Determination of Periodic Solution for Tapered Beams with Modified Iteration Perturbation Method

Document Type: Research Paper

Authors

1 Dept. of Mechanical Engineering, Babol University of Technology, Babol, Iran

2 Assistant Professor, Department of Mechanical Engineering, Babol University of Technology, Babol, Iran

3 Department of Mechanical Engineering, Semnan University, Iran

Abstract

In this paper, we implemented the Modified Iteration Perturbation Method (MIPM) for approximating the periodic behavior of a tapered beam. This problem is formulated as a nonlinear ordinary differential equation with linear and nonlinear terms. The solution is quickly convergent and does not need complicated calculations. Comparing the results of the MIPM with the exact solution shows that this method is effective and convenient. Also, it is predicated that MIPM can be potentially used in the analysis of strongly nonlinear oscillation problems accurately. 

Keywords

Main Subjects

[1] Sarma, B.S., Varadan, T.K., “Lagrange-type formulation for finite element analysis of nonlinear beam vibrations”, J. Sound Vib, Vol. 86, pp. 61–70, 1983.

[2] Shi, Y., Mei, C.A., “Finite element time domain model formulation for large amplitude free vibrations of beams and plates”, J. Sound Vib, Vol. 193, pp. 453–465, 1996.

[3] Azrar, L., Benamar, R., White, R.G., “A semi-analytical approach to the non-linear dynamic response problem of S-S and C-C beams at large vibration amplitudes part I: general theory and application to the single mode approach to free and forced vibration analysis”, J. Sound Vib, Vol. 224, pp. 183–207, 1999.

[4] Qaisi, M.I., “Application of the harmonic balance principle to the nonlinear free vibration of beams”, Appl. Acoust, Vol. 40, pp. 141–151, 1993.

[5] Mashinchi Joubari, M., Asghari, R., “Analytical Solution for Nonlinear Vibration of Micro-Electro-Mechanical System (MEMS) by Frequency-Amplitude Formulation Method”, Journal of Mathematics and Computer Science, Vol. 4, No. 3, pp. 371 – 379, 2012.

[6] Baghani, M., Fattahi, M., Amjadian, A., “Application of the variational iteration method for nonlinear free vibration of conservative oscillators”, Scientia Iranica B, Vol. 19, No. 3, pp. 513-518, 2012.

[7] Rafei, M., Ganji, D.D., Daniali, H., Pashaei, H., “The variational iteration method for nonlinear oscillators with discontinuities”, Journal of Sound and Vibration, Vol. 305, pp. 614-620, 2007.

[8] He, J.H., “Some Asymptotic Methods for Strongly Nonlinear Equations”, International Journal of Modern Physics B, Vol. 20, No. 10, pp. 1141-1199, 2006.

[9] He, J.H., “Iteration perturbation method for strongly nonlinear oscillations”, Journal of Vibration and Control, Vol. 7, pp. 631-642, 2001.

[10] Özis, Turgut., Yıldırım, Ahmet., “Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method”, Nonlinear Analysis: Real World Applications, Vol. 10, pp. 1984–1989, 2009.

[11] Younesian, Davood., Yazdi, Mohammad Kalami., Askari, Hassan., Saadatnia, Zia., “Frequency Analysis of Higher-order Duffing Oscillator using Homotopy and Iteration-Perturbation Techniques”, 18th Annual International Conference on Mechanical Engineering-ISME, 2010.

[12] Ganji, D.D., “The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer” Physics Letters A, Vol. 355, pp. 337-341, 2006.

[13] He, J.H., “The homotopy perturbation method for non-linear oscillators with discontinuities”, Appl Math Comput, Vol. 151, pp. 287–92, 2004.

[14] Barari, A., Kaliji, H.D., Ghadimi, M., Domairry, G., “Non-linear vibration of Euler-Bernoulli beams”, Latin American Journal of Solids and Structures, Vol. 8, pp. 139 – 148, 2011.

[15] Ganji, S.S., Barari, A., Ganji, D.D., “Approximate analyses of two mass-spring systems and buckling of a column”, Computers & Mathematics with Applications, Vol. 61, No. 4, pp. 1088–1095, 2011.

[16] Azami, R., Ganji, D.D., Babazadeh, H., “He’s Max-Min method for the relativistic oscillator and high order duffing equation”, International journal of modern physics B, Vol. 32, pp. 5915-5927, 2009.

[17] He, J.H., “Max-Min Approach to Nonlinear Oscillators”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 9, No. 2, pp. 207-210, 2008.

[18] Wu, B.S., Sun, W.P., Lim, C.W., “An analytical approximate technique for a class of strongly non-linear oscillators”, International Journal of Non-Linear Mechanics, Vol. 41, pp. 766-774, 2006.

[19] Mashinchi Joubari, M., Asghari, R., Zareian Jahromy, M., “Investigation of the Dynamic Behavior of Periodic Systems with Newton Harmonic Balance Method”, Journal of Mathematics and Computer Science, Vol. 4, No. 3, pp. 418 – 427, 2012.

[20] Joneidi, A.A., Ganji, D.D., Babaelahi, M., “Differential Transformation Method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity”, International Communications in Heat and Mass Transfer, Vol. 36, pp. 757–762, 2009.

[21] Momeni, M., Jamshidi, N., Barari, A., Ganji, D.D., “Application of He’s Energy Balance Method to Duffing Harmonic Oscillators”, International Journal of Computer Mathematics, Vol. 88, No. 1, pp. 135–144, 2010.

[22] Mehdipour, I., Ganji, D.D., Mozaffari, M., “Application of the energy balance method to nonlinear vibrating equations”, Current Applied Physics, Vol. 10, pp. 104–112, 2010.

[23] He, J.H., “Asymptotic Methods for Solitary Solutions and Compactons”, Abstract and Applied Analysis, Article ID 916793, 2012.

[24] Goorman, D.J., “Free vibrations of beams and shafts”, ASME Appl. Mech, Vol. 18, pp. 135–139, 1975.

[25] Evensen, D.A., “nonlinear vibration of beams with various boundary conditions”, AIAAJ, Vol. 6, pp. 370-372, 1968.

[26] Pillai, S.R.R., Rao, B.N., “On nonlinear free vibrations of simply supported uniform beams”, J. Sound Vib, Vol. 159, pp. 527–531, 1992.

[27] Barari, A., Kimiaeifar, A., Domairry, G., Moghimi, M., “Analytical evaluation of beam deformation problem using approximate methods”, Songklanakarin Journal of Science and Technology, Vol. 32, No. 3, pp. 207–326, 2010.

[28] Klein, L., “Transverse vibrations of non-uniform beam”, J. Sound Vib, Vol. 37, pp. 491–505, 1974.

[29] Sato, K., “Transverse vibrations of linearly tapered beams with ends restrained elastically against rotation subjected to axial force”, Int. J. Mech. Sci, Vol. 22, pp. 109–115, 1980.

[30] Marinca, V., Herisanu, N., “A modified iteration perturbation method for some nonlinear oscillation problems”, Acta Mechanica, Vol. 184, pp. 231-242, 2006.

[31] Lim, C.W., Wu, B.S., “A modified Mickens procedure for certain non-linear oscillators”, J. Sound Vib, Vol. 257, pp. 202-206, 2002.

[32] Shahidi, M., Bayat, M., Pakar, I., Abdollahzadeh, G.R., “Solution of free non-linear vibration of beams”, International Journal of Physical Sciences, Vol. 6, No. 7, pp. 1628-1634, 2011.

[33] Bayat, M., Pakar, I., M., Bayat, “Analytical study on the vibration frequencies of tapered beams”, Latin American Journal of Solids and Structures, Vol. 8, pp. 149–162, 2011.