Application of Matrix Iteration for Determining the Fundamental Frequency of Vibration of a Continuous Beam

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1 Iteratal Jural f Egeerg Research ad Develpet e-issn: 78-67, p-issn : 78-8, Vlue 4, Issue (Nveber ), PP. -6 Applcat f Matrx Iterat fr Deterg the Fudaetal Frequecy f Vbrat f a Ctuus Bea S. Sule, T.C. Nwfr Departet f Cvl ad Evretal Egeerg, Uversty Of Prt Harcurt, P.M.B. 5, Rvers State, Ngera. Abstract:-I the prevus paper, the use f atrx terat t detere the atural frequeces f vbrat f ctuus bea syste usg the ccept f wave prpagat a prsatc bar s reprted. The atural frequeces f vbrat btaed fr the frulated del were cpared wth the values btaed fr lterature ad thse f the exact slut. A errr f 8% ad % was bserved whe the fudaetal frequecy f vbrat btaed fr the prevus del was cpared wth that btaed fr lterature ad the exact slut. I the preset study, a prved atheatcal del s frulated t detere the fudaetal frequecy f vbrat f a ctuus bea as a structural syste wth dstrbuted ass usg de shape ccept. I the curse f the syste s scllat, the dsplaceets prduced by the frce f erta s assued t have the shape f a partcular vbrat de ad harc wth the partcular dal frequecy. A uercal exaple was gve t destrate the applcablty f the preset del. The fudaetal frequecy value btaed fr the preset del was fud t prve by 7.4% ad alst detcal t the exact fudaetal frequecy value ad that btaed lterature. Keywrds: Matheatcal del, atural frequeces, fudaetal frequecy, vbrat de, erta I. INTRODUCTION Resace s a very dagerus phee that ccurs f the fudaetal value f the atural frequeces f vbrat s exceeded by the frequecy f exctat. Resace results large apltude dsplaceet f structures leadg t develpet f large stresses ad stras the affected structure that ay evetually result t structural falure thereby affectg the perfrace f the structure r structures servce. The dyac aalyss f a ctuus bea as a syste wth dstrbuted ass t get the fudaetal vbrat frequecy s always cuberse because f the dffculty vlved the atheatcal apulats. The dffculty that s ecutered the atheatcal prcess s due t fte uber f degrees f freed [-9]. The dyac aalyss s ade spler usg luped ass ccept. The rgal bea wth dstrbuted ass s w cverted t a weghtless syste wth asses luped at chse pts called the dal pts. The weghtless bea syste w has a fte uber f degrees f freed []. The degree f freed s uercally equal t the uber f depedet geetrc paraeters that descrbes the psts f all asses fr all pssble dsplaceets f the structural syste at ay pt te. The preset del s sad t be defed f the dal luped asses ad ther crdates are kw []. I ths paper, atrx terat s eplyed t detere the fudaetal frequecy f vbrat f a ctuus bea syste udergg self excted vbrat. The algrth vlved s sple ad ca be acheved aually st especally whe fte uber f degrees f freed s vlved. II.. Frulat f Matheatcal Mdel Fr a udaped MDOF bea syste (Fgure ) wth degrees f freed, dsplaceets are assued t be lear fucts f the frces f erta, the equats f t f the luped asses at chse dal pts are gve by: Ierta frce geerated by a th partcular scllatg ass., j dsplaceet prduced at th de by erta frce geerated by a th scllatg ass. drect ad drect flexblty ceffcet respectvely. ()

2 Applcat f Matrx Iterat fr Deterg the Fudaetal The flexblty ceffcets resultg fr the frces f erta at the dvdual dal pts are gve by: N Fgure : Luped asses at bea des. L j jdx () The equat fr udaped atural vbrat frequecy s gve by : K () Fr a th vbrat de equat () trasfrs t: K (4) Multplyg bth sdes f equat (4) by a th dsplaceet caused by a th frce f erta trasfrs equat (4) t: K (5) The dsplaceet that s prduced by a th scllatg ass has the cfgurat f a th vbrat de ad als harc wth a th dal frequecy. Therefre, K (6) F Equat (5) w beces: F (7) Fr equat (6), F F K K (8) F Frce f erta at th de K Iverse f stffess atrx Substtutg fr F equat (8) usg equat (7) gves: K (9) Let the rw vectr f dess T,,..., () represet the vectr f dsplaceet the th vbrat de. Fr structural echacs, K () flexblty ceffcet atrx. Wthut lss f geeralty, K ad Kj j () Equat (7) w trasfrs t: (4) Let h (5) Equat () represets the dyac charcaterstcs f the weghtless bea subjected t scllatg asses. Therefre, fr equat (), h (6) Fr the frst vbrat de =, ad equat (5) beces: N ()

3 Applcat f Matrx Iterat fr Deterg the Fudaetal h (7) T ( ) (8) represets a arbtrarly chse tal dsplaceet vectr. Let y represet -zer th rder dsplaceet vectr ad let ax y ax represet ts largest dsplaceet eleet. Let be the vectr btaed by whe the etres f Usg equat (6), y y are scaled by y ax. (9) y ax The geeralzed sequece f prved dsplaceet vectrs are gve by equats (8) ad (9). h yk ; k,,... () y, k ax k T h k h yk k Let y k be a sequece f apprxats t y wth l ( yk ) () k (,,...) () Fr equats (9) ad (), the atural frequecy crrespdg t a gve vbrat de s gve by:. 5 y k (4) ad the fudaetal value (=) s gve by. 5 y k (5) Fr statcal csderat, the th dal ass at th dal pt ver a th weghtless bea seget s gve by: l l j = dstrbuted ass testy f the bea Kg/. III. CONCLUSION AND RESULTS A Exaple fr Nuercal Study A uercal exaple s used t destrate the applcablty f the preset frulat. A sple supprted ufr bea havg a dstrbuted ass testy f 4.75Kg/ as shw Fgure s used fr ths uercal study[]. () (6) Fgure : A degrees f freed bea syste fr uercal study.

4 Applcat f Matrx Iterat fr Deterg the Fudaetal P P P The flexblty atrx s syetrc. Therefre, j j Fgure : Dervat f flexblty factrs Usg equat (), the flexblty factrs are btaed as fllws: dx = l E dx = l l dx = E

5 Applcat f Matrx Iterat fr Deterg the Fudaetal = l dx = l l = dx Fr equat (6), 4

6 Applcat f Matrx Iterat fr Deterg the Fudaetal The flexblty ceffcets at the three dal pts are w arraged atrx fr as fllws: Fr equat (5), the dyac atrx s gve by: h h Usg equat (8), ultplcat f dyac atrx wth assued tal ut dsplaceet vectr gves: Usg gve by: as the axu dsplaceet the zer dsplaceet vectr, the prved dsplaceet vectr s.59,6.,. 59 T Aga, the axu dsplaceet s6.. The ew prved dsplaceet s: Usg equat (),.575,6.9, Fudaetal frequecy.45 T Table : Cpars f results Preset del Sule 9 Osadebe 999 Exact slut.45 IV DISCUSSION OF RESULTS.948 Table shws the cpars f result btaed fr the preset del wth Sule [], Osadebe [] ad the exact slut []. The percetage errrs f 7.5% ad.45% the prevus del[] cpared wth the ctrl pts [] ad exact slut [4] have reduced t.7% ad.58% the preset frulat shwg the effectveess f the preset del the deterat f the fudaetal frequecy f vbrat f a ctuus bea. The dsparty betwee the result f the preset del ad the prevus del[], Osadebe [] ad the exact slut[] ay be due t dfferece the assupts used the del frulat. V. CONCLUSION I cclus, the preset del prduces a result that s alst detcal wth thse f the ctrl pts [] ad [4] ad prves the prevus result f fudaetal frequecy [] by 7.4%, shwg hgher predctve ablty f the preset del. The preset del ca be used t predct the fudaetal frequecy f vbrat f a ult-strey buldg.

7 Applcat f Matrx Iterat fr Deterg the Fudaetal REFERENCES []. Osadebe N.N. (999). A prved MDOF del sulatg se syste wth dstrbuted ass. Jural f Uversty f Scece ad Techlgy, Kuas, vlue 9, N., ad. []. Sule, S. (). Apprxate ethd fr the deterat f atural frequeces f a ult-degree f freed bea syste. Ngera Jural f Techlgy, vl., N., Jue, pp []. Ra, S.S. (). Mechacal vbrats, 4 th edt, Pretce Hall, Ic. [4]. Ths, W.T. (988). Thery f Vbrats wth applcats. rd edt, CBS Publshers New Delh. [5]. Huar, J.L. (99). Dyacs f Structures, Pretce Hall, Ic. [6]. De Slva, C.W. (999). Vbrat Fudaetals ad Practce, CRC Press. [7]. Chpra, A.K. (). Dyacs f Structures, thery ad applcats t earthquake egeerg, d edt, Pretce Hall Ic. [8]. Clugh R.W. ad Peze, J. (98). Dyacs f Structures, McGraw-Hll It. Studet Edt, Ty. [9]. Sth, B.S. ad Cull, A. (999). Tall buldg structures: Aalyss ad Desg, Wley, New Yrk. []. Brggs, I.M. (964). Itrduct t structural dyacs McGraw-Hll Bk Cpay, New Yrk. []. De Hartg, J.P. (956). Mechacal vbrats, McGraw Hll Bk Cpay, New Yrk, pp