COMPLEX MODE SUPERPOSITION METHOD CONSIDERING THE EFFECT OF MULTIPLE FOLD EIGENVALUE IN SEISMIC DESIGN

Transcription

1 The 4 th Wold Confeence on Eathquae Engineeing Octobe -7, 008, Being, China COMPLEX MODE SUPERPOSITION METHOD CONSIDERING THE EFFECT OF MULTIPLE FOLD EIGENVALUE IN SEISMIC DESIGN Ruifang Yu,, Xiyuan Zhou Intitute of Geophyic, China Eathquae Adminitation, Mingu Daxue South Road No. 5, Haidian Ditict, Being, 0008, P.R. China. Being Laboatoy of Eathquae Engineeing and Stuctual Retofit, Being Univeity of Technology, Chaoyang Ditict, Pingleyuan No. 00, Being 000, P.R.China. yfang6@6.com, houxy@bjut.edu.cn ABSTRACT: When tuctue poe multiple fold eigenvalue, the othogonality among diffeent mode no longe exit in mot cae. Theefoe, in thi pape, tanfe function method independent of othogonal elation i adopted to analye the dynamic epone baed on theoy of linea algeba and complex vaiable function, the dynamic epone analyi method in time domain i deived, which i uitable fo both non-claically and claically damped linea ytem with multiple fold eigenvalue. In addition, the tuctual epone pectum i intoduced uccefully and the CCQC algoithm i deduced, which can conide the effect of multi-fold-eigenvalue. The applicability of the deduced fomula i veified though Newma integation computation of line tuctue ubjected to pecibed eathquae motion. Meanwhile, the eult how that, fo the tuctue with multiple fold eigenvalue, the calculation eo will be faily lage if imply neglecting the effect of multiple fold eigenvalue. Finally, it i pointed out that the method deived in thi pape i uitable fo computing eimic epone of MDOF claical o non-claical damping linea ytem with o without multiple eigenvalue. KEYWORDS: Complex mode upepoition, multiple fold eigenvalue, tanfe function method. INTRODUCTION Mode upepoition analyi method i widely ued in eimic deign of tuctue to implify the dynamic analye though decoupling the vibation equation baed on othogonal popety and in eult the concened complex MDOF ytem can be tuned into linea upepoition of independent dynamic epone of a eie of SDOF ytem ubjected to identical gound motion. Baed on tationay andom poce theoy and compaative analye of time hitoy, the quae oot of the um of quae (SRSS and complete quadatic combination (CQC method fo claical damping linea ytem have poved to be an effective mean fo pediction of maximum epone of tuctue ubjected to eathquae excitation (Caughey, 960. Recently the dynamic analyi of non-claically damped linea ytem ha been paid moe attention becaue it i noticed that thee ae many tuctue whoe damping ae non-unifom, fo intance, oil-tuctue inteacting ytem and tuctue equipped with upplemental linea vicou dampe uch a oil dampe. Fo the non-claical damping ytem, the taditional modal decompoed method can not mae the motion equation decoupled; o, many eeache mae effot to dicu the modal decamped method baed on complex mode (Igua, et el. 984, Sinne, et el, 993. Zhou and Yu (004 deived the complex complete quadatic combination (CCQC method fo the non-claically damped linea ytem, which i completely in eal fom, and the complex quae oot of the um of quae (CSRSS method if coelation among modal epone ae ignoed. Howeve, fo eithe claical damping ytem o non-claical damping ytem, multiple fold eigenvalue poblem, which i fequently emeged in banch ytem o ymmetical tuctue of lage cale, i not yet attacted ufficient attention in eathquae engineeing community. When tuctue poe multi-fold-eigenvalue, the othogonality among diffeent mode no longe exit in

2 The 4 th Wold Confeence on Eathquae Engineeing Octobe -7, 008, Being, China mot cae. Theefoe, in thi pape, tanfe function method (Chen and Zhu, 990; Zhen, 00, the theoy of line independent of othogonal elation i adopted to analye the dynamic epone. By uing compehenively the theoy of linea algeba and complex vaiable function, the dynamic epone analyi method in time domain i deived, which i uitable fo both non-claically and claically damped linea ytem with multiple fold eigenvalue. The new algoithm not only ha explicit phyical meaning, but alo enable to conide the effect of multiple fold eigenvalue. In addition, the epone pectum i intoduced uccefully and CCQC method i popoed, which can conide the effect of multi-fold-eigenvalue. The validity and coection of fomula ae veified though Newma integation computation of line tuctue ubjected to pecibed eathquae motion. Meanwhile, the eult how that, fo the tuctue with multiple fold eigenvalue, the calculation eo will be faily lage if imply neglecting the effect of multiple fold eigenvalue. Finally, it i pointed out that the method deived in thi pape i uitable fo computing eimic epone of MDOF claical o non-claical damping linea ytem with o without multiple fold eigenvalue,.. ESTABLISHMENT AND EXTENSION OF TRANSFER FUNCTION MATRIX FOR LINEAR SYSTEM A it ha been nown, fo a dicete ytem, having N degee of feedom, the equation of motion in tem of nodal diplacement ae expeed a: Mx + Cx + Kx = f ( t (. whee M, C and K ae the N N ma, damping and tiffne matice, x i N nodal diplacement vecto which decibe the dynamic epone of the tuctue, N i an abitaily lage intege, and f ( t i N nodal load vecto. If damping matix doe not atify the decoupling condition baed on eal mode, that i, when ytem i not the claical damping ytem, the Eq.(. ha to be olve by tate pace method. Let y T = [ x x] T, Eq. (. can be ewitten into a goup of linea diffeential equation of one ode and the coeponding eigenvalue poblem can be defined a non-eo olution of following equation: ( A+ B Φ= 0 (. in which 0 M A = M C, M 0 B = 0 K Obviouly, it i equivalent to the eigenvalue poblem a follow ( M + C + K φ = 0 (.3 Φ= λφ φ. and [ ] When tuctue poe multi-fold-eigenvalue, the othogonality among diffeent mode no longe exit in mot cae, o tanfe function method i ued to analye thi cae. Uing Laplace tanfe, Eq.(. can become into the equation in the complex-field (hotened field baed on the paamete = α + iβ, i.e. ( + + = M C K X F (.4 that i Z X = F (.5 in which Z i the impedance matix in -field of ytem, which i noningula and ymmetic matix fo the etaint ytem and ha invee matix, hence we can get X = ( Z F = H F (.6 whee

3 The 4 th Wold Confeence on Eathquae Engineeing Octobe -7, 008, Being, China adj( Z J H = ( Z = = (.7 det ( Z D i called a tanfe function matix. When extenal excitation and initial condition of ytem ae definitive, the dynamic epone in -field fo evey genealied coodinate of ytem will depend on matix H, whoe popety eflect the dynamic pefomance of ytem. In Eq.(.7, D = det ( Z i the deteminant of matix Z, which can be denoted a the N -ode polynomial with eal coefficient concening paamete, that i D( = b = 0 In addition, J = adj( Z i the companion matix of matix the element J ( can be witten a ( Then, the element ( N N = b0 + b + b + + b N (.8 Z, which i a N N ymmetic matix and N ode polynomial with paamete, that i N N ( = = N = 0 J a a a a a (.9 H of tanfe function matix can be denoted a J a0 + a+ a + + an H = = D b0 + b + b + + bn It can be named a the tanfe function in -field of ytem. N Becaue D ( i the polynomial with eal coefficient, though equality = 0 N (.0 D, we can get N oot in the complex-field. If uppoe ditinct oot,,,, the ovelapped numbe of evey multi-fold-oot ae epectively,,,, and = N. Then Eq. (0 can be expeed a J p p p H = = (. = ( ( ( ( = in which, p, p,, p ae undetemined contant. We will dicu thee contant accoding to the two cae a follow.. Single Eigenvalue When eigenvalue a Multiple by ( i the ingle-oot, accoding to the Heaviide expanion theoem, Eq.(. can be witten H p p p p = (. l l ll l l= ( l ( l ( l l in both ide of Eq. (., and let, then the evey paamete, beide the p, in the ight ide of equality ae all eo, and we can obtain J J ( p = lim( H = lim ( = (.3 l l ( ( ( l l l= l= l l.. Multiple fold Eigenvalue When eigenvalue i the multi-fold-oot, the ode numbe of pole coeponding to the tanfe function

4 The 4 th Wold Confeence on Eathquae Engineeing Octobe -7, 008, Being, China H ( i geate than. Let u define the undetemined contant p, p,, p. When ovelapped numbe of eigenvalue i the, we can eaily appove that the elational expeion followed by i valid, that i ' ( ( ( J = J = = J ( = 0 (.4 J ( = (! G( (.5 Multiple by ( in both ide of Eq. (., and let ( V = ( l l= l l (.6 then J ( V ( = p ( p ( p (.7 = Let in Eq.(.7 and ue the elation in Eq.(.4, we can get p = 0 (.8 Ue Leibnit ule and calculate the fit deivative in both ide Eq.(.7, and let and conide the elation of Eq.(.4, we can get p = 0 (.9 By thi pocedue,the undetemined coefficient can be obtained. That i 3 p = p = = p = 0 (.0 p J = (! l = l ( ( ( Subtitute the Eq.(. into Eq.(., and uing ymbol p to eplace the ymbol J p H = = = ( = l l p, then we can get (. (. Fom complex vaiable function, the expeion of undetemined coefficient p i the eidue in pole fo function H (, that i p = Re H, (.3 The peceding analyi how that, if the eigenvalue ae diffeent, the evey element in tanfe function matix can be expeed a the um of imple faction accoding to the diffeent eigenvalue. The numeato of evey imple faction can be expeed by the eidue coeponding to eigenvalue (pole. Since the evey element in tanfe function matix can be expeed a the um of imple faction accoding to the diffeent eigenvalue, then tanfe function matix i alo expanded a the um of imple faction accoding to the diffeent eigenvalue, that i H = P (.4 = in which, P i the eidue matix coeponding to eignvalue (pole, which can be detemined by Eq.( DYNAMIC RESPONSES IN TIME DOMAIN FOR LINEAR SYSTEM In pactice, the eigenvalue nomally occu in complex conjugate pai fo the damped ytem, but fo highly damped ytem, an even numbe of them can be eal (Inman and Andy j., 980, which mean the

5 The 4 th Wold Confeence on Eathquae Engineeing Octobe -7, 008, Being, China chaacteitic equation of the ytem compie ove-citical damping (Clough and Penien, 993. We dicu the cae in pape by Yu and Zhou (006, o the cae will not be handled in thi aticle. Baed on the analyi a above, we uppoe m diffeent complex eigenvalue, then Eq.(. can be ewitten a m p p H = + = (3. in which and epeent a pai of conjugate complex eigenvalue, p and p ae the coeponding eidue which can be calculated by the Eq.(.3 and Eq.(. accoding to the pactice cae. In fact, if the ovelapped numbe of eigenvalue i equal to, the Eq. (. will become to Eq. (.3. Similaly, the tanfe function matix may be expeed by a pai of conjugate eigenvalue, that i m H = P + P (3. = in which, P and P ae the a pai of eidue matice in complex pole and conjugate pole Subtitute the Eq.(3. into Eq.(.6, the following equation can be obtained m X = H F = P + P F (3.3 = The invee tanfomation of Laplace i ued in both Eq.(3.3, we can get m t t ( t τ ( t τ x( t = P e f ( τ dτ + P e f ( τ dτ (3.4 = 0 0 Suppoe: = α + iβ, = α iβ (3.5 whee α = ζω and β = ωd = ω ζ ae damping coefficient and damped fequency of the -th mode epectively, and the fee vibation fequency ω and the coeponding citical damping atio ζ can be deduced fom the geneal othogonality elation. Sepaate eal and imaginay pat of eidue matix and combine the contibution of a pai of conjugate value, then the tuctue epone howed by Eq.(3.4 can be obtained ( whee u = Re( P MI and = Im ( m t m t α ( t τ α ( t τ ( g ( g = 0 = 0 ( ( x t = u e coβ t τ y τ dτ v e inβ t τ y τ dτ (3.6 v P MI ae the the eal and imaginay pat of eidue matix P, epectively.. N vecto, in which Re( P and Im( P epeent If the Duhamel integation fo co β ( t i ubtituted by ine Duhamel integation (Zhou and Yu, 004, the Eq.(3.6 can be witten a m ( t = { q ( t + q ( t } x A B (3.7 = in which A = ( ζω u ωdv (3.8 B = u (3.9 t can be expeed a olution of the following equation. q ( q ( t + ζωq ( t + ωq ( t = y ( t (3.0 g

6 The 4 th Wold Confeence on Eathquae Engineeing Octobe -7, 008, Being, China 4. CALCULATION METHOD BASED ON RESPONSE SPECTRA The deviation o mean quae epone of x( t in Eq. (3.0 i: ( i j i( j( i j i( j( i j i( ( x = AA < >+ BB < >+ BA < > j i= j= E t q t q t q t q t q t q t (4. in which the ymbol < > epeent opeation of calculation of aveage. Let u calculate covaiance < q ( t q ( t > < q ( t q ( t > and < q ( t q ( t >, and ubtitute into Eq.(4., we can get i j i j i j dd vv vd / / = i j ρ + i jωω i j ρ + i jωi ρ < i > < j > i= j= E[ x ( t] [ AA BB B A ] q ( t q ( t (4. dd in which the calculation and dicuion of diplacement coelation coefficient ρ, velocity coelation vv vd coefficient ρ and diplacement-velocity coelation coefficient ρ can be een in efeence (Zhou and Yu, 004. If we aume a uual that the maximum epone x ( t max i popotional to the oot of the mean quae epone, the following cloed-fom fomula of complex mode epone-pectum upepoition which conide the effect of multiple fold eigenvalue, i.e. the complex complete quadatic combination (CCQC fomula i deduced: / dd vv vd x( t = [ ] ( ( max AA i jρ + BB i jωω i jρ + BiA jωiρ qn t q max m t max (4.3 i= j= 5. NUMERICAL EXAMINATION AND APPLICATIONS A ingle-phee netwo hell i conideed in thi pape, whoe diamete i 0m and height i 5m. The lattice layout of netwo hell i the unflowe fom. The mateial i the eamle teel pipe whoe ectional dimenion i expeed a: Ф35 (the ba in ing dimenion of netwo hell, Ф5 (diagonal bace. Young modulu i. 0 N/m of the mateial and unifomly ditibuted load i 00Kg/m. In addition, the beaing of netwo i the fixed hinge beaing, and the plain and thee-dimenional view ae figued in Fig. and Fig., epectively. The eial numbe of tuctual node ae hown in Fig.. Six feedom degee ae conideed fo evey node. Theefoe, thee ae 36 feedom degee in total. Define the damping matix a C = α M + β K, in which M and K ae the tuctual ma and tiffne matice, epectively, and let α = (/, β = (, then the fit two damping atio ae equal to 0.0. Fig. plain view and node layout Fig. Thee-dimenion tuctual view The NS component of the El-Cento eathquae acceleation ecoded on May 8, 940 eathquae in Califonia, which contain enegy ove a boad ange of fequencie, i ued a a gound motion input. Via mode analyi pocedue and tanfe function method, the dynamic epone analyi i caied out in Matlab platfom. The modal popetie of the tuctue ae given in Table 5.. It can be een that 4 pai of multiple fold eigenvecto appea fo the ingle netwo hell, in which the fit two mode ae multiple fold. Then the mode

7 The 4 th Wold Confeence on Eathquae Engineeing Octobe -7, 008, Being, China coeponding to the multiple fold eigenvalue do not poe othogonality. Table 5. how the diplacement pea value of X and Y diection, in which the value in the column i the nodal diplacement calculated by Newma- β numeical method, and the value in column 3 i the eult calculated by the Eq. (3.7 deduced in thi pape, which coincide with the eult calculated fom Newma numeical method. The compaion of two calculation method veified the coection of Eq. (3.7. Calculation eult coming fom imple neglecting the effect of multiple fold eigenvalue i hown in column 4, which indicate that faily lage eo, a hown in column 5, can be poduced by the imple calculation. Table 5. Modal popetie of the tuctue Mode numbe fequency Damping atio Mode numbe fequency Damping atio Table 5. The maximum diplacement in the X and Y diection (*0-3 (unit:cm X diection Node Newma β Eq.(3.7 MSM * eo(% Node Y diection * MSM: Modal upepoition method 7. CONCLUSIONS Accoding to theoetical analyi and numeical examination in thi pape, ome concluion ae obtained:

8 The 4 th Wold Confeence on Eathquae Engineeing Octobe -7, 008, Being, China Fo the linea ytem, the complex modal upepoition method, Eq.(3.7, completely in eal fom, i deduced by uing tanfe function method, in which the effect of multiple fold eigenvalue on tuctual epone i conideed. The new algoithm i not only concie, but alo convenient to be undetood and gaped by the enginee. The validity of fomula i veified though Newma integation computation of line tuctue ubjected to pecibed eathquae motion. In addition, the epone pectum i intoduced uccefully and CCQC method i popoed, which can conide the effect of multi-fold-eigenvalue. Fo the linea ytem with multiple fold eigenvalue, the calculation eo will be faily lage if imply neglecting the effect of multiple fold eigenvalue. 3 It i pointed out that the method deived in thi pape i uitable fo computing eimic epone of MDOF claical o non-claical damping linea ytem with o without multiple eigenvalue. Acnowledgement:Thi wo get financial uppot fom the poject of DQJB06B0, IGPCEA. REFERENCE Caughey T. K. (960. Claical nomal mode in damped linea dynamic ytem. J. Appl. Mech. 7: 3, Chen F. X. and Zhu W. N. (990. Contol theoy of linea ytem. Wuhan univeity of technology pe. Clough R.W. Penien J. (993. Dynamic of tuctue. econd edition, McGaw-Hill,Inc. Igua T., Kiueghian A. D. and Sacman J. L. (984. Modal decompoition method fo tationay epone of non-claically damped ytem. Eathq. Eng. Stuct. Dyn. :, -36. Inman, D. J. and Andy J., A. N. (980. Some eult on the natue of eigenvalue of dicete damped linea ytem. J. App. Mech., ASME 47, Sinne, R. I, Robinon, W. H. and McVey, G..H. (993. An intoduction to eimic iolation. Jhon Wiley & Son Ltd. Yu R.F. and Zhou X.Y. (006. Complex mode upepoition method fo non-claically damped linea ytem with ove-citical damping peculiaity. Jounal of Building Stuctue. 7:, Zhen D. Z. (00. Theoy of linea ytem. Tinghua Univeity Pe. Being. Zhou X.Y., Yu R.F. and Dong D.(004. Complex mode upepoition algoithm fo eimic epone of non-claically damped linea MDOF ytem, Jounal of Eathquae Engineeing, 8:4,