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Nonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System

Published online by Cambridge University Press:  09 January 2015

H. Askari ,
D. Younesian  and
Z. Saadatnia
H. Askari
Affiliation:
Center of Excellence in Railway Transportation, School of Railway Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran
D. Younesian*
Affiliation:
Center of Excellence in Railway Transportation, School of Railway Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran
Z. Saadatnia
Affiliation:
Center of Excellence in Railway Transportation, School of Railway Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran
*
*Email:[email protected](D. Younesian)
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Abstract

This paper aims to investigate nonlinear oscillations of an elevator cable in a drum drive. The governing equation of motion of the objective system is developed by virtue of Lagrangian’s method. A complicated term is broached in the governing equation of the motion of the system owing to existence of multiplication of a quadratic function of velocity with a sinusoidal function of displacement in the kinetic energy of the system. The obtained equation is an example of a well-known category of nonlinear oscillators, namely, non-natural systems. Due to the complex terms in the governing equation, perturbation methods cannot directly extract any closed form expressions for the natural frequency. Unavoidably, different non-perturbative approaches are employed to solve the problem and to elicit a closed-form expression for the natural frequency. Energy balance method, modified energy balance method and variational approach are utilized for frequency analyzing of the system. Frequency-amplitude relationships are analytically obtained for nonlinear vibration of the elevator’s drum. In order to examine accuracy of the obtained results, exact solutions are numerically obtained and then compared with those obtained from approximate closed-form solutions for several cases. In a parametric study for different nonlinear parameters, variation of the natural frequencies against the initial amplitude is investigated. Accuracy of the three different approaches is then discussed for both small and large amplitudes of the oscillations.

Keywords

70K3070E55Nonlinear oscillationperturbation methodsnon-perturbative approachfrequency-amplitude relationship
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 1 , February 2015 , pp. 43 - 57
Copyright
Copyright © Global Science Press Limited 2015 

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References

[1]Nayfeh, A. H., Problems in Perturbation, Wiley, New York, 1985.Google Scholar
[2]Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973.Google Scholar
[3]Nayfeh, A. H. AND Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.Google Scholar
[4]Hinch, E. J., Perturbation Methods, Cambridge University Press, London, 1991.CrossRefGoogle Scholar
[5]Younesian, D.Esmailzadeh, E. AND Sedaghati, R., Existence of periodic solutions for generalized form of Mathieu equation, Nonlinear Dyn., 39 (2005), pp. 335–348.Google Scholar
[6]Kovacic, I. AND Brennan, M. J., The Duffing Equation Nonlinear Oscillators and Their Behavior, Wiley, New York, 2011.Google Scholar
[7]Younesian, D.Esmailzadeh, E. AND Sedaghati, R., Asymptotic solutions and stability analysis for generalized non-homogeneous Mathieu equation, Commun. Nonlinear Sci. Numer. Sim., 12 (2007), pp. 58–71.Google Scholar
[8]Belhaq, M. AND Lakrad, F., The elliptic multiple scales method for a class of autonomous strongly non-linear oscillators, J. Sound Vibration, 234 (2000), pp. 547–553.Google Scholar
[9]Arafat, H. N. AND Nayfeh, A. H., Non-linear responses of suspended cables to primary resonance excitations, J. Sound Vibration, 266 (2003), pp. 325–354.Google Scholar
[10]Nayfeh, A. H., The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance, J. Sound Vibration, 90 (1983), pp. 237–244.Google Scholar
[11]Arafat, H. N. AND Nayfeh, A. H., Non-linear responses of suspended cables to primary resonance excitations, J. Sound Vibration, 266 (2003), pp. 325–354.CrossRefGoogle Scholar
[12]Nayfeh, A. H., The response of non-linear single-degree-of-freedom systems to multifrequency excitations, J. Sound Vibration, 102 (1985), pp. 403–414.Google Scholar
[13]Rand, R. H.Sah, S. M. AND Suchorsky, M. K., Fractional Mathieu equation, Commun. Nonlinear Sci. Numer. Sim., 15 (2010), pp. 3254–3262.Google Scholar
[14]Storti, D. W. AND Rand, R. H., Dynamics of two strongly coupled van der pol oscillators, Int. J. Non-Linear Mech., 17 (1982), pp. 143–152.Google Scholar
[15]Rand, R.Guennoun, K. AND Belhaq, M., Resonance in the quasi-periodic Mathieu equation, Nonlinear Dyn., 31 (2003), pp. 367–374.Google Scholar
[16]Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), pp. 501–544.CrossRefGoogle Scholar
[17]He, J. H.Wu, G. CH. AND Austin, F., The variational iteration method which should be followed, Nonlinear Sci. Lett. A, 1 (2010), pp. 1–30.Google Scholar
[18]Liao, S. J., Beyond Perturbation-Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, 2003.Google Scholar
[19]Liao, S. J., On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), pp. 499–513.Google Scholar
[20]Liao, S. J., A kind of approximate solution technique which does not depend upon small parameters: a special example, Int. J. Non-Linear Mech., 30 (1995), pp. 371–380.Google Scholar
[21]Younesian, D.Askari, H.Saadatnia, Z. AND Yildirim, A., Analytical solution for nonlinear wave propagation in shallow media using the variational iteration method, Waves Random Complex Media, (2012), doi:10.1080/17455030.2011.633578.CrossRefGoogle Scholar
[22]Younesian, D.Saadatnia, Z. AND Askari, H., Analytical solutions for free oscillations of beams on nonlinear elastic foundation using the variational iteration method, J. Theor. Appl. Mech., 50 (2012), pp. 639–652.Google Scholar
[23]Wazwaz, A. M., The variational iteration method for solving linear and nonlinear Volterra integral and integro-differential equations, Int. J. Comput. Math., 87 (2010), pp. 1131–1141.CrossRefGoogle Scholar
[24]Chen, H.Shen, J. AND Zhou, Z., On the application of the homotopy analysis method to limit cycles approximation in planar self-excited systems, Commun. Nonlinear Sci. Numer. Sim., 17 (2012), pp. 4494–4507.Google Scholar
[25]Biazar, J.Gholami Porshokuhi, M. AND Ghanbari, B., Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method, Comput. Math. Appl., 59 (2010), pp. 622–628.Google Scholar
[26]Xu, L., Variational iteration method for solving integral equations, Comput. Math. Appl., 54 (2007), pp. 1071–1078.Google Scholar
[27]He, J. H., Preliminary report on the energy balance for nonlinear oscillations, Mech. Res. Commun., 29 (2002), pp. 107–111.Google Scholar
[28]He, J. H., Hamiltonian approach to nonlinear oscillators, Phys. Lett. A, 374 (2010), pp. 2312–2314.CrossRefGoogle Scholar
[29]Kalamiyazdi, M.Khan, Y.Madani, M.Askari, H.Saadatnia, Z. AND Yildirim, A., Analytical solutions for autonomous conservative nonlinear oscillator, Int. J. Nonlinear Sci. Numer. Sim., 11 (2010), pp. 979–984.Google Scholar
[30]He, J. H., Max-min approach to nonlinear oscillators, Int. J. Nonlinear Sci. Numer. Sim., 9 (2008), pp. 207–210.Google Scholar
[31]He, J. H., Variational approach for nonlinear oscillators, Chaos Solitons Fractals, 34 (2007), pp. 1430–1439.Google Scholar
[32]He, J. H., An Improved amplitude-frequency formulation for nonlinear oscillators, Int. J. Nonlinear Sci. Numer. Sim., 9 (2008), pp. 211–212.Google Scholar
[33]Askari, H.Yazdi, M. K. AND Saadatnia, Z., Frequency analysis of nonlinear oscillators with rational restoring force via He’s energy balance method and He’s variational approach, Nonlinear Sci. Lett. A, 1 (2010), pp. 425–430.Google Scholar
[34]Askari, H.Saadatnia, Z.Younesian, D.Yildirim, A. AND Kalamiyazdi, M., Approximate periodic solutions for the Helmholtz-Duffing equation, Comput. Math. Appl., 62 (2011), pp. 3894–3901.Google Scholar
[35]Cveticanin, L.Kalamiyazdi, M. AND Askari, H., Analytical approximations to the solutions for a generalized oscillator with strong nonlinear terms, J. Eng. Math., DOI 10.1007/s10665-012-9542-4.Google Scholar
[36]Yildirim, A.Askari, H.Saadatnia, Z.Kalamiyazdi, M. AND Khan, Y., Analysis of nonlinear oscillations of a punctual charge in the electric field of a charged ring via a Hamiltonian approach and the energy balance method, Comput. Math. Appl., 62 (2011), pp. 486–490.Google Scholar
[37]Cveticanin, L.Kalamiyazdi, M.Saadatnia, Z. AND Askari, H., Application of Hamiltonian approach to the generalized nonlinear oscillator with fractional power, Int. J. Nonlinear Sci. Numer. Sim., 11 (2010), pp. 997–1002.Google Scholar
[38]Yildirim, A.Askari, H.Kalami Yazdi, M. AND Khan, Y., A relationship between three analytical approaches to nonlinear problems, Appl. Math. Lett., doi:10.1016/j.aml.2012.02.001.Google Scholar
[39]Yildirim, A.Saadatnia, Z. AND Askari, H. , Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms, Math. Comput. Model., 54 (2011), pp. 697–703.CrossRefGoogle Scholar
[40]Askari, H.Saadatnia, Z. AND Yildrim, A., Application of higher order Hamiltonian approach to nonlinear vibrating systems, J. Theor. Appl. Mech., (2012), in Press.Google Scholar
[41]Younesian, D.Saadatnia, Z. AND Askari, H., Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the variational iteration method, J. Theor. Appl. Mech., 50 (2012), pp. 639–652.Google Scholar
[42]Mohammadpour, A.Rokni, E.Fooladi, M. AND Kimiaeifar, A., Approximate analytical solution for Bernoulli-Euler beams under different boundary conditions with non-linear Winkler type foundation, J. Theor. Appl. Mech., 50 (2012), pp. 339–355.Google Scholar
[43]Marincaa, V. AND Heriṣanua, N., Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, J. Sound Vibration, 329 (2010), pp. 1450–1459.CrossRefGoogle Scholar
[44]Wu, B. S.Lim, C. W. AND He, L. H., A new method for approximate analytical solutions to nonlinear oscillations of nonnatural systems, Nonlinear Dyn., 32 (2003), pp. 1–13.Google Scholar
[45]Bayat, M.Pakar, I. AND Bayat, M., Analytical study on the vibration frequencies of tapered beams, Latin American J. Solids Structures, 8 (2011), pp. 149–162.Google Scholar
[46]Salem, M. I.Mucino, V. H. Saunders, E. AND Gautam, M., Lateral sloshing in partially filled elliptical tanker trucks using a trammel pendulum, Int. J. Heavy Vehicle Syst., 16 (2009), pp. 207–224.Google Scholar
[47]Fu, Y.Zhang, J. AND Wan, L., Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS), Current Appl. Phys., 11 (2011), pp. 482–485.Google Scholar
[48]Herisanu, N. AND 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 (2010), pp. 847–855.Google Scholar
[49]Hamdan, M. N. AND Dado, M. H. F., Large amplitude free vibrations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia, J. Sound Vibration, 206 (1997), pp. 151–168.Google Scholar
[50]Mirabdollah Yani, R.Ghodsi, R. AND Darabi, E., A closed-form solution for nonlinear oscillation and stability analyses of the elevator cable in a drum drive elevator system experiencing free vibration, Commun. Nonlinear Sci. Numer. Sim., 17 (2012), pp. 4467–4484.Google Scholar
[51]Shou, D. H. AND He, J. H. , Application of parameter-expanding method to strongly nonlinear oscillators, Int. J. Nonlinear Sci. Numer. Sim., 8 (2007), pp. 121–124.Google Scholar