Abstract
This paper is concerned with analytical treatment of non-linear oscillations of planar, flexural large amplitude free vibrations of a slender, inextensible cantilever beam carrying a lumped mass with rotary inertia at an intermediate position along its span. An analytic approximate technique, namely Optimal Homotopy Asymptotic Method (OHAM) is employed for this purpose. It is proved that OHAM provide accurate solutions for large amplitudes and large modal constants in the considered nonlinear equations, when other classical methods fail. Our procedure provides us with a convenient way to optimally control the convergence of solution, such that the accuracy is always guaranteed. An excellent agreement of the approximate frequencies and periodic solutions with the numerical results and published results has been demonstrated. Two examples are given and the results reveal that this procedure is very effective, simple and accurate. This paper demonstrates the general validity and the great potential of the OHAM for solving strongly nonlinear problems.
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References
Trendafilova E, Manoach I (2008) Large amplitude vibrations and damage detection of rectangular plates. J Sound Vib 315:591–606
Kanaka RK, Venkateswara RG (2005) Towards improved evaluation of large amplitude free-vibration behaviour of uniform beams using multi-term admissible functions. J Sound Vib 282:1238–1246
Wu BS, Sun WP, Lim CW (2007) Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations. J Sound Vib 307:953–960
Cveticanin L, Kovacic I (2007) Parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity. J Sound Vib 304:201–212
Kovacic I, Brennan M, Lineton B (2008) On the resonance response of an asymmetric Duffing oscillator. Int J Non-Linear Mech 43:858–867
Hamdan MN, Shabaneh NH (1997) On the large amplitude free vibrations of a restrained uniform bean carrying an intermediate lumped mass. J Sound Vib 199:711–736
Hamdan MN, Dado MHF (1997) Large amplitude free vibrations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. J Sound Vib 206:151–168
Qaisi MI, Al-Huniti NS (2001) Large-amplitude free vibration of a conservative system with inertia and static nonlinearity. J Sound Vib 242:1–7
Wu BS, Lim CW, Ma YF (2003) Analytical approximation to large-amplitude oscillation of a non-linear conservative system. Int J Non-linear Mech 38:1037–1043
Wu WP, Li PS (2001) A method for obtaining approximate analytical periods for a class of nonlinear oscillators. Meccanica 36:167–176
Marinca V, Herişanu N, Bota C (2008) Application of the variational iteration method to some nonlinear one-dimensional oscillations. Meccanica 43:75–79
Herişanu N, Marinca V (2010) A modified variational iteration method for strongly nonlinear problems. Nonlinear Sci Lett A 1:183–192
He JH, Wu GC, Austin F (2010) The variational iteration method which should be followed. Nonlinear Sci Lett A 1:1–30
Herişanu N, Marinca V (2009) An iteration procedure with application to Van der Pol oscillator. Int J Nonlinear Sci Numer Simul 10:353–361
Belendez A, Pascual C, Neipp C et al. (2008) An equivalent linearization method for conservative nonlinear oscillators. Int J Nonlinear Sci Numer Simul 9:9–17
JI Ramos (2008) Linearized Galerkin and artificial parameter techniques for the determination of periodic solutions of nonlinear oscillators. Appl Math Comput 196:483–493
Belendez A, Hernandez A, Belendez T et al. (2007) Application of He’s homotopy perturbation method to the Duffing-harmonic oscillator. Int J Nonlinear Sci Numer Simul 8:79–88
He JH (2005) Homotopy perturbation method for bifurcation of nonlinear problems. Int J Nonlinear Sci Numer Simul 6:207–208
Yildirim A (2010) Determination of periodic solutions for nonlinear oscillators with fractional powers by He’s modified Lindstedt-Poincaré method. Meccanica 45:1–6
JI Ramos (2009) An artificial parameter–Linstedt–Poincaré method for oscillators with smooth odd nonlinearities. Chaos Solitons Fractals 41:380–393
Herişanu N, Marinca V, Marinca B (2007) An analytic solution of some rotating electric machines vibration. Int Rev Mech Eng (IREME) 1:559–564
Marinca V, Herişanu N (2008) Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int Commun Heat Mass Transf 35:710–715
Marinca V, Herişanu N, Nemeş I (2008) Optimal homotopy asymptotic method with application to thin film flow. Cent Eur J Phys 6:648–665
Herişanu N, Marinca V, Dordea T, Madescu G (2008) A new analytical approach to nonlinear vibration of an electrical machine. Proc Romanian Acad Ser A 9:229–236
Marinca V, Herişanu N, Bota C, Marinca B (2009) An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Appl Math Lett 22:245–251
Marinca V, Herişanu N (2010) Determination of periodic solution of a particle on a rotating parabola by means of the optimal homotopy asymptotic method. J Sound Vib 329:1450–1459
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Herişanu, N., Marinca, V. Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. Meccanica 45, 847–855 (2010). https://doi.org/10.1007/s11012-010-9293-0
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DOI: https://doi.org/10.1007/s11012-010-9293-0