Wavs and Vibration An ntrodction to Wavs and Vibration in ivil Enginring ntrodction to spctral lmnts and soil-strctr intraction Matthias Baitsch Vitnams Grman Univrsity Ho hi Min ity Yvona olová lova Tchnical Univrsity Bratislava Mira Ptronijvić Univrity of Blgrad Günthr chmid Rhr Univrsity Bochm Dcmbr 0 - -
Wavs and Vibration ntrodction n th last dcads lctrs as vibration of strctr and soil dynamics hav bn introdcs in many crricla of ivil Enginring. Mostly ths lctrs ar offrd from diffrnt chairs whr vibrations of strctrs concntrats on strctrs rigidly basd on th sb-grad. pcial attntion is givn to modal analysis and stp by stp procdrs to analyz gomtrical and physical non-linar bhavior. Th sbjct soil dynamics on th othr hand invstigats vibration of bloc fondations on visco-lastic soil whr spcial attntion is givn to th propagation of soil wav as thortical basis for th dynamic stiffnss of th sb-grad fondation from which th motion of th fondation bloc is drivd. inc th dynamic rspons in both cass ar drivd from th wav qation and sinc any strctr rsts on mor or lss lastic matrial thr is no rason to sparat th sbjct in two diffrnt filds. n this introdction to wav propagation and vibration w considr ampls with harmonic citations of linar systms. This is a spcial cas of arbitrary citation in tim wr th citation is transfrrd by Forir transformation from th tim domain into th frqncy domain and th rspons in th frqncy domain is transformd into th tim domain by invrs Forir transformation. For ths cass th Finit Elmnt Mthod, familiar to th strctral nginr, can b applid for dynamic problms, inclding problms of soil-strctr intraction, as in static problms. Mchanical wavs convrt potntial nrgy into intic nrgy. Th wavs travl with vlocity c throgh lastic mdia. Th vlocity of propagation is largr in stiffr and lss inrtial mdia; it is smallr in softr and mor inrtial mdia. This is dmonstratd for slctd mdia in Tabl. Flids c = /ρ Air 340 m/s Elastic mdims Watr P [m/s] 450 m/s c = E /ρ sand 450 30 P [m/s] granit 4000 400 c = G/ρ concrt 3800 00 stl 500 500 c wav vlocity comprssion modls ρ dnsity c p comprssion wav vlocity E sid constraint E-modls c shar wav vlocity Tabl : Propagation vlocitis in homognos contina - -
Wavs and Vibration Wav in a bam. Dfinitions = 0 d = Fig.: Elmnt with mass dnsity ρ, cross sction A, lasticity modls E. W assm along th bam, shown in Fig., an aial motion (, whr is th position in spac and t is th tim. n ach lmnt d with mass ρ Ad th lastic forc and th inrtia forc ar in qilibrim, s Fig., rslting in th qation Adσ = ρ Ad& ρ Ad& σ A ( σ + d σ ) A d Fig.: Dynamic qilibrim at infinitsimal lmnt with lngth d () For linar lastic matrial th rlation btwn strssσ and dformationε = ʹ is prssd by Hoo s law. Eq () rslts in th wav qation d Ad E d = ρ Ad& () d or for constant matrial proprtis along Eʹ ʹ = ρ &. (3) n q (3) th prim stands for drivativ with rspct to and th dot for drivativ with rspct to tim. Eqation (3) is th diffrntial qation for an initial-bondary val problm which can b solvd for sitabl bondary vals at th bam s nds = 0, = and initial vals for th displacmnt = (,0) and vlocity = (,0), 0. 0 0-3 -
Wavs and Vibration. On-dimnsional wav.. Dformation along th bam W assm that th motion along th bam at a slctd tim t = t can b dscribd by th spacharmonic fnction ( t ),. ψ λ π = 0 (, t ) = cos( κ + ψ ) α π κ = λ POT Fig.3: napshot of th longitdinal harmonic dformation along th bam at timt = t ; spacharmonic wav with amplitd, wav lngth λ and phas shift ψ. W considr th spac-harmonic wav (, t ) = cos( κ + ψ ) (4) with amplitd, wav lngth λ and phas shift ψ. Th (circlar) wav nmbr is dfind as π =. (5) λ From trigonomtric rlation w hav and for = α (6) α = π w obtain λ = π. (7) Using compl notation, s Appndi A, th wav along th bam at tim nmbr, amplitd and phas shift ψ,can b writtn as t = t with wav iκ i( κ+ ψ ) (, = = (8) n q (8) th tild dnots a compl val. n gnral is a fnction of th wav nmbr. Th rlation btwn location and wav nmbr κ for th snapshot at tim t is, s Apndi A, { ( i R m ) } = cos sin = cos( +ψ ) ( ) = R (9) = + m and ψ = tan. R R m whr ( ) ( ) - 4 -
Wavs and Vibration.. Vibration of a bam particl at th location = W assm that th vibration at a slctd position fnction (,. = can b dscribd by th tim-harmonic W considr th tim-harmonic vibration with priod T, amplitd and phas shift ϕ. T π ϕ = 0 cos( ω t + ϕ) ω = π T t β POT Fig. 4: Tim-harmonic vibration of th particl = with amplitd and priodt. Th (circlar) frqncy is dfind as ω = π (0) T From trigonomtric rlation w hav ω t = β () and for β = π w obtain ωt = π. () n compl notation th vibration of th particl bcoms iωt i( ωt + ϕ ) ( ω, = = (3) n gnral is a fnction of th frqncyω. Th rlation btwn tim t and frqncyω for th particl is, s Appndi A { iωt R m } = cosωt sinωt = cos( ω + ϕ) ( = R t.. ongitdinal wav along a bam W assm that th motion ( harmonic fnctions, s Fig. 4. (4), as a fnction of spac and tim t can b dscribd by - 5 -
Wavs and Vibration POT t (, = 0(cos ) (cosω 0 Fig. 5: ongitdinal harmonic dformation in spac and tim of a bam; longitdinal harmonic dformation along th bam at tim t = t ; harmonic wav with amplitd = 0, wav lngth λ and frqncy ω. From qs (5,) w hav th idntity or or λ = ωt (5) λ ω = λ f = = c T ω = c ; (6) ω = (7) c Eqations (7), hr bing trivial, ar calld disprsion rlation of th wav. f c is a fnction of ω (or of λ, s qation (5) thn th wav is calld disprsiv. ω c c ω Fig. 6: Disprsion rlation of a non-disprsiv wavs; c = const. (not dpndnt of ω or ). - 6 -
Wavs and Vibration Th qotint ω in q (6) has th dimnsion s m. As w will s latr th qotint c rprsnts th propagation vlocity of th harmonic wav ω i( + ct ( ) ( ) ( i iωt,, ) i + t = ω 8) From q (8) and Fig. 5 w ddc that th harmonic wav with amplitd propagats with vlocity c in th ngativ -dirction, sinc for ct th val stays constant. Th harmonic wav propagating in th positiv -dirction wold hav th form i( c Not that any fnction (, = f ( ± c) satisfis th wav qation (3) and thrfor rprsnts a wav propagating in th ngativ or positiv -dirction, rspctivly.. Eampl : A machin sitatd on a long rigid fondation on sandy soil, rotating with 60 rvoltion pr mint, vibrats in vrtical dirction with an amplitd of cm. Th cratd wavs can b assmd as -dimnsional (plan wav front paralll with th long sid of th fondation. Damping in th soil is nglctd. Dtrmin th corrsponding wav qation a) in th wav nmbr frqncy domain, (, ω ). b) in th spac-tim domain, (,. Tas: Rpat th problm givn in Eampl if th machin is sitatd on roc (granit). a) Plot th wav fild in i ( ) (, ) = 0 ct b) how th corrsponding animation. ω in diagram as shown in Fig. 5..3 oltion of th longitdinal (comprssion) wav in a bam Th soltion of th wav qation may b obtaind by sparation of th variabls in q (3): (, = X ( ) T( (9) lading to E X ʹ ʹ ρ X T = =, whr is a constant. (0) T - 7 -
Wavs and Vibration With th assmption of a tim-harmonic soltion w writ for th scond part of q (0) in iωt compl form, indicatd throgh tild, T ( = and obtain or iω t ω = iωt = ω. () i With th assmption of spac-harmonic fnction, X ( ) = A, w obtain from th first part of q (0): or with () E i i A = ω A (3) ρ E c = (4) ρ c = ω (5 From whr w obtain th wav nmbrs ω ω = = = =. (6) c c i i X = A + B hav for th spac and tim transformd fnction. ; Whnc w ( ) T ( ω ) = iωt and according to q (9) i i i t. ω (, = ( A + B ), (8) whr th ral constant has bn incldd in th constants A and B. Th wav qation is than, s also Apndi A, i (, ) R R ( ) ( ) i iωt i + ωt i ωt t = A + B = A + B (9) {( ) } { } A and B ar compl conjgats which hav to b dtrmind from th bondary condition at th bam s nds. (7) Eampl For th -dimnsional infinit stl bar, shown in Fig.7, calclat th vlocity of propagation, th wav lngth and th wav nmbr if th point O at th origin ndrgos a harmonic motion with frqncy f = = Hz and amplitd 0 = 0. 0cm. Giv th graph for (, for a slctd tim T span and a slcd distanc along th bam. - 8 -
Wavs and Vibration 0 O = 0 E = 80 A = 0cm 9 N / m ρ = 7850 g / m 3 Fig. 7: nfinit stl bar oltion With th dfinition E c = ρ W can writ th wav qation (3) as c (30) ʹ ʹ = &. ((3) Any fnction = f ( c satisfis q (3). Th vlocity of wav propagation is E 8 3 c = = 0 = 888.68 m/s ρ 7.85 Th prscribd circlar frqncy is π 6.8 ω = π f = = = 3.4 rd/s T Th wav nmbr is, q (7), ω = = c Wav qation 3.4 888.868 Frqncy domain i ( ) (, 0 ct = = = = 0.0066 rd/m. iκ ict 0 = iωt i( c 0 = i 0 = = (3) Tim domain (, = 0 cos( c (33) - 9 -
Wavs and Vibration (, (, = 0 cos( c t POT Fig. 8: Wav (, in an infinit bam with ( 0,0) = 0. 0m; plottd for 0 00 m, 0 t 0 s. 3 Finit Elmnt in pac and Tim 3. pctral Bam lmnt; aial motion n sction w considrd an infinit aial bam sbjctd to a harmonic wav. Nt w will considr a finit aial bam sbjctd to gnral bondary conditions. (, d Fig. 9: Bam with inmatic vals ( ) and ( ) at its nd cross sctions t onsidr th bam lmnt in Fig. 9 with constant mass distribtion m = ρ A, lngth and aial stiffnss EA. Th longitdinal forcs P ( and P ( act on th nd sctions and of th lmnt. Th corrsponding displacmnts ar ( and ( rspctivly. Th dynamic qilibrim lads to th partial diffrntial qation (3) now writtn in th form m = EAʹ ʹ (34) with m = ρ A (35) Th wav qation (8) w writ now as i t ω (, = ( ) (36) whr = i (37) and E EA c = =. (38) ρ m Thn th linar combination of th spctral displacmnts is writtn as = +, (39) () t E ρ A= m () t - 0 - t
Wavs and Vibration with m, = ± i = ± i ; = ω with wav nmbr (40) EA whr th constants and in qation (39) can b dtrmind sing th inmatic (displacmn bondary conditions (0) = and ( ) = : = + (4) = + (4) Eprssing th constants, throgh th inmatic vals,,, at th bam s nd cross sction and insrting ths rlations into qs (4,4) rslts in th dynamic shap fnctions ) + ( ) ( ( ) = = ( ) ( ) + (43) Th strain along th bam, prssd throgh th ban s nd displacmnts, is ( ( ) ) ( ) ε = ʹ = + ( ) (44) Th normal strss bcoms ( ( ) ) ( ) σ = Eε = ʹ = + ( ). (45) inc th shap fnctions ar act w can also actly calclat th dynamic forcs, P, P, corrsponding to and : EA EA P = Aσ ( 0) = (46) P ( ) EA EA = Aσ ( ) = + (47) t is convnint to writ th ssntial qations abov in matri notation: Trial soltion of th homognos wav qation ( ) = = [ ] = ( ) (48) onstraint qation d to B.. s = (49) or = f() = f ( ) ; (50) hap fnctions or ( ) = ( ) = ( ) f ( ) = N( ) ( ) = ( ) ( ) = [ N ( ) N ( ) ] = N ( ) (5) (5 - -
Wavs and Vibration train [ ] = B ( ) ε ( ) = (53) trss ( σ ) = Eε ( ) = E [ ] = EB ( ) (54) Wav qation for spctral bar or P = P EA + + ; with m = iω (55) EA P =. (56) EA + = 57) + is th spctral stiffnss matri of th lmnt. Not: Matrial damping can b incldd by introdcing a compl Yong s modls, according to th corrsponding principl for visco-lastic matrial: E = E( + iη). (58) For linar visco-lastic matrial th damping cofficintη is a linar fnction of th frqncyω. From qation (57) on obtains for any frqncyω th corrct rlation btwn th corrsponding nd displacmnts and nd forcs of th bam lmnt. Not that for ω=0 (static cas), sing l Hopital s rl, th th stiffnss matri of a bar lmnt is obtaind: EA =. (59) Eqation (55) can b sd to formlat an initial-bondary val problm for on strctral mmbr with lngth for givn nd displacmnts %and %or a givn nd displacmnt%on on nd and a spcifid aial forc P % on th othr nd, citd by a tim harmonic fnction with frqncy ω. Th soltion of this bondary val problm givs th (in gnral compl) frqncy rspons for th considrd frqncy rang. f, for ampl, i t ω ( = is a bas citation with amplitd % ( ω) is th frqncy rspons at th fr nd. and frqncyω than - -
Wavs and Vibration Th amplitd frqncy rspons and th phas shift ψ with rspct to th citation is thn R ( ) ( ) % ( ) = +, tan ψ( ω) =. 60) ω R ( ) Th rspons in tim domain for a slctd frqncy ω is obtaind as t i ω, ) = R (, ω ) R (, ω) t. (6) { } { } ( = Eqation (6) is bst and most instrctivly rprsntd as animation. Not: Only on lmnt is ndd for on strctral lmnt, vn for vry high citation frqncis. Eampl 3 E g η A ρ rigid M O Fig. 0: Towr-li strctr citd by a vrtical harmonig bas roc motion ( = cosωt. g g alclat analytically possibly with a comptr program li Mathmatica - for th strctr shown in Fig. 8 a) th ignfrqncis ωi and ignmods φi () nglcting damping, i =,,3. b) th frqncy rspons of th top of th strctr, ( ω ), nglcting matrial damping; 0 ω < ω 4 c) th frqncy rspons of th top of th strctr, ( ω ), with strctral damping η = 0.0; 0 ω < ω4 d) th dformation (, with th frqncy ω as paramtr ) allat with Mathmatica th dformation (, t, ω) with damping cofficintη = 0. 0. f) Plot snap shots of th dformation (, for slctd frqncis ω and tim instants t with and withot damping and show ths rslts as animation - 3 -
Wavs and Vibration oltion (slctd itms) a) Eignvibration Th ignfrqncy ar obtaind from th soltion of th homognos qation (5) for th considrd bondary val problm with P = 0 = EA + 0 + = g = 0 P = ; with = = i = iω m EA.(6) Th only inmatic nnown in q (6) is th displacmnt. Ths only th scond qation ot th matri qation (6) is of intrst, rslting in + = cos( ) = 0. (63) Th non trivial soltions of q (63) ar π n = (n ) (64) m which givs with n = ωn (s q (9) th ignvals of th bar lmnt, fid on on nd EA as π EA π E ωn = (n ) = (n ) (65) m 3 ρ Th corrsponding ignmods follow from qation (6) scald with n ( ) = 0, n n n n( ) = [ N ( ) N, n( ) ] ( ) sin n = = (67), n n n n n n( ) n n, sin n n n = φ = sin n n, = (68) n( ), n sin n π φ n = sin n = sin(n ), n =,,3,... (69) π EA ω= m π φ( ) = sin π EA ω= m 3π φ ( ) = sin Fig.: First 3 vibration mods of a bar lmnt Not: f matrial damping is nglctd th rslts (, ω) and (, can b obtaind also with a linar combination of ral trial fnctions sin and cos in q (39). Tas: Rpat th analysis with ral trial fctions, = {( cos sin )}cosωt ( + π EA ω3= m 3π φ 3( ) = sin - 4 -
Wavs and Vibration b) Frqncy rspon of top of strctr d to harmonic bas motion s ( s (, = cos + sin = cos(, ω ) = cos + sin = cos( ). onfirm with ral analysis that )cosωt g ( ω) = m (70) cos ω EA Th frqncy rspons at th top of a bam citd at is bas in aial dirction, is shown in Fig. 8. As no damping is assmd th rspons is ral, ithr in phas or in opposit phas with rspct to th citation. Matrial damping can b incldd throgh a compl modls, E = E( + iη), whr th damping cofficint η is linarly dpndnt on th frqncy for viscos matrial. n ordr to flfil th rqirmnt of casality η has to b fnction of ω. Not: Th so calld hystrtic damping with th damping cofficint bing constant, mostly chosn by civil nginrs for nmrical invstigations in th frqncy domain, violats casality [4]. ω/ω Fig. : Amplitd s of th top of a bam citd in aial dirction by a harmonic bas ω π EA motion g as fnction of ; ω = is th first ignfrqncy of th bam. m ω f ω approachs ω th nmrical rsonanc frqncy approachs th first analytical ignfrqncy. - 5 -
Wavs and Vibration ) Dformation along th bam Th motion along th bam is obtaind as (, R (, ω) i ω = R (,ω) t (7) = { } { } Dformation (, ω) along th bam follows from th trial fnction abov m cos ω ( ) EA (, ω) = g (7) m cos ω EA Ths th dformation along th bam as a fnction of timt is m cos ω ( ) EA (, = g cosωt (73) m cos ω EA n figrs 3 and 4 snap shots of th bam motion ar shown. n figr 3 th citation is blow th first ignfrqncy, in figr 4 btwn th first and scond ignfrqncy. inc no matrial damping is prsnt vibration nods ist..00 Undmpd Displacmnt.50.00 0.50 0.00-0.50 -.00 -.50 t=0,0 t=0,t t=0,t t=0,3t t=0,4t t=0,5t t=0,6t t=0,7t t=0,8t t=0,9t -.00 0 0 0 30 40 50 60 70 80 90 00 Points of th lmnt Fig. 3: Dformation of th bam dring on priod T; ω=30 rad/sc; ω < ω, - 6 -
Wavs and Vibration Undmpd 5.00 4.00 Displacmnt 3.00.00.00 0.00 -.00 -.00-3.00-4.00-5.00 0 0 0 30 40 50 60 70 80 90 00 t=0,0 t=0,t t=0,t t=0,3t t=0,4t t=0,5t t=0,6t t=0,7t t=0,8t t=0,9t Points of th lmnt Fig. 4: Dformation of th bam dring on priod T; ω=6,839 rad/sc; ω, < ω < ω, Ral displacmnt [Ur/Ug].50.00 0.50 0.00-0.50 -.00 -.50 REA 0 0 0 30 40 50 60 70 80 90 00 t=0.0 t=0.t t=0.t t=0.3t t=0.4t t=0.5t t=0.6t t=0.7t t=0.8t t=0.9t t=,0t Points of th lmnt [m] Fig. 5: napshots of th dformation of th bam dring on priod T; ω = 7.656 rad/sc.viscos damping: E % ω = E( + i0.0 β), β = ω ; ω, < ω < ω,3. Th rslts in th Figs. 3-5 ar tan from []., - 7 -
Wavs and Vibration - 8-3. pctral bam lmnt; bnding motion Fig. 6: Brnolli bam lmnt Th spctral bam lmnt, in bnding, shown in Fig., is obtaind, in th sam procdr as in sction 3., by transforming th diffrntial qation = 0 + v m Ev V (74) into th frqncy domain: 0 = v m v E V ω (75) Th soltion ) ( v of th 4th-ordr diffrntial qation (33) can b prssd as linar combination of th 4 ponntial fnctions j. [ ] ) ( ) ( i i = = = (76) Th ponnts j, j=,,3,4 ar obtaind as charactristic roots ± =,, i ± =,4 3 ; 4 E m ω = (77) of th wav qation (75). Thy dpnd on th proprtis of th bam and on th frqncy ω. W giv hr only th matri qation for th frthr drivation and rfr th intrstd radr for mor dtails to [3,4,5]. Trial soltion of th homognos wav qation [ ] ) ( ) ( 4 3 4 3 v i i = = = (78) onstraint qation d to B.. s ) ( 4 3 4 3 4 3 4 3 4 3 4 3 v v v v = = (79) v() E,, m v% v y 3 v 4 v v
Wavs and Vibration or v = f( ) = f ( ) v. (80) hap fnctions ( v = ( ) = ( ) f ( ) v = N ( ) N ( ) N ( ) N ( ) = N ( ) [ ] v 3 (8) rvatr κ ( ) = v ʹ ʹ ( ) = B ( ) v (8) Momnt M ( ) = Eκ ( ) = EB ( ) v (83) Wav qation for spctral bnding bam lmnt or P P = P3 P4 3 4 3 4 3 3 33 43 4 4 43 44 v v v 3 v4 (84a) P =. (84b) Tass: alclat with Mathmatica th shap fnctions ( ), j =,,3, 4and plot ths for slctd frqncis forω. alclat with Mathmatica th dynamic stiffnss matri For ω = 0 th dynamic stiffnss matri rdcs to th stiffnss of th static bam lmnt. f th matrial has no damping th trms of th shap fnctions and of th stiffnss matri ar ral and may b prssd by sin-, cos-, sinh-, cosh-fnctions. N j Eignvibration Th bondary val problm,.g. for a towr-li strctr, rprsntd as on bam lmnt fid at its lowr nd, Fig. 3, is prssd as v 3 v 4 P 3 4 v = 0 P = = 3 4 v 0 Fig.7: 3 3 3 33 43 (85) P v3 Towr-li stctr P4 4 4 43 44 v4 v v Th ignvals and ignvctors (ignfrqncis and ignmods) ar obtaind from th homognos qations in (44) for th displacmnts v 3 and v 4 : - 9 -
Wavs and Vibration 33 43 v3 0 = (86) 43 44 v4 0 From th condition 33 43 = 3344 4334 = 0 (87) 34 44 on obtains th vals for which dtrmin th ignfrqncis of this bam. Ths ignfrqncis sd in on of th qation (86) givs th corrsponding ignmods. f matrial damping is nglctd soltion of th bam s wav qation can b writtn in ral fnctions [ ] v ( ) = sin cos sinh cosh (88) Tas: a) alclat analytically (with Matmatica) th first thr ignfrqncis and ignmods and compar it with th rslts givn in []; sction 8, p.3-34; st dition). b) calclat analytically (with Matmatica) th stiffnss matri of th bam lmnt. Tas Fig. 8: Towr-li strctr citd by a horizontal harmonig bas roc motion, v ( = v cosωt, v = 0. g g rigid g,r E η ρ M, v, h v, r v g,r alclat analytically possibly with a comptr program li Mathmatica - for th strctr shown in Fig. 8 a) th ignfrqncis ωi and ignmods φi (),,,3 nglcting damping, b) th frqncy rspons of th top of th strctr, v, h and v, r, nglcting matrial damping, c) th frqncy rspons of th top of th strctr, ( ω ), with strctral damping η = 0.0, d) th dformation (, with th frqncy ω as paramtr ) th dformation (, t, ω) with damping cofficintη = 0. 0. f) Plot snap shots of th dformation (, for slctd frqncis ω and tim instants t with and withot damping and show ths rslts as animation v g,h - 0 -
Wavs and Vibration 4 trctr and b-grad 4. Dynamic systm A ivil Enginring strctr is, in contrast to a flying objct, always fondd on th soil (sbgrad). Th strctr is in its tnsion finit, th sb-grad gnrally tnds to infinity. Th dynamic nrgy introdcd into th strctr is containd in it sinc th nrgy is rflctd at its bondaris. Th dynamic nrgy introdcd into th sb-grad is dissipatd and radiats to infinity. Thrfor a strctr fondd on th sb-grad loss nrgy throgh th common soilsstrctr intrfac. Th sb-grad acts as a dampr for th strctr. This is vn th cas if all matrial damping in th systm is nglctd. Using th mastr-slav concpt th dynamic stiffnss matrics, which ar compl, frqncy dpndnt and contain stiffnss, damping and mass proprtis, can b condnsd to th ssntial DOFs withot loss of accracy. Any linar rhological matrial bhavior can b modlld with frqncy-dpndnt compl matrial fnctions. By copling th sb-strctrs, on obtains th discrt qation of th soil-strctr systm in total displacmnts (lastic dformation pls rigid body motion): 0 V P F F + G V = P F F 0 G GG VG PG (89) whr and F ar th dynamic stiffnss matrics of th sb-strctrs (ppr ind for strctr, F for fondation or soil). Th systm s N inmatic DOFs V ar partitiond in thos on th intrfac, V, th rmaining ons on th strctr, V, and som on th sb-grad, V, whr trnal forcs may act. P, P, P ar th dynamic forcs corrsponding to th inmatic G DOFs, indicatd in Fig.5. G trctr ntrfac Prscribd motion G V V = V V G Fondation F Fig. 9: Dfinition of inmatic DOFs - -
Wavs and Vibration Thy ar in gnral compl and frqncy-dpndnt. W assm that a sismic vnt is prodcd at th bondary G of th considrd rgion throgh a spcifid tim-harmonic grond motion V i t G ω with frqncyω. W assm frthr that no othr trnal loads act on th strctr ( P = P = 0). Thn th nnown strctral displacmnts, V and V, ar obtaind from th first two qations of th matri qation (49) rslting in V 0 F = F + V GVG n qation (50) th trm F GVG = P ff, is ndrstood as th driving forc of th motion acting at th intrfac. Th sismic sorc may b so far away that it can not b incldd in th modl. Thn as an altrnativ dscription of th sismic vnt, th fr fild motion, Th rlation btwn fr fild motion V ʹ and prscribd motion G trms of th strctr ar omittd in th systm qation Ths F F G G (90) ' V, can b sd. V can b obtaind if th Vʹ = V (9) F F dfins th fr fild motion Vʹ = V in trms of th prscribd motion V G and dfins at th sam tim th driving forc F F ff, G G G G P = Vʹ = V (9) acting at th intrfac. Ths th final systm qation can b writtn as V 0 F = (93) + V P ff, f a rigid bas plat with mass M and mass momnt of inrtia r with rspct to its cntr of gravity is considrd in th analysis its dynamic stiffnss (impdanc) is addd to. n this cas qation (54) rslts in V ω M V F + Pff, = 0, (94) whr M is a diagonal matri with ntris M and r for translational and rotational DOFs, rspctivly, of th fondation plat. - -
Wavs and Vibration 4. llstrating ampl To simplify th formlation w considr as strctr a Brnolli bam fondd on a rigid bas plat. As sb-grad w assm a homognos lastic halv-spac, s Fig. 6. Firstly, all matrial damping is nglctd. W assm th systm is sbjctd to a Rayligh-fr fild wav. half-spac half-spac G, ρ, ν G, ρ, ν Fig. 0: trctr-oil-ystm, lft; fr-fild citation, right. Th dynamic qation in th frqncy domain for th systm with 5DOGs is with ( ω) V( ω) = P( ω), (95) Vh V v V = V, h ; V,v V, r 0 0 P = Pff, h ; Pff,v 0 = + V ar th dynamic DOFs of th systm, P ar th corrsponding loads and is th sm of th dynamic stiffnss matri of th strctr and th dynamic stiffnss matri of th soil-strctr F intrfac. For a srfac fondation can b assmd as diagonal so that P V ff = h F,h ʹ ; P V ff = h F, ʹ v. (97), h,v inc th horizontal (anti-symmtric) and vrtical (symmtric motions ar dcopld w may analyz th two motions sparatly. For ach dgr of frdom: R Pi = Pi + i Pi R = + i i i i R ( P ) ( P ( ) P i i + i = ; ( R ) ( ( ) i i + i = ; tan ψ = m V,v M, 0 tan ϕ = i R i i R i P P 0 V,v E A, V,r V,h V,h rigid F O 0 a Vʹ,v Vʹ,r a Vʹ,h (96) - 3 -
Wavs and Vibration M m V 0 V E A half-spac G, ρ, ν rigid m O M, 0 0 E, half-spac G, ρ, ν V,r V V,h rigid Fig. : trctr-oil-ystm for vrtical motion, lft; for horizontal motion, right. 4.. Vrtical citation trctr W assm that th strctr is modl (ppr ind ) is rprsntd throgh on bar lmnt with point masss m and M at th top and th bas of th towr-li strctr, Fig.8,. n th frqncy domain th dynamic stiffnss matri of th strctr tas th form m 0 = ( ω) ω M; with M = (98) 0 M is th dynamic stiffnss of th bar lmnt, sction 3., and M rprsnts th inflnc to th dynamic stiffnss of th point masss. Th towr-li strctr is sbjctd to th vrtical componnt of a harmonic Rayligh-wav. m V V E A M rigid intrfac Fig. : b-strctr strctr for vrtical motion: strctr b-grad W assm that th sb-grad is a visco-lastic half-spac. W assm frthr that th intrfac btwn strctr and sb-grad is constraint throgh th stiffnss of th fondation bloc. Th intrfac is thrfor also considrd as rigid. Whnc th sbstrctr sb-grad has only th on dynamic DOF V. - 4 -
Wavs and Vibration intrfac V rigid, masslss R 0 R O Fig. 3: b-strctr sb-grad for vrtical motion F Th corrsponding dynamic stiffnss is dfind as ( ω ). t dpnds on th gomtry of th intrfac and of th gomtric and matrial proprtis of th sb-grad. n cas of a homognos half-spac and a circlar fondation, Fig. 9. th dynamic stiffnss matri can b obtaind analytically. Mor complicatd cass ar fond in th litratr [7] or hav to b obtaind throgh advancd nmrical mthods. For illstrating prposs w assm hr a circlar fondation with radis R sitatd on top of a homognos half-spac with Poisson s nmbr ν 0. 3 (sand). A good approimation is thn for th dynamic stiffnss (impdanc) in [7]: F F 4GR ( ω ) = 0 + iω = + i0.85 (99) ν F whr 0 is th static stiffnss and is th viscos damping constant. Ecitation W assm a fr-fild Rayligh wav with frqncyω. W assm that th wav lngth of th Rayligh wav is mch largr than th radis R of th fondation and nglct thrfor th rotation of th fr-fild for th intrfac. Th ffctiv arthqa forc is thn, according to q (5) F Pff = 0 ʹ whr ʹ is th amplitd of th fr-fild at th soil-strctr intrfac. half-spac G, ρ, ν ystm qation Th discrtizd wav qation for th soil-strctr systm, accoding q () is ss ( ω) ω m s ( ω) s ( ω) ( ω) ω M + F v ( ω) v = F 0 ( ω) vʹ (00) - 5 -
Wavs and Vibration 5 as stdis 5. trt scn in HM-ity A strt vndor in HM ity carris hr load of frits on a bamboo stic throgh th city. n ach of hr basts sh carris a load of 5 g. Whn th vndor stands qitly th displacmnts at th stic s nds ar cm. Whn sh wals, sh mas stps pr scond. n ach stp hr sholdr ndrgos a half sinsoidal vrtical motion with an amplitd of 5 cm. Assm that th strings which carry th basts ar rigid in longitdinal dirctions and that th mass of th strings and th basts can b nglctd. Assm th damping ration for th stic as ξ = 5%. a) alclat th ndampd and dampd ignfrqncy of th bam with th basts. b) ppos th vndor wals stadily. alclat th rspons of th basts. Plot th displacmnts of th basts and show th motion of th vndor s sholdr in th sam plot. c) alclat th forc on th sholdr and plot it. d) Discss: Which stp frqncy of th strt vndor wold crat th largst forc on hr shold; with which stp frqncy shold th vndor wal to rdc th forc on hr sholdr. - 6 -
Wavs and Vibration 5. tdy rlatd to th dynamic soil-strctr intraction in cas of arthqa Towr li strctr Antnna: Bam: Bam: =80m =6m E=80000 6 Pa η = 0.0 E=0000 6 Pa ; η = 0.0 A=m A=0.075m =57 m 4 =.9 m 4 ρs =3500 g/m 3 ρs =7850 g/m 3 Fondation bloc: R=7.5m c=.5m ρf=400g/m 3 oil: G=40 6 Pa ρ=900g/m 3 R Fig.0: Data of two strctr on sam fondation R Establish a mchanical and nmrical modl for th dynamic analysis if th strctr shown in Fig. is sbjctd to a Rayligh wav. Fig. : Raligh wav: horizontal, R, and vrtical, v R, displacmnts, scald with th Rayligh wav lngth λ R. 4GR Hint: Us from [7]: Horizontal static stiffnss: 0 F = ; damping constant: = 0. 58. ν - 7 -
Wavs and Vibration 6 itratr [] iril tojanovsi, Analysis of a strctr ndr vrtical low and high frqncy citation considring soil-strctr intraction, Diplom Thsis, s. iril and Mthodis Univrsity opj, 006 [] logh and Pnzin: Dynamics of strctrs, p.393 [3] Ptronijvic, M., G. chmid, Y. olova: "Dynamic soil-strctr intraction of fram strctrs with spctral lmnts Part ", GNP008, Žablja 3-7 Mart, 008 [4] Hillmr, P., G. chmid, alclation of fondation plift ffcts sing a nmrical aplac transform, Earthqa nginring and strctral dynamics, vol.6, 789-80 (988) [5] Hillmr, P. Brchnng von tabtragwrn mit loaln Nichtlinaritätn ntr Vrwndng dr aplac-transformation, TWM-Hft Nr.87-,nstitt für onstrtivn ngnirba, Rhr-Univrsität Bochm, 987 [6] Domingz, J. Bondary lmnts in dynamics, omptational Mchanics Pblications, othampton, Boston, 993 [7] ifrt, J.G., Handboo of impdanc fnctions, Ost Editions,,r dl la No, Nats, 99-8 -
Wavs and Vibration Appndi A ompl rprsntation of harmonic vibration Harmonic motion in th compl plan. Elr s pair of qations srvs to transform trigonomtric to ponntial fnctions: iωt = cos(ω + i sin(ω; -iωt = cos(ω - i sin(ω iω t iω t iω t iω t cosωt = ( + ); sinωt = ( ) i t sin ωt+ cos ωt = ω = Fig.A: Rprsntation of harmonic vibration in compl plan Using th ponntial fnction w can writ th displacmnts v(as: iωt % %, whr R v( = v v= % v + iv is a compl qantity. Or sing th modls (amplitd) and th argmnt (phas angl) of th motion i(ωt+θ) R v( % = v% with amplitd v= % (v ) + (v) and phas angl θ obtaind from v tan θ =. v R Fig.: A: ompl plan rprsntation of th vctor v. Using th Figr A, on can asily imagin th vibration as a projction of th rotating vctor v% onto th rl ais: v( = R{ v( % } = ( v( % + v( % ), whr v%is th compl conjgat of v%. iωt Also t { v R v( ) = R } = v cosωt v sinωt m v v v R v( v v v R Fig. A3: Rprsntation of v( in compl plan - 9 -