Analytical Approximation of Nonlinear Vibration of Euler-Bernoulli Beams

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1 15 Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas Absrac In his paper, he Hooopy Analysis Mehod (HAM) wih wo auxiliary paraeers and Differenial ransfor Mehod (DM) are eployed o solve he geoeric nonlinear vibraion of Euler- Bernoulli beas subjeced o axial loads. A second auxiliary paraeer is applied o he HAM o iprove convergence in nonlinear syses wih large deforaions. he resuls fro HAM and DM are copared wih anoher popular nuerical ehod, he shooing ehod, o validae hese wo analyical ehods. HAM and DM show excellen agreeen wih nuerical resuls (he axiu errors in our calculaions are abou.%), and hey addiionally provide a siple way o conduc a paraeric analysis wih differen physical paraeers in Euler-Bernoulli beas. o show he benefis of his ehod, he effec of differen physical paraeers on he apliude is discussed for a canilever bea wih a cyclically varying axial load. Keywords Nonlinear vibraion, Euler-Bernoulli bea, Hooopy Analysis Mehod (HAM), wo auxiliary paraeers, Differenial ransfor Mehod (DM). S.S. Jafari a M.M. Rashidi b, c S. Johnson d a Young Researchers & Elie Club, Haedan Branch, Islaic Azad Universiy, Haedan, Iran. E-ail: sjd.jafari@yahoo.co, sjd.jafari@iuh.ac.ir b Shanghai Key Lab of Vehicle Aerodynaics and Vehicle heral Manageen Syses, ongji Universiy, Address: 48 Cao An Rd., Jiading, Shanghai 184, China. c ENN-ongji Clean Energy Insiue of advanced sudies. d Universiy of Michigan-Shanghai Jiao ong Universiy Join Insiue, Shanghai Jiao ong Universiy, Shanghai, Peoples Republic of China. hp://dx.doi.org/1.159/ Received In revised for Acceped Available online INRODUCION he governing equaions of bea vibraion are generally non-linear aking i difficul o solve he nonlinear proble via analyical ehods. Because of his, any researchers use nuerical ehods o solve hese equaions. In recen years, soe approxiae analyical soluions have been offered o invesigae hese non-linear equaions. Wu and Liu (1999) used differenial quadraure o solve he

2 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas 151 single-span Bernoulli Euler bea's buckling equaion. He (6) suggesed ha he Paraeerized- Perurbaion Mehod (PPM) be used o solve srongly nonlinear equaions. Qaisi (1993) deerined he vibraion odes of geoerically nonlinear beas under he various edge condiions by he haronic balance principle. Moeenfard e al. (11) developed he hooopy perurbaion ehod (HPM) o analyze he nonlinear free vibraion of ioshenko beas. hey also convered he nonlinear parial differenial governing equaion o a non-linear ordinary differenial equaion using Galerkin s projecion ehod. Sfahani e al. (11) used he energy balance ehod (EBM) o sudy he dynaic response of inexensible beas (neural axis lengh is preserved during vibraion). Barari e al. (1) used he Hooopy-Perurbaion ehod o analyze he bea deforaion. Pillai and Rao (199) applied several ehods o sudy he large apliude free vibraions of siply suppored prisaic beas wih fixed ends. he ehods ha hey used in heir sudy consis of he ellipic funcion ehod, haronic balance ehod and ehods considering siple haronic oscillaions. Kopaza and Gündogdub (3) developed he relaion beween curvaure of an Euler- Bernoulli bea and bending oen. Soldaos and Selvadurai (1985) used a perurbaion ehod o survey he saic flexure of a Bernoulli-Euler bea resing on a nonlinear Winkler-ype foundaion. Ahadi e al. (14) devoed o he new classes of analyical echniques called he Ieraion Perurbaion Mehod (IPM)and Hailonian Approach (HA) for solving he equaion of oion governing he nonlinear vibraion of Euler-Bernoulli beas. Liu and Gurra (9) eployed He s Variaional Ieraion Mehod (VIM) o solve free vibraion probles for an Euler Bernoulli bea wih various supporing condiions. hey copared VIM resuls wih Adoian decoposiion ehod resuls. hey concluded ha VIM is accurae and i provides a siple and efficien approach for solving vibraion of unifor Euler Bernoulli beas. Rashidi e al. (1) derived he equaions of he oion for a recangular isoropic plae. hey considered he effec of shear deforaion and roary ineria and used he Hooopy Perurbaion Mehod (HPM) o solve he nonlinear equaion. In 199, Liao (199) proposed he hooopy analysis ehod (HAM) o solve nonlinear equaions. Many researchers used HAM o solve various nonlinear probles. Hoseini e al. (8) sudied he nonlinear free vibraion of a conservaive oscillaor via HAM. hey showed ha he HAM leads o an accurae analyical soluion, which is valid for a wide range of syse paraeers. Mohaadpour e al. (1) sudied deflecion of beas wih nonlinear Winkler ype foundaions using HAM. hey showed ha he HAM iproves convergence of nonlinear probles. he Differenial ransfor Mehod (DM) was developed by aylor's series expansion. DM applied o solve linear and nonlinear differenial equaions. his ehod can be used for soluion of ordinary and parial differenial equaions. Ayaz (3) solved he iniial value proble for parial differenial equaions (PDE) using a wo-diensional differenial ransfor ehod. Using his ehod reduces copuaional ie. Ayaz (4) sudied PDE by a hree-diensional differenial ransfor ehod. Hassan () solved eigenvalue probles by he DM ehod. Kuang Chen and Shin Chen (4) applied DM o solve he free vibraions of a conservaive oscillaor and also copared he resuls wih he Runge-Kua ehod. In addiion, Shin Chen and Kuang Chen (9) invesigaed he free vibraions of srongly non-linear oscillaors by DM and proposed his ehod o solve he non-linear proble because of DMs iproved accuracy. Lain Aerican Journal of Solids and Srucures 13 (16)

3 15 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas In his paper, wo new analyical ehods are proposed o solve he nonlinear vibraion of an Euler-Bernoulli bea. Firs, he governing equaion is obained, and hen he Galerkin Mehod is used o obain he corresponding ordinary differenial equaion. hen, he HAM wih wo auxiliary paraeers and DM are eployed o solve his equaion. Finally, he resuls of HAM and DM are copared wih nuerical ehod resuls and good agreeen is achieved. GOVERNING EQUAION In order o obain he governing equaion, we consider hree assupions: he plane deforaion is negligible so i is negleced, ransverse shear srains are sall so hey are negleced, and roaion of he cross secion is only due o bending. he scheaic of Euler-Bernoulli bea subjeced o he axial loading is shown in Fig. 1. W x, F Figure 1: A scheaic of an Euler-Bernoulli bea subjeced o an axial load. he equaion of oion including he effecs of id-plane sreching is inroduced as (Rao, 7): 4 W W W EAb W æ W ö EI + F +, 4 b + dx L ò = ç X X X è X (1) ø where L is he lengh of bea, A b is he cross-secional area, I is he oen of ineria, E is he young odulus, b is he ass per uni lengh of bea and. F. is agniude of axial force. For siplificaion, he following non-diensional variables are used: L R 4 L EI b L X W EI FL X =, W =, =, F =, () where R = I Ab is he radius of gyraion of he cross-secion. herefore, Eq. (1) can be wrien as follows: Lain Aerican Journal of Solids and Srucures 13 (16)

4 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas W W W 1 W æ W ö + F + - dx. 4 ò = X X X ç è X (3) ø Assuing ha W( X, ) = ( ) w( X), where ( ) w X is he firs Eigen-ode of he bea. By applying he Galerkin ehod, he equaion of oion is obained as follows:.. 3 ( ) b ( ) l ( ) + + =, (4) where b = b1 + Fb. he above equaion is he governing equaion of nonlinear vibraion of Euler- Bernoulli beas. he b1, b and l paraeers are as follows: b = ò ò b = ò ò dx, dx 1 iv '' 1 dx, dx 1æ 1 '' ' ö -.5 ò ç dx dx çè ò ø l =, 1 dx ò (5a) (5b) (5c) he cener of he bea are subjeced o he following iniial condiions: ( ) ( ) = A, =, (6) where A denoes he non-diensional axiu apliude of oscillaion. 3 APPLICAION OF HAM Consider he suiable iniial approxiaion, in order o solve Eq. (4). (for ore inforaion abou HAM seps, see Refs. (Abolbashari e al. (14), Abolbashari e al. (15), Jafari and Freidooniehr(15)): ( ) ( a ) = ACos, (7) where a is he secondary auxiliary paraeer, which is used o accelerae he convergence of he becoes: series. he auxiliary linear operaor ( ) Lain Aerican Journal of Solids and Srucures 13 (16)

5 154 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas ( ) = + a, (8) x which he following relaion is saisfied ( ( a ) ( a )) Sin + Cos =, (9) c 1 c where c 1 and c are he arbirary consans. According o he Eq. (4), he nonlinear operaor is inroduced as: ( ) ˆ ; p 3 é ˆ( ; ) ˆ( ; ) ( ˆ p ù ê = ë úû + b p + l ( ; p) ), (1) he zero-order deforaion equaion is inroduced as: p ˆ ˆ f ; p p ; p 1, (11) where p is an ebedded paraeer and is an auxiliary nonzero paraeer. he auxiliary funcions are inroduced as: Due o he boundary condiions 1. (1) ˆ ;p=a, ˆ d ;p =. d (13) Where By aylor's heore, we have ˆ ; p p, (14) 1 1! ˆ ; p p p. (15) Consider ha is chosen such ha he aylor series expansion of Eq. (14) has converged a p 1:, (16) 1 Lain Aerican Journal of Solids and Srucures 13 (16)

6 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas 155 χ 1,, R (17) where 1 j 1 R, 1 1, j i j i (18) j i and χ, 1, 1, >1. (19) he h order deforaion equaions (Eq. (17)) were solved using MAHEMAICA sofware. he resul of firs order soluion of HAM is deerined as: A Sin( )[ (4 3 4 ) Sin( )] 1 A A. () 16 I is iporan o selec a proper value of auxiliary paraeer o conrol he speed of convergence of he approxiaion series by he help of he so-called curve. he opial values of are seleced fro he valid region in sraigh line. he obained by he differen curve of order of approxiaion is shown in Fig.. he averaged residual error is inroduced as Eq. (1): d Res d In order o selec he opial value of he auxiliary paraeer, he averaged residual error (for ore deails, see Refs. (Liao (1), Rashidi and Abbasbandy(11)) is defined as: 3. (1) K 1 Res ix j K i j, () where x 5 K and K 1. For a given order of approxiaion, he opial value of is given by he iniu of, corresponding o nonlinear algebraic equaions d dh. (3) In order o check he accuracy of he HAM ehod wih wo auxiliary paraeers, he residual errors for 5 h order HAM soluions of Eq. (1) is shown in Fig. 3. Lain Aerican Journal of Solids and Srucures 13 (16)

7 156 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas -6-8 n=3 n=4 n=5 n=6 n=7 () -1 Figure : he h '' curve of obained by differen orders of approxiaion of HAM when A, 1, 1 and E-5 5E-6 h = -.9 h = -1 (Opial value) h = -1.1 Residual error -5E-6-1E Figure 3: he residual error for Eq. (1) using 5h-order of approxiaions when A 1,.5 and.1. 4 APPLICAION OF DM aking he differenial ransfor of Eq. (4), one can obain (for ore deails, see Refs. (Freidooniehr (15), Rashidi e al. (15), Rashidi and Freidooniehr (14)): Lain Aerican Journal of Solids and Srucures 13 (16)

8 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas 157 where k r k 1 k k k s r s k r, (4) rs k is he differenial ransfors of and displayed as: k, = k (5) k= A 1, Eq. (6) provides he ransfored boundary condiions. By subsiuing Eq. (6) ino Eq. (4) and cobining wih Eq. (5), he recursive ehod yields he values of : æ ö 3 4 ( ) = A+ (-ba- la ) + - b( -ba-la )- la (-ba- la ) +. 1ç è (7) ø I is iporan o noe ha he nuber of required ers is deerined by he convergence of he nuerical values up o one s desired accuracy. (6) 5 RESULS AND DISCUSSIONS Firs of all, coparisons have been done beween he resuls of he curren sudy; he resuls of nuerical soluion via he shooing ehod are used o validae he proposed analyical ehods (HAM & DM) in Fig. 4. Excellen agreeen can be observed beween he E-5 5E-5 Nuerical HAM DM.5E-5 () -.5E-5-5E-5-7.5E Figure 4: Deflecion versus ie resuls coparing HAM, DM, and a nuerical ehod a A.1, 1 and.1. Lain Aerican Journal of Solids and Srucures 13 (16)

9 158 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas Fig. 5 displays he resul of HAM soluion wih and wihou considering he second auxiliary paraeer. As enioned previously, he second auxiliary paraeer is used o increase he convergence perforance of he HAM. hus, he HAM wih wo auxiliary paraeers provides soluions ha have good agreeen wih he nuerical soluion. In addiion, i is obvious ha by increasing he value of, which represens he value of non-lineariy of he Eq. 4, he HAM wih one auxiliary paraeer diverge faser. hus for large values of he non-linear paraeer, he HAM wih wo auxiliary paraeer us be applied o ensure he series convergence. ables 1-3 presen he coparison beween he HAM resuls wih one and wo auxiliary paraeers wih he nuerical ehod resuls for he various values of he apliude,, and ie. As i is obvious for hese ables, good agreeen can be seen beween he resuls of he HAM wih wo auxiliary paraeers and he nuerical ehod resuls. a) b) () () -4 Nuerical soluion HAM wih wo auxiliary papraeer HAM wih one auxiliary papraeer -4 Nuerical HAM wih wo auxiliary papraeer HAM wih one auxiliary papraeer Figure 5: Apliude oscillaion coparison of HAM and nuerical soluion for he one and wo auxiliary paraeers a A 1, 1 a) 1 and b). A=.1 A=1 (s) Nuerical HAM I HAM II Nuerical HAM I HAM II able 1: Coparison beween HAM I & HAM II wih nuerical ehod for various apliude ( A) and ie a.5 and.5. Lain Aerican Journal of Solids and Srucures 13 (16)

10 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas (s) Nuerical HAM I HAM II Nuerical HAM I HAM II able : Coparison beween HAM I & HAM II wih nuerical ehod for various and ie a A 1 and (s) Nuerical HAM I HAM II Nuerical HAM I HAM II able 3: Coparison beween HAM I & HAM II wih nuerical ehod for various and ie a A 1 and.5. A paraeric sudy was conduced on a canilever bea wih cyclically varying axial load. he nonlinear vibraion analysis of deflecion is shown in Figs. 6-7 for a wide range of apliudes (A) and ie ( ) and differen values of he consan physical paraeers via wo auxiliary paraeer of HAM soluion. he difference beween hese wo figures is in heir physical consan and, where in Fig.6,.15,.15 and.5 in Fig. 7. Lain Aerican Journal of Solids and Srucures 13 (16)

11 16 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas Figure 6: 3D plo of deflecion () for a wide range of apliude A and ie( ), where and.15. Figure 7: 3D plo of deflecion () for a wide range of apliude A and ie( ), where.15 and.5. Influence of on behavior is deonsraed in Fig. 8, where A and are equal o 1. Also, he effec of on he behavior of is shown in Fig. 9, where A and are equal o 1. I is worh enioning ha he wo auxiliary paraeers of HAM soluion are used in Figs Lain Aerican Journal of Solids and Srucures 13 (16)

12 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas =.15 =.5 =.5 =1 () Figure 8: HAM soluion o show he effec of on () behavior a A 1 and =.15 =.5 =.5 =1 () Figure 9: HAM soluion o show he effec of on () behavior a A 1 and 1. he DM response depends on he order of soluions. he DM response a A.1,.5 and.5 is shown in Fig. 1 for differen orders of soluions. According o Fig. 1 here is good agreeen wih nuerical resuls a he h order of soluion. he soluions based on DM and nuerical ehod for 1 are copared in able. 4. In his able, he values of he Lain Aerican Journal of Solids and Srucures 13 (16)

13 16 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas and are equal o.5. I can be observed ha he axiu errors of HAM and DM soluions wih nuerical resuls are less han.%..1 () nuerical n=5 n=1 n= Figure 1: he () behavior a A.1,.5 and.5, based on DM for differen order of soluions. A Nuerical DM HAM Error (DM) Error (HAM) E E E E E E-7 able 4: Coparison beween DM & HAM wih nuerical ehod for various apliude (A) a 1s,.5 and.5. 6 CONCLUSIONS In his paper, nonlinear vibraion of an Euler-Bernoulli bea subjeced o a cyclically varying axial load is invesigaed by analyical ehods. he governing equaion of nonlinear vibraion of he bea is in he for of a parial differenial equaion. he Galerkin ehod is eployed o reduce he governing equaion o ordinary differenial equaion. he hooopy analysis ehod (HAM) wih wo auxiliary paraeers and differenial ransfor ehod (DM) are used o express he response of he axially loaded bea. In his proble, he second auxiliary paraeer is used o accelerae he convergence of he aylor series expansion, and i was shown ha his paraeer i- Lain Aerican Journal of Solids and Srucures 13 (16)

14 S.S. Jafari e al. / Analyical Approxiaion of Nonlinear Vibraion of Euler-Bernoulli Beas 163 proves he analysis of highly nonlinear syses. Excellen agreeen was found in coparing he HAM, DM and nuerical soluion resuls. References Abdel-Hali Hassan, I. H., (). On solving soe eigenvalue probles by using a differenial ransforaion. Applied Maheaics and Copuaion, 17:1-. Abolbashari, M. H., Freidooniehr, N., Nazari, F., Rashidi, M. M., (14). Enropy analysis for an unseady MHD flow pas a sreching pereable surface in nano-fluid. Powder echnology, 67: Abolbashari, M. H., Freidooniehr, N., Nazari, F., Rashidi, M. M., (15). Analyical odeling of enropy generaion for Casson nano-fluid flow induced by a sreching surface. Advanced Powder echnology, 6: Ahadi, M., Hashei, G., Asghari, A., (14). Applicaion of ieraion perurbaion ehod and Hailonian approach for nonlinear vibraion of Euler-Bernoulli beas. Lain Aerican Journal of Solids and Srucures, 11: Ayaz, F., (3). On he wo-diensional differenial ransfor ehod. Applied Maheaics and Copuaion, 143: Ayaz, F., (4). Soluions of he syse of differenial equaions by differenial ransfor ehod. Applied Maheaics and Copuaion, 147: Barari, A., Kiiaeifar, A., Doairry, G., Moghii, M., (1). Analyical evaluaion of bea deforaion proble using approxiae ehods. Songklanakarin Journal of Science and echnology, 3:7-36. Chen, C. o.-k., Chen, S.-S., (4). Applicaion of he differenial ransforaion ehod o a non-linear conservaive syse. Applied Maheaics and Copuaion, 154: Chen, S.-S., Chen, C. o.-k., (9). Applicaion of he differenial ransforaion ehod o he free vibraions of srongly non-linear oscillaors. Nonlinear Analysis: Real World Applicaions, 1: Freidooniehr, N., Rashidi, M. M., Jalilpour, B., (15). MHD sagnaion-poin flow pas a sreching/shrinking shee in he presence of hea generaion/absorpion and cheical reacion effecs. Journal of he Brazilian Sociey of Mechanical Sciences and Engineering:1-1. He, J. H., (6). Soe asypoic ehods for srongly nonlinear equaions. Inernaional Journal of Modern Physics B, : Hoseini, S. H., Pirbodaghi,., Asghari, M., Farrahi, G. H., Ahadian, M.., (8). Nonlinear free vibraion of conservaive oscillaors wih ineria and saic ype cubic nonlineariies using hooopy analysis ehod. Journal of Sound and Vibraion, 316: Jafari, S. S., Freidooniehr, N., (14). Second law of herodynaics analysis of hydro-agneic nano-fluid slip flow over a sreching pereable surface. Journal of he Brazilian Sociey of Mechanical Sciences and Engineering, 37: Kopaza, O., Gündogdub, Ö., (3). On he curvaure of an Euler Bernoulli bea. Inernaional Journal of Mechanical Engineering Educaion, 31:. Liao, S. (199). he proposed hooopy analysis echnique for he soluion of nonlinear probles. Ph. D. hesis, Shanghai Jiao ong Universiy, China. Liao, S., (1). An opial hooopy-analysis approach for srongly nonlinear differenial equaions. Counicaions in Nonlinear Science and Nuerical Siulaion, 15:3-16. Liu, Y., Gurra, C. S., (9). he use of He s variaional ieraion ehod for obaining he free vibraion of an Euler Bernoulli bea. Maheaical and Copuer Modelling, 5: Moeenfard, H., Mojahedi, M., Ahadian, M., (11). A hooopy perurbaion analysis of nonlinear free vibraion of ioshenko icrobeas. Journal of Mechanical Science and echnology, 5: Lain Aerican Journal of Solids and Srucures 13 (16)

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