Eigenfrequencies and Critical Speeds on a Beam due to Travelling Waves

Transcription

1 The Ope echaic Joural, 9,, -9 Ope cce igefrequecie ad Critical Speed o a Bea due to Travellig Wave T.G. Kotatakopoulo, I.G. Raftoiai ad G.T. ichalto* Departet of Civil gieerig, Natioal Techical Uiverit of the, the 578, Greece btract: daic load, uddel applied at a poit of a bea, produce a local diturbace that propagate or diffue to the ret of the bea. Thi propagatio take place with a peed depedig o the aterial ad geoetrical characteritic of the bea. It ha bee deotrated that a ipulive diturbace ivolvig hear ad oet will reult i two wave tpe, oe that propagate with the hear wave velocit ad a ecod that propagate with a oet-wave velocit. It i oberved that taperig with the cro-ectio of the bea a reult to equal hear wave ad oet-wave velocitie ad the two tpe of diturbace will travel together alog with additioall iterferig hear wave fro bea' ed reflectio. I thi paper, the effect of the travelig wave o the daic characteritic of a bea i tudied. coplete bea odel i preeted, which otio i govered b the Tioheko equatio. Two ai cae are eaied, ael a ipl upported bea, ad a bea retig o a Wikler-tpe elatic foudatio. altical reult are preeted i graphical for howig the ifluece of the travelig wave o the eigefrequecie ad critical peed of uch a bea ad ueful cocluio are draw. INTRODUCTION daic load, uddel applied at a poit of a bea, produce a local diturbace that propagate or diffue to the ret of the bea. Thi propagatio take place with a peed depedig o the bea characteritic a well a the aterial characteritic fro which the bea i ade fro. The above iple fact i the bai for the tud of the ubject kow a wave propagatio. Thi pheoeo i failiar to everoe i variou for uch a the traiio of oud i air, or the preadig of ripple o water urface, the traiio of a earthquake wave, or the radio wave. The phical bai of the propagatio of a diturbace i fiall caued b the iteractio of the dicrete ato withi a olid. I olid ad fluid echaic, the ediu i aued to be cotiuou ad thu, it phical characteritic uch a deit or odulu of elaticit are alo coidered to be cotiuou fuctio repreetig average value. diturbace to a a particle i traitted to the et particle b a iagiar iterveig prig. Through thi wa, the diturbace i traitted to a reote a particle. The aterial characteritic uch a deit ad elatic cotat affect the peed of propagatio. Icreaig the elatic cotat (or i other word the prig cotat, the peed of propagatio icreae ad vice-vera. I a olid, two differet actio are poible for wave propagatio. I the firt cae, the olid will trait tree (teile or copreive, ad otio of the particle will take place *ddre correpodece to thi author at the Departet of Civil gieerig, Natioal Techical Uiverit of the, the 578, Greece; -ail: ichalt@cetral.tua.gr alog the directio of the wave otio. The behavior i aalogou to that of fluid, ad olid of thi tpe do ot have a reitace to bedig. characteritic cae of thi behavior i the fleible trig. olid though a alo trait hear tree, ad the otio of particle i travere to the propagatio directio. Such a behavior doe ot eit i fluid. Durig their otio, the wave ecouter the boudarie of the olid bod ad, evidetl, a iteractio betwee wave ad boudarie will be ievitable. The behavior of wave i a olid differ fro that i a fluid. I tructural echaic, the otio of rod, bea, ad plate ca be decribed without eedig to coider the propagatio ad reflectio of wave withi the tructure. The o-called tregth of aterial theorie a be derived o the bai of variou auptio regardig deforatio. Wave propagatio i tructure ha bee tudied over a coiderable period of tie b a igificat uber of reearcher. a olutio techique have bee reported [-] for tructure with pecific geoetrical characteritic ad fiite, periodic, or ei-ifiite boudar coditio. og a frequec doai ethod, the pectral eleet ethod [5] ha bee proved uitable for aali of wave propagatio i real egieerig tructure. The pectral eleet ethod ue the eact olutio of the differetial equatio which gover the proble. Flügge deotrated [6,7] that a ipulive diturbace o a bea ivolvig both hear ad oet will reult i two wave trai, i.e., oe that propagate with the hear wave velocit kg / ad aother that propagate with a 87-58/9 9 Betha Ope

2 igefrequecie ad Critical Speed The Ope echaic Joural, 9, Volue oet-wave velocit / alog the bea ( beig the a per uit volue. It ha bee oberved that if the cro-ectio of the bea i uch a to have k G, the the velocitie ad will be equal ad the two tpe of diturbace will travel together. I geeral, thee two tpe of diturbace will travel with differet velocitie. oreover, ice the reflectio of oet wave fro the bea boudarie will reult i additioal hear wave trai, the oet ad hear iterferig wave will oo create a ver coplicated ituatio. For tudig the aforeetioed ver iteretig proble, uerou echaical odel of bea have bee preeted. I Fig. (, three echaical odel of bea are preeted, i which the ai paraeter are the luped ae. Thee odel were firt propoed ad tudied b Schirer [8]. the lupig of the ae becoe aller ad approache i ie the uiforl ditributed a ad retoratio eleet of the bea, the wave travel velocitie i the odel will approach the liitig value ad. Fig. (. uped a ad prig odel with higed joit for the wave travel propertie: (a Shear tiffe, ad tralatio iertia ol, Fleural tiffe, ad tralatio iertia ol, ad (c Fleural ad hear tiffe a well a tralatio ad rotator iertia. I thi paper, the effect of the travelig wave o the daic characteritic of a bea i tudied. For thi tud, the coplete odel (ee Fig. c i ued, the otio of which i govered b the Tioheko equatio. fter epreig the Tioheko equatio i it coplete for ad evaluatig the cotributio of each ter i the equatio (ad ubequetl, the eceit of keepig or igorig each ter, two ai cae are eaied, ael: the igle-pa ipl upported bea, ad the fiite bea o a Wikler-tpe elatic foudatio. Graphical reult howig the ifluece of the travelig wave o the eigefrequecie ad critical peed of a bea are preeted ad ueful cocluio are gathered. (a (c THTIC FORUTION et u coider the oe-pa ipl upported bea of Fig. (a, ubjected to the ditributed load p(,t. fter it deforatio, the ifiiteial part d of the bea take firtl the poitio abcd (caued b the bedig of the bea ad fiall the poitio BCD becaue of the hear ifluece.,w,w d (a p(,t X p(,t j w dj ~ j d d g dg g d d a d d B d.. b w d d d V.. r w dv V d c d C d d d D d d Fig. (. The defored tate of the ifiiteial part d. The fial deforatio of a fiber i a ditace fro the cetral ai i: (, t (, t ( becaue, a the coordiate icreae the agle decreae. O the other had we have: d (, t ( d ad fiall: (, t ( quilibriu of oet give: d V ( d

3 The Ope echaic Joural, 9, Volue Kotatakopoulo et al. where i the iertia oet produced b the rotatio of the ifiiteial part d. quilibriu of force give: V w p(, t (5 where, w i the iertia force becaue of the oveet of d i parallel to O ai.. For the oet we fid the followig epreio: (, t d d J (6. For the hear force V, we have: V k G (7 where k i the corrective factor of Tioheko, epreig the o-uifor ditributio of hear tree alog the height of the cro-ectio. Fro Fig. (b we have the relatio: w (8 ad q. (7 becoe: w V k G (9. The iertia oet i produced b the iertia force of the fiber d i parallel to O ai i a ditace fro the eutral ai: F d d d d where i the a per uit volue. Therefore the produced oet will be: d d d d So, q. ( becoe: V J d J d ( ( Itroducig ad V fro q. (6 ad (9 repectivel ito q. ( ad (5 we obtai the followig differetial te: J w k G J w k G w p t ( Fro q. (b, we deterie,,, t that we itroduce ito the equatio gettig after differetiatio of q. (a. So, we coe to the followig equatio: J w w w J t k G t J w J p p p kg t kg kg The equatio of the free vibratig bea i: w w w t kg t uig a olutio of the for: w(, t i t w kg t ( ( X( e (5 we get the followig equatio: X X kg kg X ad ice it i, the above equatio becoe: X X X (6 kg k G The characteritic equatio of the above differetial oe i: μ μ kg k G The root of the above equatio are: μ, ± ad μ, ± i (7 where: kg kg kg kg kg kg Hece, the olutio of q. (6 i give b the followig relatio: ( c i c co c ih c coh (8 X For the cae of a ipl upported oe-pa bea, the followig boudar coditio are valid: X ( X( X( X( (9 Uig the above coditio we arrive at the te: c c c c c i c co c ih c coh c i c co c ih c coh

4 igefrequecie ad Critical Speed The Ope echaic Joural, 9, Volue 5 I order for the te to have ot ol trivial olutio, the followig coditio ut be fulfilled: coh ih co i coh ih co i which iplifie to the followig eigefrequecie equatio: ( i ih ad ice it i ( ih, it will be: i ( The above equatio ha the olutio:,,,, or ad takig ito accout q. (7, we have: k G k G k G or fiall: k G G k ( Flügge [6], deotrated i 9, that a ipulive diturbace ivolvig hear ad oet will reult i two wave trai, oe that propagate with the oet-wave velocit: (a ad aother that propagate with the hear-wave velocit: k G (b O the other had we have: r (c where r i the radiu of gratio i parallel of O ai ad the ledere of the bea. Itroducig q. (a,b,c ito q. (, we obtai: ( ad after oe aipulatio: ( ( The above equatio ha the olutio: } ] [( ] [( { (5 Neglectig the ter k G / ( a beig ver all copared to the other ter, we arrive at the followig approiate relatio: ( (6 TH IGNFRUNCIS CHNG The epreio givig the eigefrequecie pectru b the claical theor i: o r or fiall: o (7 The followig ratio i of great iteret: o ( (8 TH CRITIC SPDS CHNG factor of great iteret for the bridge, i the o called critical peed. critical peed, we defie the peed that a vehicle eed for croig a bea i tie equal to the correpodig half-period of the bea. Thu for the th period the critical peed will be: cr T (9 Itroducig fro q. (6 ito q. (9 we get: cr ( ( Callig ocr the critical peed foud b the claical theor, we fiall obtai: o ocr cr ( Fro the pectru of the above critical peed give b q. (, the firt oe cr i the ot iteretig, which fro ow o we hall call cr.

5 6 The Ope echaic Joural, 9, Volue Kotatakopoulo et al. FINIT B ON STIC FOUNDTION et u coider ow the bea B i Fig. (, which i baed o a Wikler-tpe elatic foudatio. ccordig to the claical theor, the force per uit legth of the bea reactig to the eteral loadig i: P k w ( where k i the o-called Wikler factor. We hall proceed deteriig thi factor. Fig. (. Fiite bea o elatic foudatio. The reduced deforatio of a earthe ifiiteial part d d d which i uder three-aial loadig are: W [ [ [ ( ( ( ] ] ] ( where i the oil odulu of elaticit ad the Poio ratio (with value fro. to.. If the lateral iflatio i retraied (i.e.,, it will be: ( ad itroducig,, ito the third of q. (: ( ( (5 ( d It i valid that: ad thu will be: d ( ( d d d. ( uig additioall that ad reai cotat alog the depth H (ee Fig. we will have: ( ( w d ( H ad the force reactig per uit legth of the bea will be: ( P b b w (6 H ( ( B Fig. (. Stre ditributio uder a bea of width b. We et et: h H ( ( ( (7 where i the a per uit volue of the earth, ad h i the height of the equivalet orthogoal cro-ectio that ha the ae area with the oe of the bea B. Thu q. (6 becoe: ( P w h H ( ( or fiall: P w ad thu: k (8 Itroducig the reactio of the foudatio ito q. ( we get: J w w w J t k G t w p where the higher order ter have bee eglected. The equatio of the free vibratig bea i: d (9 w w w w ( t kg t d Pp.b Followig a iilar proce, like the oe for q. (, we fiall obtai the equatio: h b H

6 igefrequecie ad Critical Speed The Ope echaic Joural, 9, Volue 7 or fiall: kg kg ad ( (a [( ] ad (b where i the peed of the oud i the earth (depeded o the characteritic of the groud, ad, are give b q. (a,b. Thu the critical peed will be: ( cr ( T ( Fro q. ( ad (, arie the followig iteretig ratio betwee the cr of the oe-pa ipl upported bea ad the cr of the ae bea but laid o a elatic foudatio: bcr cr ( NURIC RSUTS ND DISCUSSION The aterial ued i egieerig tructure have odulu of elaticit wavig fro. dn/ to. dn/, while their a per uit volue tart fro kg/ to 8 kg/. uig that the Tioheko coefficiet k. to. we get the followig value for the peed of the travelig wave: a. Speed of the oet wave:.5 k/ec to 5.5 k/ec, ad b. Speed of the hear wave:. k/ec to. k/ec For differet oil qualitie we have: a. odulu of elaticit:.5 6 dn/ to 5 6 dn/ b. a per uit volue: kg/ to kg/, ad thu we ca deterie the wave peed:.5 k / ec to.5k / ec. The Oe-Spa Sipl Supported Bea et u coider firtl a bea of teel i which the peed of oet wave i 5.7 k / ec. For value of Tioheko factor k. to. we get peed of hear wave fro. to.9 k/ec repectivel. I the diagra of Fig. (5a it i plotted the ratio / o veru, for 5.7 k / ec,. k / ec, ad,,,, 5. I the diagra of Fig. (5b it i plotted the ratio / o veru, for 5.7 k / ec,.9 k / ec, ad,,,, 5. w /wo w /w o Fig. (5. a. Ratio / o veru, for 5.7 k / ec ad. k / ec. b. Ratio / veru, for o 5.7 k / ec ad.9 k / ec ( 6 8 We et coider a bea ade fro cocrete, i which the peed of oet wave i.8 k / ec. For value of Tioheko factor fro k. to. the peed of hear wave var fro, to,9 k/ec repectivel. I Fig. (6, we ee the ae a above diagra correpodig to a cocrete bea. 5 5 Y

7 8 The Ope echaic Joural, 9, Volue Kotatakopoulo et al. w /w o v cr u.5 w /w o ( 5 v cr u.5 ( Fig. (6. (a The ratio / veru, for o.8 k / ec ad.k / ec. The ratio / o veru, for.8k / ec ad.9k / ec I Fig. (7a, the diagra of the critical peed cr of the bea veru the ledere for 5.5 k / ec ad variou value of are draw. I Fig. (7b, the ae a i Fig. (7a diagra are draw for.5 k / ec ad variou value of. Fiite Bea o latic Foudatio The cool ued aterial for uch a bea i cocrete. For cocrete, it i,8 k/ec while the value of var fro, k/ec up to,9 k/ec for k, to,, repectivel. For oil, the correpodig value for the peed of wave are.5 k/ec up to.5 k/ec. Coiderig a bea with legth, b. ad h. up to., ad H varig fro to, the coefficiet,, ad take the followig value:.55.. to to to.5.5. ad wave fro ~ to ~9. I Fig. (8, o ca ee the diagra bcr / cr veru for differet, ad for,, ad 8. Fig. (7. (a cr veru for 5,5 k/ec ad variou value of. cr veru for,5 k/ec ad variou value of. CONCUSIONS Fro the reult obtaied ad the diagra how i the precedig aali, we ca draw the followig cocluio: Sigle-Spa Sipl Supported Bea The ue of the eact theor give eigefrequecie, which are igificatl aller tha the oe obtaied uig the claical iplified theor. a. For teel bea ad for peed of the hear wave. k / ec, thi effect i igificat ad aout: for, ad 5 to 8%, for, ad to 8%, for, ad 5 to 55%, ad for, ad to 7%. For >, the effect i till igificat for the eigefrequecie ad take it iia (~% for ad. For peed of the hear wave.9 k / ec, the above value decreae a follow: for, ad 5 the ifluece i 55%, for, ad it i 5%, while for it i 5% (for 5 ad.5% (for, repectivel. For bea ade fro cocrete, the effect i ore igificat, i.e., for. k / ec, it i 55% for, ad,9% for, ad 5,% for, ad,75% for, ad 5,% for 8, ad,55% for 8, ad 5,5% for, ad,5% for, ad 5, while for.9k / ec, it i % for, ad,85% for, ad 5,% for, ad, 55% for, ad 5,5% for 8, ad, 5% for 8, ad 5,% for, ad, 5% for, ad 5.

8 igefrequecie ad Critical Speed The Ope echaic Joural, 9, Volue 9 v b,cr /v,cr v b,cr /v,cr v b,cr/v,cr v b,cr/v,cr V v ( v ( The critical peed are epeciall affected b the peed of the hear wave for ledere <. The peed (trogl depeded o the cro-ectioal hape i poible to reult (for the ae ledere differece aouted to about % (for to % (for, etc. Fiite Bea o latic Foudatio The proble of how the travelig wave affect the critical peed i ver coplicated i thi cae. The characteritic of the oil that, accordig to q. ( ad (, affect the critical peed are two: the factor ad the peed of oud i the oil. Higher value of correpod to a ore coheret oil. The critical peed of a bea retig o a elatic foudatio, copared to the oe of the correpodig oe-pa ipl upported bea with the ae characteritic, are igificatl higher. For differet value of the ledere, we oberve that the priar critical peed of a fiite bea o elatic foudatio i uch higher tha the oe of the correpodig ipl upported bea. Ol for tiff bea ad epeciall oft oil (ee Fig. 8a, for ad there i a iigificat differece. For > ad uual oil tpe, the priar critical peed i higher fro tie (Fig. 8a up to tie (Fig. 8d. RFRNCS [] Bathe KJ. Fiite eleet procedure i egieerig aali. Pretice-Hall: glewood Cliff, NJ 98. [] Redwood. echaical wave guide. Pergao Pre: New York, NY 96. [] Che YK. Fiite trip ethod i tructural aali. Pergao Pre: New York, NY 997. [] Graff FK. Wave otio i elatic olid. New York, NY; Dover Publicatio 99. [5] Dole JF. pectrall forulatio fiite eleet for logitudial wave propagatio. It J al p odal al 988; : -5. [6] Flügge W. Die aubreitug vo biegugwelle i täbe. Z gew ath ech 9; : -8. [7] Flügge W. Vicoelaticit. d ed. Spriger Verlag: Berli 975. [8] Shirer H. Über biegewelle i täbe. rchiv ppl ech 95; : ( Fig. (8. Velocit ratio bcr / cr v oil peed..5 v Received: arch, 8 Revied: Jue 5, 8 ccepted: Jul, 8 Kotatakopoulo et al.; iceee Betha Ope. Thi i a ope acce article liceed uder the ter of the Creative Coo ttributio No-Coercial icee ( which perit uretricted, o-coercial ue, ditributio ad reproductio i a ediu, provided the work i properl cited.