Chapter 11.00C Physical Problem for Fast Fourier Transform Civil Engineering
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1 haptr. Physical Problm for Fast Fourir Trasform ivil Egirig Itroductio I this chaptr, applicatios of FFT algorithms [-5] for solvig ral-lif problms such as computig th dyamical (displacmt rspos [6-7] of sigl dgr of frdom (SOF watr towr structur will b dmostratd. Fr Vibratio Rspos of Sigl gr-of- Frdom, (SOF Systms Figur SOF dyamic (watr towr structur systm..b.
2 .. haptr. Figur Watr towr structur subjctd to dyamic loads. a Watr towr structur, Idalizd as SOF systm. b Impuls blast loadig F (t, or arthquak groud acclratio g (t. Th dyamical quilibrium for a SOF systm (show i Figur ca b giv as: m y + cy + ky t F si( wt ( m, c ad k mass, dampig ad sprig stiffss, rspctivly (which ar rlatd to irtia, dampig ad sprig forcs, rspctivly. y, y, y displacmt, vlocity, ad acclratio, rspctivly. Practical structural modls such as th watr towr structur subjctd to applid blast loadig (or arthquak groud acclratio tc. ca b covitly modld ad studid as a simpl SOF systm (show i Figur. For fr vibratio rspos, Equatio ( simplifis to m y + cy + ky t ( Th solutio (displacmt rspos y of Equatio ( ca b xprssd as pt y( t Q displacmt (3 Hc pt dy y Qp vlocity (4 dt pt d y y Qp acclratio (5 dt Substitutig Equatios (3-5 ito Equatio (, o obtais mp + cp + k (6 Th two roots of th abov quadratic quatio ca b obtaid as
3 Physical Problm for FFT: ivil Egirig.B.3 c ± c 4( m( k p (7 m c ± m c m ritical ampig ( cr k m I this cas, th trm udr th squar root i Equatio (8 is st to b zro, hc cr k (9 m m or cr km ( sic k w ( m Hc cr mw ( k w Th two idtical roots of Equatio (8 ca b computd as p, cr p (3 m ad th solutio y (t i Equatio (3 ca b giv as pt pt y( t Q + Q t (4 cr t m + Qt ( Q (5 which ca b plottd as show i Figur 3. (8
4 ..4 haptr. Ovr dampig ( > cr Figur 3 Fr vibratio with critical dampig. I this cas, o has k > (6 m m Th solutio of y (t from Equatio (3 ca b giv as pt pt y( t Q + Q (7 Th rspos of ovr dampig systm is similar to Figur 3. Udr ampig ( < cr I this cas, o has k < m m ad th two complx roots from Equatio (8 ca b giv as, k p p ± i (9 m m m θ Substitutig Equatio (9, ad usig Eulr s quatio [ i cos( θ + i si( θ ], Equatio (3 or Equatio (7 bcoms ( c / m t y( t ( Acos w t B w t + si ( k w s Equatio (9 ( m m (8
5 Physical Problm for FFT: ivil Egirig.B.5 ξ w ξ ( cr (3 km Usig th iitial t ; y y ; y v (4 Th, th two costats ( A ad B ca b solvd, ad Equatio ( bcoms + ξwt v yξw y( t y w t + w t cos si (5 w Equatio (.6 ca also b xprssd as: ξwt y( t K cos( wt α (6 K ( v + y ξw y + (7 w ta ( α v + y ξw (8 w y Equatio (6 ca b plottd as show i Figur 4.
6 ..6 haptr. Figur 4 Fr vibratio of SOF udr dampd systm. Forc Vibratio Rspos of SOF Systms For forc vibratio problm, th right-had-sid (RHS of Equatio ( t, ad th gral solutio for Equatio ( ca b giv as y( t yc ( t + y p ( t (9 th complimtary solutio y c (t ca b obtaid as (s. Equatio ( assumd udr-dampd ( < cr cas ( / m t y ( t ( Acos w t Bsi w t (, rpatd c + k t * k m m ( Acos w t + B si w ( k / m t / km ( Acos wt + B si wt Usig Equatios ( ad (, Equatio (3 bcoms wt / cr yc( t ( Acos wt + Bsi wt Usig Equatio (3, th abov quatio bcoms wt y ( t ξ ( Acos w t Bsi w t c + t (3 (3
7 Physical Problm for FFT: ivil Egirig.B.7 Th particular solutio y p (t, associatd with th particular si trm forcig fuctio F ( t F si( wt s Equatio ( ca b giv as y p ( t si( wt + cos( wt (3 Th ukow costats ad ca b foud by substitutig Equatio (3 ito Equatio (, ad quatig th cofficits of th si ad cosi fuctios. Usig Eulr s idtity, o has i wt cos( wt + i si( wt (33 Thus, th RHS of Equatio ( ca b xprssd as iwt m y + cy + ky F Imagiary portio of (34 Hc, th rspos will cosist of OLY th imagiary portio of Equatio (9. Th particular solutio y p (t, show i Equatio (3, ca b mor covitly xprssd as * iwt y p ( t (35 Substitutig Equatio (35 ito Equatio (34, o gts * iwt * iwt * iwt iwt m { i w } + c{ iw } + k{ } F (36 or * { k + icw mw } F (37 Hc * F (38 k mw + icw Substitutig Equatio (38 ito Equatio (35, o obtais F iwt y p ( t (39 k mw + icw I Equatio (39, th complx umbr d ( k mw + i( cw (4 ca b symbolically xprssd as d ( d R + i( d I (4 or i polar coordiats, o has (s Figur 5 iθ d d d cos( θ + i si( θ (4 { } si( θ ta( θ (43 cos( θ cw k mw d R k mw (44 d I cw (45 ( d ( R d I d + (46
8 ..8 haptr. ( k mw + ( cw (47 Thus, Equatio (39 ca b r-writt as: iwt F y p ( t (48 i k mw + ( cw ( θ F i( wt θ ( k mw + ( cw (49 Figur 5 Polar coordiats. Th imagiary portio of Equatio (49 ca b giv as F si( wt θ y p ( t (5 ( k mw + ( cw fi F Y amplitud of th stady stat motio (5 k mw + ( cw ( F y st static dflctio of a sprig actd by th forc F k (5 w r frqucy ratio (of applid load/structur w (53 Th, Equatios (43 ad (5 bcom
9 Physical Problm for FFT: ivil Egirig.B.9 ξr ta( θ ; also rfr to Equatio (3 r (54 y p ( t Y si( wt θ (55 y st si( wt θ (56 ( r + (ξr Th complimtary (or trasit solutio y c (t show i Equatio (3, ad th particular solutio y p (t show i Equatio (56 ca b substitutd ito th gral solutio (s Equatio (9 to obtai ξwt yst si( wt θ y( t ( Acos wt + Bsi wt + ( r + (ξr (57 fi Y y st (58 ( r + (rξ yamic Magificatio Factor (59 yamical Rspos by Fourir Sris, FT ad FFT. Th dyamic load F (t actig o th SOF systm ca also b xprssd i Fourir sris as + a cos( wt + b si( wt F ( t a (6 th ukow Fourir cofficits ca b computd as a a T T t + T t t + T t t dt tcos( wt dt b tsi( wt dt T If th forcig fuctio cotais oly si trms, th th particular (stady stat solutio ca b foud as (s Equatio (56: y y p (6 b si( wt θ k r + r ξ ( ( (6
10 .. haptr. b si( wt cos( θ si( θ cos( wt (63 k Rcalld Equatio (54, o has si( θ ta( θ cos( θ r ξ r Hc si ( θ si ( θ x cos ( θ si ( θ [ ] cos ( θ [ cos ( θ y ] ( ξr ( r ( r + (r ξ Solvig Equatio (64 for ( x si( θ ad ( y cos( θ, o gts x siθ ξr r + ξr ( ( y cosθ (65 r ( r + ( ξr Substitutig Equatio (65 ito Equatio (63 to obtai: y ( t (66 y p ( r si( wt ( ξr cos( wt ( r + ( ξr b k Similarly, if th forcig fuctio cotais oly th cosi trms, th th particular (stady stat solutio ca b foud as: y ( t (67 y p ( r cos( wt + ( ξr si( wt ( r + ( ξr a k Fially, if th forcig fuctio cotais both si ad cosi trms, th th total rspos ca b computd by combiig both quatios (66 ad (67, icludig th costat forcig trm a, as followig (64
11 Physical Problm for FFT: ivil Egirig.B. y( t + k y b ( t a k ( r + a ( ξr ( r + ( ξr a si( wt + ( r b ( ξr ( r + ( ξr cos( wt (68 Rmarks Usig Eulr s rlatioships, th dyamic load F (t as show i Equatio (6, ca also b xprssd i xpotial form as iwt ( (8, h.. F t T iwt t dt (, h.. T For FT, dfi T t ; with t, t, t,... t (69 t j j t (7 Th, th FT pairs of Equatios (,, h..4 bcoms: ad ~ k t j j t t t ~ j π i j π ik w t T π ik t t π i j ; with,,,... (7 ; with j,,,... (7 Sic both Equatios 7 ad 7 do hav similar opratios, with th xcptios of th factor ad th sig ( or + of th xpotial trm, both ths quatios ca b hadld by th sam gral_dft program giv at Itroduc th uit amplitud xpotial forcig fuctio F iw t ( t ( F (73
12 .. haptr. ito RHS of Equatio (, th stady stat solutio ca also b obtaid as (s Equatio 39: iwt y( t y p ( t k mw icw (39, rpatd + Usig th otatios dfid i Equatios (3 ad (53, th abov quatio ca b writt as, for a harmoic forc compot of amplitud ~. ~ i( w w ( t j j t y ( t j k( r + iξr ~ π i w j t T k( r + iξr ~ π i j t t k( r + iξr ~ ij π / (74 k( r + iξr ad th total (stady stat rspos du to harmoic forc compots ca b calculatd as ~ ij π / y( t j (75 k r + iξr ( yamic Rspos of Watr Tak Structur by FFT. Th dyamic rspos y( t j i frqucy domai of a gral SOF systm (such as th watr tak structur ca b obtaid by Equatio (75, ad th rquird cofficits c ~ ca b computd by Equatio (7. Both of ths quatios ca b rprstd (xcpt for th sig, by th followig gral xpotial fuctio ( j A ( j factor * A ( W (76 sig * iπ / W (77 If Equatio (7 ds b computd for ~, th o should dfi factor, sig -, ( ad A t j. Howvr, if Equatio (75 ds b computd for y ( t j, th o should ~ ( dfi factor, sig +, ad A k( r + iξr. It is importat to otic that Equatio (76 has th sam form as show i th arlir Equatio (74. Howvr, th dfiitio of W i Equatio (77 is diffrt from th o show i Equatio (4, h..5 by a gativ sig i th powr of W. Thrfor, fficit
13 Physical Problm for FFT: ivil Egirig.B.3 FFT subrouti (with usr s spcifid SIG, or - ca b utilizd, as giv at Rfrcs [] E.Ora Brigham, Th Fast Fourir Trasform, Prtic-Hall, Ic. (974. [] S.. hapra, ad R.P. aal, umrical Mthods for Egirs, 4 th Editio, Mc-Graw Hill (. [3] W.H. Prss, B.P. Flary, S.A. Tkolsky, ad W.T. Vttrlig, umrical Rcipis, ambridg Uivrsity Prss (989, haptr. [4] M.T. Hath, Scitific omputig, Mc-Graw Hill (997. [5] H. Josph Wavr, Applicatios of iscrt ad otiuous Fourir Aalysis, Joh Wily & Sos, Ic. (983. [6] Mario Paz, Structural yamics: Thory ad omputatio, d Editio, Va ostrad Ic. (985. [7] R.W. lough, ad J. Pzi, yamics of Structurs, Mc-Graw Hill (975.
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