NUMERICAL RESEARCH ON THE EQUIVALENT TRANSFORMATION BETWEEN STRUCTURAL DYNAMIC ANALYSIS IN TIME-DOMAIN AND FREQUENCY DOMAIN

Transcription

1 he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi NUMERICAL RESEARCH ON HE EQUIVALEN RANSFORMAION BEWEEN SRUCURAL DYNAMIC ANALYSIS IN IME-DOMAIN AND FREQUENCY DOMAIN Yuelig Jig,Jibo Li d Go Li 3,, 3 Ist. Of Erthquke Egg., Dep. Of Civil d Hydr. Egg., Dli Uiv. of ech., Dli, Chi Emil: yljig@6.com, jigyue@mst.edu ABSRAC : Numericl lysis methods i time-domi d frequecy-domi re commoly cosidered s two importt wys for seismic evlutio of structure resposes. I terms of the seismic wve excittio d the structurl output respose, the expressios of time histories re usully pplied i the time-domi method, while the complex hrmoic wves or their summtio geerlly used i the frequecy-domi method, it is focus i the field of structure egieerig to costruct equivlet expressios of seismic wve excittio d structurl resposes betwee time-domi d frequecy-domi solutio methods. I this pper, s fr s geerl dymic lysis of structure is cocered, formul of trigoometric coefficiets i time domi is deduced to compute frequecy spectrum vlues t rbitrry frequecy poits for time history dt, which voids the disdvtge of the covetiol discrete Fourier trsform (DF) method tht merely suitig for the discrete frequecy poits. Hece, ccordig to the seismic lysis of structurl resposes, qutified ssessmet of equivlet expressios for vrious wve sigls re give i detil, which builds trsformtio bridge betwee time-domi d frequecy-domi solutio methods. Filly, the vlidity d fesibility of the trsformtio lgorithm i time-domi d frequecy-domi re umericl verified by the dymic respose lysis of multiple prticles dmp system d log sp structure with seismic wve excittios. It s lso show from the results, s the trigoometric coefficiet method (CM) is cocered, oly iputted the rel prt or the imgiry prt of the seismic wve exctio time history, the equivlece respose of structure i frequecy domi c be obtied by simple combitio of trigoometric coefficiets i time domi. KEYWORDS: Seismic lysis of structurl resposes, Seismic wve, rigoometric series, Fourier trsform. INRODUCION Numericl lysis methods i time d frequecy domi re powerful tools for the dymic lysis of structure. Sice i frequecy domi, the mplitudes of the structure dymic respose t differet frequecies c be ccurtely described i simple formultio, d it is coveiet to express the frequecy cotet of the groud motio; so it is commoly used i the derivtio of structure respose lysis, such s soil-structure iterctio lysis [] d groud respose []. I geerl, the structure respose i frequecy domi c be cosidered s the stedy-stte of the respose i time domi. With the developmet of computer, more d more dymic lysis methods i time domi re used i erthquke egieerig. Compred with the methods i frequecy domi, methods i time domi re ble to ccout for the chrcteristics of olier behvior of the structure [, 3, 4, 5], d it c be express the dymic respose of structure t y time. So my methods of structure dymic lysis re trsferred from frequecy domi to time domi, for exmple, structurl dmge detectio [6]. As result, it is ecessry to prctice the umericl reserch o the equivlet trsformtio betwee structurl dymic lysis i time-domi d frequecy-domi. As we ll kow, vibrtory motio is physicl qutity vried with time, it is c be described i frequecy domi d time domi. So it is possible for vibrtory motio processig betwee frequecy d time domi. I frequecy domi, reserchers usully used the discrete Fourier trsform (DF) to obti the mplitude of the vibrtory motio t the specific frequecies, but DF method limits the output frequecies becuse of the

2 he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi frequecy itervl Δ.So it is impossible for DF method to get the spectrum t y frequecies. While i this pper, bsed o the ottios of the vibrtory motio: trigoometric form i time domi d complex expoetil form i frequecy domi, the equivlet trsformtio coditio of vibrtory motio formultio betwee frequecy domi d time domi is derived. he equivlet trsformtio reltioship c be expressed by trigoometric coefficiets, which is described by lyticl formultio. Ad therefore, the spectrum t y frequecies c be obtied by usig the methods i time domi. Furthermore, how to costruct the equivlet seismic wve excittio d how to uderstd the respose of the structure i time d frequecy domis? It is iterestig d ttrctive topic i civil egieerig. I this pper, with respect to the dymic respose lysis of dmped system subjected to hrmoic lodig, the respose solutios i time d frequecy domi re derived. A ssessmet o reltioship of respose solutio betwee time d frequecy domis is crried out. From the ssessmet, it is obvious tht, oly iput the rel prt or imgiry prt of the excittio seismic lodig, the respose i frequecy domi c be obtied by the combitio of trigoometric coefficiets, which c be clculted by the respose of structure i time domi. Filly, the vlidity d fesibility of the equivlet trsformtio coditio betwee time-domi d frequecy-domi re umericl verified by the dymic respose lysis of multiple prticles dmp system d log sp structure subjected to seismic wve excittio. he umericl results showed tht the trigoometric coefficiet method (CM) d DF method re ideticl i clcultig the spectrum of frequecy. Furthermore, the dvtge of the former is the cotiuity of output frequecies.. EQUIVLEN CONDIION OF VIBRAORY MOION FORMULAION BEWEEN FREQUENCY DOMAIN AND IME DOMAIN Vibrtory motio c be described i terms of displcemet x ( by three wys, which will be preseted s follows: (A) he motio x( c be expressed by trigoometric ottios: x( = = A cos( t ϕ ) (.) Where A represets the displcemet mplitude, is the circulr frequecy, d ϕ is the phse gle. (B)he motio x( c be cosidered s summtio of simple hrmoic fuctios; it c be expressed by usig trigoometric ottios: x( = + = ( cos t + b si (.) Where the trigoometric coefficiets re = x( dt (.3) = x( cos tdt (.3b) b = x( si tdt (.3c)

3 he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi Ad = π /, is the durtio of the motio, represets the verge vlue of x ( over the rge t = tot =. (C) he motio x( c lso be expressed i expoetil form. ( ) (.4) π i t x t = c e d Where c is the spectrum of circulr frequecy frequecy, it c be preseted s, c is the spectrum mplitude of the circulr c = / / x( τ ) e i τ dτ (.5) Equtio (.), d equtio (.4) is differet formultio of x (, they re equivlet i the frmework of mthemtics [7]. he reltioship betwee usig Euler s lw. c d, b c be derived directly from the expoetil form of the motio by c = i b (.6) Form the equtio (.6), we c coclude tht the trigoometric coefficiets d b reflect the rel prt d imgiry prt of c, respectively. Hece, except for DF method, we itroduce ew method to obti the spectrum; it is referred s the trigoometric coefficiet method (CM). I fct, t the specific frequecies, we my be ot got its spectrum by DF method becuse of the discrete frequecies itervl Δ. I order to remedy this poit, it is usul to dd log segmet of zero vlue t the ed of motio durtio, d the cpture the spectrum pproximtively. I doig so, the iformtio of the time history of the motio will be destroyed to certi extet. Altertively, there re lyticl formultios for db, so the spectrum of y circulr frequecy c be obtied ccurtely by CM. 3. FORMULAION OF DYNAMIC RESPONSE ANALYSIS OF SRUCURE IN FREQUENCY DOMAIN AND IME DOMAIN I the process of dymic respose lysis of structure, the time history of the motio mily cosists of seismic wve excittio d output structure respose. I the frequecy domi, the seismic wve excittio c be described s complex expoetil form or their summtio; ccordigly, the output respose of the structure c lso be described s complex expoetil form. I the time domi, the seismic wve excittio is expressed i the form of time history, bsed o tht, how to costruct the equivlet seismic wve excittio betwee time domi d frequecy domi d how to uderstd the output respose of the structure, it is oe of most importt d iterestig topics i erthquke egieerig. I geerl, this problem c be summrized s two poits. Oe poit is: oly the rel prt or imgiry prt of the seismic wve excittio is iputted, the complete respose of structure i the frequecy domi c be obtied. he other poit is: the rel prt d imgiry prt of the iput motio eed to iput, respectively, d the the complete respose of structure i the frequecy domi c be obtied by

4 he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi combiig the respose of the structure i time domi. I the ext sectio, we will tke dmped system for exmple d discuss this problem i detil. 3.. Respose of dmped system subject to periodic lodig i frequecy domi I order to evlute the dymic respose of dmped system, the differetil equtio of motio must be solved. First, the equtio of motio c be writte s i & t + & (3.) u ζu + u = P e Where represets the turl circulr frequecy of the system, the respose of the system c be relted to the lodig by u( = U e (3.) Where U is the mplitude of respose i frequecy domi, substitutig equtio (3.) ito the equtio of motio gives u( = P ~ (siθ i cosθ ) e (3.3) he mplitude of respose i frequecy domi is U = P siθ i P ~ ~ cosθ (3.4) Where ~ ˆ = =, ζ ~ ˆ siθ =, cosθ = (3.5) ~ ~ 3.. he reltioship betwee Respose of dmped system i time domi d i frequecy domi here re two types of exterl lodig i time domi uder which the dymic respose of the structure is equivlet to the respose i frequecy domi. i t First, the equtio of motio of the dmped system subject to the rel prt of the exterl lodig P e is writte s u& + ζu& + u = P cos t (3.6) he geerl solutio to the equtio of motio for dmped forced vibrtio c be obtied by combiig the complemetry d prticulr solutios. Note tht the complemetry solutio, which describes trsiet respose cused by the requiremet of stisfyig the iitil coditios, decys with time. After the trsiet dies out, oly the stedy-stte respose described by the prticulr solutio remis, so oly stedy-stte respose solutio is produced s follows P u( = (siθ cost + cosθ s) (3.7) ~ he trigoometric coefficiets of equtio (3.7) d b re obtied by the CM

5 he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi P P = siθ, b ~ = cosθ (3.8) ~ Compred with the mplitude of respose solutio i frequecy domi, which is writte s equtio (3.4), it is obvious tht the formultio of i b is cosisted withu, is the rel prt of theu, -b is the imgiry prt of theu. Secod, the equtio of motio of the dmped system subject to the imgiry prt of the exterl i t lodig P e is writte s u& + ζu& + u = P si t (3.9) he stedy-stte respose solutio of equtio (3.9) is writte s P u( = (cosθ cost siθ s) (3.) ~ I the sme mer, the trigoometric coefficiets of equtio (3.) d b re expressed s P P = cosθ, = siθ ~ ~ b (3.) Compred d b with equtio (3.4), the formultio of i( i b ) is the sme s the mplitude of respose solutio i frequecy domi, where is the imgiry prt of theu, b is the rel prt of theu. From the compriso metioed s bove, if both the rel prt d imgiry prt of the seismic wve excittio re iputted, it is ot correct to obti the mplitude of respose solutio i frequecy domi by combiig the respose of the structure i time domi. I relity, either the rel prt or the imgiry prt of the seismic wve excittio is iputted i time domi, it is coveiet to obti the mplitude of respose solutio i frequecy domi from the time history of respose of structure i time domi. As result, it lso supplies tool to verify dvced time domi lgorithms i frequecy domi. I sectio 4., the vlidity of CM is proved through dymic respose lysis of multiple prticles dmp system. I the sme wy, through CM, the respose of structure, which is subjected to the seismic wve excittio icluded phse differece d mplitude decy i frequecy domi, c lso be obtied. It lso provides theoreticl foudtio for dymic lysis of log sp structure subjected to multi-support seismic wve excittio. I order to verify its vilbility d ccurcy, the CM is prcticed o sigle sp bridge structure i sectio NUMERICAL EXAMPLES 4. Dymic respose lysis of multiple prticles dmp system he multiple prticles dmp system show i Figure 4. is t rest whe the hrmoic lodig F = P e is pplied, where P is 4.5N, is rd/s. Determie the mplitude of respose of prticle. he Ryleigh dmpig scle Figure 4. clcultio model

6 Displcemet(m) Spectrum he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi coefficiet is ζ =. 5, ccordigly, the turl circulr frequecy of first mode shpe is =3.48rd/s,the turl circulr frequecy of secod mode shpe is =.5rd/s. First, the equtio of motio is writte s u&& u&& u& u& u 3456 u 5 i P e = t (4.) he solutio to this equtio c be represeted s obtied s u = ue, u = ue, the lyticl solutio of prticle u = i (4.) So the mplitude of the respose of prticle is u =.53m. Secod, referrig to sectio 3., the equtio of motio of the dmped system subject to the imgiry prt of i t the exterl lodig P e i t c be writte s (4.), where the P e is replced by 4.5si(t ). Furthermore, he solutio of equtio c be obtied by the Precise ime-itegrtio Method [8], where u ( ) is plotted i Figure 4.. t ime(s) CM DF Circulr Frequecy(rd/s) Figure 4. Respose of prticle Figure 4.3 Frequecy spectrum of prticle Usig CM, the trigoometric coefficiet d b of u ( t ) c be obtied, bsed o tht, the mplitude of respose i frequecy domi c be got by i( i b ). At the sme time, the DF [9] method is used to get the mplitude of respose. he compriso of the results of spectrum is displyed i the Figure 4.3; the vlue of mplitude of respose is compred i the ble 4.. ble 4. Compriso tble of results i differet methods Rel prt (m) Spectrum (m) Alyticl solutio trigoometric coefficiet method () Reltive Error -4.6% -.3% trigoometric coefficiet method (4) Reltive Error -3.47% -.87% FF(9.8946) Reltive Error -8.7% -6.55% DF(9.8946) Reltive Error % -6.48%

7 he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi he rel prt d spectrum of the respose of the structure i frequecy domi t the specific circulr frequecy of rd/s re listed i tble 4., the i the brcket mes the durtio of u ( t ) i time domi is s, i the sme mer, the 4 i the brcket mes the durtio of u ( t ) i time domi is 4s, the i the brcket is pproximte to the specific circulr frequecy of rd/s, becuse the DF method c ot cpture the frequecy of rd/s ccurtely. he lyticl solutio is obtied by equtio (4.). From the results show i ble 4., the CM grees with the lyticl solutio very well, d s the durtio of the u ( t ) icrese, the reltive error is decrese. Compred with DF, there re lyticl formultios for db, so the spectrum of y circulr frequecy c be obtied ccurtely by the CM. While there is certi itervl of the circulr frequecy i the DF method, sometimes, it is ievitble to pproximte the specific circulr frequecy. So the CM is better th the DF method i the output of the frequecy iformtio. 4.. Dymic respose lysis of log sp structure For erthquke egieerig problems, dymic lodig ofte results from vibrtio of the supports of system rther th from dymic exterl lods. o evlute the vilbility of the CM i such systems, exmple of dymic respose lysis of log sp structure is preseted i this sectio. I Figure 4.4 is simple sigle sp structure. A EA EI l B EA EI l EA EI l vg C D vg Figure 4.4 clculte model Where EI =, EI =, EA =, EA =, l =, l =, m 5 = m =. the turl circulr frequecy of first mode shpe is =.447rd/s,the turl circulr frequecy of secod mode shpe is =.83rd/s. he i t prticle C d D re subjected to horizotl excittio e i( t ϕ ) d Ae, respectively, i which A is.6, is rd/s, ϕ is.. Determie the respose of prticle B. First, for this cse of bse shkig the equtio of motio c be expressed s m u& + cu& + ku = mrv& g (4.3) i(t ) i(t ) Where r k k g v& g = e, e, u = { u }, u, u 3, u 4.he expressio v& & g i the right of equtio (4.3) represets the free-field iput ccelertio pplied t the bse of the structure. I more geerl cse, where the reltive displcemets re ot ll mesured prllel to the groud motio, the totl displcemet my be expressed s the sum of the reltive displcemet d the qusi-sttic displcemet tht would result from sttic-support displcemet. Accordig to theory of dymics of structure, the solutio of equtio (4.3) is writte s =, { } u = ue, u = ue, u3 = u3e, u4 = u4e, (4.4)

8 Displcemet/m Spectrum/m he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi Substitutig equtio (4.4) ito equtio (4.3), the horizotl respose of prticle B i frequecy domi is obtied s u = i.accordigly, the spectrum vlue of the prticle B is 3 + u 3 =8m Secod, the imgiry prt of the complex horizotl excittio is represeted t the bse of the sigle sp structure, the equtio of motio c be writte s (4.3), where v& { } g = sit,.6si(t -), through the coefficiet.6, the decy of the mplitude of the seismic wve is cosidered. he time lg betwee C d D is s, which reflects the excittio time is differet t differet support CM DF ime/s Figure 4.5 ime history of horizotl displcemet of prticle B Circulr Frequecy/rd.s - Figure 4.6 he horizotl spectr of prticle B he solutio of equtio is obtied by the Precise ime-itegrtio Method [3], where u3 ( is plotted i Figure 4.5. d b of u 3 ( c be obtied by the CM. Ad the respose solutio i frequecy domi is expressed s i( i b ). I other sides, the spectrum t the specific frequecy is lso got by the DF. he compriso betwee the spectr curve is show i the Figure4.6. he vlue of mplitude of respose is compred i the ble.4. ble 4. Compriso betwee theory solutio d the umericl solutio o the horizotl spectr of prticle B Durtio of time history of respose 4s 48s heory solutio 8 8 trigoometric coefficiet method Reltive Error -.95% -.89% From the ble 4., the horizotl spectr of prticle B obtied by the CM grees well with the theory solutio, the reltive error does ot exceed %. Ad the ccurcy of the clcultio will be improved if the durtio of time history of respose is loger. As show i Figure 5, the differece betwee the spectr which clculted by the DF d the CM is very smll, but i the method of DF, the durtio of the respose is divided ito N equl itervls Δ t, s result, the spectr c oly be obtied t the discrete frequecies. Compred with DF, the CM c be output the spectr t the cosecutive frequecies. 5 CONCLUSION I this pper, through derivtio d the umericl exmple of the equivlet coditio betwee the CM d DF, the coclusio is summrized s follows:

9 he 4 th World Coferece o Erthquke Egieerig October -7, 8, Beijig, Chi () he spectrum t y frequecies c be obtied by the CM, ccordigly, the reltioship betwee trigoometric coefficiets d spectr c is writte s ib = c / (5.) () I relity, the CM is powerful tool to trsfer respose of structure from time domi to frequecy domi. As fr s the dymic respose lysis of structure is cocered, the rel prt or imgiry prt of the exterl lodig c oly be pplied o the structure, d the clculted the time history of the structure respose, the d b of the respose is obtied by the CM. Accordigly, the respose of the structure i the frequecy domi c be expressed s i b or i i b ). ( (3) We compute the umericl results of the dymic respose lysis of multiple prticles dmp system d log sp structure d compre them with the results obtied by the DF s well s the theory solutio. he compriso verifies the vilbility d ccurcy of the CM. Ad it is lso show tht the trvelig wve effect d mplitude decy of the wve c be cosidered i this method. I future, this method c be used to do the dymic respose lysis of log sp structure subjected to multi-support excittio (4) I the previous methods of geertig rtificil groud motios comptible with respose spectrum, the trget spectrum is usully obtied by the sttisticl method bsed o phse-differece spectrum d Fourier mplitude spectrum []. he CM itroduced i this pper c be used to get spectrum of rbitrry frequecy, so the spectrum of the specific frequecy c be set s trget prmeter, it is possible to evlute the costrit equtio to produce some rtificil groud motios o bsis of the prmeter optimiztio. ACKNOWLEDGEMENS he support provided by the Nturl Sciece Foudtio of Chi (958 d 55783) is grtefully ckowledged. REFERENCES []Li, Jibo (5). Numericl Alysis of the Ifiite Soil-structure Dymic Iterctio Withi the Frmework of ime-domi Substructure Method. PhD Disserttio. Dli Uiversity of echology. []Krmer, Steve Lwrece (c996). Geotechicl erthquke egieerig. Upper Sddle River, N.J.: Pretice Hll. [3]Wolf J.P. (998). Soil-structure iterctio i time domi [M].Pretice-Hll, Eglewood Cliffs, N.J. [4]Wolf J.P.,Motosk (989). Recursive evlutio of iterctio forces of ubouded soil i the time domi. Erthq.egg struct.dy 8, [5]Xiog Zhg, Weger J L. (999). hree-dimesiol dymic soil-structure iterctio lysis i the time domi. Erthq. egg struct.dy 8:, [6]Li, Guoqig, Li, Jie (). heory d Applictio Reserch of the dymic dmge Idetifictio of the structure, Sciece Press, Beijig. [7]Mthemtics echig d Reserch Group i Njig techicl college (3). Itegrl rsformtio, High Eductio Press, Beijig. [8] Zhog, Wxie (). Dulity System of Applied Mechics, Sciece Press, Beijig. [9]Leg, Jihu (4). Fourier rsform, sighu Uiversity Press, Beijig. [] R. W. Clough, J. Pezie (975). Dymics of Structures, Mc-Grw Hill Ic., New York. []Yg,Qigsh, Jig, Hipeg (). Geertio of respose-spectrum-comptible groud motios bsed o phse-differece spectrum, Erthquke Egieerig d Egieerig Vibrtio :, 3-38.