbe ay cofgurao usg also ad. Noe ha whle ad ca o be measured, hey are coordaes ha descrbe he cofgurao ad are jus as vald as he physcal coordaes. Geeralzed coordaes are ay se of parameers ha fully descrbed he cofgurao of he sysem. HAMILTON'S PRINCIPLE Cosder a parcle ha has moved from locao o locao uder he aco of a resula force F (). The rue pah ad alerave (correc pah) are oed he fgure. Vared Pah F() δr m True pah (Newoa Pah) r r δ r From Newo's Law we ca wre: or, placg he form of a eulbrum euao we ge; F ( ) m r () F ( ) m r () Applyg he prcple of vrual work we ca wre (oe ha hs s doe wh me froze); ( F ( ) m r ) δ r (3) 6

62 Cosder he followg dey; d d from where s evde ha we ca wre subsug e.5 o e.3 we ge [( m r ) δ r ] m r δ m r δ r r (4) d m r δ r [( mr ) δ r ] mr δ r (5) d Nog ha; d F () δ r d [( m r )] δ m r δ r r (6) δ m rr m ( r δ r )( r δ r ) m r r (7) or wha s he same; ä m r r m r r m r δ r m δ r r m δ r δ r m r r (8) allows oe o wre, eglecg hgher order erms; δ ( m r ) r m r δ r (9) subsug e.9 o e.6 gves d F ( ) δ r [( m r ) δ r ] δ mr r () d We recogze he erm F () δ r as he work doe by F o he arbrary vrual dsplaceme a me, we desgae hs erm δ W. Furhermore, he hrd erm s he varao of he kec eergy whch we desgae as δ T. I euao form, we have; 6

63 ad δ W F ( ) δ () r δ T δ mr r () Subsug es. ad oe ca wre e. as; δ W d δt [ m r δ r ] (3) d Mulplyg boh sdes of e.3 by d ad egrag from o oe ges; ( W δt) δ d m r δ r (4) where he eualy o zero s jusfed because δr δr a ad (recall ha a reureme mposed a he sar of he dervao s ha he vared pah sars ad eds o he acual pah). I summary; ( W δ ) δ T d (5) whch s he mahemacal saeme of Hamlo's Prcple. As ca be see, he prcple saes ha he varao of he work of all he forces acg o he parcle ad he varaos of he kec eergy egrae o zero bewee ay wo pos ha are locaed o he rue pah. Provded ha he vared pahs cosdered are physcally possble (do o volae cosras) he varao ad he egrao processes ca be erchaged. For hs codo e.5 ca be wre as; ( W ) δ T d (6) whch s Hamlo's prcple s mos famlar form. The prcple saes: The acual pah ha a parcle ravels reders he value of he defe egral e.6 saoary wh respec o all arbrary varaos bewee wo sas ad, provded ha he pah varao s zero a hese wo mes. 63

64 Illusrave Example Cosder he case of a mass ha falls freely. r r g/ Free fallg True pah Cosder a famly of pahs ha coas he correc soluo. say r g where we mpose he lmao ad o esure ha he vared pahs ad he rue pahs cocde a he lms of he me spa; dffereag we ge; r g he work ad kec eergy expressos are, herefore; g W m ad g T m ( ) subsug e.6 δ ( W ) he defe egral euals T d δ g g 4 ( ) m d m d 64

65 ( W ) T d mg 4( -) I hs smple case he reureme ha he varao of he egral be zero s euvale o reurg ha he dervave wh respec o be zero. Takg a dervave ad euag o zero oe fds whch s, of course, he correc soluo. Assume ow r g ( ) ( e ) e whch, of course, s a famly ha does o coa he correc pah, we have; herefore r g e ( ) e ad he defe egral s herefore; mg W T m g ( e ) ( e ) ( e ) 4 e H m g ( e ) e ( e ) 8 A plo of hs fuco from - o s show he fgure below, as expeced, o saoary pos are foud

66 Lagrage's Euaos: The dervao of Lagrage's euaos sars by cosderg Hamlo's Prcple, amely; ( W ) ä ät d (7) For coveece assume ha W s separaed o Wc Wc where Wc s he work doe by coservave forces, (gravy feld for example) ad Wc s he work from o-coservave forces, whch s ypcally ha whch comes from appled loads ad dampg. Defg Wc -V, where V s called he poeal, oe ca wre; ( äv ) äwc ät d (8) The key sep he dervao of Lagrage's euaos s he assumpo ha he erms sde he parehess Hamlo's euao ca be expressed as fucos of geeralzed coordaes he followg way, T T(,,...,,,,..., ) (9) V V(,,..., ) () ad ä Wc δ () Q where we oe ha he kec eergy s o jus a fuco of veloces bu may also have erms ha deped o dsplacemes. Takg he varao of T ad V oe ges; ä T N T δ T δ () 66

67 N V ä V δ (3) subsug es. ad 3 o 7 gves; δ δ - δ Qδ d N T T V (4) we wsh o elmae he dervave wh respec o he varao of he velocy o allow he varao wh respec o dsplaceme o be ake commo facor of he erms he parehess; we ca do hs wh a egrao by pars of he egral T d δ (5) we ake U T ad herefore dv δ d ad du d d V δ T d ad we ge T δ d d T δ d (6) where he frs erm s zero because he varaos vash a ad. Subsug e.6 67

68 o e.4 oe ges; N d d T T V Q δ d (7) Sce he varaos δ are depede we ca choose all of hem o eual zero excep for oe. I hs way we reach he cocluso ha he oly way he egral ca be zero s f each of he dvdual erms he bracke vash. I parcular, we mus reure ha; d d T T V Q (8) for every. Es.8 are he much celebraed Lagrage's Euaos. Example Cosder he SDOF sysem show he fgure; K M P() he ecessary expressos for obag he euao of moo usg e.8 are: Wc T m K d V K K δwc P() δ 68

69 The mahemacal operaos are sraghforward ad are preseed ex whou comme. T m Subsug e.8 oe ges; d d T T m V K m K whch s he euao of moo for he sysem. P() Example Cosder he pedulum show he fgure. l x ls y l l cos x l cos y l s M L() x y The expressos for he kec eergy, he poeal, ad he work of he o-coservave forces are; T mx my m( l cos l s ) V mgl( cos) ml 69

70 δwc L() δx L() l(s( δ) s ) L() l(s cos cos s δ s ) δwc L() l cosδ herefore, T ml d d T ml T V mgls Q L() l cos or, devdg by he legh l mlgs L() l cos m m l mgs L()cos Noe ha he above euao s olear. A learzed euao, however, ca be obaed by resrcg he valdy of he resuls o small agles. I hs case we ca subsue he se of he agle by he agle self ad he cose by oe, we ge; m l mg L() whch s a smple secod order lear dffereal euao. The perod of he pedulum for small ampludes of vbrao s readly obaed as; ϖ T g l π g l 7

71 Soluo of Lear ad Nolear Dffereal Euaos by Coverso o Frs Order As he example of he pedulum llusraes, he euaos of moo are geerally olear ad ca oly be learzed by resrcg he soluo o small dsplacemes aroud he eulbrum cofgurao. Aalycal soluos o olear dffereal euaos are o geerally possble ad oe mus almos always voke a umercal echue o oba he respose for a parcular se of codos. I order o make fuure dscussos more eresg s useful o acure he capably for solvg, albe umercally, he olear euaos ha we oba. A powerful soluo sraegy cosss casg he euaos frs order form. The procedure s readly udersood by specg he example preseed ex. Cosder he case of he smple pedulum, he euao of moo was foud o be; m l mgs L()cos we ake x x ad wre he euao as; ml x mgs( x ) L( ) cos( x) from hs euao ad he fac ha x x we ca wre he dervave of he sae as; ad x x x ( L( )cos( x ) mgs( x)) ml Clearly, kowg he grade of he sae oe ca esmae he creme he respose ad proceed a sep by sep fasho. I MATLAB he soluo ca be obaed usg he umercal egraor ODE45 (here are ohers MATLAB ha may perform beer for cera problems). The form of he commad s: 7