Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 1, pp. 215-224, Warsaw 2013 WAVELET-BASED SOLUTION FOR VIBRATIONS OF A BEAM ON A NONLINEAR VISCOELASTIC FOUNDATION DUE TO MOVING LOAD Zdzisław Hryniewicz, Piotr Kozioł Koszalin University of Technology, Department of Civil and Environmental Engineering, Koszalin, Poland e-mail: zdzislaw.hryniewicz@tu.koszalin.pl; piotr.koziol@wbiis.tu.koszalin.pl The paper presents a new analytical solution for the dynamic response of an infinitely long Timoshenko beam resting on a nonlinear viscoelastic foundation. Vibrations of the beamare analysed by usingAdomian’s decompositionmethod combinedwithwavelet based approximation alleviating difficulties related to Fourier analysis and numerical integration. The developed approach allows various parametric analyses leading to full characteristics of the investigated dynamic system. Key words: Timoshenko beam, Adomian’s decomposition, coiflet expansion, moving load 1. Introduction Thedevelopment ofmodern high speed trains and continuously growing influence of train trans- portation on the surrounding environment, especially buildings and road constructions, have led to the necessity of better understanding and modelling of dynamic phenomena related to beam-soil-loads interactions (Bogacz and Frischmuth, 2009; Bogacz andKrzyzynski, 1991). The model of a beam on a nonlinear foundation is of great theoretical and practical significance in railway engineering as, in practice, the foundation can be strongly nonlinear. One can find in the literature many interesting results concerning moving load problems in mostly linear cases (Fryba, 1999), whereas the nonlinear approach is still open for investigation. The lack of effective methods prevents analytical solving of nonlinear problems that can be used for parametric analysis of systems. Many results obtained by using numerical simulations have already been published showing better modelling possibilities when using nonlinear cases. Dahlberg (2002) obtained some partial results and found that the nonlinear model simulates the beamdeflection fairlywell, comparedwithmeasurements, whereas the linearmodel did not. More recently,WuandThompson(2004)useda similar nonlinearmodel andstudied thedynamic response using FEM. A relatively small number of papers regarding analytical approaches can be found, highlighting their importance for the subject. In the paper by Kargarnovin et al. (2005), the governing nonlinear equations of motion are solved by using a perturbationmethod in conjunction with the Fourier transform and Cauchy’s residue theorem. The present paper shows a new approach for dynamic analysis of the Timoshenko beam on a nonlinear viscoelastic foundation. This new method is based on Adomian’s decomposition method (Adomian, 1989) combined with wavelet-based expansion of functions (Koziol, 2010; Wang et al., 2003) that allows one to omit numerical integration, leading to effective evaluation of approximating functions. The presented analytical solution, together with the developed in- novative method, is the main novelty of the article. The question of stability for the obtained solution is left as an open problem and it is not discussed in the paper. The wavelet-based method of the system dynamic solution, using the coiflet expansion of functions, allowed one to formulate a special methodology for vibration analysis of beam- foundation structures subject to moving loads. The basics of the developed methodology are 216 Z. Hryniewicz, P. Kozioł presented in Koziol (2010), along with a number of mathematical examples showing main fe- atures of the coiflet-based method. In that monograph, a few models related to moving load problems were analysed. Amodel of the Euler-Bernoulli beam placed inside a viscoelastic layer (or on its surface) shows how the applied semi-analytical method improves efficiency and effec- tiveness of the parametric analysis compared to numerical computations (Koziol et al., 2008; Koziol, 2010; Koziol andMares, 2010). The wavelet-based expansion combined with Bourret’s approximation gave a new solution in the case of a dynamic response of the Euler Bernoulli and Timoshenko beam resting on a viscoelastic foundationwith randomvariationsmodelled by a stochastic subgrade reaction. The obtained result extended a class of solutions for problems described by stochastic differential equations with differentiable correlation functions (Koziol and Hryniewicz, 2006; Hryniewicz and Koziol, 2009). The analysis of the coiflet-based solution for the Euler-Bernoulli beam placed inside a vi- scoelastic medium subject to moving point load carried out in the frequency domain allowed one to estimate critical velocities for specific sets of parameters (Koziol and Mares, 2010). The Timoshenko beamwas analyzed as a better representation of the track response tomoving load (Hryniewicz, 2009; Hryniewicz and Koziol, 2009). The present paper extends a class of models analyzed by using the developed coiflet-based methodology for the analysis of dynamic models. The wavelet expansion combined with Ado- mian’s decomposition is proposed as one of possible semi-analytical approaches for the nonlinear model solution. The nonlinear term included in the Timoshenko beam equation leads to com- plexity of computations, and a relatively small number of analytical solutions for thismodel can be found in the literature so far. These solutions are usually insufficiently exact (Kargarnovin et al., 2005) andmay give wrong results for some sets of parameters. 2. Formulation of the system The Timoshenko beam is considered by many authors as a more relevant modelling approach for moving load problems analysis due to its specific form taking into account both the shear deformations and the rotary inertia of cross sections. The dynamic equations for the beam on a nonlinear viscoelastic foundation can be written as follows mb ∂2W ∂t2 +Cd ∂W ∂t −S ∂2W ∂x̃2 +S ∂Ψ ∂x̃ +kLW +kNW 3 =−P(x̃, t)eiΩt J ∂2Ψ ∂t2 −EI ∂ 2Ψ ∂x̃2 +SΨ −S∂W ∂x̃ =0 (2.1) where W is the transversedisplacementof theneutral axis, Ψ is theangular rotation of the cross- section, x̃ is the space coordinate in the direction along the beam, t represents time, kN is the nonlinear part of the foundation stiffness, kL is the linear coefficient of the foundation stiffness, kL = k̃(1+ 2ζi), i = √ −1 and ζ is the hysteretic damping ratio. The used parameters are: Young’s modulus E, shearmodulus G, mass density ρ, viscous damping of the foundation Cd, cross-section area A, moment of inertia I, shear correction factor κ, mass distribution per unit length mb = ρA, shear rigidity S = κAG, beam flexural rigidity EI, mass moment of inertia J = ρI, load frequency Ω =2πfΩ and the load velocity V . The cubic representation of the nonlinear term in Eq. (2.1)1 was already used in previous papers (e.g. Kargarnovin et al., 2005; Sapountzakis andKampitsis, 2011). Some authors investi- gate theoretically more complex representations of the nonlinear factor. An analysis of how the order of polynomial representing the nonlinear spring force influences the solution for vibrations of the infinite Euler-Bernoulli beam resting on a nonlinear elastic foundationwas carried out by Wavelet-based solution for vibrations of a beam... 217 Jang et al. (2011). The authors assumed that the nonlinear restoration is analytic and therefore can be expanded by the Taylor series. Their newly developed iterative method allowed them to obtain the nonlinear deflection of the beam and to analyse the solution parametrically. It is shown that the contribution from the nonlinear spring becomes smaller when the order of nonlinearity increases. In this paper, the distributedmoving load is considered, being amore realistic representation of loading related to train motion P(x̃, t)= P0 2a cos2 π 2a (x̃−V t)H[a2− (x̃−V t)2] (2.2) where H(·) is the Heaviside step function and 2a is the span of moving load. In order to analyze the steady-state response of the beam, theGalilean co-ordinate transfor- mation x = x̃−V t can be used. In the case of an infinitely long beam, the boundary conditions must reflect the fact that the displacement, the slope of the beam curvature, the shear force and the bendingmoment tend to zero when the variable x tends to the infinity. The following representation of the response can be assumed for analytical solution of the system W(x,t)= w(x)eiΩt Ψ(x,t)= ψ(x)eiΩt (2.3) The use of the chain rule applied to these equations leads to following expressions ∂W ∂x̃ = dw dx eiΩt ∂2W ∂x̃2 = d2w dx2 eiΩt ∂W ∂t = ( −V dw dx +iΩw ) eiΩt ∂2W ∂t2 = ( V 2 d2w dx2 −2iΩV dw dx −Ω2w ) eiΩt (2.4) and similar formulas can be obtained for the function Ψ. Substitution of Eqs. (2.4) to system (2.1) yields a0 d2w dx2 +a1 dw dx +a2w+S dψ dx =−P(x)−kNw3e2iΩt b0 d2ψ dx2 + b1 dψ dx +b2ψ−S dw dx =0 (2.5) where new coefficients are introduced: a0 = mbV 2−S a1 =−V (2iΩmb +Cd) a2 = kL −mbΩ2+iCdΩ b0 = JV 2−EI b1 =−2iΩV J b2 =S −Ω2J (2.6) 3. Adomian’s decomposition method Adomian’s decomposition assumes that the solution can be represented by an infinite serieswith one linear term, and the rest of them as a set of functions related to the nonlinear factor. Thus, the nonlinear part of Eq. (2.5)1 can be described by a series w3(x)= ∞∑ j=0 Aj(x) (3.1) with the Adomian polynomials (Pourdarvish, 2006) Aj(x)= 1 j! [ dj dλj ( ∞∑ k=0 λkwk(x) )3] λ=0 j =0,1,2, . . . (3.2) 218 Z. Hryniewicz, P. Kozioł Adomian’s decomposition assumes the specific form of solution w(x) = ∞∑ j=0 wj(x) ψ(x)= ∞∑ j=0 ψj(x) (3.3) and the explicit forms of the Adomian polynomials for the term w3(x) can be found as follows (Pourdarvish, 2006) A0 = w 3 0 A1 =3w 2 0w1 A2 =3(w0w 2 1 +w 2 0w2) A3 = w 3 1 +6w0w1w2+w 2 0w3 A4 =3(w 2 1w2+w0w 2 2 +2w0w1w3+w 2 0w4) . . . (3.4) Equations (2.5)1 can be rewritten by using the operator notation as follows Lww+S dψ dx =−P(x)−kNw3(x)e2iΩt Lψψ−S dw dx =0 (3.5) with the differential operators Lw = a0 d2 dx2 +a1 d dx +a2 Lψ = b0 d2 dx2 + b1 d dx + b2 (3.6) The convergence condition for series (3.1) can be formulated as the system of inequalities 0¬ αj < 1, (j =0,1,2, . . .) for the parameter αj defined as follows αj =    ‖wj+1‖ ‖wj‖ for ‖wj‖ 6=0 0 for ‖wj‖=0 (3.7) with the norm ‖wj‖=maxx |Re[wj(x)]|. One can show (Adomian, 1989) that lim n→∞ Swn (x)= limn→∞ n∑ j=0 wj(x)= w(x) (3.8) and the effective analysis of solution can be carried out by using the n-th order approximation Swn (x). The same assumptions can be used when dealing with the terms ψj(x). The classical Fourier transforms are used for derivation of the solution ŵ(ω)= F[w(x)] = ∞∫ −∞ w(x)e−iωx dx ψ̂(ω)= F[ψ(x)] = ∞∫ −∞ ψ(x)e−iωx dx (3.9) and w(x) = F−1[ŵ(ω)] = 1 2π ∞∫ −∞ ŵ(ω)eiωx dω ψ(x) = F−1[ψ̂(ω)] = 1 2π ∞∫ −∞ ψ̂(ω)eiωx dω (3.10) Formulas (3.1) and (3.2) lead to the following system of equations Lww0+S dψ0 dx =−P(x) Lψψ0−S dw0 dx =0 (3.11) Wavelet-based solution for vibrations of a beam... 219 and Lwwj +S dψj dx =−kNAj−1(x)e2iΩt Lψψj −S dwj dx =0 j =1,2, . . . (3.12) where the formsof wj(x) and ψj(x) canbe foundusing recursively the above systemof formulas. Applying the Fourier transform to both sides of Eqs. (3.11), one obtains (−a0ω2+ia1ω+a2)ŵ0+iSωψ̂0 =−P̂(ω) (−b0ω2+ib1ω+b2)ψ̂0− iSωŵ0 =0 (3.13) where P̂(ω)= P0 sin aω 2 aω [ 1− (aω π )2] Solving the system of equations (3.13) leads to the transformed solution for w0 and ψ0 ŵ0 = P̂Hψ S2ω2−HwHψ ψ̂0 = iSωP̂ S2ω2−HwHψ (3.14) where Hw =−a0ω2+ia1ω +a2, Hψ =−b0ω2+ib1ω + b2. In order to find the solution in the physical domain, it is enough to derive the inverse Fourier transforms (3.10) of Eqs. (3.14). 4. Coiflet-based wavelet approximation The specific form of integrands (3.14) prevents effectively analytical evaluation of the integrals. On the other hand, numerical evaluations lead to computational difficulties resulting in inexact solutions. Therefore, one must apply an alternative method of derivation, giving results precise enough that can be used for further approximation of the higher order (i.e. calculation of w1, ψ1, w2, ψ2 and so on). One of the analytical methods whose efficiency for analysis of dynamic systems has already been proved before, is the approximation which uses the wavelet expansion of functions (Koziol, 2010; Wang et al., 2003). This approach is based on approximation of multiresolution coefficients that can be relatively easily done in the case of wavelet functions defined by coiflets (Monzon et al., 1999) ΨC(x)= 3N−1∑ k=0 (−1)kp3N−1−kΦC(2x−k) ΦC(x)= 3N−1∑ k=0 pkΦC(2x−k) (4.1) ΨC and ΦC are the wavelet and scaling function, respectively, and N is treated as a degree of accuracyof the coiflet filter (pk).Thepropertyof vanishingmoments (Koziol, 2010;Mallat, 1998; Monzon et al., 1999;Wang et al., 2003) of coiflets for both, wavelet and scaling functions, allows one to estimate analytically the wavelet coefficients (Wang et al., 2003), leading to formulas used in an effective approximation of the Fourier integrals. Using the refinement equations of ΨC and ΦC in the transform domain (Mallat, 1998; Monzon and Beylkin, 1999; Wang et al., 2003) and assuming Φ̂C(0) = 1 leads to an efficient algorithm for the approximation of the Fourier transform and the inverse Fourier transform f̂(ω)= lim n→∞ f̂n(ω)= lim n→∞ 2−(n+1) ∞∏ k=1 (3N−1∑ k=0 pke ikω2−(n+k) ) +∞∑ k=−∞ f((k+M)2−n)e−iωk2 −n f(x)= lim n→∞ fn(x)= lim n→∞ 1 2n+2π ∞∏ k=1 (3N−1∑ k=0 pke −ikx2−(n+k) ) +∞∑ k=−∞ f̂((k+M)2−n)eixk2 −n (4.2) 220 Z. Hryniewicz, P. Kozioł Thehigh order of coiflets accuracy, although givingbetter regularity of the generatedwavelet function, leads to computational difficulties (Monzon et al., 1999), and therefore, a good balance between the efficiency and cost-effectiveness must be found. The criterion for the choice of the approximation order is the stabilisation of solutions above some value of the parameter n (Koziol, 2010). 5. Wavelet-based solution The coiflet expansion applied to formulas (3.14) gives the following representation of the solution for the first terms of sought series (3.8) w0(x)= 1 2n+1π ϕC(−x2−n) kmax∑ k=kmin ŵ0((k+M)2 −n)eikx2 −n (5.1) where ϕC(−x2−n)= kp∏ k=1 3N−1∑ j=0 pje jx2−(n+k) (5.2) and the approximated solution for ψ0(x) can be written similarly. In order to obtain the next terms of series (3.8), onemust apply the Fourier transform to Eqs. (3.12). Assuming j =1, one obtains ŵ1(ω)= kNHψÂ0(ω) S2ω2−HwHψ e2iΩt ψ̂1(ω)= iSωkNÂ0(ω) S2ω2−HwHψ e2iΩt (5.3) where the term Â0(ω) is the Fourier transform of the Adomian polynomial A0(x) = [w0(x)] 3, and it can bederived by using the inverse coiflet expansion (4.2)1. The inverse Fourier transform applied to ŵ1(ω) and ψ̂1(ω) gives terms w1(x) and ψ1(x). Applying the same procedure for evaluation of terms with arbitrarily chosen index j, leads to a general form of the Adomian polynomials Aj and the solution represented by series (3.8) Âj(ω)= 2 −nϕ̂C ( ω 2n ) kmax∑ k=kmin Aj (k+M 2n ) e−ikω2 −n ŵj(ω)= kNHψÂj−1(ω) S2ω2−HwHψ e2iΩt ψ̂j(ω)= iSkNωÂj−1(ω) S2ω2−HwHψ e2iΩt wj(x)= 1 2n+1π ϕC ( − x 2n ) kmax∑ k=kmin ŵj (k+M 2n ) eikx2 −n ψj(x)= 1 2n+1π ϕC ( − x 2n ) kmax∑ k=kmin ψ̂j (k+M 2n ) eikx2 −n (5.4) The range of summation kmin(n)= ωmin2 n −3N −2, kmax(n)= ωmax2n−1must be found on the basis of information about the transformed function (Koziol, 2010; Wang et al., 2003). The interval [ωmin,ωmax] must cover the set of variable ω having strong influence on the behaviour of the original function. Wavelet-based solution for vibrations of a beam... 221 6. Numerical examples Theobservationpoint x canbe chosen arbitrarily and thepoint x̃ =0(i.e. x =−V t) is assumed for numerical examples presented in this paper. The following set of parameters (Kargarnovin et al., 2005; Kim, 2005) is adopted for nume- rical simulations ζ =0.02 kN =5 ·1011N/m4 k̃ =2 ·107N/m2 P0 =5 ·104N/m EI =3 ·105Nm2 a =0.075m Cd =0 J = ρI =0.24kgm S =2 ·108N mb =300kg/m The values used for numerical examples are chosen on the basis of analyses of systems presented in the literature and are accordingly adjusted in order to present important features of the developed method of solution. One should note that the present paper shows theoretical investigations. The aim of the authors is to present a new semi-analytical method of solution for the considered nonlinearmodel. The performed simulations show strong potential of the coiflet- basedmethod for the analysis of nonlinearmodels.However, furthermoredetailed investigations should be carried out for better physical interpretation of the solution and determination of stability domains. Significant changes in the shape of plots can be observed for some systems of parameters compared with the linear model, especially in the area of critical velocities. Figure 1 shows the vertical displacement of the beam for the linear case and fourth orderAdomian’s approximation w ≈ w0+w1+ . . .+w4 for sub-critical ((a) and (b)) and super-critical velocities ((c) and (d)) (Hryniewicz, 2009). Fig. 1. The vertical displacement in the case of the linear (solid) and nonlinear (dashed) system (fΩ =22Hz): (a) V =55m/s; (b) V =65m/s; (c) V =85m/s; (d) V =105m/s 222 Z. Hryniewicz, P. Kozioł Figure 2 indicates that the effect of higher order terms of Adomian’s series (3.8) on the vibratory characteristics of the beamvanishes and they can be neglected above some value of n. The influence of nonlinear terms of Adomian’s estimation on the obtained solution becomes stronger along with increasing velocity, with the strongest impact in the case of velocities near critical values (Fig. 3). Fig. 2. The first four terms of Adomian’s decomposition for the vertical displacement w (V =10m/s and fΩ =22Hz) Fig. 3. The first four terms of Adomian’s decomposition for the vertical displacement w (V =62m/s and fΩ =22Hz Wavelet-based solution for vibrations of a beam... 223 7. Conclusions The new solution for vibrations of an infinite Timoshenko beam resting on a nonlinear visco- elastic foundation as a result of moving load is obtained. Adomian’s decomposition method combined with the wavelet approximation is adopted in order to obtain an effective solution allowing parametric analysis of the dynamic system. The performed simulations show strong efficiency of the developed approach compared with methods applied by other authors. The presentedmethod can be used for further investigations in order to carry out deeper analysis of physical properties of this model and similar ones. References 1. 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Part F: Rail Rapid Transit, 218, 1-12 Falkowe rozwiązanie problemu drgań belki spoczywającej na nieliniowym lepkosprężystym podłożu generowanych przez poruszające się obciążenie Streszczenie Artykuł prezentuje nowe analityczne rozwiązanie problemu dynamicznej odpowiedzi nieskończenie długiej belki Timoshenki spoczywającej na nieliniowym lepkosprężystympodłożu, poddanej ruchomemu obciążeniu rozłożonemunaodcinku i harmonicznemuwczasie.Analizadrgańbelki zostałaprzeprowadzo- naprzyużyciudekompozycjiAdomianapołączonej z aproksymacją falkowąpozwalającąominąć trudności związane z numerycznym całkowaniem oraz zminimalizować niedogodności analizy Fouriera. Uzyskane rozwiązaniedajemożliwośćparametrycznej analizybadanegoukładudynamicznegoprowadzącejdo opisu jego fizycznych własności. Opracowana metoda wykorzystująca filtry falkowe typu coiflet może być za- stosowana do rozwiązania nieliniowych równań różniczkowych opisujących inne układy dynamiczne typu belka-podłoże. Manuscript received November 15, 2010; accepted for print May 24, 2012