An Exact Solution for the Free Vibration Analysis of Timoshenko Beams

Transcription

1 Review of Applied Physics Volue 3, 4 A Exac Soluio for he Free Vibraio Aalysis of Tiosheko Beas Raaza A. Jafari-Talookolaei *, Marya Abedi School of Mechaical Egieerig, Babol Noshirvai Uiversiy of Techology, , Babol, Mazadara Provice, Ira * raazaali@gail.co, aryaabedy@yahoo.co Received 7 May 3; Acceped 8 Jue 3; Published 3 March 4 4 Sciece ad Egieerig Publishig Copay Absrac This work preses a ew approach o fid he exac soluios for he free vibraio aalysis of a bea based o he Tiosheko ype wih differe boudary codiios. The soluios are obaied by he ehod of agrage ulipliers i which he free vibraio proble is posed as a cosraied variaioal proble. The egedre orhogoal polyoials are used as he bea eigefucios. Naural frequecies ad ode shapes of various Tiosheko beas are preseed o deosrae he efficiecy of he ehodology. Keywords Tiosheko Bea; Naural Frequecies; Mode Shapes; egedre Polyoials; agrage Mulipliers Iroducio Beas play a ipora role i he creaio of echaical, elecroechaical, ad civil syses. May of hese syses are subjeced o dyaic exciaio. As a cosequece, he exac deeriaio of he aural frequecies ad ode shapes of liear elasic beas have bee sudied by ay researchers. I has bee kow for ay years ha he classical Euler-Beroulli bea heory is able o predic he frequecies of flexural vibraio of he lower odes of hi beas wih adequae precisio. The vibraory oio of hick beas is described by he Tiosheko bea heory, as hey icorporae he effecs of roary ieria ad deforaio due o shear. Durig he pas decades, he free vibraios of Euler Beroulli beas have received cosiderable aeio of ay researchers, bu oly few publicaios were devoed o icludig he effecs of shear deforaio ad roary ieria. The ode shape differeial equaio describig he rasverse vibraios of a hagig Euler Beroulli bea uder liearly varyig axial force has bee derived by Schafer (985). ee ad Ng (994) have copued he fudaeal frequecies ad he criical bucklig loads of siply suppored beas wih sepped variaio i hickess usig wo algorihs based o he Rayleigh-Riz ehod. The firs algorih which has bee used exesively i aalyzig beas wih o-uifor hickess, ivolves usig a series of assued fucios ha saisfy oly he exeral boudary codiios ad disregard he presece of he sep. The secod algorih cosiders a bea wih a sep as wo separae beas divided by he sep. Two differe ses of adissible fucios which saisfy he respecive geoeric boudary codiios have bee assued for hese wo ficiious sub-beas. Geoeric coiuiies a he sep have bee eforced by iroducig arificial liear ad orsioal sprigs. ee ad Kes (99) have coduced a sudy o deerie he aural frequecies of o-uifor Euler beas resig o a o-uifor foudaio wih geeral elasic ed resrais. The free vibraio respose of a Euler-Beroulli bea suppored by a ierediae elasic cosrai has bee sudied by Riedel ad Ta (998) usig he rasfer fucio ehod. Rosa ad Maurizi (998) have ivesigaed he ifluece of coceraed asses ad Paserak soil o free vibraio of beas ad gave exac soluios for Beroulli Euler beas based o he bea heory. A odified fiie differece ehod has bee preseed by Che ad Zhao (5) o siulae rasverse vibraios of a axially ovig srig. i ad Tsai (7) have deeried he aural frequecies ad ode shapes of Beroulli Euler

2 Review of Applied Physics Volue 3, 4 uli-spa bea carryig uliple sprig-ass syses. A aalyical soluio has bee preseed for he aural frequecies, ode shapes ad orhogoaliy codiios of a arbirary syse of Euler Beroulli beas iercoeced by arbirary jois ad subjec o arbirary boudary codiios by Wiedea (7). Failla ad Saii (8) have addressed he eigevalue proble of he Euler Beroulli discoiuous beas. A siulaio ehod called he differeial rasfor ehod (DTM) has bee eployed o predic he vibraio of a Euler Beroulli bea (pipelie) resig o a elasic soil by Balkaya ad Kaya (9). Ali ad Akkur () have ivesigaed he free vibraio aalysis of sraigh ad circular beas o elasic foudaio based o he Tiosheko bea heory. Ordiary differeial equaios i scalar for obaied i he aplace doai are solved uerically usig he copleeary fucios ehod. He ad Huag (987) have used he dyaic siffess ehod o aalyze he free vibraio of coiuous Tiosheko bea. The full develope ad aalysis of four odels for he rasversely vibraig uifor bea have bee preseed by Ha e al. (999). The four heories aely he Euler-Beroulli, Rayleigh, shear ad Tiosheko have bee cosidered. Zhou () has sudied he free vibraio of uli-spa Tiosheko beas by he Rayleigh-Riz ehod. The saic Tiosheko bea fucios have bee developed as he rial fucios i he aalysis which are he coplee soluios of rasverse deflecios ad roaioal agles of he bea whe a series of saic siusoidal loads acs o he bea. A sudy of he free vibraio of Tiosheko beas has bee preseed by ee ad Schulz (4) o he basis of he Chebyshev pseudospecral ehod. Che e al. (4) have proposed a ixed ehod, which cobies he sae space ehod ad he differeial quadraure ehod, for bedig ad free vibraio of arbirarily hick beas resig o a Paserak elasic foudaio. The aplace rasfor has bee used o obai a soluio for a Tiosheko bea o a elasic foudaio wih several cobiaios of discree ispa aaches ad wih several cobiaios of aaches a he boudaries by Magrab (7). I he prese paper, a ovel approach is ade o he proble of he free vibraios of a Tiosheko bea, i which he orhogoal egedre polyoials i cojucio wih agrage ulipliers are used. The frequecies ad he correspodig ode shapes for coo ypes of boudary codiios are copared exreely well wih he available soluio. Proble Forulaio Cosider a sraigh Tiosheko bea of legh, a uifor cross-secioal area A(=b h), he ass per ui legh of, he secod oe of area of he crosssecio I, Youg s odulus E, ad shear odulus G. I is assued ha he bea is ade of a hoogeous ad isoropic aerial. The kieic eergy T ad he srai eergy U of he vibraig bea ca be wrie as: = {, + ψ, } { ψ, xˆˆ γ} ˆˆ { ψ, x γ } T w r dx ˆ () U = o M + V dx = EI + kag dx () Where γ represes he shear agle ( γ = w, xˆ ψ ), wx (,) ˆ ad ψ (,) x ˆ are rasverse displacee ad he cross-secio roaio due o he bedig oe, ˆx is he axial coordiae of he bea, r is he radius of gyraio ( = I / A ) ad k is he bea cross secioal shape facor. Also M ad V are he bedig oe ad shear forces, respecively. Coa deoes differeiaio wih respec o ˆx or. Applyig Hailo's priciple, he goverig equaios of oio ad boudary codiios are obaied as follows: w, ( kag( w, xˆˆ ψ )), x = (3) r ψ kag( w ψ) ( EI ψ ) = (4),, xˆˆˆ, x, x δψ δ ( M ) =, ( V w ) = (5) i oher words, a he eds ˆx = ad, we have: eiher M = or ψ = is specified, (6) eiher V = or w= is specified. The equaio (6) gives he boudary codiios of he prese case. Aalyical Soluio I he prese work, series of soluios i cojucio wih he agrage ulipliers are used o sudy he free vibraio characerisics of he bea. The ai advaage of he agrage uliplier echique is ha he choice of he assued displacee fucio is easy because hey do o have o saisfy he boudary codiios of he proble. I he prese sudy, he siple egedre polyoials are chose as displacee fucios, ad his siplifies he proble furher sice he orhogoaliy properies 3

3 Review of Applied Physics Volue 3, 4 lead o siple eergy expressio. Haroic soluios for he variables ψ (,) x ˆ are assued as: wx (,) ˆ ad i (,) ˆˆˆˆ () ω iω w x = W x e, ψ (,) x = Ψ() x e (7) i which variables W( x ˆ) ad Ψ( x ˆ) are he displacee fucios ad ω is he circular frequecy. As eioed above, he displacee fucios ca be expressed i ers of he siple egedre polyoials ad are give by: Ψ = = W( x) = W P ( x), Ψ ( x ) = P ( x) (8) Here, P is he siple egedre polyoial of degree. I should be eioed ha he axial coordiae is rasfored o he ierval x by leig ˆ / x = x. / We have four boudary codiios for each bea, i.e. wo boudary codiios for wo eds. These four boudary codiios which are o saisfied by he assued series, are iposed as cosrais. For four coo boudary codiios (B.C.s), hese cosrais ca be wrie as follows: Claped-Claped Bea (C-C): Claped-Higed Bea (C-H): Higed-Higed Bea (H-H): Claped- Free Bea (C-F): W ( ) =, Ψ( ) = W () =, Ψ () = W ( ) =, Ψ( ) = W () =, Ψ () = W ( ) =, Ψ ( ) = W () =, Ψ () = W ( ) =, Ψ( ) = W () Ψ () =, Ψ () = (9a) (9b) (9c) (9d) i which prie deoes differeiaio wih respec o x. The boudary codiios yield liear cosrais relaed o he liear cobiaios of he egedre polyoials ad subjec o he degree of approxiaio, i.e. he uber of polyoials ivolved. By subsiuig equaios (8) i equaio (9), he cosrais ca be rewrie as: Claped-Claped Bea (C-C): W = = ( ) =, ( ) Ψ =, W =, Ψ = = = Claped-Higed Bea (C-H): W = = = = k = ( ) =, ( ) Ψ = ( ) (a) (b) W =, Ψ 4k = Higed-Higed Bea (H-H): = = k = = = k = ( ) ( ) ( ) W =, Ψ ( ) 4k = k W =, Ψ 4k = Claped- Free Bea (C-F): W = = W ( k ) = k = = Ψ ( 4k ) = = k = ( ) =, ( ) Ψ = 4 Ψ = (c) (d) I above equaio, we have used he followig properies of egedre polyoial (Gradshey ad Ryzhik, 7): ( ) k k = P ( x) = 4k P ( x) ( ) i which sigifies he iegral par of (-)/ (Gradshey ad Ryzhik, 7). A variaioal priciple is forulaed based o he kieic ad srai eergies by a procedure siilar o oe followed by Washizu (98). This variaioal priciple alog wih he cosrai codiios is used o solve he vibraio proble. The fucio o be exreized is give by he expressio: 4 F = U + T α i (cosrais equaio) i= () where α i ( i = 4) are he agrage ulipliers. Subsiuig he assued series for W( x)a d Ψ( x ) i equaio () ad siplifyig yields: 4

4 Review of Applied Physics Volue 3, 4 k k = k = = k = EI F = Ψ ( 4k ) P ( x) Ψ ( 4k ) P ( x) dx kag + Ψ 4 = + k k k = k = = k = kag + W ( 4k ) P ( x) W ( 4k ) P ( x) dx k = = k = kag Ψ P ( x) W ( 4k ) P ( x) dx ( W r ) 4 ω + Ψ αi(cosrais equaio) 4 = + i= () The ecessary exreizig codiios are give by: F F = =, =,,,... W Ψ (3) Usig equaio (3) i cojucio wih equaio () resuls i a syse of liear algebraic equaios which, i arix for, ca be wrie as: {,,...,,,,..., } AW W W Ψ Ψ Ψ = B (4) i which he righ had side of equaio (4) cosiss of agrage ulipliers. Solvig equaio (4) for Wad Ψ ( =,,..., ) ad subsiuig io he cosrai equaios () resuls i a syse of hoogeous liear algebraic equaios wih he agrage ulipliers as ukows. The syse of equaios is give by: { } T C α, α, α3, α 4 = (5) The aural frequecies ad correspodig ode shapes of beas ca be calculaed usig equaios (4) ad (5). I calculaig he aural frequecy, he deeria of he coefficie arix i equaio (5) is copued for various values of frequecy sarig fro a ear zero value. Deeria chage of sig fucio is ideified ad he correspodig value of frequecy is he aural frequecy of he bea i quesio. Resuls ad Discussio I order o deosrae he high accuracy of he prese ehod, he covergece ad copariso sudies are carried ou. Uless eioed oherwise, i all of he followig aalysis he recagular crosssecioal beas wih shear correcio facor k=5/6 ad T he Poisso raio υ=.3 are cosidered. The firs five diesioless frequecies ( Ω= 4 EIπ 4 ω ) of higedhiged (H-H) ad claped-claped (C-C) beas are give i Table. The uber of ers of he egedre polyoial seadily icreases fro 6 o. Oe ca see ha he covergece is very rapid. I geeral, ers of he egedre polyoial are eough o give saisfacory resuls. TABE THE CONVERGENCE STUDY ON THE FIRST FIVE DIMENSIONESS FREQUENCIES OF H-H AND C-C TIMOSHENKO BEAMS FOR (/H=) B.C. Ω Ω Ω3 Ω4 Ω5 H-H C-C The copariso sudy has bee give i Table for he firs five diesioless frequecies of Tiosheko beas by usig he prese ehod, he dyaic siffess ehod (DSM) (He ad Huag, 987) ad saic Tiosheko bea fucio (STBF) (ee ad Schulz, 4). Three ypes of boudary codiios: H- H, C-H ad CC have bee cosidered. Excelle agreee has bee observed for all cases, which shows ha he prese ehod has very high accuracy. TABE THE COMPARISON STUDY OF THE FIRST FIVE DIMENSIONESS FREQUENCIES OF H-H, C-H AND C-C TIMOSHENKO BEAMS FOR (/H=6.667) B.C. Mehods Ω Ω Ω3 Ω4 Ω5 Prese H-H DSM STBF Prese C-H DSM STBF Prese C-C DSM STBF Fig. shows he effec of legh-o-hickess raio (/h), where varies while h keeps cosa, o he diesioless fudaeal frequecy of he bea wih C-C, C H, H-H ad C-F boudary codiios. I 5

5 Review of Applied Physics Volue 3, 4 is clear ha for he bea wih /h<3, he variaio of /h has drasic effec o Ω. While for higher values of /h, he fudaeal frequecy eds o be cosa, ha is, he ifluece of /h is pracically egligible. As expeced, he curve shows clearly ha he saller he legh-o-hickess raio is, he lower he frequecy will be. MODE FIG. EFFECT OF ENGTH-TO-THICKNESS RATIO ON THE DIMENSIONESS FUNDAMENTA NATURA FREQUENCIES OF THE BEAM WITH VARIOUS BOUNDARY CONDITIONS Fig. shows he variaio of he displacee Wad he bedig slope Ψ alog he bea legh for he firs hree odes of vibraio of Tiosheko bea wih hickess raio /h= ad C-C boudary codiio. Sice he coveioal bea heories ca o ivolve he effec of Poisso s raio, i is raher ieresig o ake a deep isigh io i usig he prese approach. Table 3 gives he variaio of he firs hree aural frequecy paraeers (Ω) of H-H beas wih he Poisso s raio. I is show ha he aural frequecy decreases gradually wih he icreasig of Poisso s raio. We ca see ha he aural frequecies for υ=.5 have a appare deviaio fro ha for υ=.. Fro his poi of view, he Poisso s raio is of grea sigificace i srucural desig especially for coposie aerial beas. MODE 3 FIG. VARIATION OF DISPACEMENT AND THE BENDING SOPE AONG THE BEAM ENGTH (MODES -3) TABE 3 EFFECT OF POISSON S RATIOS ON THE FIRST THREE NATURA FREQUENCY PARAMETERS (Ω) OF H-H BEAMS /h Poisso s raio (υ) Ω Ω Ω Ω Ω Ω Coclusios MODE The free vibraio of he Tiosheko beas is ivesigaed usig a assued series soluio i cojucio wih agrage ulipliers. I is observed ha he prese ehod is a copuaioally efficie 6

6 Review of Applied Physics Volue 3, 4 ool i predicig he aural frequecies of he beas. This ehod is paricularly aracive because of he ease wih which oe ca choose he geeralized displacee fucios. This fac is show by choosig egedre Polyoials whose orhogoal properies siplify eergy expressio cosiderably. The aural frequecies of he Tiosheko bea obaied by his ehod copare exreely well wih he available exac soluio. I should be eioed ha by applyig his ehod, he covergece is very rapid. REFERENCES Ali, F.F., Akkur, F.G., Saic ad free vibraio aalysis of sraigh ad circular beas o elasic foudaio. Mechaics Research Couicaios 38 (): Balkaya Müge, Kaya Mei O. ad Sa glaer Ahe, Aalysis of he vibraio of a elasic bea suppored o elasic soil usig he differeial rasfor ehod. Archive of Applied Mechaics 79(9): Che i-qu, ad Zhao Wei-Jia, A uerical ehod for siulaig rasverse vibraios of a axially ovig srig. Applied Maheaics ad Copuaio 6 (5): 4. Che W.Q., u C.F., Bia Z.G., A ixed ehod for bedig ad free vibraio of beas resig o a Paserak elasic foudaio. Applied Maheaical Modellig 8 (4): Failla Giuseppe ad Saii Adolfo, A soluio ehod for Euler Beroulli vibraig discoiuous beas. Mechaics Research Couicaios 35 (8): Gradshey I.S. ad Ryzhik I.M., Table of Iegrals, Series, ad Producs, Seveh Ediio, Elsevier Ic. 7. Ha Seo M., Bearoya Hay ad Wei Tiohy, Dyaics of Trasversely Vibraig Beas Usig Four Egieerig Theories. Joural of Soud ad Vibraio 5 (999): He Y.-S. ad Huag T. C., 987 Advaced Topics i Vibraios: preseed a Aerica Sociey of Mechaical Egieers Desig Techology Cofereces-h Bieial Coferece o Mechaical Vibraio ad Noise, 43-48, New York. Free Vibraio aalysis of coiuous Tiosheko beas by dyaic siffess ehod. ee H. P., ad Ng T. Y., Vibraio ad Bucklig of a Sepped Bea. Applied Acousics 4 (994) ee J. ad Schulz W.W., Eigevalue aalysis of Tiosheko beas ad axisyeric Midli plaes by he pseudospecral ehod. Joural of Soud ad Vibraio 69 (4): 69. ee S.Y., ad Kes H.Y., Free vibraios of o-uifor beas resig o o-uifor elasic foudaio wih geeral elasic ed resrais. Copu. Sruc. 34 (99): 4 9. i HY ad Tsai YC, Free vibraio aalysis of a uifor uli-spa bea carryig uliple sprig-ass syses. Joural of Soud ad Vibraio 3 (7): Magrab Edward B., Naural Frequecies ad Mode Shapes of Tiosheko Beas wih Aaches. Joural of Vibraio ad Corol, 3(7): Riedel C.H., ad Ta C.A., Dyaic Characerisics ad Mode ocalizaio of Elasically Cosraied Axially Movig Srigs ad Beas. Joural of Soud ad Vibraio 5 (998): Rosa M.A. De, ad Maurizi M.J., The ifluece of coceraed asses ad Paserak soil o he free vibraios of Euler beas-exac soluio. Joural of Soud ad Vibraio (998): Schafer B., Free vibraios of a graviy-loaded claped-free bea. Igeieur-Archiv 55 (985): Washizu, K. 98, Variaioal Mehods i Elasiciy ad Plasiciy, New York: Pergao Press. Wiedea S.M., Naural frequecies ad ode shapes of arbirary bea srucures wih arbirary boudary codiios. Joural of Soud ad Vibraio 3 (7): 8 9. Zhou D., Free Vibraio of Muli-Spa Tiosheko Beas Usig Saic Tiosheko Bea Fucios. Joural of Soud ad Vibraio 4 ():