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http://dx.doi.org/10.12989/sem.2012.43.3.337

Accurate analytical solution for nonlinear free vibration of beams  

Bayat, M. (Department of Civil Engineering, Mashhad Branch, Islamic Azad University)
Pakar, I. (Department of Civil Engineering, Mashhad Branch, Islamic Azad University)
Publication Information
Structural Engineering and Mechanics / v.43, no.3, 2012 , pp. 337-347 More about this Journal
Abstract
In this study, Hamiltonian Approach (HA) is applied to analysis the nonlinear free vibration of beams. Two well-known examples are illustrated to show the efficiency of this method. One of them deals with the Nonlinear vibration of an electrostatically actuated microbeam and the other is the nonlinear vibrations of tapered beams. This new approach prepares us to achieve the beam's natural frequencies and mode shapes easily and a rapidly convergent sequence is obtained during the solution. The effects of the small parameters on the frequency of the beams are discussed. Some comparisons are conducted between the results obtained by the Hamiltonian Approach (HA) and numerical solutions using to illustrate the effectiveness and convenience of the proposed methods.
Keywords
non-linear vibration; hamiltonian approach; electrostatically actuated microbeam; tapered beams;
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Times Cited By KSCI : 3  (Citation Analysis)
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1 Liu, J.F. (2009), "He's variational approach for nonlinear oscillators with high nonlinearity", Comput. Math. Appl., 58(11-12), 2423-2426.   DOI   ScienceOn
2 Bayat, M., Pakar, I. and Bayat, M. (2011), "Analytical study on the vibration frequencies of tapered beams", Latin Am. J. Solids Struct., 8(2), 149-162.   DOI
3 Bayat, M., Shahidi, M., Barari, A. and Domairry, G. (2011a), "Analytical evaluation of the nonlinear vibration of coupled oscillator systems", Zeitschrift fur Naturforschung Section A-A J. Phys. Sci., 66(1-2), 67-74.
4 Bayat, M., Barari, A. and Shahidi, M. (2011b), "Dynamic response of axially loaded euler-bernoulli beams", Mechanika, 17(2), 172-177.
5 Bayat, M. and Pakar, I. (2011c), "Application of He's energy balance method for nonlinear vibration of thin circular sector cylinder", Int. J. Phy. Sci., 6(23), 5564-5570.
6 Bayat, M., Bayat, M. and Bayat, M. (2011d), "An analytical approach on a mass grounded by linear and nonlinear springs in series", Int. J. Phy. Sci., 6(2), 229-236.
7 Bayat, M., Shahidi, M. and Bayat, M. (2011e), "Application of iteration perturbation method for nonlinear oscillators with discontinuities", Int. J. Phy. Sci., 6(15), 3608-3612.
8 Bayat, M., Pakar, I. and Shahidi, M. (2011f), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
9 Bayat, M. and Abdollahzadeh, G.R. (2011g), "Analysis of the steel braced frames equipped with ADAS devices under the far field records", Latin Am. J. Solids Struct., 8(2), 163-181.   DOI
10 Bayat, M. and Abdollahzadeh, G.R. (2011h), "On the effect of the near field records on the steel braced frames equipped with energy dissipating devices", Latin Am. J. Solids Struct., 8(4), 429-443.   DOI
11 Burgreen, D. (1951), "Free vibrations of a pin-ended column with constant distance between pin ends", J. Appl. Mech., ASME, 18, 135-139.
12 Dumir, P. and Bhaskar, A. (1988), "Some erroneous finite element formulations of non-linear vibrations of beams and plates", J. Sound Vib., 123(3), 517-527.   DOI   ScienceOn
13 Fu, Y.M., Zhang, J. and Wan, L.J. (2011), "Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)", Curr Appl. Phys., 11, 482-485.   DOI   ScienceOn
14 Ganji, D.D., Kachapi, H. and Seyed, H. (2011), "Analytical and numerical methods in engineering and applied sciences", Prog. Nonlin. Sci., 3, 1-579.
15 Ganji, D.D., Bararnia, H., Soleimani, S. and Ghasemi, E. (2009), "Analytical solution of magneto-hydrodinamic flow over a nonlinear stretching sheet", Modern Phys. Lett. B, 23, 2541-2556.   DOI   ScienceOn
16 Ghasemi, E., Bayat, M. and Bayat, M. (2011), "Visco-elastic MHD flow of walters liquid B fluid and heat transfer over a non-isothermal stretching sheet", Int. J. Phy. Sci., 6(21), 5022-5039.
17 Goorman, D.J. (1975), "Free vibrations of beams and shafts", Appl. Mech., ASME, 18, 135-139.
18 He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillations", Mech. Res. Comm., 29, 107-111.   DOI   ScienceOn
19 He, J.H. (2010), "Hamiltonian approach to nonlinear oscillators", Phys. Lett. A., 374, 2312-2314.   DOI   ScienceOn
20 He, J.H., Zhong, T. and Tang, L. (2010), "Hamiltonian Approach to duffing-harmonic equation", Int. J. Nonlin. Sci. Numer., 11(S), 43-46.
21 Hoseini, S.H., Pirbodaghi, T., Ahmadian, M.T. and Farrahi, G.H. (2009), "On the large amplitude free vibrations of tapered beams: an analytical approach", Mech. Res. Comm., 36, 892-897.   DOI   ScienceOn
22 Jamshidi, N. and Ganji, D.D. (2010), "Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire", Current Appl. Phys., 10(2), 484-486.   DOI   ScienceOn
23 Lou, C. and Sikarskie, D. (1975), "Nonlinear vibration of beams using a form-function approximation", J. Appl. Mech., 42(1), 209-214.   DOI
24 Mehdipour, I., Ganji, D.D. and Mozaffari, M. (2010), "Application of the energy balance method to nonlinear vibrating equations", Current Appl. Phys., 10(1), 104-112.   DOI   ScienceOn
25 Oztur, B. and Coskun, S.B. (2011), "The Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation", Struct. Eng. Mech., 37(4), 415-425.   DOI
26 Pakar, I., Bayat, M. and Bayat, M. (2011a), "Analytical evaluation of the nonlinear vibration of a solid circular sector object", Int. J. Phy. Sci., 6(30), 6861-6866.
27 Pakar, I. and Bayat, M. (2011b), "Analytical solution for strongly nonlinear oscillation systems using energy balance method", Int. J. Phy. Sci., 6(22), 5166-5170.
28 Pakar, I. and Bayat, M. (2012), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. Vibroengineering, 14(1), 216-224.
29 Pakar, I., Bayat, M. and Bayat, M. (2012), "On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroengineering, 14(1), 423-429.
30 Prathap, G. and Varadan, T. (1978), "The large amplitude vibration of tapered clamped beams", J. Sound Vib., 58(1), 87-94.   DOI   ScienceOn
31 Sathyamoorthy, M. (1982), "Nonlinear analysis of beams, Part-I: A survey of recent advances", Shock Vib. Dig, 14, 19-35.
32 Xu, L. and He, J.H. (2010), "Determination of limit cycle by hamiltonian approach for strongly nonlinear oscillators", Int. J. Nonlin. Sci., 11(12), 1097-1101.
33 Shahidi, M., Bayat, M., Pakar, I. and Abdollahzadeh, G.R. (2011), "On the solution of free non-linear vibration of beams", Int. J. Phy. Sci., 6(7), 1628-1634.
34 Singh, G., Sharma, A. and Venkateswara Rao, G. (1990), "Large-amplitude free vibrations of beams--A discussion on various formulations and assumptions", J. Sound Vib., 142(1), 77-85.   DOI   ScienceOn