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Glbal Jural Sciece Frtier Research Matheatics & Decisi Scieces Vlue 1 Issue 3 Versi 1.

(USA Olie ISSN: 49-466 & Prit ISSN: 975-5896 O The Respse a N -Uir Bea Trasvered by Mbile Distributed ads By Oguyebi S.

The urth rder partial dieretial equati gverig the prble is slved whe the bea is trasverse by bile distributed lads.

It is bserved that the delecti -uir bea uder the acti vig asses is higher tha the delecti vig rce whe ly the rce eects the vig lad are csidered.

1 Glbal Jural Sciece Frtier Research Matheatics & Decisi Scieces Vlue 1 Issue 3 Versi 1. March 1 Type : Duble Blid Peer Reviewed Iteratial Research Jural Publisher: Glbal Jurals Ic. (USA Olie ISSN: & Prit ISSN: O The Respse a N -Uir Bea Trasvered by Mbile Distributed ads By Oguyebi S. N & Suday J Uiversity Ad-Ekiti, Ekiti State, Nigeria Abstract The prble beig ivestigated i this paper is that the respse -uir bea uder tesile stress ad restig a elastic udati. The urth rder partial dieretial equati gverig the prble is slved whe the bea is trasverse by bile distributed lads. The elastic prperties the bea, the leible rigidity, ad the ass per uit legth are epressed as uctis the spatial variable usig Struble s ethd. It is bserved that the delecti -uir bea uder the acti vig asses is higher tha the delecti vig rce whe ly the rce eects the vig lad are csidered. Fr the aalysis, the respse aplitudes bth vig rce ad vig ass prbles decrease with icreasig udati cstat. Keywrds : Distributed ad, N-uir, Elastic Fudati, vig Mass. GJSFR-F Classicati : FOR Cde: 199 Strictly as per the cpliace ad regulatis : 1. Oguyebi S. N & Suday J.This is a research/review paper, distributed uder the ters the Creative Cs Attributi-Ncercial 3. Uprted icese perittig all cercial use, distributi, ad reprducti i ay ediu, prvided the rigial wrk is prperly cited.

Re. several vig 3. Klusek V. et al (1967: Civil egieerig Structures Subjected t Dyaic ads (i slvak SVT, Bratislava. 1. Oi, S.T (1996: Respse a - uir bea restig a elastic udati t asses.

N & Suday J Abstract - The Prble beig ivestigated i this paper is that the respse -uir bea uder tesile stress ad restig a elastic udati.

The elastic prperties the bea, the leible rigidity, ad the ass per uit legth are epressed as uctis the spatial variable usig Struble s ethd.

Fr the aalysis, the respse aplitudes bth vig rce ad vig ass prbles decrease with icreasig udati cstat. Keywrds : Distributed ad, N-uir, Elastic Fudati, vig Mass. I.

e vibratis turbies, hulls shills ad bridge girders variable dept etc. Bea elastic udati subjected t vig asses have received etesive atteti i the literature.

2 Re. several vig 3. Klusek V. et al (1967: Civil egieerig Structures Subjected t Dyaic ads (i slvak SVT, Bratislava. 1. Oi, S.T (1996: Respse a - uir bea restig a elastic udati t asses. Abacus Jural Matheatical Assciati Nigeria. Vl. 1,Pp O the Respse a N-Uir Bea Trasvered by Mbile Distributed ads Oguyebi S. N & Suday J Abstract - The Prble beig ivestigated i this paper is that the respse -uir bea uder tesile stress ad restig a elastic udati. The urth rder partial dieretial equati gverig the prble is slved whe the bea is trasverse by bile distributed lads. The elastic prperties the bea, the leible rigidity, ad the ass per uit legth are epressed as uctis the spatial variable usig Struble s ethd. It is bserved that the delecti -uir bea uder the acti vig asses is higher tha the delecti vig rce whe ly the rce eects the vig lad are csidered. Fr the aalysis, the respse aplitudes bth vig rce ad vig ass prbles decrease with icreasig udati cstat. Keywrds : Distributed ad, N-uir, Elastic Fudati, vig Mass. I. INTRODUCTION Structural egieers usually ecutered prble that arises especially whe a bea is beig trasverse by a vig lad. The thery vibrati structures has treated se these prble i.e vibratis turbies, hulls shills ad bridge girders variable dept etc. Bea elastic udati subjected t vig asses have received etesive atteti i the literature. Klusek et al [3] used ral de aalysis t address the prble leible vibrati -uir bea. This was llwed by Sadiku ad eiphlz [6] wh ly studied the dyaics a uir bea by csiderig the iertia eect a vig ass ad later develped the Gree s ucti the assciated dieretial prble thereby btaied a clsed r sluti. I a later develpet, Oi [1] preseted the prble dyaic aalysis a uir bea t several vig asses uder ccetrated lad. The bea csidered is uder tesile stress ad by the ethd Galarki, the result is btaied r the irst de respse the bea. Chau ad Seg [8] wrked the static respse beas -liear elastic udati where the dered shape the structure was represeted by a Furier series, ad thereater, the givig equati is reduced t a set secd rder siultaeus equatis usig Galarki s ethd. I all the areetied wrks, the practical cases where the elastic systes are variable crss secti ad distributed vig lads use t csidered. The paper therere presets the prble dyaic respse a -uir bea t vig asses elastic udati traversed by bile distributed lad. Authr : Departet Matheatical Scieces, Uiversity Ad-Ekiti, Ekiti State, Nigeria. Authr : Departet Matheatical Scieces, Adaawa State Uiversity Mubi, Adaawa State, Nigeria. 11 March Glbal Jural Sciece Frtier Research F Vlue XII Issue III V ersi I 1 Glbal Jurals Ic. (US

II. DERIVATION AND ASSEMBY OF THE GOVERNING EQUATION Csider a vig lad ( t, ass M actig a Berulli-Euler bea (Nuir uirly laded ad ve at a cstat velcity c as shw belw: Ntes Jural Sciece Frtier Research

3 II. DERIVATION AND ASSEMBY OF THE GOVERNING EQUATION Csider a vig lad ( t, ass M actig a Berulli-Euler bea (Nuir uirly laded ad ve at a cstat velcity c as shw belw: Ntes Jural Sciece Frtier Research Vlue XII Issue III V ersi I Glbal March 1 Figure 1: Uirly distributed lad siply supprted bea. I the structure abve, the displaceet is gvered by the equati U(, t U(, t U(, t EI ( ( N k( U(, t (, (, 1 t U t U (, t t g where U(, t is trasverse displaceet, E is the Yug dulus, I( is variable et iertia, EI( is leible rigidity,, is the substative accelerati peratr, g is the accelerati due t gravity. Fr the -uir bea such as abve, its prperties such as et iertia I ad the ass per uit legth the bea vary alg the spa the bea. The structure uder csiderati is siply supprted ad carryig a arbitrary uber asses M vig with cstat velcities. The Operatr is deied as F ad the lad ( t, is give as c c c t t t (. (, t MH ( ct g c c t t (.3 where H( ct is the Heaviside ucti. Furtherre, the budary cditi r the dyaical syste is take t be arbitrary ad the iitial cditi the ti is U(, t U(, t (.4 t Substitutig equatis (., (.3, it (.1, the gverig ti takes the r EI ( U(, t ( U(, t N U(, t K( U(, t t MH ( ct c c U(, t MgH ( ct t t (.5 1 Glbal Jurals Ic. (US

4 Equati.5 ca be urther be sipliied t give urther sipliicati yields; N 1 Si U(, t N 1 Si Cs U(, t U(, t N3 1 Si Cs N4(1 si Si N5 Re. U(, t M U(, t U(, t 1 Si N 6U(, t H ct U( ct c c t t t where, g ( H ct (.6 31 March 7. Wh, J.S et al (1987:The Dyaic aalysis a lat plate uder a vig lad by a deiite eleet ethd. Iteratial Jural Nuerical ethds i Egieerig 4, EI 6 EI 6 EI 3 EI N K N N N N N N (.7 1,, 3, 4 5, 6 Equati (.6 is a -hgeus partial dieretial equati with variable ceiciets. Clearly, it is see that the clsed r sluti des t eists. III. SOUTION PROCEDURE T slve equati (.6, a appriate sluti is sught. Oe the appriate ethds best suited t slve diverse prbles i dyaics structures is the Galarki s ethd [7]. This ethd requires that the sluti equati (.6 be the r U Y ( t X ( (3.1 1 where X( is chse such that all the budary cditis are satisied. Equati (3.1 whe substituted it equati (.6 yields; 1 3 IV III N11 Si Y ( t X ( N1 Si Cs Y ( t X ( II N3 1 Si Cs N4(1 si Si N5 Y( t X ( M I II 1 Si Y ( t X ( N6Y ( t X ( H ct Y ( t X ( cy ( t X ( c Y ( t X ( Mg ( H ct (3. Glbal Jural Sciece Frtier Research F Vlue XII Issue III V ersi I 1 Glbal Jurals Ic. (US

5 I rder t deterie Y ( t, it is required that the epressi the let had side equati (3. be rthgal t ucti X (. Hece, 3 IV III N11 Si Y ( t X ( N1 Si Cs Y ( t X ( 1 II N3 1 Si Cs N4(1 si Si N5 Y( t X ( Ntes 1 March 4Glbal Jural Sciece Frtier Research Vlue XII Issue ersi I M I II 1 Si Y ( t X ( N6Y ( t X ( H ct Y ( t X ( cy ( t X ( c Y( t X ( Mg ( ( H ct X k d Sice ur dyaical syste has siple supprts at the edges ad, we chse; V ( X Si (3.3 III Csequetly, usig (3.4 i (3.3 gives M Mg HaY ( t HbY ( t Hc( t Y ( t ch d ( t Y ( t c He( t Y ( t H ( t 1 (3.5 where F k Ha 1Si Si Si d, Hb Q Q Q Q Q Q k Hc( t H( ct Si Si d, H ( t H( ct Si Si d, e k k Hd ( t H( ct Cs Si d k H ( t H( ct Si d (3.6 ad k k Q1 N si Si Si d, Q N 3 1 si Cs Si Si d k k Q3 N 3 1 si Cs Si Si d, Q 4 N 4 1 si Si Si Si d k Q5 N5 Si Si d, Q 6 N6 si si k d (3.7 Whe the itegrals (3.6 ad (3.7 are evaluated, the result is a series cupled dieretial equatis called Galarki s equatis r -degree reed syste gverig 1 Glbal Jurals Ic. (US

6 the ceiciets all lwer ad higher des the bea. Thus, restrictig urselves t the aalysis the irst de respse, we set 1ad 1i equati (3.5 r aalytical appriati. Fllwig the ethd [9] where Heaviside ucti is epresses as Furier csie series. Thus, equati (3.5 leads t Re. 1 ct H Y ( t H Y ( t I Cs I C I Y ( t a b C 4C ct CC I ( 4 Cs I 5 I 6 Y t 1 C C ct C C ct I 3 7 Cs I 3 8 I 9 Y ( t P Cs C s 1 ( March 6. Sadiku S eiphtz H.H.E (1987: O the dyaics elastic systes with vig ccetrated asses. Ig. Archiv 57, Sith, J.W (1988: ibrati structures. Applicati i Civil Egieerig Desig. Chapa ad hall td d. M Mg where 1 ad P (3.9 which is the trasred equati the dyaical syste. IV. ANAYTICA APPROXIMATE SOUTION a Siply Supprted Traversed By Mvig Frce A appriate del the syste, whe the iertia eect the vig ass is eglected, is the vig rce prble assciated with the syste. Settig, we have 1 where H H b a Subjectig equati (4. t aplace trasr deied by where S is a aplace trasr. It yields, ct Y t Y t P Cs C ( ( s ( CsZ 1 kt Cst Cs t U (, t P E( Si 1 Zk (4.4 ct where Zk ad E( Cs which is the respse t vig rce sluti the elastic syste at cstat velcity. b Siply Supprted Traversed By Mvig Mass Fr the vig ass sluti, we set 1, i this case, the etire sluti t the prble is sught. T this ed, a diicati the asypttic ethd Struble[6] te used r treatig weakly hgeeus ad -hgeus -liear syste is eplyed. Further arrageet equati (3.8 yields e st dt (4.1 (4. (4.3 Glbal Jural Sciece Frtier Research F Vlue XII Issue III V ersi I 1 Glbal Jurals Ic. (US

7 Jural Sciece Frtier Research F Vlue XII Issue III V ersi I 6Glbal March 1 At this jucture, we seek the diied requecy crrespdig t the requecy the ree syste due t the presece vig ass [8].T this ed, the sluti t equati (4.5 ca be writte as Therere whe the ass the particle is csidered, the irst appriati t the hgeeus syste is give as where 1 ct H I Cs I C I Y t a 1 ( C 4C ct CC ( 1 I 4 Cs I 5 I 6 Y t 1 C C ct C C ct Hb 1 I 3 7 Cs I 3 8 I 9 Y ( t P Cs C s 1 (4.5 Y ( t N(, t t (, t (4.6 where t ad ( t, are cstats. Y ( t D (, t jjt (, t Equati (4.8 is called the diied requecy crrespdig t the requecy the ree syste due t the presece the vig ass. Thus, the etire equati (4.5 takes the r d P j ct Y ( ( s t Y t Cs C dt H a (4.9 which is a prttype equati (4.1ad whe iverted we have Equati (4.1 is the trasverse displaceet respse t vig ass sluti r siply supprted bea elastic udati. V. DISCUSSION OF RESUTS (4.7 j 1 C jj 1 I1 C I3 I7 I9 (4.8 Ha P 1 j CsZkt Cs jjt Cs jjt U (, t E( Si 1 Ha jj Zk jj Resace cditi It is desirable t ispect clsely the respse aplitude the dyaical syste. Fllwig [1], the vig rce i equati (4.8 attais a resace wheever c (5.1 while whe c jj (5. (4.1 Re. 8. Chau, F.W ad Seg, O. (198: Beas -liear elastic udati. Jural Applied Mechaics. Vl., Pp Oi S.T ad Oguyebi S.N (8: Dyaical aalysis a prestressed elastic bea with geeral budry cditis uder the acti uir distributed asses. 1 Glbal Jurals Ic. (US

8 gives r the vig ass prble. Re-writte equati (4.8 i the r jj 1 1 C j I1 C I3 I7 I9 Ha (5.3 which iplies Ntes 1 1 c C j I1 C I3 I7 I9 Ha VI. CONCUSION I view the cditi r resace established abve, it is deduced that r the sae atural requecy, the critical speed r the vig rce siply supprted bea is greater tha that the vig ass prble. Thus r the sae atural requecy, resace is reached earlier i the vig ass syste tha i the vig rce syste. Fr practical purpses, a e diesial structures (Bea are used as atheatical dels i the buildigs ad bridges cstructi. Hece apprpriate precauti ay w be take by the structural egieers t restall the ccurrece resace i the structure by itegratig the ecessary vibrati absrber it the del. 1. Saistic M.M et al (1974: O a thery ccerig the dyaic behavir structures carryig vig asses.ig Archiv. 43, Dig Z. (1993: A geeral sluti t vibratis beas variables. Wiker elastic udati, Cputers ad structures, Vl. 47, Klusek V. et al (1967: Civil egieerig Structures Subjected t Dyaic ads (i slvak SVT, Bratislava. 4. Esailzadeh E. ad Ghrashi M. (1995: Vibrati aalysis bea traversed by uir distributed vig ass, Jural sud ad vibrati, 184 (1, i, Y.H (1996: Cets vibrati aalysis bea traversed by Uir partially distributed vig ass. Jural sud ad vibrati; 199(4, Sadiku S eiphtz H.H.E (1987: O the dyaics elastic systes with vig ccetrated asses. Ig. Archiv 57, Wh, J.S et al (1987:The Dyaic aalysis a lat plate uder a vig lad by a deiite eleet ethd. Iteratial Jural Nuerical ethds i Egieerig 4, Chau, F.W ad Seg, O. (198: Beas -liear elastic udati. Jural Applied Mechaics. Vl., Pp Sith, J.W (1988: ibrati structures. Applicati i Civil Egieerig Desig. Chapa ad hall td d. 1. Oi, S.T (1996: Respse a - uir bea restig a elastic udati t several vig asses. Abacus Jural Matheatical Assciati Nigeria. Vl. 1,Pp Kerr, Arbld D (1964: Elastic ad Viscelastic udati dels. Jurals Applied Mechaics, Trasactis the ASME Oi S.T ad Oguyebi S.N (8: Dyaical aalysis a prestressed elastic bea with geeral budry cditis uder the acti uir distributed asses. 13. Jural the Nigeria Assciati Matheatical physics. Vl. 1,Pp REFERENCES RÉFÉRENCES REFERENCIAS ( March Glbal Jural Sciece Frtier Research F Vlue XII Issue III V ersi I 1 Glbal Jurals Ic. (US

9 8Glbal Jural Sciece Frtier Research Vlue XII Issue ersi I Ntes F III V March 1 This page is itetially let blak 1 Glbal Jurals Ic. (US

A New Method for Finding an Optimal Solution. of Fully Interval Integer Transportation Problems

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