Dynamic response of axially loaded Euler-Bernoulli beams

172ISSN 1392 - 1207. MECHANIKA. 2011. 17(2): 172-177Dynamic response of axially loaded Euler-Bernoulli beamsM. Bayat*, A. Barari**, M. Shahidi****Civil Engineering Department, Shomal University, Amol, Iran, P.O.Box731, E-mail: mbayat14@yahoo.com**Department of Civil Engineering, Aalborg University, Sohngårdsholmsvej 57, 9000 Aalborg, Aalborg, Denmark,E-mail: ab@civil.aau.dk, amin78404@yahoo.com***Civil Engineering Department, Shomal University, Amol, Iran, P.O.Box73, E-mail: mehran.shahidi13@gmail.com1. IntroductionThe Euler–Bernoulli theory of beams provides areasonable explanation of the bending behavior of longisotropic beams. It is based on the assumption that a relationshipbetween bending moment and the beam curvatureexists.Kopmaz et al. [1] considered different approachesto describing the relationship between the bending momentand curvature of a Euler-Bernoulli beam undergoing alarge deformation. Then, in the case of a cantilevered beamsubjected to a single moment at its free end, the differencebetween the linear theory and the nonlinear theory basedon both the mathematical curvature and the physical curvaturewas shown. Biondi and Caddemi [2] studied the problemof the integration of static governing equations of theuniform Euler-Bernoulli beams with discontinuities, consideringthe flexural stiffness and slope discontinuities.Many researchers have addressed the nonlinearvibration behavior of beams, theoretically [3-6]. The vibrationproblems of uniform Euler-Bernoulli beams can besolved by analytical or approximate approaches [7, 8].Failla and Santini [9] presented the eigenvalue problem ofEuler-Bernoulli discontinuous beams. Specifically, forstepped beams with internal translational and rotationalsprings, they proved that a formulation of well-establishedlumped-mass methods employing exact influence coefficientsis readily feasible, based on appropriate Green’sfunctions yielding the response of the discontinuous beamto a static unit force. Yeih et al. [10] obtained the naturalfrequencies and natural modes for an Euler-Bernoulli beamusing a dual multiple reciprocity method (MRM) and thesingular value decomposition method. Yeih's method wasable to avoid the spurious eigenvalue problem and modesresulted from applying the conventional MRM.A recent innovative method in solving these problemsis presented by Lai et al. [11]. Through their contribution,the Adomian Decomposition Method was employedto obtain the natural frequencies and mode shapes for theEuler- Bernoulli beam under various supporting conditions.The technique used is based on the decomposition ofa solution of nonlinear operator equation in a series offunctions. Each term of the series is obtained from a polynomialgenerated from an expansion of an analytic functioninto a power series. Liu and Gurram [12] utilizedvariational iteration method (VIM) to solve free vibrationof Euler-Bernoulli beam under various supporting conditions.The technique they used is based on the use of restrictedvariations and correction functionals which hasfound a wide application for the solution of nonlinear ordinaryand partial differential equations. The proposedmethod does not require the presence of small parametersin the differential equation, and provides the solution (oran approximation to it) as a sequence of iterates.Recently, researchers have been concentrated onapproximate analytical methods such as Parameter ExpansionMethod [13,14], Adomian Decomposition Method[15], Differential Transform Method [16], VIM [17,18],Homotopy Perturbation Method [19-24], Max-Min Approach[25-27] and other analytical techniques [28-30].He [31] gave a comprehensive review of the recentlydeveloped nonlinear analytics techniques for solvingnonlinear oscillations problems, which comprise the relativelynewer family of solutions which lie within theframework of periodic analytical solutions. Other methodshave also been developed in recent years which seem to bejust as promising in obtaining accurate solutions to generallymore difficult nonlinear problems. Energy balancemethod [32] is one such method, which is actually a heuristicapproach valid not only for weakly nonlinear systems,but also for strongly nonlinear ones [33-35].The main objective of this study is to obtain analyticalexpressions for geometrically nonlinear vibration ofEuler-Bernoulli beams. First, the governing nonlinear partialdifferential equation is reduced to a single nonlinearordinary differential equation. It is assumed that only thefundamental mode is excited. The latter equation is solvedanalytically in time domain using Energy Balance Method(EBM).2. Mathematical formulationConsider a straight Euler-Bernoulli beam oflength L , a cross-sectional area A , the mass per unitlength of the beam m, a moment of inertia I, and modulusof elasticity E that is subjected to an axial force of magnitudeP as shown in Fig. 1. The equation of motion includingthe effects of mid-plane stretching is given by2 4 2 2 2∂ w′ ∂ w′ ∂ w′ EA∂ w′ L ⎛∂w′⎞m + EI + P − dx 02 2 2 2 ⎜0 2 ⎟ ′∂t′ ∂x′ ∂x′ 2L∂x′ ∫ ⎝ ∂x′= ⎠(1)For convenience, the following nondimensionalvariables are used( ) 12= ′ = ′ = ′=4 2x x L, w w ρ , t t EI ml , P PL EIwhere ρ = ( I A) 1/2is the radius of gyration of the crosssection.As a result Eq. (1) can be written as follows2 4 2 2 2∂ w ∂ w ∂ w 1 ∂ w L⎛∂w⎞+ + P − d 02 2 2 2 ⎜0 2 ⎟ x =∂t ∂x ∂x 2 ∂x ∫ ⎝ ∂x(2)⎠22

173buckling load can be determined asP =− α α(8)c1 2 .3. Basic idea of energy balance methodFig. 1 A schematic of an Euler-Bernoulli beam subjectedto an axial loadAssuming wxt ( , ) = Wt ( ) φ ( x)where φ ( x)isthe first eigenmode of the beam [36] and applying the Galerkinmethod, the equation of motion is obtained as follows2dWt2()3+ ( α1 + Pα2) W() t + α3W () t = 0 (3)dtEq. (3) is the differential equation of motion governingthe nonlinear vibration of Euler-Bernoulli beams.The center of the beam is subjected to the following initialconditionsdW (0)W(0) = Wmax, = 0(4)dtwhere W maxdenotes the nondimensional maximum amplitudeof oscillation.Under the transformation τ = ω t , the Eq. (3) canbe written asW 3+ + P W + W = 0 (5)2ω ( α1 α2)α3where ω is the nonlinear frequency and double-dot denotesdifferentiation with respect to τ and α1,α2and α3are as follows⎛⎜α =⎝∫101 1⎛⎜α =⎝∫4⎛∂φ( x)⎞ ⎞φ( x)dx4⎜ x ⎟ ⎟⎝ ∂ ⎠ ⎠2φ ( xdx )10∫02 12⎛∂φ( x)⎞ ⎞⎜ φ( x)dx2 ⎟ ⎟⎝ ∂x⎠ ⎠∫2φ ( xdx )00(6a)(6b)2 22⎛⎛ 1 ⎞⎛ 1 ∂ φ( x) 1 ⎛∂φ( x)⎞ ⎞ ⎞⎜⎜−⎟ dx φ( x)dx0 2 ⎜0 2 ⎟⎟⎜ 2∫ ⎜ ∂x∫⎝ ⎠ ⎝ ∂x⎠ ⎟ ⎟⎝α⎝⎠3=⎠1(6c)2φ ( xdx )∫Post-buckling load-deflection relation for theproblem can be obtained from Eq. (5) by substitutingω = 0 as2( α1 α3)P = − − W α2(7)Neglecting the contribution of W in Eq. (7), theIn the present paper, we consider a general nonlinearoscillator in the form [32]u′′ + f( u( t)) = 0(9)In which u and t are generalized dimensionlessdisplacement and time variables, respectively. Its variationalprinciple can be easily obtainedt ⎛ 1 2 ⎞J ( u) = ∫ − u′+ F( u)0⎜⎟⎝ 2 ⎠dt (10)2πwhere T = is period of the nonlinear oscillator,ωF ( u) = ∫ f ( u) du.formorIts Hamiltonian, therefore, can be written in the1= ′ + ( ) + ( )(11)22H u F u F A1 2R() t = − u′+ F( u) − F( A) = 0 (12)2Oscillatory systems contain two important physicalparameters, (i.e., the frequency ω and the amplitude ofoscillation A ). So let us consider such initial conditionsu(0) = A, u′ (0) = 0(13)We use the following trial function to determinethe angular frequency ωut () = Acosωt(14)Substituting (14) into u term of (12), yield1 2 2 2Rt () = ω Asinω t+ F( Acosωt) − F( A) = 0 (15)2If, by chance, the exact solution had been chosenas the trial function, then it would be possible to make Rzero for all values of t by appropriate choice of ω . SinceEq. (14) is only an approximation to the exact solution,R cannot be made zero everywhere. Collocation atω t = π /4 gives2 F( A) − F( Acosωt)ω = (16)2 2AsinωtT =Its period can be written in the form2π2 F( A) − F( Acosωt)2 2Asinωt.(17)

1744. Application of the energy balance methodConsider the Eqs. (3) and (4) for the vibration ofan Euler-Bernoulli beam. Free oscillation of the systemwithout damping is a periodic motion and under the transformationW() t = V( τ ), Eqs. (3) and (4) become as follows22 dV( τ )3ω + ( α2 1+ Pα2) V( τ) + α3V( τ) = 0 (18)dτdV (0)V(0) = Wmax, = 0(19)dτIts variational formulation can be readily obtainedas followsformandJ ( V)⎛ 1 dV ( τ ) 1 ⎞− ω + ( α + Pα) × ⎟ dτ⎟⎝⎠2=t⎜ 0 1 2∫ 2 dτ20 ⎜⎜× 2 4V ( τ) + α3V( τ)(20)Its Hamiltonian, therefore, can be written in the1 2 dV ( τ ) 1 20 (1 2 ) ()4H =− ω + α + Pα V τ + α3V() τ (21)2 dτ21 21 4Ht= 0= Wmax( α1+ Pα2)+ α4W max(22)2 41 2 dV ( τ ) 1 2Ht− Ht=0= ω0 + ( α1+Pα2) V ( τ) +2 dτ24 1 2 1 4+ α3V ( τ) − Wmax( α1 + Pα2)− α4Wmax2 4(23)According to Eq. (14) and Eq. (27), we can obtainthe following approximate solution⎛υ () t = Wmaxcos⎜⎝TEBM4( α + Pα ) + 3αW1 2 32ω02maxIts period can be written in the form=4πω4( α + Pα ) + 3αW01 2 35. Results and discussions2max⎞⎟⎠(28)(29)The simply supported and clamped beams areused to demonstrate the accuracy and effectiveness of theEnergy Balance Method, as the procedure explained inprevious sections. Table shows the comparison of nonlinearto linear frequency ratio ( ωNL ωL) with those reportedin the literature. It has illustrated that there is an excellentagreement between the results obtained from the energybalance method and those reported by Azrar et al. [37] andTableThe comparison of nonlinear to linear frequency ratioω ω )(NL LSimply supportedClampedW max Azrar[36]Qaisi[30]PresentstudyAzrar[36]Qaisi[30]Presentstudy1 1.0891 1.0897 1.0897 1.0221 1.0628 1.05722 1.3177 1.3229 1.3228 1.0856 1.2140 1.21253 1.6256 1.6394 1.6393 1.1831 1.3904 1.43444 - - 1.9999 1.3064 1.5635 1.6171We will use the trial function to determine the angularfrequencyω , i.e.V ( τ ) = Acosωτ(24)If we substitute Eq. (24) into Eq. (23), it resultsthe following residual equation1 2 2 1ω0( − Wmaxωsin( ωt)) + ( α1+Pα2 2)× 22 14× ( Wmaxcos( ωt)) + α3( Wmaxcos( ωt))−21 2 1 4− Wmax( α1+ Pα2) − α4Wmax= 02 4πIf we collocate at ω t = we obtain4(25)a1 2 2 2 1 2 3 4ω0Wmax ω − Wmax ( α1+ Pα2)− α3Wmax= 0 (26)4 4 16The nonlinear natural frequency and deflection ofthe beam centre become as followsω4( α + Pα ) + 3αW2NL=1 2 3 max2ω0(27)bFig. 2 Variation of the nondimensional amplitude ratioversus τ for Wmax= 1 and Wmax= 2

175Qaisi [30]. The difference between the nonlinear frequencyand linear frequency increases when the amplitude of vibrationis increased. In general, large vibration amplitudewill yield a higher frequency ratio. It can be easily seenthat for high-amplitude ratios the present method overestimatesthe frequencies of clamped beams but gives closeagreement with published results for simply supportedbeams. The reason is because of using the trigonometricbase functions in the application of energy balance method,which means that we assumed the general form of solutionis a combination of trigonometric functions. Since the eigenmodesfor simply supported beams involve only thesinusoidal component, the energy balance method givesmore accurate results in comparison with clamped beamswhich have hyperbolic component in their eigenmodes. Todemonstrate the accuracy of the obtained analytical resultswe also calculate the variation of nondimensional amplituderatio versus τ for the beam center using fourth-orderRunge-Kutta method. Fig. 2 illustrates the comparison betweenthese results. As can be seen in the figure, the resultsobtained using the energy balance method have a goodagreement with numerical results.6. ConclusionsIn this study, the energy balance method was employedto obtain analytical expressions for the nonlinearfundamental frequency and deflection of Euler-Bernoullibeams. These expressions are valid for a wide range ofvibration amplitudes, unlike the solutions obtained by theother analytical techniques such as perturbation methods.The energy balance method solution converges quickly andits components can be simply calculated. Also, comparedto other analytical methods, it can be observed that theresults of energy balance method require smaller computationaleffort and only a first-order approximation leads toaccurate solutions. Beside all the advantages of the energybalance method, there are no rigorous theories to direct usto choose the initial approximations, auxiliary linear operators,auxiliary functions, and auxiliary parameter. However,further research is needed to better understand theeffect of these parameters on the solution quality.References1. Kopmaz, O.; Gündogdu, Ö. 2003. On the curvature ofan Euler–Bernoulli beam, International Journal of MechanicalEngineering Education, v.31, No.2: 132-142.2. Biondi, B.; Caddemi, S. 2005. Closed form solutionsof Euler-Bernoulli beams with singularities, InternationalJournal of Solids and Structures, v.42, No9-10:3027-3044.3. Jalili, N.; Esmailzadeh, E. 2002. Adaptive-passivestructural vibration attenuation using distributed absorbers.Proceedings of the Institution of MechanicalEngineers, Part K: Journal of Multi-body Dynamics,v.216, No3: 223-235.4. Sfahani, M.G.; Barari, A.; Omidvar, M.; Ganji,S.S.; Domairry, G. 2010. Dynamic response of inextensiblebeams by improved energy balance method.Proceedings of the Institution of Mechanical Engineers,Part K: Journal of Multi-body Dynamics, in press, DOI:10.1177/2041306810392113.5. Tumonis, L.; Schneider, M.; Kačianauskas, R.;Kačeniauskas, A. 2009. Comparison of dynamic behaviourof EMA-3 railgun under differently inducedloadings, Mechanika 4(78): 31-37.6. Mazeika, D.; Bansevicius, R. 2009. Study of resonantvibrations shapes of the beam type piezoelectric actuatorwith preloaded mass, Mechanika 2(76): 33-37.7. Meirovitch, L. 2001. Fundamentals of Vibrations. InternationalEdition. McGraw-Hill. 806p.8. Dimarogonas, A. 1996. Vibration for Engineers. 2nded. Prentice-Hall, Inc. 815p.9. Failla, G.; Santini, A. 2008. A solution method forEuler-Bernoulli vibrating discontinuous beams, MechanicsResearch Communications, v.35, No.8: 517-529.10. Yeih, W.; Chen, J.T.; Chang, C.M. 1999. Applicationsof dual MRM for determining the natural frequenciesand natural modes of an Euler-Bernoulli beamusing the singular value decomposition method, EngineeringAnalysis with Boundary Elements, v.23, No.4:339-360.11. Lai, H.Y.; Hsu, J.C.; Chen, C.K. 2008. An innovativeeigenvalue problem solver for free vibration of Euler_Bernoullibeam by using the Adomian decompositionmethod, Computers and Mathematics with Applications,v.56, No.12: 3204-3220.12. Liu, Y.; Gurram, C.S. 2009. The use of He's variationaliteration method for obtaining the free vibrationof an Euler_Bernoulli beam, Mathematical and ComputerModelling, v.50, No.11-12: 1545 -1552.13. Wang, S.Q.; He, J.H. 2008. Nonlinear oscillator withdiscontinuity by parameter-expansion method, ChaosSolitons and Fractals, v.35, No.4: 688-691.14. Shou, D.H.; He, J.H. 2007. Application of parameterexpandingmethod to strongly nonlinear oscillators, InternationalJournal of Nonlinear Sciences and NumericalSimulation, v.8, No.1: 121-124.15. Mirgolbabaei, H.; Barari, A.; Ibsen, L.B.; Sfahani,M.G. 2010. Numerical solution of boundary layer flowand convection heat transfer over a flat plate, Archivesof Civil and Mechanical Engineering, v.10, No.2: 41-51.16. Ganji, S.S.; Barari, A.; Ibsen, L.B.; Domairry, G.2010. Differential transform method for mathematicalmodeling of jamming transition problem in traffic congestionflow, Central European Journal of OperationsResearch, in press, DOI: 10.1007/s10100-010-0154-7.17. Barari, A.; Omidvar, M.; Ganji, D.D.; Tahmasebipoor, A. 2008. An approximate solution for boundaryvalue problems in structural engineering and fluid mechanics,Journal of Mathematical Problems in Engineering,Article ID 394103: 1-13.18. Fouladi, F.; Hosseinzadeh, E.; Barari, A.; DomairryG. 2010. Highly nonlinear temperature dependent finanalysis by variational iteration method, Journal ofHeat Transfer Research, v.41, No.2: 155-165.19. Barari, A.; Omidvar, M.; Ghotbi, Abdoul, R.;Ganji, D.D. 2008. Application of homotopy perturbationmethod and variational iteration method to nonlinearoscillator differential equations, Acta ApplicandaMathematicae, v.104: 161-171.20. Omidvar, M.; Barari, A.; Momeni, M.; Ganji, D.D.2010. New class of solutions for water infiltration problemsin unsaturated soils, Geomechanics and Geoengineering:An International Journal, v.5, No.2: 127-135.

17621. Bayat, M.; Shahidi, M.; Barari A.; Domairry, G.2010. The approximate analysis of nonlinear behaviorof structure under harmonic loading, InternationalJournal of the Physical Sciences, v.5, No.7: 1074-1080.22. Miansari, M.O.; Miansari, M.E.; Barari, A.; Domairry,G. 2010. Analysis of Blasius equation for flatplateflow with infinite boundary value, InternationalJournal for Computational Methods in EngineeringScience and Mechanics, v.11, No.2: 79-84.23. Ghotbi, Abdoul R.; Barari, A.; Ganji, D.D. 2008.Solving ratio-dependent predator–prey system withconstant effort harvesting using homotopy perturbationmethod, Journal of Mathematical Problems in Engineering,Article ID 945420: 1-8.24. Ganji, S.S.; Barari, A.; Domairry, G.; TeimourzadehBaboli, P. 2010. Consideration of transientstream/aquifer interaction with the non-linear Boussinesqequation using HPM, Journal of King Saud University,Science, in press, DOI:10.1016/j.jksus.2010.07.011.25. Ibsen, L.B.; Barari, A.; Kimiaeifar, A. 2010. Analysisof highly nonlinear oscillation systems using He’smax-min method and comparison with homotopy analysisand energy balance methods, Sadhana, v.35, No.4:1-16.26. Babazadeh, H.; Domairry, G.; Barari, A.; Azami, R.and Davodi, A.G. 2010. Numerical analysis of stronglynonlinear oscillation systems using He’s max-minmethod, Frontiers of Mechanical Engineering in China,DOI 10.1007/s11465-009-0033-x.27. Sfahani, M.G.; Ganji, S.S.; Barari, A.; Mirgolbabaei,H.; Domairry, G. 2010. Analytical solutions tononlinear conservative oscillator with fifth-order nonlinearity,Earthquake Engineering and Engineering Vibration,v.9, No.3: 1-9.28. Evensen, D.A. 1968. Nonlinear vibrations of beamswith various boundary conditions, AIAA Journal, v.6,No.2: 370-372.29. Pillai, S.R.R.; Rao, B.N. 1992. On nonlinear free vibrationsof simply supported uniform beams, Journal ofSound and Vibration, v.159, No.3: 527-531.30. Qaisi, M.I. 1993. Application of the harmonic balanceprinciple to the nonlinear free vibration of beams, AppliedAcoustics, v.40, No.2: 141-151.31. He, J.H. 2008. An elementary introduction to recentlydeveloped asymptotic methods and nanomechanics intextile engineering, International Journal of ModernPhysics B, v.22, No.10: 3487-3587.32. He, J.H. Preliminary report on the energy balance fornonlinear oscillations. -Mechanics Research Communications,2002, v.29, No.2-3, p.107-111.33. Bayat, M.; Shahidi, M.; Barari A.; Domairry, G.2011. Analytical Evaluation of the Nonlinear Vibrationof Coupled Oscillator Systems, Zeitschrift für NaturforschungA, v.66a, No.1: 1-16.34. Ganji, S.S.; Ganji, D.D.; Ganji, Z.Z. 2009. Periodicsolution for strongly nonlinear vibration systems byHe's energy balance method, Acta ApplicandaeMathemathicae, v.106, No.1: 79-92.35. Momeni, M.; Jamshidi, N.; Barari, A.; Ganji, D.D.2010. Application of He’s energy balance method toDuffing harmonic oscillators. -International Journal ofComputer Mathematics, in press,DOI:10.1080/00207160903337239.36. Tse, F.S.; Morse, I.E.; Hinkle, R.T. 1978. MechanicalVibrations: Theory and Applications. Second ed. Allynand Bacon Inc. Bosto. 449p.37. Azrar, L.; Benamar, R.; White, R.G. 1999. A semianalyticalapproach to the nonlinear dynamic responseproblem of S–S and C–C beams at large vibration amplitudespart I: general theory and application to thesingle mode approach to free and forced vibration analysis,Journal of Sound and Vibration, v.224: 183-207.M. Bayat, A. Barari, M. ShahidiAŠINE KRYPTIMI APKRAUTŲ EULERIO IRBERNULIO SIJŲ DINAMINIS ATSPARUMASR e z i u m ėNagrinėjamas naujo analitinio metodo, vadinamoenergijos balanso metodu (EBM), taikymas netiesiniamsuždaviniams spręsti. Energijos balanso metodas taikomasnetiesinių svyravimų poveikio Eulerio ir Bernulio sijoms,apkrautoms ašiniais krūviais, analitiniam sprendimui gauti.Pasiūlytos sijų geometriškai netiesinių virpesių analitinėsišraiškos. Aptarta virpesių amplitudės įtaka netiesiniamdažniui. Energijos balanso metodo rezultatų palyginimassu literatūros rezultatais patvirtina šio metodo tikslumą.Šiuo metodu, priešingai nei įprastiniais metodais, tik vienupriartėjimu gaunamas labai tikslus sprendinys, galiojantisplačiame svyravimo amplitudžių intervale.M. Bayat, A. Barari, M. ShahidiDYNAMIC RESPONSE OF AXIALLY LOADEDEULER-BERNOULLI BEAMSS u m m a r yThe current research deals with application of anew analytical technique called Energy Balance Method(EBM) for a nonlinear problem. Energy Balance Method isused to obtain the analytical solution for nonlinear vibrationbehavior of Euler-Bernoulli beams subjected to axialloads. Analytical expressions for geometrically nonlinearvibration of beams are provided. The effect of vibrationamplitude on the nonlinear frequency is discussed. Comparisonbetween Energy Balance Method results and thoseavailable in literature demonstrates the accuracy of thismethod. In Energy Balance Method contrary to the conventionalmethods, only one iteration leads to high accuracyof the solutions which are valid for a wide range ofvibration amplitudes.

177М. Баиат, А. Барари, М. СхахидиДИНАМИЧЕСКОЕ СОПРОТИВЛЕНИЕ БАЛОКЭУЛЛЕРА-БЕРНУЛИ НАГРУЖЕННЫХ ОСЕВОЙНАГРУЗКОЙР е з ю м еНастоящее исследование рассматривает применениенового аналитического метода, называемогометодом баланса энергии (МБЭ) для решения нелинейныхпроблем. Метод баланса энергии применяетсядля аналитического решения влияния нелинейных колебанийбалок Эуллера-Бернули, нагруженных осевойнагрузкой. Составлены аналитические зависимости длягеометрически нелинейных колебаний балок. Рассмотреновлияние амплитуды колебаний на нелинейнуючастоту. Сопоставление результатов метода балансаэнергии с результатами литературы подтверждает точностьэтого метода. Этот метод в противоположностьизвестным методам, уже после одного приближениядает точные результаты для широкого интервала амплитудколебаний.Received August 30, 2010Accepted April 11, 2011