Transverse Vibrations of Elastic Thin Beam Resting on Variable Elastic Foundations and Subjected to Traveling Distributed Forces.

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1 Trasverse Vibraios of Elasic Thi Beam Resig o Variable Elasic Foudaios ad Subjeced o Travelig Disribued Forces. B. Omolofe ad S.N. Oguyebi * Deparme of Mahemaical Scieces, Federal Uiversiy of Techology, Akure, Nigeria. Deparme of Mahemaical Scieces, Uiversiy of Ado-Ekii, Nigeria. * seguoguyebi@yahoo.com ABSTRACT I his paper, he problem of he forced vibraios of elasic Beroulli-Euler beam resig o variable elasic foudaios ad raversed by uiformly disribued load is ivesigaed. The beam is assumed o be uiform ad has simple suppor a boh eds. The movig disribued force is assumed o move wih cosa velociy. The robus echique called Galerki s mehod i cojucio wih iegral rasform mehod are used o rea he fourh order parial differeial equaios describig he moio of he beam-load sysem. Resuls show ha, icreases i he values of beam parameers, aial force N, ad foudaio modulus K, sigificaly reduce he deflecio profile of he vibraig beam. I is equally foud ha, icorporaig aial force N, foudaio modulus K, ad a dampig erm io he goverig equaio of moio icreases he criical velociy of dyamical sysem hereby reducig he risk of resoace. (Keywords: dampig, disribued forces, foudaio siffess, respose of srucures, movig masses) INTRODUCTION This paper is cocered wih assessig he problem of he compleiy of he ierracio of deformable elasic sysem (beam or plae) ad he dyamic subsysem (movig force or movig mass) raversig i. Due o is very wide rage of applicaios i diverse areas, researchers from he field of egieerig, mahemaical physics, ad applied physics have made sigifica coribuios i his area of sudy i he pas few years [-]. I is well kow ha whe loads move o srucural members ha offer resisace o bedig, i produces wo effecs which cause he srucure o vibrae coiously. These wo effecs i he field of srucural dyamics are ermed he movig force effecs ad movig mass effec [,]. I imes pas, aalyical sudies cocerig he vibraios of elasic srucures uder he passage of fas ravellig loads have bee eclusivelly reserved i he lieraure for he case of srucureload ierracios where he movig load is assumed o be coceraed loads. These ypes of loads are oly useful i mahemaical idealizaios, bu i realiy cao be foud i he real world. The more pracical cases where he movig load is a disribued ype is very seldom foud i he lieraure. Amog he few sudies abou he respose of elasic srucures o movig disribued loads kow i he lieraure are he works of Gbadeya ad Dada who sudied he dyamic respose of plaes o paserak foudaio o disribued movig load. I his sudy he movig force plae model is cosidered ad i is foud ha a icrease i he area of he disribuio of he movig mass causes a reducio i he maimum dyamic deflecio. I all of he aforemeioed sudies, he auhors egleced he dampig erm i he goverig differeial equaio of moio ad effecs of elasic foudaio of o- uiform siffess was o ivesigaed. This sudy herefore ivesigaes he rasverse displaceme respose of aially presressed hi beams resig o variable elasic foudaio o movig disribued loads. FORMUATION OF TE BOUNDARY VAUE PROBEM Cosider a prismaic Beroulli-Euler beam of legh resig o a Wikler foudaio ad The Pacific Joural of Sciece ad Techology hp:// Volume. Number. November 9 (Fall)

2 raversed by uiformly disribued load movig a cosa velociy. The vibraioal behaviour of he beam-load sysem is described by he fourh order parial differeial equaio wih variable coefficie give as: 4 w (, ) w (, ) w (, ) w (, ) EJ N + ε 4 + μ + K( ) w(, ) = P(, ) () where: w (, ) EJ E J K g ε μ is he laeral deflecio of he beam measured from is equilibrum posiio is he fleural rigidiy of he beam is he modulus of elasiciy is he mome of ieria is he coefficie of Wikler foudaio is he acceleraio due o graviy is he fied legh of he beam is he spaial coodiae is mass per ui legh of he beam is he ime The movig load o he beam uder cosideraio has mass commesurable wih he mass of he beam ad he load o he beam is assumed o be of mass m movig wih cosa velociy c. A variable elasic foudaio of he form: K = K + () (4 3 ) K where is he foudaio cosa, is cosidered. The ravelig ime of he movig load is assumed o be limied o ha ierval of ime wihi which he he movig load is o he beam, ha is: c (4a) ad he movig force P (, ) is assumed o be a uiformly disribued sigle poi load give as: P (, ) = P ( c) (4b) Furhermore, we assume ha he beam has simple suppors a he ed = ad a he ed =, so ha boh he bedig mome ad he deflecio vaish a boh eds. Thus, he perie boudary codiios ha perai a hese wo eds is give as: W(,) = = W(,), W(,) = = W(,) ad he iial codiios are: (3a) W(,) = = W(,) (3b) P where is he mass of he load muliply he acceleraio due o graviy g ad ( c) is he heaviside ui sep fucio defied as:, for < ( c) =, for > wih he properies, (5) (i) (ii) d (i) [ ( c )] = δ ( c ) d (6) (iii), for < c f( ) ( c) = f ( ), for c (7) (ii) The Pacific Joural of Sciece ad Techology 3 hp:// Volume. Number. November 9 (Fall)

3 Figure: A Disribued oad o Elasic Beam. where δ ( c) represes he Dirac dela fucio. ( c) is a ypical egieerig fucio made o measure egieerig applicaios which ofe ivolved fucios ha are eiher off or o. Subsiuig (3), (4), ad (5) io (), oe obais: w (, ) w (, ) w (, ) w (, ) EJ N K K w Mg (8) 4 + ε 4 + μ + = (4 3 + ) (, ) = ( c) SOUTION PROCEDURES: where is give as: I his secio, we proceed o solve he above iiial-boudary value problem (8) by employig Galerki mehod. The mehod epresses he soluios of he equaio (8) respecively as: W (, ) = Y ( ) U ( ) (9) m m= m m ( ) mπ Um Si = () The fucio U ( m ) is choose o saisfy he perie boudary codiios. Thus subsiuig (9) ad () ad is derivaives io (8), we have: 4 mπ mπ mπ mπ mπ EJ Ym() Si + N Ym() Si + ε Y & m() Si m= m= m= mπ mπ + μ + + = Y && m( ) Si K (4 3 ) Ym( ) Si Mg( c) m= m= () which afer some re-arragemes gives: 4 mπ mπ mπ mπ mπ mπ EJ Ym() Si + N Ym() Si + εy& m() Si + μy&& m() Si m= mπ + K (4 3 + ) Ym ( ) Si = Mg ( c) () I order o deermie Y (), i is required ha he equaio (.4) be orhogoal o he fucio U ( ). ece: m k The Pacific Joural of Sciece ad Techology 4 hp:// Volume. Number. November 9 (Fall)

4 4 mπ mπ mπ mπ mπ mπ EJ Ym() Si + N Ym() Si + εy& m() Si + μy&& m() Si m= mπ + K (4 3 + ) Ym ( ) Si = Mg ( c) Si d (3) Equaio (3) ca furher be re-arraged as: m= m Y && π m() + QY & m() + QYm() = Q3 ( c) Si d (4) where, Q = + +, Q 5 = Q = (5) ad 4 EJ mπ mπ Si Si d μ (6) = 3 4 N mπ mπ Si Si d μ (7) = ε mπ μ = Si Si d (8) mπ = Si Si d (9) K mπ Si Si d μ () 3 5 = (4 3 + ) 6 mg = () μ Epressig he eaviside ui sep fucio as a fourier cosie series i equaio (.5) o (.7) he we have: Thus, πc π ( c) = Cos Cos d + = () c = ( c) = Cos + Cos (3) The Pacific Joural of Sciece ad Techology 5 hp:// Volume. Number. November 9 (Fall)

5 Cosiderig oly he mh paricle of he dyamical sysem, hus, oe obais, k c Y&& π m() + QY& m() + QYm() = Q3 Cos + Cos (4) To obai soluio o he secod ordiary differeial equaio (4) above, we subjeced i o a aplace rasform defied as: () = () e s d % (5) Thus, Y() s = S Q Cos ( ) 3 S S + S QS Q (6) which reduces o: a s Y() s = Q3 + b s s + η ( s α)( s α) (7) where, a = Cos, b = ad Furher simplificaio yields: α Q + Q 4Q Q Q 4Q =, α = (8) = Q a a + s s 3 Y() s b b α α s s α s s α s + η s α s + η s α (9) I order o obai he aplace iversio of (9), we make he followig represeaios: g() = a, g() = Cosη, f ( ) e, f ( ) e α α = = (3) so ha he aplace iversio of (9) is he covoluio of fi s ad gi s defied by: (3) f g = f ( u) g ( u) du r =,,3... i i i i Thus he aplace iversio of equaio (9) is give by: The Pacific Joural of Sciece ad Techology 6 hp:// Volume. Number. November 9 (Fall)

6 Q3 a a ηsiη αcosη ηsiη αcosη U j () = + + b b α α α α α η α η (3) where, α αu IA = ae e du α αu IB = ae e du αu α IC = ae e Cosη udu αu α I D = ae e Cosη udu (33) (34) (35) (36) Usig (33) o (36) i (3), oe obais, Q3 ηsiη αcosη ηsiη αcosη mπ U j () = a + b b Si α α α α α η α η (37) which o iversio yields: Q3 ηsiη αcosη ηsiη αcosη mπ wm (, ) = a + b b Si α α m= α α α η α η (38) NUMERICA RESUTS AND DISCUSSION For he purpose of Numerical aalysis of our dyamical sysem, he uiform hi beamof legh.9m is cosidered. Also EI 4 m / s, μ = speed of he mass is 8.8m/s ad he raio of he mass of he load o he beam is.. The rasverse deflecio of he beam are calculaed ad ploed agais he ime for various values of aial force N ad subgrade K. Values of N bewee ad were used while he values of K were varied bewee N/m 3. The resuls are as show o he various graphs below. Figure displays rasverse displaceme respose of a simply suppored uiform beam uder he acio of disribued forces movig a a variable velociies for various values of aial force N for fied values of foudaio moduli K=4,. The figure shows ha as N icreases, he deflecio of he uiform beam decreases. I a similar way, for various ime, he deflecio profile of he beam for various values of foudaio moduli K ad for fied aial force N are show i Figure 3. I is observed ha higher values of foudaio moduli reduce he deflecio profile of he beam. CONCUSION I his paper he dyamic behavior of fiie elasic beam resig o elasic foudaio ad subjeced o movig disribued forces is ivesigaed. The Pacific Joural of Sciece ad Techology 7 hp:// Volume. Number. November 9 (Fall)

7 Figure : The Deflecio Profile of he Simply Suppored Thi Beam Uder he Acio of Cosa Velociy for Various Values of Aial Force N ad for Fied Value of Foudaio Modulus K (4). Figure : The Deflecio Profile of he Simply Suppored Thi Beam Uder he Acio of Cosa Velociy for Various Values of Foudaio Moduli K for Fied Value of Aial Force N (). The Pacific Joural of Sciece ad Techology 8 hp:// Volume. Number. November 9 (Fall)

8 The bea is assumed o be uder esile sress ad he moio of he movig load is assumed o be a cosa velociy ype of moio. Specral Galerki s mehod i cojucio wih iegral rasform mehod is used o obai a closed form soluio o his dyamical problem. The resuls show ha wih Icrease i he values of beam parameers such as aial force ad foudaio siffess, he respose ampliude of he vibraig beam reduces. Furhermore, Icrease i he values of hese parameers ad he dampig coefficie produces a sigifica effec o he criical velociy of he beam-load sysem ad he risk of resoace is sufficiely reduced. This resul is i perfec agreeme wih eisig resul [, ]. REFERENCES. Milormir, M., Saisic, M.M., ad ardi, J.C O he Respose of Beam o a Arbirary Number of Coceraed Movig Masses. Joural of he Frakli Isiue. 87().. Saisic, M.M, ardi, J.A., ad ou, Y.C O he Respose of he Plae o a Muli-Mass Movig Sysem. Aca Mechaical. 5: Wilso, J.F Dyamic Whip of Elasically Resraied Plae Srip o Rapid Trasi oad. Trasacio of America Sociey of Mechaical Egieers, series G. 96: Seele C.R The Fiie Beam wih a Movig oad. Joural of Applied Mechaics. 34():- 8.. Shadam, M.R., Rofooeei, F.R., Mofid, M., ad Mehri, B.. Periodiciy i he Respose of No-iear Movig Mass. Thi-Walled Srucures. 4: Oi, S.T. 3. Fleual Moio of a Uiform Beam uder Acios of a Coceraed Mass Travelig wih Variable Velociy. Abacus, Joural of Mahemaical Associaio of Nigeria.. Oi S.T.. Fleural Vibraios uder Movig oads of Isoropic Recagular Plaes o a No- Wikler Elasic Foudaio. Joural of he Nigeria Sociey of Egieers. 35(): 8-7, Dada, M.S.. Vibraio Aalysis of Elasic Plaes uder Uiform Parially Disribued oads. Ph.D. Thesis. Uiversiy of Ilori: Ilori, Nigeria. 4. Wu Jia-Jag. 5. Vibraio Aalysis of a Parial Frame uder he Acio of a Movig Disribued Masses usig Movig Mass Eleme. I. Joural for Numerical Mehods i Egieerig. 6:8-5. SUGGESTED CITATION Omolofe, B. ad S.N. Oguyebi. 9. Trasverse Vibraios of Elasic hi Beam Resig o Variable Elasic Foudaios ad Subjeced o Travellig Disribued Forces. Pacific Joural of Sciece ad Techology. ():-9. Pacific Joural of Sciece ad Techology 5. Timosheko, S., Youg, D.., ad Weaver, W Vibraio Problem i Egieerig. Wiley: Sadiku, S. ad eipholz,..e. 98. O he Dyamics of Elasic Sysems wih Movig Coceraed Masses. Archiv. 43: Gbadeya, J.A. ad Oi, S.T Dyamic Behaviour of Beams ad Recagular Plae uder Movig oads. Joural of Soud ad Vibraio. 8(5): Oi, S.T. 99. O he Dyamic Respose of Elasic Srucures o Movig Muli-Mass Sysem. Ph.D. Thesis. Uiversiy of Ilori: Ilori, Nigeria. 9. Savi, E.. Dyamic Amplificaio Facors ad Respose Specrum for he Evaluaio of Soud ad Vibraio. 48(): The Pacific Joural of Sciece ad Techology 9 hp:// Volume. Number. November 9 (Fall)