Improving the Mathematical Model of the Tacoma Narrows Bridge

Transcription

1 Rose-Hulman Undergraduae Mahemaics Journal Volume 8 Issue 2 Aricle 7 Improving he Mahemaical Model of he Tacoma Narros Bridge Brian Fillenarh Universiy of Evansville, bf36@evansville.edu Follo his and addiional orks a: hps://scholar.rose-hulman.edu/rhumj Recommended Ciaion Fillenarh, Brian (2007 "Improving he Mahemaical Model of he Tacoma Narros Bridge," Rose-Hulman Undergraduae Mahemaics Journal: Vol. 8 : Iss. 2, Aricle 7. Available a: hps://scholar.rose-hulman.edu/rhumj/vol8/iss2/7

2 Improving he Mahemaical Model of he Tacoma Narros Bridge Brian A. Fillenarh Deparmen of Mahemaics Universiy of Evansville Advisor: Dr. Taliha M. Washingon

3 1 Inroducion Since he opening day on July 1, 1940, he Tacoma Narros Bridge has been a opic of ineres for physiciss, engineers, and mahemaicians. Immediaely folloing opening day, he bridge could be seen oscillaing verically up and don [6]. On November 7, 1940, he verical movemen of he Tacoma Narros Bridge changed o a violen orsional roaion [14]. This ising moion coninued for abou 45 minues unil he bridge ulimaely collapsed [5]. Afer he collapse of he Tacoma Narros Bridge, i became imporan o deermine ha facors caused his caasrophic failure so ha hese facors ould be aken ino consideraion for he design of fuure suspension bridges. Alhough debae persiss abou he exac reason for he Tacoma Narros Bridge failure, mahemaical models have been developed o illusrae ho he bridge behaved during is final momens. There are models ha illusrae boh he verical moion, as ell as he orsional moion exhibied by he bridge. Since he orsional roaion of he bridge is ha ulimaely caused is failure [7], he model of he orsional roaion ill be analyzed in deph. The base mahemaical model ha ill be used for all modificaions presened in his paper as derived in [9] by P.J. McKenna, and is & lk θ = δθ& cosθ + f ( (1.1 m here δ = damping coefficien l = idh of half he bridge k = spring consan m = mass per foo of bridge θ = angle of roaion f ( = exernal forcing erm = ime For all mahemaical models presened in his paper, Neon s noaion for differeniaion is used in hich a single do represens he firs derivaive aken ih respec o ime, and a double do represens he second derivaive aken ih respec o ime. In Equaion 1.1, he firs erm, δ & θ, represens he ind resisance of he bridge and he second lk erm, cosθ, represens he cable resisance. A deailed derivaion of Equaion 1.1 can be found in [3]. m McKenna s model as chosen because his model as one of he firs ha correced small angle linear approximaions ha had been made in previous models of he Tacoma Narros Bridge, such as he model presened in [5]. The small angle linear approximaions are simply sin θ = θ (1.2 and cos θ = 1 (1.3 Unforunaely hese approximaions are only valid for small angles of displacemen. Since he angular displacemens of he Tacoma Narros Bridge ere very large, he use of he small angle approximaions is no valid. In McKenna s model (Equaion 1.1, he exernal forcing erm as assumed o be periodic. This may be represened by f ( = λ sin( μ (1.4 here λ is he ampliude of he forcing erm, and μ is he period of his erm. The problem ih his forcing erm is ha i may no accuraely depic exacly ho he Tacoma Narros Bridge behaved in orsion. The exernal forcing erm represened in Equaion 1.4 ill be modified in all bu he las case presened o see ho he bridge ould have responded due o various ypes of physical assumpions hich ill be explained in subsequen secions. All he models presened in his paper are numerical, and he ode15s solver as used in Malab o solve and graph hese soluions. This solver is a variable-sep solver, hich simply means he sep size is varied during he simulaion of he model. This solver reduces he sep size o increase he accuracy hen he model s sae changes rapidly, and increases he sep size o reduce he simulaion ime hen he model s sae changes sloly [11]. 1

4 Iniially a fixed-sep solver as used, bu afer preliminary ess, his as found o be impracical due o he large ime inerval ha as used and he small sep size ha as required for he model o be accurae. For all cases presened, iniial condiions represening a large orsional displacemen ere used and are θ ( 0 = 1. 2 and & θ ( 0 = 0. The iniial condiions used can change he behavior of he model, and a deailed look a he iniial condiions can be found in [8]. 2 Deriving he Consans We approximae values found in [1] of he characerisics of he Tacoma Narros Bridge o deermine he consans for Equaion 1.1. I has been esimaed ha he bridge eighed 5,000 lbs per foo, so e se m in Equaion 1.1 o be 2,500 kgs. The bridge defleced abou 0.5 meers per foo of he bridge ih a 100 kg load applied. From his informaion e can deermine he spring consan knoing ha he cables ac as springs in ension and here are o resising cables [14], [8], one on each side of he bridge. This can be shon as 2 kδ = mlg (2.1 here k represens he spring consan, Δ is he change in lengh of he cables (0.5 meers, m L is mass of he applied load (100 kgs, and g is acceleraion due o graviy (9.8 m/s 2. If e solve for k e ge approximaely 1,000 N/m. I is also knon ha he idh of half he bridge (l as 6 meers, and i has been deermined ha he damping coefficien (δ should be abou 0.01 [1]. Wih all of hese consans in place, Equaion 1.1 becomes & θ = 0.01 & θ 2.4 cosθ + f ( (2.2 For he forcing erm shon in Equaion 1.4, McKenna deermined ha he μ value should be somehere beeen 1.2 and 1.6 [9]. In his paper e ill assume ha he value is 1.3. The λ value has been se o cause a small forcing erm, and his value ill be 0.05 for all he cases explored in his paper. Wih he consans in place, he exernal forcing erm becomes f ( = 0.05 sin(1.3 (2.3 This forcing erm assumes ha he forcing parameers on he bridge do no change ih respec o ime. Tha is, he ampliude and period of he forcing erm remain consan he enire ime he bridge is in orsional roaion. Hoever, he ampliude or period of he exernal force may have varied ih ime. The forces on he bridge for a specified ime, say 0 < 8, may be compleely differen from forces on he bridge a anoher ime such as 8 < 10. In addiion, he forces applied o he bridge may have no been periodic. Also, i is unlikely ha he curren mahemaical model accuraely depics he orsional behavior of he Tacoma Narros Bridge since he roaion of he bridge ould have caused faigue on he hanger cables and bridge deck. This faigue ould allo he bridge o roae furher he longer ha he bridge sayed in moion. Faigue in he lk bridge cables ould no be accouned for in he exernal forcing erm bu raher in he cosθ erm ha m represens he force due o he cables. The firs modificaion ha e presen invesigaes an exernal forcing erm on he Tacoma Narros Bridge ih varying ampliudes on boh sides of he bridge. We hen discuss he effecs of an exernal forcing erm on he bridge having differen periods on each side. The nex modificaion involves ho he bridge reacs o a change in he ampliude of he periodic exernal forcing erm for a large block of ime. We ill also look a he effecs of a consan force for shor ime inervals as ell as for a large block of ime. The final modificaion o McKenna s model ha ill be presened invesigaes he bridge response as a resul of eakening hanger cables. Reasons for hese modificaions ill be given and from he resuls, a more accurae mahemaical model ill be presened. 3 Modifying he Mahemaical Model 3.1 Sandard Bridge Response The sandard response of he bridge ill be defined as he mahemaical model creaed by McKenna, using he consans presened in Secion 2. Wih he consans placed ino Equaion 1.1, he sandard bridge response may be shon as & θ = 0.01 & θ 2.4 cosθ (3.1 2

5 Figure shos he roaion of he bridge as a resul of he mahemaical model developed by McKenna. Figure The sandard response of he bridge Our simulaion shos ha he roaion of he bridge does no sop, hich is consisen ih observaions made he day ha he bridge failed. 3.2 Modificaions ih a Periodic Forcing Term The firs modificaion o he mahemaical model is varying he period or ampliude of he forcing funcion for specified ime inervals. We ill keep he modified periodic forcing erms in he same form as he original forcing erm (Equaion 1.1, bu he ampliude or period of hese erms ill be varied. The effecs of changing hese erms in shor bu frequen ime inervals as ell as a single long coninuous block of ime ill be invesigaed in deph. Shor and Frequen Periodic Forces Firs e changed he exernal force by adding anoher periodic force for many shor bu frequen ime inervals. To illusrae he effec of increasing he ampliude of he original force for shor and frequen imes, his addiional force ill have he same period as he original force. This model can be illusraed ih 0.01 & 2.4 cos sin sin( & θ θ θ λ sin(1.3 10n + 9 < 10( n θ = ( & θ 2.4 cosθ 10( n < 10( n here λ 2 represens he ampliude of he addiional forcing erm and n denoes a posiive ineger. This model ill illusrae he response of he bridge if differen periodic forces ere acing on each side. If he exernal forces on one side ere larger han he side ih he original exernal force on i, he addiional forcing erm ould be posiive. Similarly, if he force on one side ere smaller han he original, he addiional forcing erm ould be negaive. Unforunaely due o he complexiy of finding he rae a hich he period of his funcion varies, e ill have o sele for looking a he effecs of an addiional exernal forcing funcion over he arbirary ime inerval specified in Equaion 3.2. If e modeled he funcion ih he differen ampliude over he acual ime inerval ha he bridge ere on one side, he responses ould likely be very similar o he ones found in his example. We ill firs look a ho a posiive addiional force changes he bridge response. By seing 2 λ equal o 0.01, e find ha iniially he roaion of he bridge varies significanly, bu hen he roaion levels ou and becomes fairly consan (Figure In his Figure, he peaks of he maximum ampliude occur a a laer ime han hey did in he sandard response, and he ampliude in his response flucuaes a lile more afer a ime of abou

6 Figure The bridge response of a small addiional periodic forcing erm Thus ih a slighly larger ampliude of forcing on one side, here does no appear o be a significan change in he response of he bridge hen compared o he sandard response. By adding a larger force, such as by doubling he original forcing erm by seing λ 2 equal o 0.05, a slighly differen response can be produced. As shon in Figure 3.2-2, here is a more gradual flucuaion in he ampliude, and i never becomes consan. Figure The bridge response of doubling he original periodic forcing erm Alhough here is a noiceable change in he response as a resul of a large addiional periodic forcing erm, i is very unlikely ha he exernal forces acing on each side of he Tacoma Narros Bridge varied his significanly. We obained similar resuls hen e used a negaive λ 2 of he same magniude. By using a ampliude for he addiional forcing erm, e are modeling he bridge as if i ould lose all exernal forcing for a small period of ime. Even ih his he bridge sill remains in orsion, and he response resembles ha of Figure Exernal Forces ih Differing Periods We no consider ha ould happen if he exernal forcing funcion on he Tacoma Narros Bridge had a varying period. The model ha ill be used in his case is 0.01 & 2.4 cos sin sin( 2 & θ θ θ μ 10n + 9 < 10( n θ = ( & θ 2.4 cosθ 10( n < 10( n here μ 2 represens he ne period of he exernal forcing funcion and n is a posiive ineger. 4

7 The case illusraed by his model may correspond o one of he cables holding he bridge in place being sronger han he cable on he oher side. This may cause he bridge o resis ising on one side beer han he oher side, hich could resul in a shorer period force for one side of he bridge. Once again, due o he difficuly of deermining he rae ha he period of his funcion varies, e ill model he differen period for an arbirary lengh of ime. If e modeled he funcion over he acual ime inervals ha he bridge as on one side, he responses ould no change a significan amoun from his model. The firs case e ill look a is if he period of he exernal forcing funcion ere o change a small amoun for a shor lengh of ime. We ill se μ 2 equal o 1.32 represening a very small change in he period of he exernal forcing funcion. Figure shos a large deviaion from he sandard response in Figure Figure The bridge response for a small change o he period for a shor ime In his case, he ampliude of he bridge deck s roaion levels ou fairly quickly and remains level from a ime of abou 200 o a ime of abou 450. Afer his ime, he ampliude of he roaion begins o vary and coninues undulaing over he shon ime. Thus, a sligh change in he period of he exernal forcing funcion ill resul in a subsanial change in he response of he bridge. If he μ 2 erm is furher increased o 1.35, an unexpeced bridge response occurs (Figure Figure The bridge response for μ 2 = Figure shos he ampliude of roaion in he Tacoma Narros Bridge decreasing oards zero. For his reason, i is unlikely ha he exernal forces on he bridge had a varying period for repeaed shor ime inervals. Furher invesigaion of he μ 2 erm resuled in finding ha he responses of he bridge coninued o change beeen remaining in orsion (responses similar o Figure 3.2-3, and having he orsion diminish (responses similar o Figure

8 Addiional Periodic Force for a Single Block of Time The nex model describes ho he Tacoma Narros Bridge response is affeced by a change of he ampliude in he exernal forcing erm for a single large block of ime. Similar o he firs models, he period of he addiional forcing erm ill be he same as he period in he original forcing erm (Equaion 2.3. Thus, e have he folloing equaion 0.01 & 2.4 cos sin + 2 & θ θ θ λ sin(1.3 S < E θ = ( & θ 2.4 cosθ S, > E here λ 2 sill represens he ampliude of he addiional force, S is he saring ime for he addiional force, and E is he end ime of he addiional force. One reason his may have occurred in he bridge ould be due o a minor srucural failure ih he bridge in orsion. A minor srucural failure may cause he exernal forcing erm on he Tacoma Narros Bridge o induce a larger roaion in he bridge. The larger roaion in he bridge ill be modeled by simply modifying he exernal forcing erm o have a differen ampliude han he original forcing erm. From all he cases esed ih his model, e found ha here he ime inerval sars can significanly change he resuls. For his reason, he criical values for modeling his case ere found, and he model as esed a hese locaions. We found hese criical values o be around imes = 50 and = 90. These represen he ime a hich he lo and high poins of he original funcion occur, and are idenified in Figure Figure Three criical imes in he original funcion All cases presened ih his model ill also be looked a for a hird criical sar ime of = 600, since a his ime he orsional moion of he sandard bridge response is fairly sable. For many of hese cases e found ha if he ime inerval of he differen forcing erm ere sufficienly large, he ising bridge response ould end. We sho he responses of he bridge for he ime inerval direcly prior o he ime inerval a hich he ising bridge response sops. In oher ords, e ill sho he bridge response here he ampliude of roaion does no approach zero since e kno ha he roaion of he Tacoma Narros Bridge did no sop. The firs case of his model predics ho cuing he general forcing funcion s ampliude in half changes he response. Tha is, e ill se he λ 2 equal o The criical cases for his forcing erm are shon in Figures

9 Figure The bridge response ih λ 2 = for S = 50 and E = Afer many numerical experimens ih he sar ime a 50, e deermined ha for any block of ime ha ended a a ime greaer han abou 61.4, he ampliude of he bridge roaion ould evenually approach zero. Since he roaion of he Tacoma Narros Bridge did no sop, Figure shos he case ih an ending ime for he addiional force of 61.4, hich is he ime direcly prior o he ampliude going o zero. From his Figure i can be seen ha a large bu gradual variance in he roaion occurs iniially from a ime of abou 50 o a ime of abou 125. From he ime of 125 o abou 300, he ampliude of roaion varies more significanly, bu levels ou a a ime of around 500. Figure shos he case ih a sar ime of 90, and unlike he sar ime of 50, he roaion did no sop even as he ime inerval for he force approached infiniy. Iniially he ampliude of roaion varies a significan amoun, and a a ime of abou 125, his variance becomes more gradual unil i levels ou a a ime of abou 500. Figure The bridge response ih λ 2 = for S = 90 and E The final criical case here he 2 λ equals is shon in Figure and has a sar ime of 600. In Figure 3.2-8, here is a sligh dip in he limis of he orsional roaion afer a ime of 600 as a resul of he loer ampliude. Similar o he case ih he sar ime of 90, he ampliude of roaion in his case does no approach zero as he ime inerval for he force approaches infiniy. 7

10 Figure The bridge response ih λ 2 = for = 600 S and E The response ha began a a ime of 50 (Figure as he only case here he orsional roaion of he bridge approached zero hen he ampliude of he forcing funcion as cu in half for a se block of ime. The end ime value a hich he response changed from large oscillaions o small oscillaions as found o be around Thus, if here as a change in he periodic forcing erm by having he ampliude reduced, he bridge s orsional roaion ould no have sopped if he roaion as already ell esablished. The consequence of he periodic forcing erm being compleely removed from he Tacoma Narros Bridge for a block of ime ill no be considered. This can be done by simply seing λ 2 equal o he opposie value of he sandard bridge response s ampliude, making λ 2 = The criical cases of his are shon in Figures , and in all of hese cases he orsional roaion sopped if he block of ime as large enough. For he firs criical case ih a sar ime of 50 (Figure 3.2-9, e found ha if he end ime as greaer han 57.3, he ampliude of roaion of he bridge deck en o zero. Figure shos ha iniially here is a large variance in he ampliude of roaion unil i levels ou a a ime of abou 500. Figure The bridge response ih λ 2 = for S = 50 and E = The second criical case ih a sar ime of 90 (Figure has he ime a hich he bridge deck roaion ould go o zero if he exernal forcing erm ere removed for any longer a abou Figure clearly illusraes he effec of removing he exernal forcing funcion in he mahemaical model. From a ime of 90 o abou 208.7, he ampliude of he roaion in he bridge deck can be seen o gradually decrease. A he ime of 208.7, he exernal forcing funcion is added back ino he model, and he ampliude of roaion of he bridge can be seen o jump back up and level ou as i has in many previous cases. 8

11 Figure The bridge response ih λ 2 = for S = 90 and E = The final case in hich he exernal forcing funcion as removed from he mahemaical model has a sar ime of 600 (Figure Afer many numerical experimens, e found ha ih he exernal force removed for any longer han a ime of 100 (end ime 700, he ampliude of roaion of he bridge deck ould go o zero. Similar o Figure , from he sar ime of 600 o he end ime of 700, he ampliude of roaion in he bridge deck gradually decrease. Figure The bridge response ih λ 2 = for S = 600 and E = 700 In summary, e found ha he ampliude of roaion in all hree of hese cases approached zero if he block of ime ha he exernal force as removed from he sysem as large enough. Once again, since e kno he roaion of he bridge did no sop, e have shon he cases direcly prior o hose ha he ampliude of roaion en o zero. The criical values for hese rials ere found o be a for he case ha began a = 50 (Figure 3.2-9, for he case ha began a = 90 (Figure , and for he case ha began a = 600 (Figure This suggess ha no only is he exernal forcing funcion needed o sar he bridge in orsional roaion, bu ihou an exernal forcing funcion he bridge ill no remain in orsional roaion. All cases esed ih an increase in he ampliude of he exernal periodic forcing funcion resuled in responses ha mainained heir large oscillaions. 3.3 Modificaions ih a Consan Forcing Term The second modificaion o he mahemaical model ha e ill consider is he effecs of he addiion of a consan forcing erm o he sandard response in shor bu frequen ime inervals and a long and coninuous inerval. 9

12 Shor and Frequen Consan Forces We no consider ho he Tacoma Narros Bridge orsional response changes due o shor bu frequen addiional exernal forces on he bridge. This ill be modeled as 0.01 & & θ 2.4 cosθ + C 10n + 9 < 10( n θ = ( & θ 2.4 cosθ 10( n < 10( n here C represens he addiional consan force on he bridge and n represens any posiive ineger. The mos obvious reason for his change o occur ould be many spikes in he ind speed on he bridge possibly resuling from gusy inds. This ould induce many addiional consan forces on i acing a various imes. If he ind guss ere greaer han he curren ind speed hen he C value ould be a posiive value, and similarly if he ind speed decreased suddenly he C value ould be negaive. If he C value ere chosen o be a large enough negaive number, his can represen a change in he ind direcion a he Tacoma Narros Bridge. The firs case ha ill be invesigaed is ho an addiional force of half he ampliude of he original forcing erm ill change he response. This means ha C ill be se o In Figure 3.3-1, hich can be seen belo, he ampliude of he roaion in he bridge deck never quie level ou as a resul of he addiional consan force added periodically. Figure The bridge response as a resul of an addiional consan force of C = From comparing Figure o Figure 3.1-1, no much change is observed in he bridge response as a resul of an addiional consan force ha is half he value of he ampliude he original forcing funcion has. When e increased he magniude of he C value, e found ha he bridge response became more chaoic, bu folloed a similar paern o ha illusraed in Figure The resul of adding a negaive consan forcing erm of also resuls in a similar response o he posiive force illusraed above. If his value increased o , a very differen response occurs (Figure Tha is, he roaion of he bridge deck is very chaoic, and he ampliude of roaion evenually goes o zero a a ime of abou

13 Figure The bridge response as a resul of an addiional consan force of C = Therefore, if a sufficienly large force opposie ha of he original force is applied o he bridge a shor and frequen inervals, he orsional moion of he bridge ill sop. Hoever, i is unlikely ha he acual ind forces on he Tacoma Narros Bridge changed direcions his violenly. Addiional Consan Force for a Single Block of Time The nex model demonsraes ho he Tacoma Narros Bridge response changes due o an addiional exernal force for a single large block of ime. This ill be modeled as 0.01 & & θ 2.4 cosθ + C S < E θ = ( & θ 2.4 cosθ S, > E here C sill represens he addiional consan force on he bridge, S is he saring ime for he addiional force, and E is he end ime of he addiional force. Similarly o he shor and frequen case of a consan force, e ill use his case o demonsrae a change in ind speed. We no focus on a consan change in he speed hich is more likely o have occurred a Tacoma Narros. No significan changes could be noed from he sandard response ih any of he cases ha ere examined ih his model. The responses observed simply shifed he ampliude of roaion of he bridge, and made he ampliude slighly more random. The case here C = , S = 600, and E can be seen in Figure This Figure shos ha a a ime of 600, hich corresponds o he sar ime of he addiional consan force, a sligh jump in he ampliude of roaion of he bridge deck happens. As ould be expeced, if a negaive C value ould be inroduced, a donard shif in he ampliude ould occur insead of a shif up in he oscillaions. Figure The bridge response due o a large addiional consan force of C = for S = 600 and E 11

14 Figure indicaes ha i as easier for he bridge o roae o one side han he oher since he amoun of roaion in he bridge shifed up. This makes sense since if a force is acing on only one side of a body, he body ill an o move in he direcion of he applied force. 3.4 Weakening Hanger Cables on he Tacoma Narros Bridge We no consider a model o illusrae ho he Tacoma Narros Bridge responds if he hanger cables holding he bridge eakened as a resul of he large forces applied on hem from he orsional roaion. This ill be modeled as & θ = 0.01 & θ (2.4 + b cosθ (3.7 ih represening a reducing facor for he amoun he cables eaken as ime passes, and b represening a rae facor. As previously saed, his ill illusrae ho he Tacoma Narros Bridge behaves as a resul of he cables eakening as ime passes. I is very likely ha his is ha acually happened o he bridge, bu he magniude of his eakening of he cables is uncerain. For his reason, he model ill be esed using differen reducing and rae facors. The firs reducing facor ha ill be esed illusraes a minor loss in srengh of he cables as ime passes. In order o accomplish his, he value ill be se o 1,000 (Figure Also for his case, he rae facor ill be se o 1.00, hich represens he cables losing srengh a a consan rae as ime passes. For his case, iniially here is a large variance in he ampliude of he roaion and he loer limi of he variance can be seen o gradually increase over he enire ime. Figure The bridge response due o eakening cables ih a reducion facor of = 1, 000 and rae facor of b = Figure shos ha a small reducion in srengh of he cables due o faigue ill cause he bridge o roae furher as ime passes. This is ha ould be expeced, hoever he process is very gradual ih a reducion facor of 1,000. We ill no look a ha happens if he hanger cables los srengh a an increasing rae. This may have happened since he more ha he bridge deck as alloed o roae, he larger he forces on he cables ould be. The reason he forces ould be larger is simply because he bridge deck ould be falling from a higher poin as i roaed more, so he deck ould gain more velociy before he cables began picking up he eigh. We can model his by seing he rae facor a a value greaer han For his case e ill se b o 1.20 represening a slo increase in he rae ha he cables los srengh, and ill remain a 1,000. The resuls of his are shon in Figure

15 Figure The bridge response due o eakening cables ih a reducion facor of = 1, 000 and rae facor of b = From Figure 3.4-2, e can see ha he response is very similar o Figure 3.4-1, bu ih he ampliude of roaion increasing a a quicker rae hich is ha as expeced. Increasing he rae facor furher simply increases he ampliude of roaion more quickly. The reducion facor ill no be se a 250 and rae facor se a 1.00 in order o sho ha happens o he bridge if he cables lose a fairly subsanial amoun of srengh due o he oscillaions. This case can be seen in Figure and e find ha he increase of he loer limi of roaion occurs faser han in he case ih he reducion facor a 1000 and rae facor a Thus, he bridge is alloed o roae furher he eaker he cables ge. Figure The bridge response due o eakening cables ih a reducion facor of = 250 and rae facor of b = Once his roaion reaches abou 2 π radians, he bridge deck compleely flipped over. The bridge deck flipping over for he case ih a reducion facor of 250 is shon in Figure In his Figure, he deck flips over once a a ime of abou 1,180, and hen flipped over again a a ime of abou 1,220. Afer he deck flips over a second ime, he ampliude of roaion decreases significanly. 13

16 Figure The Tacoma Narros deck flipping over compleely When he case ih a reducion facor of 1,000 and rae facor of 1.00 as modeled over a very large ime, e found ha he deck flipped over a a ime of abou 4,340, and his case roaed 2 π radians as opposed o he 4π radian roaion ha e see in Figure Obviously due o physical resrains, he bridge deck could no have flipped over compleely. This model does help illusrae he large impac faigue in he cables may have had on he bridge hough. From hese resuls i can be seen ha a possible reason for he final collapse of he bridge as due o faigue in he cables hich resuled in a cable or cable connecion failing. 4 Summary of Resuls By comparing Figure Figure 3.2-4, Figure , Figure Figure 3.3-3, and Figure Figure 3.4-3, ih Figure 3.1-1, e have shon various modificaions o he mahemaical model ha have a significan impac on he overall response of he Tacoma Narros Bridge. Since he roaion in mos of hese cases does no sop as a subsanial amoun of ime passes, e can conclude ha any one of hese may very ell be a more accurae represenaion of ha acually happened o he Tacoma Narros Bridge during is final momens. I ould be more likely ha he acual response of he Tacoma Narros Bridge as a combinaion of some of he cases presened. I is also possible ha as ime passed he cables acually become eaker hich enabled he bridge o is furher, unil he cables could no longer suppor he momenum of he bridge and ulimaely failed. For his reason he general direcion ha he bridge response probably mos closely folloed is he direcion presened in Figure as opposed o Figure If a fe of hese cases are combined, many very unique and ineresing resuls occur. One such example of he infinie possibiliies occurs if he cables eakened ih a reducing facor of 1,000 and rae facor of 1.00 he enire ime he bridge as in orsional roaion, he bridge had a slighly higher ampliude for he periodic force acing on i a a relaively frequen inerval for he enire duraion of roaion ( λ 2 = in his case, he bridge experienced a large increase in ind speed ( C = for he ime inerval 50 < 200, and he bridge experienced 14

17 a decrease in he ampliude of he periodic forcing funcion ( λ 3 = 0.03 in his case over he inerval 400 < 600. This can be mahemaically represened ih he folloing funcion b 0.01 & θ (2.4 + cosθ + λ 2 sin(1.3 20n 6 < 20n 1, 50 b 0.01 & θ (2.4 + cosθ 20n 1 < 20( n + 1 6, 50 b 0.01 & θ (2.4 + cosθ + λ 2 sin(1.3 + C 20n 6 < 20n 1, 50 < 200 b 0.01 & θ (2.4 + cosθ sin(1.3 + C 20n 1 < 20( n + 1 6, 50 < 200 b 0.01 & θ (2.4 + cosθ + λ 2 sin(1.3 20n 6 < 20n 1, 200 < 400 && θ = (4.1 b 0.01 & θ (2.4 + cosθ 20n 1 < 20( n + 1 6, 200 < 400 b 0.01 & θ (2.4 + cosθ + λ 2 sin(1.3 + λ 3 sin(1.3 20n 6 < 20n 1, 400 < 600 b 0.01 & θ (2.4 + cosθ + λ 3 sin(1.3 20n 1 < 20( n + 1 6, 400 < 600 b 0.01 & θ (2.4 + cosθ + λ 2 sin( n 6 < 20n 1, > 600 b 0.01 & θ (2.4 + cosθ 20 n 1 < 20( n + 1 6, > 600 here b = 1.00 = rae facor = 1,000 = reducing facor λ 2 = 0.021= ampliude of λ 3 = = decreased ampliude for he block of ime specified C = 0.05 = large addiional ind force for he block of ime specified n = any posiive ineger This case is presened in Figure 4-1 belo. he periodic forcing for he enire orsional roaion Figure 4-1 The bridge response due o a combined loading siuaion I is quie clear ha Figure 4-1 differs significanly from Figure Unforunaely, i is impossible o say exacly ha forces ere acing on he Tacoma Narros Bridge and hen hese forces ere acing on i. Hoever his does emphasize ha here is sill some ork needed o be done on he mahemaical model of he Tacoma Narros Bridge in order o pain a more accurae picure of he orsional moion of he bridge. 15

18 5 Developmen of a Theory From he resuls of he modificaions made o McKenna s mahemaical model, e ill no develop a heory as o hy he Tacoma Narros Bridge failed. Since he case demonsraing he hanger cables eakening produced he mos ineresing resuls, e ill ake a closer look a he hanger cables. Firs e should deermine ha he acual forces on he hanger cables ere. A good place o sar is by assuming ha he only force on he hangers as a resul of he bridge deck free falling from a heigh equal o he ampliude of he bridge decks roaion. From Neon s Second La of Moion e kno ha F = ma (5.1 here F is force, m is mass, and a represens acceleraion [12]. Unforunaely, he only variable knon ihou furher invesigaion is he mass of he bridge. To deermine he acceleraion of he bridge deck, e ill analyze he basic disance-velociy-acceleraion relaionships hich are a = v& (5.2 and v = x& (5.3 ih v denoing velociy and x represening disance [4]. The acceleraion ha e seek represens he acceleraion he bridge deck is subjeced o hile he cable resiss he force of he bridge deck. We ill approximae he acceleraion o be consan since e are only concerned ih obaining a general idea of ha he forces ere in he cables. In order o find his, e firs need o inegrae Equaion 5.2 dv a = a d = dv Equaion 5.2 d a a 1 0 d = v = f1 v f vo v dv a = v o1 f v o Inegraing 5.2 In Equaion 5.4, v f 1 is knon o be zero since his ould correspond o he velociy of he deck a he poin hen he deck is changing direcion from falling o being pulled up by he cable. Hoever, he ime ( i akes for he cable o sop he bridge deck, and he velociy of he deck a he poin hen he cable firs begins resising i ( v o 1 are sill unknon. When he cable begins resising, he velociy of he bridge deck ( v o1 ould be he same as he final velociy of he bridge deck as i falls from he op of he roaion o he poin here he cable begins resising ( v f 2. From furher manipulaions o Equaion 5.2 and Equaion 5.3, e can arrive a Equaion 5.5. dx dx v = d = Equaion 5.3 d v dv a = Equaion 5.2 d dv a = v Combining (5.2 and (5.3 dx a v x f xo dx = v f vo v dv a ( x 2 2 f = v 2 2 ( 2 o + a x 2 f xo f 2 v f xo = 2 2 o v 2 (5.4 Inegraing combined Equaion (5.5 16

19 In Equaions 5.5, v o 2 is zero since his is he velociy of he deck a he poin here he deck is changing direcions from rising o falling. Also in his equaion, a 2 represens he acceleraion due o graviy since e are assuming he deck is free falling from he apex of is roaion o he poin ha he cables begin resising forces. Oher variables in Equaion 5.5 are x f and x o hich represen he final and iniial posiions of he bridge deck respecively. When e subsiue v f 2 in for v o 1, Equaion 5.4 simplifies o Equaion g( x f xo a1 = (5.6 If e no combine Equaion 5.6 and Equaion 5.1, e ge 2g( x f xo F = m (5.7 hich e can use o find he force in each hanger cable ih some minor assumpions. As F. B. Farquharson as observing he bridge hile i as in orsional roaion, he noed ha The moion had a frequency of 14 cycles per minue [1]. Also, in [9], i is menioned ha he double ampliude of he bridge as abou 28 fee. From his informaion, e can conclude ha one cycle ook approximaely 4.3 seconds and had an ampliude of approximaely 4.2 meers. As menioned earlier, he bridge deck eighed approximaely 2,500 kgs. per foo of bridge [1]. The spacing beeen groups of hanger cables on he bridge as approximaely 50 fee, and here ere four cables grouped ogeher per side of he bridge [13]. If e assume ha he bridge deck eigh is evenly disribued beeen all eigh cables for each secion of bridge, e find ha m 15, 625 kgs per cable. To proceed any furher, e ill have o make he assumpion as o hen he cables began picking up load. To mainain simpliciy, e ill assume he load as picked up halfay hrough he fall of he bridge deck, and he duraion of he cables resising forces as one-forh he ime he deck ook o make once cycle. Wih hese assumpions, e obain x f x o 4. 2 meers and 1. 1 seconds. We can no subsiue hese values ino Equaion 5.7 o ge F 120, 000 N. No ha e have an approximae force in one hanger cable, e can find he sress in his cable by using F σ = (5.8 A here σ is sress and A is cross-secional area [10]. Due o he diameer of he hanger cables on he Tacoma Narros Bridge no being readily available, e ill have o assume a diameer for hese. From inspecion of many phoos of he bridge, a 1 diameer cable seems reasonable and ill be used. Wih his assumpion, e ill find ha he sress in each suspender cable is approximaely 230 MPa. Typical srucural seel ha is used oday has a yield srengh in he range of 250 MPa o 350 MPa [2], hich shos ha he forces in he hanger cables during he orsional roaion of he bridge ere probably significan. Alhough his example did no ake ino accoun all of he forces acing on he bridge, i does give an idea of ho large he forces may have been. From hese resuls, e can see ha he seel used for he hanger cables may have been repeaedly sressed up o is yield poin. If his occurred, he hanger cables may have undergone srain hardening hich ould have increased he chances of a brile failure in he hanger cables [10]. A more in deph invesigaion of he forces on he hanger cables needs o be conduced in order o give a beer idea of his possibiliy. 6 Fuure Sudy This paper concenraed on numerically analyzing various loading siuaions of he Tacoma Narros Bridge and developing a heory as o hy he bridge ulimaely failed. The nex sep ould be o verify hese cases by creaing a scale model of a secion of he bridge and simulaing hese loading condiions. Creaing a scale model and subjecing i o hese ess ould also help o find he magniude of hese various forces. From his informaion, an 17

20 even more accurae response could be modeled. Also, ih a beer idea of he forces acing on he bridge, he heory presened could be developed furher. By analyzing he forces ha have he greaes impac on he orsional roaion of he Tacoma Narros Bridge, engineers can focus on designing bridges o resis hese forces in fuure suspension bridges hich ould help o improve heir safey. This ould allo engineers o push he limis of heir designs ih more cerainy. In doing his, significan amouns of money could be saved by no over-designing hese large srucures, and engineers ould be given more freedom o innovae ne sound, srucural designs. References 1. O. H. Amann, T. von Kármán, and G. B. Woodruff, Federal Works Agency, The Failure of he Tacoma Narros Bridge ( American Insiue of Seel Consrucion (AISC, Seel Consrucion Manual. 13 ed (Chicago, Il, M. Baes and S. Donohoe, "Tacoma Narros Bridge," Suden Projecs in Differenial Equaions, hp://online.redoods.edu/insruc/darnold/deproj/sp03/seanma/paper.pdf. 4. F. P. Beer, E. R. Johnson, Jr., and W. E. Clausen, Vecor Mechanics for Engineers: Dynamics. 7 ed. (Ne York, NY: McGra-Hill, K. Y. Billah and R. H. Scanlan, Resonance, Tacoma Narros bridge failure, and undergraduae physics exbooks, Amer. J. Physics 59 (1991: P. Blanchard, R. L. Devaney, and G. R. Hall, Differenial Equaions. 3 ed. (Crasfordsville, IN: Thomson Brooks/Cole, B. C. Evans, McKenna Uses Mah o Solve Mysery of Bridge Collapse, Advance (2001, hp://advance.uconn.edu/2001/011001/ hm. 8. K. Huffman, Tacoma Narros and he Gradien Vecor, Suden Projecs in Differenial Equaions, hp://online.redoods.edu/insruc/darnold/deproj/sp05/khuffman/finalprojecpdf.pdf. 9. P. J. McKenna, "Large Torsional Oscillaions in Suspension Bridges Revisied: Fixing an Old Approximaion," The American Mahemaical Monhly 106 (1999: W. F. Riley, L. D. Surges, and D. H. Morris, Mechanics of Maerials. 5 ed. (Ne York, NY: John Wiley & Sons, Inc., Running Simulaions: Choosing a Solver The MahWorks, Inc. (2007, hp://.mahorks.com/ access/helpdesk/help/oolbox/simulink/. 12. J. Sear, Calculus: Early Transcendenals. 5 ed. (Belmon, CA: Thomson Brooks/Cole, "Tacoma Narros Bridge: 1940 Narros Bridge - The Machine," WSDOT (2005, hp://.sdo.a.gov /TNBhisory/Machine/machine2.hm. 14. D. G. Zill, A Firs Course in Differenial Equaions ih Modeling Applicaions. 8 ed. (Taunon, MA: Thomson Brooks/Cole,