Free Vibration Analysis of Beams Considering Different Geometric Characteristics and Boundary Conditions

Transcription

1 Iteratioal Joural of Mechaics ad Applicatios 1, (3): 9-1 DOI: 1.593/j.mechaics.13.3 Free Vibratio Aalysis of Beams Cosiderig Differet Geometric Characteristics ad Boudary Coditios Mehmet Avcar Civil Egieerig Departmet, Egieerig Faculty, Suleyma Demirel Uiversity, Isparta, Turkey Abstract I this study, free vibratio of square cross-sectioed alumium beams is ivestigated aalytically ad umerically uder four differet boudary coditios: Clamped-Clamped (C-C), Clamped-Free (C-F), Clamped-Simply Supported (C-SS) ad Simply Supported-Simply Supported (SS-SS). Aalytical solutio is carried out usig Euler-Beroulli beam theory ad Neto Raphso Method. First, the equatios of motio are provided. The, solutios icludig the effects of the geometric characteristics, ad boudary coditios are obtaied ad discussed for the atural frequecies of the first three modes. To cofirm the reliability of the vibratio aalysis carried out i the preset paper as ell, all the aalytical results are checked ith the correspodig umerical results obtaied from the fiite-elemet-method (FEM) based softare called ANSYS. Numerical ad aalytical results are foud to be good agreemet. Keyords Free Vibratio, Beam, Natural Frequecy, Boudary Coditios, Geometric Characteristics 1. Itroductio A beam is a sleder horizotal structural member that resists lateral loads by bedig, ad this importat elemet of egieerig structures appears i various forms ad comprises various artifacts, such as supportig members i high-rise buildigs, railays, log-spa bridges, fleible satellites, gu barrels, robot arms, airplae igs, etc. [1, ]. I may egieerig applicatios, beams are subjected to dyamic loads, hich ca ecite beam structural vibratios ad cause durability cocers or discomfort because of the resultig oise ad vibratio. I additio, if the vibratio eceeds certai limits, there is the dager of beam breakage or failure [3-6]. Due to beams are importat structural elemets, vibratio aalysis has bee a vital task i their desig for egieers ad researchers for more tha a cetury. Early ivestigatios of the theory of vibratio ere give i Refs [7-9]. The a icreasig iterest has bee observed regardig the vibratio of beams, ad several studies have appeared i the geeral literature, some of hich are provided i the Refs [1-5]. I this study, the free vibratio of square cross-sectioed alumium beams is ivestigated aalytically ad umerically uder four differet boudary coditios. Aalytical solutio is carried out usig Euler-Beroulli beam theory, i hich material is assumed to be liear-isotropic, ad Neto Raphso Method. This method is based o the * Correspodig author: mehmetavcar@sdu.edu.tr (Mehmet Avcar) Published olie at Copyright 1 Scietific & Academic Publishig. All Rights Reserved simple idea of liear approimatio, ad used for fidig the roots of equatios. It is particularly useful for trascedetal equatios, composed of mied trigoometric ad hyperbolic terms. Such equatios occur i vibratio aalysis. A eample is the calculatio of atural frequecies of cotiuous structures [6]. Solutios icludig the effects of the geometric characteristics, i.e., legth ad cross sectioal area, ad boudary coditios are obtaied ad discussed for the atural frequecies of the first three modes. Furthermore, to cofirm the reliability of the vibratio aalysis carried out i the preset paper as ell, all the aalytical results are checked ith the correspodig umerical results obtaied from Fiite Elemet Method (FEM)-based softare called ANSYS [7], here the method is established o the idea of buildig a complicated object ith simple blocks, or, dividig a complicated object ito smaller ad maageable pieces [8]. Preset aalysis ca be used as a comparative study or data for the differet solutio methods of future orks i the related field.. Basic Equatios Cosider a elastic beam of legth, Youg's modulus E, ad mass desity ρ ith uiform cross sectio A, as sho i Figure 1. Usig Euler-Beroulli beam theory, oe ca obtai the equatio of motio of a beam ith homogeeous material properties ad costat cross sectio as follos [5]: EI + ρ A ; t (1)

2 Iteratioal Joural of Mechaics ad Applicatios 1, (3): here I is the area momet of iertia of the beam cross sectio, is the trasverse displacemet, ad t is time. Figure 1. Geometry of the beam Eq. (1) ca be rearraged as follos: EI + κ () t here κ ρa is the liear mass desity of the beam. The solutio of the Eq. () is sought by separatio of variables. Assume that the displacemet ca be separated ito to parts: oe is depedig o the positio ad the other is depedig o time, as follos: t (,) Λ( ) Ψ () t (3) here Λ ad Ψ are idepedet of time ad positio, respectively. Substitutig Eq. (3) ito Eq. () ad after some mathematical rearragemets, the folloig equatio is obtaied: EI Λ( ) 1 Ψ() t () κλ( ) () t Ψ t As observed from Eq. (), the left side depeds o the variable, ad the right side depeds o the variable t, as previously oted. Cosequetly, the variables have bee separated, ad each side of () must equal a costat, deoted ω to have simple harmoic motio i the system. EI Λ( ) 1 Ψ() t ω κλ( ) () t Ψ t If the positio variable is separated Λ( ) δ Λ ( ) here κ δ EI If the time variable is separated (5) (6) ω (7) Ψ () t + ω Ψ t t () (8) Eq. (6) is solved as follos: Λ ( ) C1sihδ+ Ccoshδ+ C3siδ + C cosδ here C1,..., (9) C are costats, ad sih ad cosh are the hyperbolic si e ad cos e fuctios, respectively. Eq. (8) is solved as follos: Ψ ( t) C siωt + C cosωt (1) 5 6 here C5 ad C 6 are costats. Thus, if Eq. (9) is multiplied by Eq. (1) to obtai t (,), it yields eight combied costats as: t (,) ( C1sihδ+ Ccoshδ+ C3si δ+ Ccosδ) ( C5si ωt + C6cosωt) (11) here the costats C1, C, C3, C ca be obtaied from the boudary coditios, ad C5, C6ca be obtaied from the iitial coditios Fially, usig Eq. (7) the atural frequecy f ( Hz ) of the beam is foud as follos: ω f (1) π 3. Solutio of the Basic Equatios 3.1. Particular Solutio for C-C Beam The boudary coditios satisfied by a C-C beam are as follos: ad ad (13) (1) Whe Eqs. (13)-(1) are cosidered i Eq. (9), after some mathematical operatios, the coefficiet matri is obtaied as follos: sihδ siδ coshδ cosδ C1 (15) coshδ cosδ sihδ+ siδ C The o-trivial solutio of the determiat of the coefficiet matri is as follos: cos δ cosh δ 1 (16) here the subscript is a iteger ide.

3 96 Mehmet Avcar: Free Vibratio Aalysis of Beams Cosiderig Differet Geometric Characteristics ad Boudary Coditios Because the first three roots of Eq. (16) are calculated usig the Neto-Raphso method, the folloig eigevalues are obtaied: δ.73 for 1 δ for δ for 3 here, δ, is the atural frequecy parameter of the beam. 3.. Particular Solutio for C-F Beam (17) The boudary coditios satisfied by a C-F beam are as follos: ad ad 3 3 (18) (19) Whe Eqs. (18)-(19) are cosidered i Eq. (9), after some mathematical rearragemets, the folloig coefficiet matri is obtaied: sihδ+ siδ coshδ+ cosδ C1 () coshδ+ cosδ sihδ siδ C The o-trivial solutio of the determiat of the coefficiet matri is as follos: cosδcosh δ 1 (1) Whe the first three roots of Eq. (1) are calculated usig Neto-Raphso method, the folloig eigevalues are obtaied: δ for 1 δ.699 for δ for 3 () 3.3. Particular Solutio for C-SS Beam The boudary coditios satisfied by a C-SS beam are as follos: ad (3) ad () Whe Eqs. (3)-() are cosidered i Eq. (9), after some mathematical rearragemets, the folloig coefficiet matri is obtaied: sihδ siδ coshδ cosδ C1 (5) sihδ+ siδ coshδ+ cosδ C The solutio of the determiat of the coefficiet matri is as follos: tahδ taδ (6) Because the first three roots of Eq. (6) are calculated usig Neto-Raphso method, the folloig eigevalues are obtaied: δ for 1 δ for δ 1.1 for 3 (7) 3.. Particular Solutio for SS-SS Beam The boudary coditios satisfied by a SS-SS beam are as follos: ad ad (8) (9) Whe Eqs. (8)-(9) are cosidered i Eq. (9), after some mathematical rearragemets, the folloig coefficiet matri is obtaied: sihδ siδ C1 sihδ siδ C3 (3) The o-trivial solutio of the determiat of the coefficiet matri is as follos: si δ sih δ (31) Because the first three roots of Eq. (31) are calculated usig Neto-Raphso method, the folloig eigevalues are obtaied:

4 Iteratioal Joural of Mechaics ad Applicatios 1, (3): δ for 1 δ 6.83 for δ 9.8 for 3. Numerical Results ad Discussio (3).1. Comparative Study I this subsectio, a comparative study as performed to validate the preset umerical results. The aalytical results ere compared ith the results of the FEM -based softare called ANSYS [7]. I fiite-elemet modelig, the beam type elemet is applied ad meshed ith 5 elemets. For the first three modes 1;;3, the aalytical atural frequecies, faaly ( Hz ), ad the FEM atural frequecies, ffem ( Hz ), differ by.3%,.8%, ad 1.78% uder C-C boudary coditios,.83%,.59%, ad 1.381% uder C-F boudary coditios,.13%,.78%, ad 1.698% uder C-SS boudary coditios, ad.183%,.7%, ad 1.65% uder SS-SS boudary coditios, respectively. The percetages are faaly ffem calculated as follos: 1. f Aaly As sho i Figure, the umerical results of both methods are cosistet, hich shos the accuracy of the preset formulatio. 65 C-C Aalytical C-C FEM 6 C-F Aalytical C-F FEM 55 C-SS Aalytical C-SS FEM 5 SS-SS Aalytical SS-SS FEM Figure. Comparisos betee the aalytical atural frequecies ad those obtaied usig FEM -based softare called ANSYS versus mode 9 3 umber, A. m ) ( E 7 1 N/m ; ρ 7 kg/m ; 3m;.. Free Vibratio Aalyses of Beams I this sectio, four studies ere performed to ivestigate the free-vibratio behaviors of square cross-sectioed alumium beams ith differet geometric characteristics uder four differet boudary coditios. The atural frequecies ere obtaied ad discussed for the first three modes (1,, 3), icludig the effects of the geometric characteristics, i.e., legth ad cross sectioal area, ad the boudary coditios. Study 1: I Figure 3, the variatios i atural frequecies, f ( Hz ), of square cross-sectioed alumium beams versus the first three modes (1,, 3) uder four differet cross sectios are plotted, here A1;A; A3 deote A.5m ;. m ;.65m, ad 1; ; 3 deote.75 m; 3m; 3.5 m, respectively. Figure 3 shos that the atural frequecies icrease ith the icrease i mode umber. The atural frequecy of the first mode umber ( 1) differs from that of the secod ad third mode umbers (, 3) by -176% ad -% uder the C-C boudary coditios, -57% ad -165% uder C-F boudary coditios, -% ad -576% uder C-SS boudary coditios, ad -3% ad -8% uder SS-SS boudary coditios, respectively. The percetages ere f1 fi calculated as follos: 1; ( i,3). f1 Therefore, the variatio i mode umber has the largest effect o the atural frequecy uder C-F boudary coditios ad the smallest effect uder C-C boudary coditios. Study : Figure 3 shos that the beam has the highest atural frequecies, f ( Hz ), uder C-C boudary coditios ad the loest frequecies uder C-F boudary coditios. To ivestigate the effect of the boudary coditio o the atural frequecies, f ( Hz ), of the beam versus the mode umber, (1,,3), the C-C boudary coditios ere compared ith the other boudary coditios. From this compariso, the folloig results ere obtaied: for 1,, ad 3, i) the differeces betee C-C ad C-F boudary coditios are 8%, 6%, ad 9%, ii) the differeces betee C-C ad C-SS boudary coditios are 31%, 19%, ad 1%, ad iii) the differeces betee C-C ad SS-SS boudary coditios are 56%, 36%, ad 7%, respectively. The percetages ere calculated as follos: fc C fi 1; ( i C F; C SS; SS SS). fc C Therefore, the effect of the type of the boudary coditio decreases ith the icrease i mode umber,.

5 98 Mehmet Avcar: Free Vibratio Aalysis of Beams Cosiderig Differet Geometric Characteristics ad Boudary Coditios 95 A A1 C-C Beam A1 3 A 1 A A A3 1 A3 A A1 1 C-F Beam 5 A1 A1 3 A 1 35 A 3 A 3 A3 1 5 A3 A a) b) A1 1 A1 A1 3 A 1 A A 3 A3 1 A3 A3 3 C-SS Beam A1 1 A1 A1 3 A 1 A A 3 A3 1 A3 A3 3 SS-SS Beam 1 3 c) d) Figure 3. Variatios of the atural frequecies,, of square cross-sectioed alumium beams for the first three modes, (1,,3) 9 3 ( E 7 1 N/m ; ρ 7 kg/m ) Study 3: ( Hz) Figure shos the variatios i atural frequecy, f ( Hz ), of square cross-sectioed alumium beams versus the legth, (m), uder four differet boudary coditios, here A1;A; A3 deotes A.5m ;.m ;.65m, ad 1; ; 3 deote 1;;3, respectively. Figure obviously shos that the atural frequecies, f ( Hz), decrease ith the icrease i legth, (m). To eamie the effect of the variatio i legth of the beam o the atural frequecies, beams ith idetical cross sectioal areas ere compared. The result idicated that the variatio i legth had a costat effect o the atural frequecies: 15.97% ad 8.% for all cross sectioal areas ( A.5m ;.m ;.65m ) uder four differet boudary coditios. The percetages ere calculated as f.75 fi follos: 1; ( i 3, 3.5). Thus, f.75 the chage i legth of the beam has a costat effect o its atural frequecies for various boudary coditios ad cross sectioal areas. f Study : Figue 5 shos the variatios i atural frequecies of square cross-sectioed alumium beams versus the cross sectioal area, A (m ), uder four differet cross sectios, here 1;; 3 deote.75 m;3 m;3.5 m, ad 1; ; 3 deote 1; ;3, respectively. Figure 5 shos that the atural frequecies icrease ith the icrease i cross sectioal area. To ivestigate the effect of the variatio i cross sectioal area of the beam o the atural frequecies, beams ith idetical legths ere compared. The result idicated that variatio of the cross sectioal area had a costat effect o the atural frequecies: % ad % for.75 m;3 m;3.5 m uder four differet boudary coditios. The percetages fa.5 fai ere calculated as follos: 1; fa.5 ( i.,.65). Thus, the chage i cross sectioal area of the beam has a costat effect o the atural frequecies of the beam for various boudary coditios ad legths.

6 Iteratioal Joural of Mechaics ad Applicatios 1, (3): C-C Beam A1 1 A1 A1 3 A 1 A A 3 A3 1 A3 A (m) C-F Beam A1 1 A1 A1 3 A 1 A A 3 A3 1 A3 A (m) a) b) C-SS Beam A1 1 A1 A1 3 A 1 A A 3 A3 1 A3 A (m) SS-SS Beam A1 1 A1 A1 3 A 1 A A 3 A3 1 A3 A (m) Figure. Variatios of the atural frequecies, 9 3 ( E 7 1 N/m ; ρ 7 kg/m ) c) d) f ( Hz), of square cross-sectioed alumium beams versus the legth, (m) C-C Beam A (m ) C-F Beam A (m ) a) b) C-SS Beam A (m ) SS-SS Beam A (m ) c) d) Figure 5. Variatios of the atural frequecies of square cross-sectioed alumium beams versus the cross sectioal area, A (m ) 9 3 ( E 7 1 N/m ; ρ 7 kg/m )

7 1 Mehmet Avcar: Free Vibratio Aalysis of Beams Cosiderig Differet Geometric Characteristics ad Boudary Coditios 5. Coclusios I this study, the free vibratio of square cross-sectioed alumium beams is ivestigated aalytically ad umerically uder four differet boudary coditios. Aalytical solutio is carried out usig Euler-Beroulli beam theory, i hich material is assumed Solutios icludig the effects of the geometric characteristics, i.e., legth ad cross sectioal area, ad boudary coditios are obtaied ad discussed for the atural frequecies of the first three modes. The folloig results ere obtaied: (1) The atural frequecies icrease ith the icrease i mode umber. () The chage i mode umbers has the largest effect o the atural frequecy uder C-F boudary coditios ad the smallest effect uder C-C boudary coditios. (3) The beam has the highest atural frequecies uder C-C boudary coditios ad the loest frequecies uder C-F boudary coditios. () The effect of the type of the boudary coditio decreases ith icreasig mode umber. (5) The atural frequecies of the beam decrease ith icreasig legth. (6) The chage i legth of the beam has a costat effect o its atural frequecies for various boudary coditios ad cross sectioal areas. (7) The atural frequecies of the beam icrease ith icreasig cross sectioal area. (8) The chage i cross sectioal area of the beam has a costat effect o the atural frequecies for various boudary coditios ad legths. REFERENCES [1].H. Doel, Beams Plates ad Shells. Egieerig Societies Moographs McGraHill., [] J.M. Gere, S.P. Timosheko, Mechaics of Materials. PWS Publishig Compay, [3] W.T. Thomso, Theory of Vibratio ith Applicatios. d ed., Pretice Hall, []. Meirovitch, Fudametals of Vibratios. Iteratioal Editio, McGra-Hill, 1. [5] A.W. eissa, M.S Qatu, Vibratio of Cotiuous Systems. McGra Hill, 11. [6] E. Magrab, Vibratios of Elastic Systems: With Applicatios to MEMS ad NEMS Vol. 18 Berli: Spriger, 1. [7] J.W.S. Rayleigh, Theory of Soud. Vol. 1 1 st ed., [8] S.P. Timosheko, O the trasverse vibratios of bars of uiform cross-sectio. Philosophical Magazie p.15, 19. [9] S.P. Timosheko, Vibratio Problems i Egieerig. D. Va Nostrad, 198. [1] J.P. De Hartog, Mechaical Vibratios. McGra-Hill, [11]. Meirovitch, Aalytical Methods i Vibratios. Macmilla Publicatios Co., Ic., [1] K.R Chu, Free vibratio of a beam ith oe ed sprig-higed ad the other free, Joural of Applied Mechaics, 39, , 197. [13] D.J. Gorma, Free Vibratio Aalysis of Beams ad Shafts, Wiley, Ne York, [1] S.Y. ee, H.Y. Ke, Free vibratios of a o-uiform beam ith geeral elastically restraied boudary coditios, Joural of Soud ad Vibratio, 136, 5-37, 199. [15] S.S. Rao, Mechaical Vibratios, 3 rd editio. Readig, MA, Addiso-Wesley Publishig Compay, [16] W.. i, Free Vibratios of beams ith geeral boudary coditios, Joural of Soud ad Vibratio, 37, 79-77,. [17] H.K. Kim, M.S. Kim, Vibratio of beams ith geerally restraied boudary coditios usig Fourier series. Joural of Soud ad Vibratio, 5, , 1. [18] S. Nagulesara, Trasverse vibratio of a Euler-Beroulli uiform beam o up to five resiliet supports icludig eds, Joural of Soud ad Vibratio, 61, 37-38, 3. [19] M. Simsek, Free vibratio aalysis of beams subjected to aial load uder various boudary coditios. Sigma Joural of Egieerig ad Natural Scieces, 3, 1-1, 5. [] M.C. Ece, M. Aydogdu, V. Taski, Vibratio of a variable cross-sectio beam, Mechaics Research Commuicatios, 3,78-8, 7. [1] T. Irvie, Bedig Frequecies of Beams, Rod, ad Pipes. Revisio M, Vibratiodata, 1. [] S.B. Coşku, M.T. Atay, B. Ozturk, Trasverse Vibratio Aalysis of Euler-Beroulli Beams Usig Aalytical Approimate Techiques, Advaces i Vibratio Aalysis Research, Chapter 1, Itech, 11. [3] J. ee, Free vibratio aalysis of beams ith o-ideal clamped boudary coditios, Joural of Mechaical Sciece ad Techology, 7, 97-33, 13. [] Z. iu, Y. Yi, F. Wag, Y. Zhao,. Cai, Study o modified differetial trasform method for free vibratio aalysis of uiform Euler-Beroulli beam. Structural Egieerig ad Mechaics 8: , 13. [5] A. Prokic, M. Besevic, D. ukic, A umerical method for free vibratio aalysis of beams, ati America Joural of Solids ad Structures, 11: 13-1, 1. [6] T. Irvie, Applicatio of the Neto-Raphso Method to vibratio problems, Revisio E, Vibratiodata, 1. [7] ANSYS Academic Research, Release 1.. [8] C. Xiaoli. Y. iu, Fiite Elemet Modelig ad Simulatio ith ANSYS Workbech, CRC Press, 1.