FREE VIBRATIONS OF SIMPLY SUPPORTED BEAMS USING FOURIER SERIES
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1 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS USING FOURIER SERIES SAWA MUBARAK ABDUAH Assistat ecturer Uiversity of Mosul Abstract Fourier series will be utilized for the solutio of simply supported beams with differet loadigs i order to arrive at a free vibratio. The equatio of the y y free vibratio is c t Oe of the methods of solvig this type of equatio is the separatio of the variables which assumes that the solutio is the product of two fuctios, oe defies the deflectio shape ad the other defies the amplitude of vibratio with time. Modes of deflectio with ad without time alog the beam were draw for certai cases. Good agreemet has bee obtaied betwee the results of the preset study ad that determied by Timoshiko[]. Key words: Beams, Fourier series, Free vibratio, Structural aalysis. / / : t y c y...[] Timoshiko. : Received d May 5 Accepted 9 th Sept. 5 5
2 Al-Rafidai Egieerig Vol. No. 6 Notatio: EI Fleural rigidity of the sectio of the beam. egth of beam. w oad fuctio. Fuctio of distace. Yt Amplitude of vibratio with time. t Time Distace m Mass c Costat Natural frequecy of the beam. / c Itroductio: The study of large amplitude of simply supported beams ca be traced to the work of Kreiger[] wherei the goverig partial differetial equatios were reduced to ordiary differetial equatios, ad the solutio was obtaied i terms of elliptic fuctios usig a oe-term approimatio. Similarly, Butgree[] gave the solutio for the large amplitude vibratio problems of higed beams based o the classical cotiuum approach. Sriivasa employed the Ritz-Galerki techique to solve the goverig oliear differetial equatio of dyamic equilibrium for free ad forced vibratio of simply supported beams ad plates [3, ]. Eveese[5] eteded the study for various boudary coditios usig the perturbatio method. Ray & Bert[6] carried out eperimetal studies to verify the aalytical solutios for the oliear vibratios of simply supported beams ad compared the solutio schemes such as the assumed space mode, assumed time mode ad Ritz-Galerki methods ad cocluded that the latter two are better tha the former. Padalai & Sathyamoorthy [7] developed model equatios for the oliear vibratios of beams, plates, rigs ad shells usig agrages equatios ad highlighted the differece i the ature of the model equatios for beams ad plates, rigs ad shells. ou & Sikarskie[8] employed form-fuctio approimatios to study the oliear forced vibratios of buckled beams. Rehfield [9] used a approimate method of oliear vibratio problems with material oliear effects for various boudary coditios. Mustafa [] used aplace trasformatio method to solve the free vibratio of simply supported beams. 5
3 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS Theory ad Applicatio The partial differetial equatio p.d.e for free udamped trasverse vibratio of beams is []: y c t where c y EI m Oe method of solvig this equatio is by the separatio of variables; it assumes that: y, t Y t where is a fuctio of distace alog the beam defiig its deflectio shape whe it vibrates ad Yt defies the amplitude of vibratio with time. Substitutig equatio for equatio yields: y c Y t 3 t The equatio 3 is rewritte so that the variables ad t are collected together ito separate terms as follows: c Y t Y t t Sice each of the variables ad t are idepedet variables, the each side of equatio is equal to a costat, say It may be rewritte dow to two ordiary differetial equatios that have to be satisfied: c 5 Rearragig equatio 5 yields: c c Puttig yields:
4 Al-Rafidai Egieerig Vol. No. 6 8 Equatio 8 ca be rewritte as: D where D The auiliary equatio is: D 9 Aalysig equatio 9 yields: D D the D D ad D D i The geeral solutio is give by: c si c cos c3 sih c cosh where c, c, c3 ad c are costats ad Y Y t t Rearragig equatio yields: Y t Y t t Ad rewritig as: D ' Y t 5
5 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS where Or D ' t D ' ad ' D The D ' i The the geeral solutio is give by: Y t Acost Bsi t 3 Substitutig equatios ad 3 for equatio yields: y, t Acost Bsit c si c cos c sih c cosh 3 The complete solutio for a particular structure requires epressios for the displacemet, slope, momet ad shear at the supports which must be substituted for. This procedure will yield three coefficiets i terms of the forth ad will also yield a frequecy equatio from which may be evaluated. The fial coefficiet epressio is a magitude of vibratio that would require ackowledgig of the iitial coditios of motios. For the simply supported beams the boudary coditios are: y, t ad y EI, t 5a y, t ad y EI, t 5b Substitutig equatio a,b for equatio yields: c si c3 Ad 3 The 3 sih c si c sih c sih 55
6 Al-Rafidai Egieerig Vol. No. 6 sice sih The c 3 Hece what is left with is the relatio A o-trivial solutio which meas: Ad From equatio 7 c EI m c sih c oly eists if si si Where is the atural frequecy of the beam. Substitutig equatio 5a for equatio to obtai: y, t A coswt B si wtsi 6 Where A ad coditios: For iitial displacemet: y, f ad iitial velocity: y, g t B are costats which ca be obtaied from the iitial The costats A ad B ca be obtaied as follows: Substitutig iitial displacemet for equatio 6 yields: f A si 7 Equatio 7 is half rage sie series[] The 56
7 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS A f si d Substitutig iitial velocity ito equatio 6 yields: 8 g B si 9 The B g si d - Itermediate Cocetrated oad: If P is the cocetrated load actig at distace from the left side of the beam as show i Fig.. p.iii.i IV.II Fig. the the load fuctio is: w lim u p u u u u u The differetial equatio relatig the deflectio ad the load is: d y w d EI 57
8 Al-Rafidai Egieerig Vol. No. 6 Represetig the load w by half rage sie Fourier series [] w C si 3 where C w si d Substitutig equatio for equatio the C lim u p u si d 5 Itegratig equatio 5 gives: lim u siu u The C p si 6 Substitutig equatio 6 for equatio 3 the p w si si 7 I order to get the deflectio due to the static load, it was assumed that the deflected shape represeted by half rage Fourier series: y b si 8 Which satisfies the boudary coditios of simply supported beams. Substitutig equatios 8 ad 7 for equatio the 58
9 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS b p si si si EI 9 which yields: 3 p b si 3 Substitutig equatio 3 for equatio 8 the 3 y p si si 3 which represets the deflectio equatio due to the itermediate cocetrated load. If the load is suddely removed the beam will vibrate freely ad the iitial displacemet is the deflected shape at t=, that meas the equatio 3 gives iitial displacemet i this case, the, f 3 P si si y 3 substitutes equatio 3 for equatio 7 3 P That meas: si si A si 33 3 P A si 3 EI As the beam was at rest whe the load was suddely removed the the iitial velocity is zero. That is: y g,, 35 t If equatio 35 is substituted for equatio 6 The value of B will be zero substitutes A ad B ito equatio 6 the: 59
10 Al-Rafidai Egieerig Vol. No. 6 3 P y, t si si cost 36 EI Which is the equatio of free vibratio for simply supported beam load by cocetrated load at distace from the left ed removed suddely at time t=. If The 3 P y, t si si cost EI Which is the same result that obtaied by Timoshiko[]. - Partially Distributed Uiform oad: Assumig that w/uit legth the itesity of the uiform load actig at distace from the left side of the beam as show i Fig.. w/uit legth.vii VIII.V.IX VI The load fuctio is: Fig. w w 37 Represetig the load w by half rage sie Fourier series [] w C si 38 6
11 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS Fidig the: C C from equatio after substitutig equatio 37 for equatio wsi d Itegratig equatio 39 to obtai: 39 C w cos cos The the solutio of equatio 38 becomes: w w cos cos si Substitute equatio 8 after rearragig the equatio to obtai: b w 5 5 cos EI cos Substitutig equatio for equatio 8 the w 5 5 cos EI cos si y 3 Substitutig equatio 3 for equatio 7 gives: w A cos cos 5 5 EI If equatio 35 is substituted ito equatio 6 The value of B will be zero, substitutig A ad B for equatio 6 the: y, t w 5 5 EI cos cos t si cos 5 6
12 Al-Rafidai Egieerig Vol. No. 6 which is the equatio of free vibratio for simply supported beam loaded by uiform load actig at distace from the left side ed ad removed suddely at time t=. If ad The y, t w si cost 5 5 EI Which are the same results obtaied by Timoshiko[]. 3- Itermediate Variable oad: If Q/uit legth the itesity of the uiform load actig at distace from the left side of the beam is as show i Fig. 3. Q/uit legth.xii XIII.X.XIV XI Fig.3 the the load fuctio is: w Q From equatio The: 6
13 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS 63 si d Q C 7 Solvig equatio 7 to obtai: cos cos si si Q C 8 the the solutio of equatio 3 becomes: si cos cos si si Q w 9 From equatio 8 after rearragig the equatio to obtai: cos cos si si Q b 5 Substitutig equatio 5 for equatio 8 yields: Q y si cos cos si si Substitutig equatio 5 for equatio 7 gives: cos cos si si EI Q A 5
14 Al-Rafidai Egieerig Vol. No. 6 If equatio 35 is substituted for equatio 6, the value of give: B will be zero, the substitutig A ad B for equatio 6 to y, t 6 6 Q 5 EI si si cos cos si cost 53 which is the equatio of free vibratio for simply supported beam loaded by a uiform load actig at distace from the left side ed ad removed suddely at time t=. Numerical eample: The followig properties of a simply supported beam with uiformly distributed load will be cosidered to draw the mode shape of deflectio: =m, 3 E= MPa, I=.666* m, w=3 N/m, m=785kg/m 3. The relatioship betwee deflectio ad the distace for these eamples are show i Fig., 5 which are idetical to that obtaied by Timoskeko[]. Coclusios: The Fourier series method with separatio of variables is suitable to be used for the solutio of free vibratio of beams. As the method is trigoometric sie ad cosie, the the deflectio modes are of the same shape for differet types of loads. The solutios obtaied are idetical to those by Timoskeko[] ad Mustafa[]. 6
15 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS Mode Mode Mode 3 Fig. Modes of deflectio alog the beam 65
16 Al-Rafidai Egieerig Vol. No rad/sec Mode.77 rad / sec Mode rad/sec Mode 3 Fig.5 Modes of deflectio with time alog the beam 66
17 Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS Refereces - Kreiger, S. W. "The effect of a aial force o the vibratio of higed bars", J.Appl. Mech., ASME 7:35-36, Burgree, D. "Free vibratios of a pi-eded colum with costat distace betwee pi eds". J. Appl. Mech., ASME 8:35-39, Sirivasa, A.V. "arge amplitude free oscillatios of beams ad plates",aiaa J. 3:95-953, Sirivasa, A.V. "Noliear vibratios of beams ad plates", It. J. Noliear Mech.,:79-9, Evese, D.A. "Noliear vibratios of beams with various boudary coditios", AIAA J.6:37-37, Ray, J.D., Bert, C.W.,"Noliear vibratios of a beam with pied eds", J. Eg. Id., ASME 9:977-, Padalai, K.A.V., Sathyamoorthy, M.,"O the modal equatios of large amplitude fleural vibratio of beams, plates, rigs ad shell", It. J. Noliear Mech. 8:3-8, ou, c.l., Sikarskie, D.., "Noliear vibratio of beams usig a formfuctio approimatio", J. Appl. Mech., ASME :9:, Rehfield,.W.,"A simple, approimate method for aalyzig oliear free vibratios of elastic structures", J.Appl. Mech., ASME :59-5, Mustafa, K.K., "Solutio of free vibratio of simply supported beam by aplace trasformatio", Scietific Joural, Tikrit Uiversity Eg.Sci. sectio,vol.6, No.5, 999, PP Timosheko, S., Youg, D.H. ad Weaver, W., " Vibratio Problems i Egieerig, Jouwily ad sos, Ic, Kreyszig, E., Advaced Egieerig Mathematics", Jouwily ad sos, td,
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