An Analytical Study Of Nonlinear Vibrations Of Buckled .

Vol.123ACTA PHYSICA POLONICA A(2013)No. 1An Analytical Study of Nonlinear Vibrationsof Buckled Euler Bernoulli BeamsI. Pakar and M. Bayat Department of Civil Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran(Received May 28, 2012; in nal form October 25, 2012)The current research deals with a way of using a new kind of periodic solutions called He's max-min approachfor the nonlinear vibration of axially loaded Euler Bernoulli beams. By applying this technique, the beam'snatural frequencies and mode shapes can be easily obtained and a rapidly convergent sequence is obtained duringthe solution. The e ect of vibration amplitude on the non-linear frequency and buckling load is discussed. Toverify the results some comparisons are presented between max-min approach results and the exact ones to showthe accuracy of this new approach. It has been discovered that the max-min approach does not necessitate smallperturbation and is also suitably precise to both linear and nonlinear problems in physics and engineering.DOI: 10.12693/APhysPolA.123.48PACS: 42.65.Wi, 46.70.De, 44.05. eing axial inertia [9] and assuming linear curvature [10].The partial-di erential equations are discrete to nonlinear ordinary-di erential equations by using the Galerkinapproach and then we can apply the direct techniquesto solve them analytically in time domain. In recentyears, many approximate analytical methods have beenproposed for studying nonlinear vibration equations ofbeams and shells and etc. such as homotopy perturbation [11], energy balance [12, 13], variational approach[14, 15], max-min approach (MMA) [16], iteration perturbation method [17] and other analytical and numericalmethods [18 25].The Adomian decomposition method (ADM) was applied by Lai et al. [26] to obtain an analytical solutionfor nonlinear vibration of the Euler Bernoulli beam withdi erent supporting conditions. Naguleswaran [27] developed the work on the changes of cross-section of anEuler Bernoulli beam resting on elastic end supports.Pirbodaghi et al. [28] presented an analytical expressionfor geometrically free vibration of the Euler Bernoullibeam by using homotopy analysis method (HAM). Theypoint out that the amplitude of the vibration has a greate ect on the nonlinear frequency and buckling load ofthe beams. Liu et al. [29] applied He's variational iteration method to assess an analytical solution for anEuler Bernoulli beam with di erent supporting conditions. Bayat et al. [30, 31] applied energy balance methodand variational approach method to obtain the natural frequency of the nonlinear equation of the Euler Bernoulli beam.In this paper we used the Galerkin method for discretization to obtain an ordinary nonlinear di erentialequation from the governing nonlinear partial di erentialequation. It was then assumed that only fundamentalmode was excited. Finally, max-min approach is compared with other researcher's results. The max-min approach results are accurate and only one iteration leadsto high accuracy of solutions for whole domain.1. IntroductionInvestigating on the dynamic response of beams is oneof the most important parts in the design process ofstructures. Many researchers have addressed the nonlinear vibration behavior of beams, both experimentallyand theoretically. Burgreen [1] considered the free vibrations of a simply supported buckled beam theoretically and experimentally. He found out that the naturalfrequencies of buckled beams depend on the amplitudeof vibration. Moon [2] and Holmes and Moon [3] useda single-mode approximation to investigate chaotic motions of buckled beams under external harmonic excitations. Abu Rayan et al. [4] continue the study on thenonlinear dynamics of a simply supported buckled beamusing a single-mode approximation to a principal parametric resonance. Ramu et al. [5] used a single-modeapproximation to study the chaotic motion of a simplysupported buckled beam. Reynolds and Dowell [6, 7]used multi-mode Galerkin discretization to analyze thechaotic motion of a simply supported buckled beam under a harmonic excitation. They used Melnikov theoryin their analysis. Lestari and Hanagud [8] used a single-mode approximation to study the nonlinear vibrationsof buckled beams with elastic end constraints. They considered the beam to be subjected simultaneously to axial and lateral loads without rst statically buckling thebeam. The nonlinear vibration of beams and distributedand continuous systems are governed by linear and nonlinear partial di erential equations in space and time.Solving nonlinear partial di erential equations analytically is very di cult.It is very common to simplify the equations of motion by introducing various assumptions which allowfor the derivation of manageable governing equations.Some of the simplifying assumptions include neglect- corresponding author; e-mail:mbayat14@yahoo.com(48)

An Analytical Study of Nonlinear Vibrations . . .2. Description of the problemConsider a straight Euler Bernoulli beam of length L,a cross-sectional area A, the mass per unit length of thebeam m, a moment of inertia I , and a modulus of elasticity E that is subjected to an axial force of magnitudeP as shown in Fig. 1.Fig. 1. A schematic of an Euler Bernoulli beam subjected to an axial load: (a) simply supported beam,(b) clamped-clamped beam.The equation of motion including the e ects of mid-plane stretching is given by 2 w0 4 w0 2 w0EA 2 w0m 02 EI 04 P̄ 02 t x x2L x02Z L 2 0 2 wdx0 0.(2.1) x00For convenience, the following non-dimensional variablesare used:x x0 /L, w w0 /ρ, t t0 (EI/ml4 )1/2 ,P P̄ L2 /EI,(2.2)where ρ (I/A)1/2 is the radius of gyration of the cross-section. As a result Eq. (2.1) can be written as follows: 2Z 2w 4w 2 w 1 2 w L w P dx 0. t2 x4 x22 x2 0 x(2.3)Assuming w(x, t) W (t)ϕ(x) where ϕ(x) is the rsteigenmode of the beam [32] and applying the Galerkinmethod, the equation of motion is obtained as follows:d2 W (t) (α1 P α2 )W (t) α3 W 3 (t) 0.(2.4)dt2Equation (2.3) is the di erential equation of motion governing the nonlinear vibration of Euler Bernoulli beams.The center of the beam is subjected to the following initial conditions:dW (0)W (0) , 0,(2.5)dtwhere denotes the non-dimensional maximum amplitude of oscillation and α1 , α2 and α3 are as follows:Z 1.Z 1α1 ϕ0000 ϕ dxϕ2 dx,(2.6a)0Zα2 01ϕ00 ϕdx.Z0101ϕ2 dx,(2.6b) Z Z.Z 11 1ϕ00ϕ02 dx ϕ dxϕ2 dx. (2.6c)2 000Post-buckling load-de ection relation for the problem canbe obtained from Eq. (2.4) asα3 49 P α1 α3 W 2 /α2 .(2.7)Neglecting the contribution of W in Eq. (2.7), the buckling load can be determined asPc α1 /α2 .(2.8)3. Basic idea of max-min approachWe consider a generalized nonlinear oscillator in theform [33]:Ẅ W f (W ) 0, W (0) , Ẇ (0) 0,(3.1)where f (W ) is a non-negative function of W . Accordingto the idea of the max-min method, we choose a trial-function in the formW (t) cos(ωt),(3.2)where ω is the unknown frequency to be determined further.Observe that the square of frequency, ω 2 , is never lessthan that in the solution pfmin t ,(3.3)W1 (t) cosof the following linear oscillator:Ẅ W fmin 0, W (0) , Ẇ (0) 0,(3.4)where fmin is the minimum value of the function f (W ).In addition, ω 2 never exceeds the square of frequencyof the solution pfmax t ,(3.5)W1 (t) cosof the following oscillator:Ẅ W fmax 0, W (0) , Ẇ (0) 0,(3.6)where fmax is the maximum value of the function f (W ).Hence, it follows thatfminfmax ω2 .(3.7)11According to the Chentian interpolation [33], we obtainmfmin nfmaxω2 ,(3.8)m norfmin kfmaxω2 ,(3.9)1 kwhere m and n are weighting factors, k n/m. So thesolution of Eq. (3.1) can be expressed asrfmin kfmaxW (t) cost.(3.10)1 kThe value of k can be approximately determined by various approximate methods [34, 35]. Among others, herebywe use the residual method [34]. Substituting Eq. (3.10)into Eq. (3.1) results in the following residual:R(t; k) ω 2 A cos(ωt) [A cos(ωt)] f (A cos(ωt)) ,(3.11)q kfmaxwhere ω fmin1 k.If, by chance, Eq. (3.10) is the exact solution, thenR(t; k) is vanishing completely. Since our approach isonly an approximation to the exact solution, we setrZ Tfmin kfmaxR(t; k) costdt 0,(3.12)1 k0where T 2π/ω . Solving the above equation, we can

I. Pakar, M. Bayat50easily obtain5. Results and discussionsfmax fmink 1 qAπRπ0.(3.13)cos2 (x)f (A cos x) dxSubstituting the above equation into Eq. (3.10), we obtain the approximate solution of Eq. (3.1).4. ApplicationsWe can re-write Eq. (2.4) in the following form:Ẅ (α1 P α2 )W α3 W 0.(4.1)We choose a trial-function in the formW (t) cos(ωt),(4.2)where ω is the frequency to be determined.By using the trial-function, the maximum and minimum values of ω will beα1 P α2ωmin ,1α1 P α2 α3 .(4.3)ωmax 1So we can writeα1 P α2α1 P α2 α3 ω2 .(4.4)11According to the Chengtian inequality [34], we havem (α1 P α2 ) n (α1 P α2 α3 )ω2 m n α1 P α2 kα3 (4.5)where m and n are weighting factors, k n/m n.Therefore the frequency can be approximated asp(4.6)ω (α1 pα2 ) kα3 2 .Its approximate solution readsp(4.7)W (t) cos (α1 pα2 ) kα3 2 t.In view of the approximate solution, Eq. (4.6), we re-write Eq. (4.1) in the form3Ẅ (α1 P α2 kα3 )W (kα3 )W α3 W 3 .(4.8)If by any chance Eq. (4.6) is the exact solution, then theright side of Eq. (4.8) vanishes completely. Consideringour approach which is just an approximation one, we setZ T /4 (kα3 ) W α3 W 3 cos ωtdt 0,(4.9)0where T 2π/ω . Solving the above equation, we caneasily obtain3k .(4.10)4Finally the frequency is obtained as1p4 (α1 pα2 ) 3α3 2 .(4.11)ω 2Hence, the approximate solution can be readily obtained 1p2W (t) cos4 (α1 pα2 ) 3α3 t . (4.12)2Nonlinear to linear frequency ratio ispωNL1 4 (α1 pα2 ) 3α3 2 .(4.13)ωL2α1 pα2To illustrate and verify the results obtained by theMMA, some comparisons with the published data andthe exact solutions are presented. The exact frequencyωExact for a dynamic system governed by Eq. (2.4) canbe derived, as shown in Eq. (5.1), as follows:Z π/2 dt sin(t)(5.1)ωExact (2π/4 2 )0q 2 sin2 (t) ( 2 α3 cos2 (t) 2pα2 2α1 2 α3 ).To obtain numerical solution we must specify the parameter β α3 /(pα2 α1 ). This parameter depends onthe type of structure and boundary condition considered.The comparison of nonlinear to linear frequency ratio(ωNL /ωL ) with those reported by Azrar et al. [36] and theexact one are tabulated in Tables I and II. The maximumrelative error of the analytical approaches is 2.004109%for the rst order analytical approximations as it is shownin Tables I and II.Fig. 2. Comparison of the approximate and exact solutions for simply supported beam with 1.5, α1 2,α2 0, α3 6: (a) time history response, (b) phasecurve.Fig. 3. Comparison of the approximate and exact solutions for simply supported beam with 0.6, α1 1,α2 0, α3 3: (a) time history response, (b) phasecurve.Fig. 4. Comparison of the approximate and exact solutions for clamped-clamped beam with 0.909,α1 1, α2 0, α3 1.814: (a) time history response,(b) phase curve.