Matrix Formulation of Foundations for Vibrating Machinery in Frequency Domain

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1 Ingenería Investgacón ecnología. Vol.I. Núm , ISSN FI-UNAM (artículo arbtrado) INVESIGACIÓN ne nge ría ECNOLOGÍA Matrx Formulaton of Foundatons for Vbratng Machner n Frequenc Doman Formulacón matrcal de cmentacones para maqunara vbratora en el domno de la frecuenca Carbajal-Romero M.F. Seccón de Estudos de Posgrado ESIME Acapotalco Insttuto Poltécnco Naconal. Méxco DF E-mal: Rodrígue-Castellanos A. Insttuto Mexcano del Petróleo. Méxco DF E-mal: Rodrígue-Sánche J.E. Insttuto Mexcano del Petróleo. Méxco DF E-mal: Flores-Ménde E. Seccón de Estudos de Posgrado ESIA acatenco Insttuto Poltécnco Naconal. Méxco DF E-mal: (Recbdo: agosto de 8; aceptado: febrero de 1) Abstract A matrx formu la ton to stud the coupled response of rgd foun da tons modelled b sprngs and dashpots s presented. Sprngs and dashpots oren ta ton can be an possble, thus a general solu ton s deter mned. Response n terms of dsplace - ments and rota tons s deter mned from a matrx sstem n the complex feld. he phscs of the problem presented here has been exten svel studed and a broad range of useful formulas to deter mne sprngs and dashpots prop er tes n sol-struc ture nter ac ton s aval able, however t has also been den t fed that there are some lm ta tons on couplng varous degrees of freedom n the aval able formu la tons. hen, the novelt of the approach presented comes from the matrx manp u la ton that leads to an expres son that provdes a closer approx ma ton to the real phenom enon, because all degrees of freedom can be coupled. hs approach ma allow to the analst fndng a coupled response ncludng the cases when ether sprngs or dashpots are not orthogo nall orented. In an example at the end of ths stud, the nflu ence of one of the nvolved param e ters n the sol-struc ture anal ss s pont ed out. ewords: Sprngs, dashpots, rgd bod, coupled anal ss, frequenc doman.

2 Matrx Formu la ton of Foun da tons for Vbra tng Mach ner n Frequenc Doman Resumen Se pre sen ta una for mu la cón ma tr cal pa ra el es tu do de la res pues ta aco pla da de c men ta - co nes rí g das, d cha ma tr se mo de la con re sor tes amor t gua do res. Los re sor tes amor t - gua do res pue den ser oren ta dos ar b tra ra men te, de ahí que es ta for mu la cón ten ga un ca rác - ter ge ne ral. La res pues ta en tér m nos de des pla a men tos de ro ta co nes se de ter m na a par tr de un ss te ma ma tr cal en el cam po com ple jo. La fí s ca del pro ble ma aquí pre sen ta do se ha es - tu da do ex ten s va men te exs te una am pla ga ma de fór mu las út les pa ra de ter m nar las ca - rac te rís t cas de los re sor tes de los amor t gua do res en la n te rac cón sue lo-es truc tu ra; sn em - bar go, tam bén se han den t f ca do al gu nas l m tan tes en re fe ren ca al aco pla men to de los gra dos de l ber tad en las for mu la co nes en con tra das. Enton ces, la no ve dad del plan tea men to ma te má t co pre sen ta do ve ne da da por la ma n pu la cón ma tr cal que con du ce a una ex pre - són que pro por co na una apro x ma cón más cer ca na al fe nó me no real, da do que to dos los gra - dos de l ber tad pue den ser aco pla dos. Esta pro pues ta pue de per m tr le al ana ls ta en con trar una res pues ta aco pla da n clu en do aque llos ca sos don de los re sor tes /o los amor t gua do res no es tán oren ta dos or to go nal men te. Al f nal de es te es tu do, se nclue un ejemplo de aplcacón donde se enfata la mportanca de uno de los parámetros nvolucrados en el análss de nteraccón suelo-estructura. Des crp to res: re sor tes, amor t gua do res, cuer po rí g do, aná l ss aco pla do, do m no de la fre - cuen ca. Intro duc ton Some of the pres ent prob lems n the feld of d nam - cs of struc tures are those re lated to the re sponse of rgd foun da tons un der d namc load ngs. Usu all, the anal ss of these struc tures s car red out con sd - er ng lum ped pa ram e ter meth ods, where the sol s re placed b a ss tem of fre quenc-de pend ent sprngs and dashpots. In man aval able for mu la tons, the anal ss of the foun da ton-sol ss tem ma be done con sd er ng an un - cou pled ss tem, where onl n plane re sponses are de - ter mned. It s well-known that a foun da ton-sol ss - tem has sx de grees of free dom n an or thogo nal ref er - ence ss tem. Ex pres sons for cal cu lat ng sprngs and dashpots for the sx de grees of free dom can be ob taned from sev eral pub lshed ref er ences. We re fer the reader to see the p o - neer ng works of Rch ards et al. (197), Luco (198), Gaetas (1983, 1991), Dobr and Gaetas (1985, 1986) and Pas and ausel (1988). And more re cent works re - lated wth lumped sprngs and dashpots for foun da ton anal ss can be con sulted n Wu and Chen (), Wu and Lee (4) and Wolf and Paronesso (7). A good ref er ence to de sgn rgd foun da tons un der d namc load ng s Bowles (1996). Re cent re sults for sprngs and dashpots n la ered me da can be seen n Wolf and Deeks (4). De ter mn ng ex pres sons for sprngs and dashpots s not wthn the scope of the pres ent work. A ma trx for - mu la ton n whch sprngs and dashpots are gath ered for sx de grees of free dom s pro posed, so a cou pled re - sponse, for sprngs and dashpots act ng s mul ta neousl s ob taned. Equa ton of moton he gen eral equa ton of mo ton, n tme do man, for a rgd bod ss tem can be ex pressed as: [ M]( a t) [ C]( a t) [ ] a( t) F( t), (1) where, [M] = Mass ma trx, [] = Stff ness ma trx, [C] = Damp ng ma trx, a(t) = Ds place ment vec tor, a( t ) = Speed vec tor, a( t ) = Ac cel er a ton vec tor and F( t) = Force vec tor. Ex press ng the ds place ment and force vec tors n fre quenc do man, one has a( t) a( ) e t () 78 Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM

3 Carbajal-Romero M.F., Rodrí gue-caste llanos A., Rodrí gue-sánche J.E. and Flores-Ménde E. F( t) F( ) e t. (3) Af ter cal cu lat ng speed and ac cel er a ton vec tors us - ng equaton () and ar rang ng com mon terms, we f - nall ex press equaton (1), n fre quenc do man, as: [ M] C [ ] a( ) F ( ). (4) he so lu ton for equa ton (4) s gven b a ( [ M ] C [ ] 1 F (. (5) Equa ton (5) s the so lu ton, n fre quenc do man, for a rgd bod ss tem, whch con tans sx de grees of free dom. Mass, damp ng and stff ness ma trxes are pre - sented n the fol low ng sec tons. Stff ness matrx Let us re name the ds place ment vec tor as { d} { d x, d, d, x,, } or { d} { d, }, where d { d x, d, d } and { x,, }. When a rgd bod ex pe r ences a small ro ta ton, see fg ure 1a, ds - place ment of the sprg due to that ro ta ton can be ex - pressed as dr r, where { x,, } s the ro - ta ton vec tor and r s the po s ton vec tor of sprng. Consderng that {,, } rep re sents the sprng vec tor of d rec tor co snes of the sprng, to tal ds place ment of sprng s gven b [ d r ]. (6) hus, the force n the sprng s F [ d r ] (7) and the mo ment s M r F (8) he prod uct r n ma trx terms s [ r], where [r] s the sprngs po s ton ma trx gven as [ r]. (9) Force n the sprng can be ex pressed as fol lows F ( d ) ( [ r] ) (1) F ( d ) ( [ r] ) (11) F ( ) d ( )[ r] (1) and the mo ment s as M r F [ r] F (13) M [ r] ( ) d [ r] ( )[ r] (14) Ex press ng equa tons (1) and (14) n ma trx form, Fx F F M x M M r r r r dx d d x (15) Fgure 1. a) Rota ton of a rgd bod; b) A sprng n the refe rence sstem Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM 79

4 F M F d dd d d (16) or n ts com pact form, the well-known ex pres son of F d, where s the gen eral stff ness ma trx. Equa ton (15) or (16) rep re sents the ss tem wth ts sx de grees of free dom that de scrbes the cou pled re sponse of a rgd foun da ton sup ported b sprngs. De vel op ng the stff ness ma trx of equa ton (15), one has n the equa ton (17). Ro ta tonal sprngs should be added d rectl to sub ma trx. As sum ng or thogo nal or en ta ton of the sprngs, as shown n fg ure, stff ness ma trx of equa - ton (17) can be sm pl fed. here fore,, hx and h, rep re sent ver t cal and hor on tal sprngs, re spec tvel. Co or d nates and d rec tor co snes for each sprg are shown n ta ble 1. 8 Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM Matrx Formu la ton of Foun da tons for Vbra tng Mach ner n Frequenc Doman (17) a) b) Fgure. Ortho gonal oren ta ton of sprngs

5 Carbajal-Romero M.F., Rodrí gue-caste llanos A., Rodrí gue-sánche J.E. and Flores-Ménde E. able 1. Coor d nates and drector cosnes for ortho gonal oren ta ton of sprngs Sprng hx 1 hx hx hx h 1 h h h 1 x Stff ness ma trx for each sprng can be ex pressed as shown n equa tons (18), (19) and () for sprngs n, and d rec tons, re spec tvel. hx 1 hx h hx hx hx hx hx hx hx hx (18) 1 Rota tonal sprngs () Ro ta tonal sprngs are added d rectl to the stff ness sub-ma trx k, whch s a part of the stff ness ma trx. Sub-ma trx k s, equa ton (1). x (1) h hx 1 h h h h h h h h h h (19) Complete stff ness matrx he com plete stff ness ma trx ref er enced to the or - thogo nal ss tem s ob taned add ng the stff ness sub-ma trxes, for the ln eal and ro ta tonal sprngs, gven n equa tons (18) to (1). here fore, the whole stff ness ma trx of the ss tem can be rep re sented as n equa ton (). Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM 81

6 Matrx Formu la ton of Foun da tons for Vbra tng Mach ner n Frequenc Doman hx hx hx hx hx h h h h h v h h h h h h h x hx hx hx hx hx hx hx hx hx hx hx h h h h h hx hx hx h h () Dampng matrx Sm larl as the stff ness ma trx was formed, and con sd er ng that the dashpots are or thogo nall or ented, the com plete damp ng ma trx s as n equa ton (3). C C C hx hx hx hx hx C C C h h h h h C C v C C h h C C h h C C C h h h C Cx C hx hx C C hx hx C C C hx hx hx C C hx hx C C C C C hx hx h h h h h hx hx hx C h h (3) 8 Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM

7 Carbajal-Romero M.F., Rodrí gue-caste llanos A., Rodrí gue-sánche J.E. and Flores-Ménde E. Mass matrx and force vector For the ss tem formed b the rgd bod, sprngs and dashpots, the mass ma trx can be as n equa ton (4). M m m m m 4B A /1 m 4L A /1 m 4 4 /1 L B (4) where, m=mass of the n stru ment or equp ment on the foundaton, B=half-wdth of the foun da ton, L=halflength of the foun da ton, A=heght of the n stru ment or equp ment. Force vector he equp ment sup pler usu all gves the force vec tor n the do man fre quenc. A gen eral ex pres son for the force vec tor s gven b F Fx, F, F, Mx, M, M, where: F x =horontal force n d rec ton, F =hor on tal force n d rec ton, F =ver t cal force n d rec ton, M x =mo ment around axs, M =mo ment around axs and M =mo ment around axs. Nume rcal example In ths sec ton we used the for mu la ton pre v ousl de - vel oped n or der to show the n flu ence that one of the m por tant fac tors has n the anal ss and de sgn of rgd foundatons. hs fac tor s the depth of embedment. For sm plc t rea sons we do not pres ent re sults re lated to the n flu ence on the re sponse ne ther of sol n ter nal damp ng nor of shear modulus. and C are func tons of fre quenc, sol prop er tes and area Ac of con tact be tween sol and foun da tons. Other pa ram e ters re qured are: d men sons L and B of a rect an gle that en cr cles the foun da ton and n er ta mo ments of Ac around, and -axs. For mu las for d namc sprngs and damp ers, and C, are wdel known (e.g. Ara et al., 1979), a de scrp - ton of these for mu lae s be ond the scope of the pres ent stud. Once the val ues have been ob taned we can cal cu late the d namc sprng and damp ers as a func ton of the sol n ter nal damp ng and the fre quenc of ex c ta ton b us ng the fol low ng equa tons: ( ) C (5) C( ) C /. (6) Ex pres sons (5) and (6) are based on the vscoelas - tct prn c ple. For a gen eral case, sx pars of equa tons lke (5) and (6) should be cal cu lated to rep re sent the sx de grees of free dom of the ss tem, that s to sa, () and C(), h() and Ch(), hx() and Chx(), x() and Cx(), () and C(), and () and C(). hese val ues are the n put to ar - range the stff ness and damp ng ma tr ces shown here. Embed ment depth nfluence on rgd foun da ton response hs sec ton pres ents an ap pl ca ton of the pre v ousl de vel oped for mu la ton b do ng a sen s tv t stud show ng the ef fect of embedment depth on ds place - ment and ro ta ton re sponse. Usu all, block foun da - tons for ma chn er are em bed ded n the sol from =. m to =.5 m depth. he ef fect of embed - ment depth n the re sponse s b an n crease of r gd t and damp ng thus, ds place ment and ro ta ton am pl - tudes are re duced. In some cases, damp ng due to embedment ma be neg l g ble how ever, the ef fect of the embedment depth s rel e vant to the sprng stff ness val ues. Embedment depth could lead to op er ate a ma chne near to ts res o - nance re gon and ths could pro duce fal ure ef fects due to the force pro duced b the ro tat ng com po nents. Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM 83

8 Matrx Formu la ton of Foun da tons for Vbra tng Mach ner n Frequenc Doman Re sults of a sen s tv t stud show ng the ef fect of the embedment depth n cla us ng the for mu la ton de - vel oped here are pre sented n fg ures 3, 4 and 5. Rela - ton shps pro posed b Ara et al. (1979) be tween su per - f cal and embeded sprngs and damp ers were used and are shown n ta ble. he for mu las seen n ta ble show the re la ton shp be tween the em bed dng co ef f cents (end and Cemb) and the non- em bed dng ones (sup and Csup). he n put data for sol, foun da ton and ma - chne used n ths stud s pre sented n ta ble 3. Co ef f - cents for d namc sprngs and damp ers are also shown n ta ble 4. rep re sents the Pos son ra to used for the calculatons. able. Rela tons between stff ness and dampng for foun da tons surface overl ng and foun da tons embed dng Mode emb/sup Cemb/Csup Vertcal orontal Rockng 1 6. ( 1 v) B ( v) B 1 1. ( 1 v) B. ( v) B ( 1 v) B emb sup ( v) B emb sup 1 7. ( 1 v) 6 B. ( v) B. 5 emb sup 3 able 3. Sol, Foun da ton and Machne data for the sens t vt stud Sol Data Foundaton Data Machne Data Unts (nternal dampng)=.3 B (wdth) =.5 IG =.5 ons. (volumetrc weght)= 1.6 L (large) = 1. WEIG=.5 Meters G (shear modulus)= 146 A =.5 MASRO =.15 Seconds (Posson rato)=.33 L/B =. ECENRIC. =.8 Vs (shear speed)= (embeddng depth)= varable able 4. Coef f cents for dnamc sprngs and dampers for the embed ment depth sens t vt stud Coeffcent Machne operaton veloct (RPM) a k kh kx k k c ch cx c c Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM

9 Carbajal-Romero M.F., Rodrí gue-caste llanos A., Rodrí gue-sánche J.E. and Flores-Ménde E. Re sults ob taned from the embedment depth sen s - tv t stud are as fol lows; fg ure 3 shows the n flu ence of the embedment depth n the hor on tal ds place - ment d, t can be seen that the re sponse of the ss - tem s hghl con trolled b the embedment depth and 1.5 res o nant peaks are ob served at 5 and 3 RPM. Fg ure 4 shows ver t cal ds place ments d, t s clear that for deeper embedments, ver t cal ds place ments are d mn shed. F nall, fg ure 5 shows that the embed - ment depth also leads to re duce the ro ta tons. =. DISPLACEMEN "d" (Mlímeters) =.1 =. =.3 NOE: "" n meters OPERAION FRECUENC (R.P.M.) Fgure 3. Embed ment depth nfluence on dspla ce ment d DISPLACEMEN "d" (Mlmeters) =. =.1 =. =.3 NOE "" n meters OPERAION FRECUENC (R.P.M.) Fgure 4. Embed ment depth nfluence on dspla ce ment d ROAE IN "" (RAD) =. =.1 =. =.3 NOE: "" n meters OPERAION FRECUENC (R.P.M.) Fgure 5. Embed ment depth nfluence on rota ton around x Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM 85

10 Matrx Formu la ton of Foun da tons for Vbra tng Mach ner n Frequenc Doman Conclu sons he well known prob lem of a v brat ng struc ture and ts sol n ter ac ton re sponse has been solved wth a novel math e mat cal ma np u la ton that leads to a for - mu la ton of mul t ple ap pl ca tons. hese ex pres sons al low de ter mn ng the re sponse of rgd bod es on elas - tc damped foundatons consderng the foundaton re - strc ton ef fects act ng s mul ta neousl n all d rec tons; ths means that a cou pled ds place ment re sponse con - sd er ng sx de grees of free dom s ob taned. he re - sponse of the ss tem n terms of ds place ments and ro - ta tons s de ter mned for a de fned set of fre quen ces. hus, re sponse can be com pared wth op er a tonal ds - place ment and ro ta ton lm ts. More over, ths for mu - la ton ma be use ful for de ter mn ng cou ple re sponse of rgd foun da tons n clud ng the cases where sprngs or dashpots are not or thogo nall or ented. he for mu - la ton s ma trx based thus; t can be computatonall pro grammed for n dus tral ap pl ca tons. As an ap pl ca - ton, the ef fect of embedment depth was de ter mned n a sen s tv t stud and ts rel e vant ef fects were ponted out. Refe rences Ara S.C., Ónell M.W., Pncus G. Desgn of Struc tures and Foun - da tons for Vbra tng Machnes. ouston. Gulf Pub. Co. Books Dv son Bowles J.E. Foun da ton Analss an Desgn. Ffth Ed. he Mc. Graw-ll Compa nes, Inc Dobr R., Gaetas G. Stff ness and Dampng of Arb trar Shaped Machne Foun da tons. J. Geotech. Eng., 11: Dobr R., Gaetas G. Dnamc Stff ness and Dampng of Foun da tons b Smple Methods. On: Proc. Smpo sum. Vbra ton Problems n Geotech ncal Eng nee rng, ASCE, Gaetas G. Analss of Machne Foun da ton Vbra ton: State of the Art. J. Sol Dna mcs and Eart hquake Eng nee rng, : Gaetas G. Formulas and Charts for Impe dances of Surface and Embedded Foun da tons. J. Geotech. Engng., ASCE, 117(a): Luco J.E. Lnear Sol-Struc ture Inte rac ton: A Bref Revew. Eart hquake Ground Moton and ts Effects on Struc tures, AMD 53: Rchard F.E., all J.R., Woods R.D. Vbra tons of Sols and Foun da tons. Nueva Jerse. Pren tce all Inc, Engle wood Cltts Pas A., ausel E. Appro x mate Formulas for Dnamc Stff - nesses of Rgd Foun da tons. Sol Dna mcs and Eart hquake Eng nee rng, 7: Wolf J.P., Deeks A.J. Foun da ton Vbra ton Analss: A Strength-of-Mate - rals Approach. 1st Ed. Oxford. Else ver Mass. 4. Wolf J.P., Paro nesso A. Lumped-Para meter Model for a Rgd Cln drcal Foun da ton Embedded n a Sol Laer on Rgd Rock. Eart hquake Eng nee rng & Struc tural Dna mcs, 1: Wu W.., Chen C.. Smpl fed Sol-Struc ture Inte rac ton Analss Usng Eff cent Lumped Para meter Models for Sol. Sols and Foun da tons, 4: Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM

11 Carbajal-Romero M.F., Rodrí gue-caste llanos A., Rodrí gue-sánche J.E. and Flores-Ménde E. About the authors Manuel Faraón Carbajal-Romero. Obtaned a BSc n elec tro me cha ncal eng nee rng at Inst tuto ecno ló gco de ehuacan, a MSc n mecha ncal eng nee rng at Seccón de Estu dos de Posgrado e Inves t ga cón of Escuela Supe ror de Inge nería Mecá nca Eléc trca of IPN and a PhD n mecha ncal eng nee rng n Seccón de Estu dos de Posgrado e Inves t ga cón of Escuela Supe ror de Inge nería Mecá nca Eléc trca n IPN. e works as nves t ga tng professor n Seccón de Estu dos de Posgrado e Inves t ga cón of Escuela Supe ror de Inge nería Mecá nca Eléc trca at IPN undad Aca pot alco and he belongs to the Natonal Sstem of Inves t ga tors (SNI). Alejandro Rodrí gue-caste llanos. Obtaned a BSc n cvl eng nee rng at Inst tuto Pol téc nco Naconal, a MSc n struc tural eng nee rng n Seccón de Estu dos de Posgrado e Inves t ga cón from Escuela Supe ror de Inge nería Arqu tec tura n IPN and a PhD n mecha ncal eng nee rng n Seccón de Estu dos de Posgrado e Inves t ga cón of Escuela Supe ror de Inge - nería Mecá nca Eléc trca at IPN. e works n the tech no lo gcal area of cvl eng nee rng n Inst tuto Mex cano del Petróleo and he s nvted professor n Seccón de Estu dos de Posgrado e Inves t ga cón of Escuela Supe ror de Inge nería Mecá nca Eléc trca n IPN undad Aca pot alco, where he lectures Numerc Methods and he belongs to the Natonal Sstem of Inves t ga tors (SNI). Efraín Rodrí gue-sánche. Obtaned a BSc n Cvl Eng nee rng at Facultad de Inge nería, UNAM and a PhD n Mecha ncal Eng nee rng from Unver st College London. Profes so nall he has colla bo rated at Inst tuto Mex cano del Petróleo n the area of marne plat forms n the desgn dsc plnes, nspec ton and man te nance as well as n the plan nng of the deve lop ment of felds n deep waters. At the moment he s the respon sble for the tech ncal area of opera ton and man - te nance of sstems n deep waters and belongs to the Natonal Sstem of Inves t ga tors (SNI). Esteban Flores-Ménde. Obtaned a BSc n Phscs at Inst tuto de Físca, UNAM. a MSc n struc tural eng nee rng n Seccón de Estu dos de Posgrado e Inves t ga cón from Escuela Supe ror de Inge nería Arqu tec tura n IPN and a PhD n struc - tural eng nee rng n Seccón de Estu dos de Posgrado e Inves t ga cón from Escuela Supe ror de Inge nería Arqu tec tura n IPN. e works as nves t ga tng professor n Seccón de Posgrado de Estruc turas en la ESIA Undad aca tenco, IPN Undad aca tenco and asso cate professor n Inst tuto de Geofí sca de la UNAM. Inge ne ría Inves t ga cón ec no lo gía. Vol. I. Núm , ISSN FI-UNAM 87

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