Journal of Applied and Computational Mechanics
Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term
Document Type : Research Paper
Authors
1 Shahid Chamran University, Faculty of Engineering, Mechanical Engineering Department, Ahvaz, Iran
2 Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6
Abstract
This paper presents the high order frequency-amplitude relationship for nonlinear transversely vibrating beams with odd and even nonlinearities, using Homotopy Perturbation Method with an auxiliary term (HPMAT). The governing equations of vibrating buckled beam, beam carrying an intermediate lumped mass, and quintic nonlinear beam are investigated to exhibit the reliability and ability of the proposed asymptotic approach. It is demonstrated that two terms in series expansions are sufficient to obtain a highly accurate periodic solutions. The integrity of the analytical solutions is verified by numerical results.
Keywords
- Homotopy Perturbation Method with an auxiliary term
- Non-linear vibrating beams
- Frequency-amplitude relationship
Main Subjects
[1] Love, A.E.H.; 1927. A treatise on the Mathematical theory of Elasticity. New York: Dover Publications, Inc.
[2] Sedighi, H.M., Shirazi, K.H., and Noghrehabadi, A. Application of Recent Powerful Analytical Approaches on the Non-Linear Vibration of Cantilever Beams, Int. J. Nonlinear Sci. Numer. Simul., 2012; 13(7–8): 487-494, DOI: 10.1515/ijnsns-2012-0030.
[3] Barari, A.; Kaliji, H.D.; Ghadami, M.; Domairry, G.; 2011. Non-Linear Vibration of Euler-Bernoulli Beams. Latin American Journal of Solids and Structures. 8: 139-148.
[4] Sedighi, H.M.; Reza, A.; Zare, J.; 2011. Dynamic analysis of preload nonlinearity in nonlinear beam vibration, Journal of Vibroengineering. 13: 778-787.
[5] Sedighi, H.M.; Reza, A.; Zare, J.; 2011. Study on the frequency – amplitude relation of beam vibration, International Journal of the Physical Sciences. 6(36): 8051-8056.
[6] Sedighi, H.M.; Shirazi, K.H.; 2012. A new approach to analytical solution of cantilever beam vibration with nonlinear boundary condition, ASME Journal of Computational and Nonlinear Dynamics. 7: 034502. DOI:10.1115/1.4005924.
[7] Sedighi, H.M.; Shirazi, K.H.; Noghrehabadi, A.R.; Yildirim, A.; 2012. Asymptotic Investigation of Buckled Beam Nonlinear Vibration. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 36(M2): 107-116.
[8] Sedighi, H.M.; Shirazi, K.H.; Zare, J.; 2012. Novel Equivalent Function for Deadzone Nonlinearity: Applied to Analytical Solution of Beam Vibration Using He’s Parameter Expanding Method. Latin American Journal of Solids and Structures, 9(4), 443-451.
[9] Sedighi, H.M.; Shirazi, K.H.; Reza, A.; Zare, J.; 2012. Accurate modeling of preload discontinuity in the analytical approach of the nonlinear free vibration of beams. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 226(10), 2474–2484, DOI: 10.1177/0954406211435196.
[10] Nikkhah Bahrami, M.; Khoshbayani Arani, M.; Rasekh Saleh, N.; 2011. Modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform beams. Scientia Iranica B, 18(5):1088–1094.
[11] Arvin, H.; Bakhtiari-Nejad, F.; 2011. Non-linear modal analysis of a rotating beam. International Journal of Non-Linear Mechanics, 46: 877–897.
[12] Zohoor, H.; Kakavand, F.; Vibration of Euler–Bernoulli and Timoshenko beams in large overall motion on flying support using finite element method, Scientia Iranica B, in press, doi:10.1016/j.scient.2012.06.019.
[13] Freno, B.A.: Cizmas, P.G.A.; 2011. A computationally efficient non-linear beam model. International Journal of Non-Linear Mechanics, 46: 854-869.
[14] Awrejcewicz, J.; Krysko, A.V.; Soldatov, V.; Krysko, V.A.; 2012. Analysis of the Nonlinear Dynamics of the Timoshenko Flexible Beams Using Wavelets. ASME Journal of Computational and Nonlinear Dynamics, 7(1): 011005.
[15] Andreaus, U.; Placidi, L.; Rega, G.; 2011. Soft impact dynamics of a cantilever beam: equivalent SDOF model versus infinite-dimensional system. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(10): 2444-2456. doi: 10.1177/0954406211414484.
[16] Chen, J.S.; Chen, Y.K.; 2011. Steady state and stability of a beam on a damped tensionless foundation under a moving load. International Journal of Non-Linear Mechanics, 46: 180–185.
[17] Sapountzakis, E.J.; Dikaros, I.C.; 2011. Non-linear flexural-torsional dynamic analysis of beams of arbitrary cross section by BEM. International Journal of Non-Linear Mechanics, 46: 782–794.
[18] Jang, T.S.; Baek, H.S.; Paik, J.K.; 2011. A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation. International Journal of Non-Linear Mechanics, 46: 339–346.
[19] Campanile, L.F.; Jähne, R.; Hasse, H.; 2011. Exact analysis of the bending of wide beams by a modified elastica approach, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(11): 2759-2764. Doi: 10.1177/0954406211417753.
[20] He, J.H., 2008. Max-Min Approach to Nonlinear Oscillators, International Journal of Nonlinear Sciences and Numerical Simulation, 9(2), 207-210.
[21] Liao, S.J. 2004. An analytic approximate approach for free oscillations of self-excited systems, International Journal of Non-Linear Mechanics, 39, 271-280.
[22] Sedighi, H.M.; Shirazi, K.H.; Zare, J.; 2012. An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method. International Journal of Non-Linear Mechanics, 47: 777- 784, DOI: 10.1016/j.ijnonlinmec.2012.04.008.
[23] Ghaffarzadeh, H; Nikkar, A; 2013. Explicit solution to the large deformation of a cantilever beam under point load at the free tip using the Variational Iteration Method-II, Journal of Mechanical Science and Technology, 27(11) 3433-3438.
[24] Bagheri, S.; Nikkar, A; Ghaffarzadeh, H; 2014. Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques, Latin American Journal of Solids and Structures, 11: 157-168.
[25] Shadloo, M.S.; Kimiaeifar, A.; 2011. Application of homotopy perturbation method to find an analytical solution for magneto hydrodynamic flows of viscoelastic fluids in converging/diverging channels. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225: 347-353.
[26] Soroush, R.; Koochi, A.; Kazemi, A.S.; Noghrehabadi, A.; Haddadpour, H.; Abadyan, M.; 2010. Investigating the effect of Casimir and van der Waals attractions on the electrostatic pull-in instability of nano-actuators, Phys. Scr., 82: 045801. doi:10.1088/0031-8949/82/04/045801.
[27] Bayat, M.; Shahidi, M.; Barari, A.; Domairry, G.; 2011. Analytical evaluation of the nonlinear vibration of coupled oscillator systems. Zeitschrift fur Naturforschung A-A Journal of Physical Sciences, 66(1-2): 67–74.
[28] He, J.H., 2002. Preliminary report on the energy balance for nonlinear oscillations. Mech. Res. Commun., 29, 107-111.
[29] Evirgen, F.; Özdemir, N.; 2011. Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System. ASME Journal of Computational and Nonlinear Dynamics, 6(2): 021003. Doi:10.1115/1.4002393.
[30] He, J.H; 2011. A short remark on fractional variational iteration method , PHYSICS LETTERS A, 375(38), 3362-3364, dio: 10.1016/j.physleta.2011.07.033.
[31] Khosrozadeh, A.; Hajabasi, M.A.; Fahham, H.R.; 2013. Analytical Approximations to Conservative Oscillators With Odd Nonlinearity Using the Variational Iteration Method, Journal of Computational and Nonlinear Dynamics, 8, 014502, DOI: 10.1115/1.4006789.
[32] Hasanov, A.; 2011. Some new classes of inverse coefficient problems in non-linear mechanics and computational material science, International Journal of Non-Linear Mechanics, 46(5): 667-684.
[33] He, J.H.; 2010. Hamiltonian approach to nonlinear oscillators, Physics Letters A, 374(23): 2312-2314.
[34] Baferani, A.H.; Saidi, A.R.; Jomehzadeh, E.; 2011. An exact solution for free vibration of thin functionally graded rectangular plates. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(3): 526-536. doi: 10.1243/09544062JMES2171.
[35] Naderi, A.; Saidi, A.R.; 2011. Buckling analysis of functionally graded annular sector plates resting on elastic foundations. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(2): 312-325.
[36] He, J.H.; 2002. Modified Lindstedt-Poincare methods for some strongly non-linear oscillations Part I: expansion of a constant, International Journal of Non-linear Mechanics, 37(2): 309-314. DOI: 10.1016/S0020-7462(00)00116-5.
[37] He, J.H., Homotopy Perturbation Method with an Auxiliary Term, Abstract and Applied Analysis, 2012, Article ID 857612, doi:10.1155/2012/857612.
[38] M.N. Hamden and N.H. Shabaneh, On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass, Journal of Sound and Vibration, 199(5) (1997), 711–736.
[39] M. R. M. Crespo da Silva & C. C. Glynn, Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. I. Equations of Motion, Journal of Structural Mechanics, Volume 6, Issue 4, 1978, DOI: 10.1080/03601217808907348.
[40] Lacarbonara, W. (1997). A theoretical and experimental investigation of nonlinear vibrations of buckled beams. Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg Virginia.
[2] Sedighi, H.M., Shirazi, K.H., and Noghrehabadi, A. Application of Recent Powerful Analytical Approaches on the Non-Linear Vibration of Cantilever Beams, Int. J. Nonlinear Sci. Numer. Simul., 2012; 13(7–8): 487-494, DOI: 10.1515/ijnsns-2012-0030.
[3] Barari, A.; Kaliji, H.D.; Ghadami, M.; Domairry, G.; 2011. Non-Linear Vibration of Euler-Bernoulli Beams. Latin American Journal of Solids and Structures. 8: 139-148.
[4] Sedighi, H.M.; Reza, A.; Zare, J.; 2011. Dynamic analysis of preload nonlinearity in nonlinear beam vibration, Journal of Vibroengineering. 13: 778-787.
[5] Sedighi, H.M.; Reza, A.; Zare, J.; 2011. Study on the frequency – amplitude relation of beam vibration, International Journal of the Physical Sciences. 6(36): 8051-8056.
[6] Sedighi, H.M.; Shirazi, K.H.; 2012. A new approach to analytical solution of cantilever beam vibration with nonlinear boundary condition, ASME Journal of Computational and Nonlinear Dynamics. 7: 034502. DOI:10.1115/1.4005924.
[7] Sedighi, H.M.; Shirazi, K.H.; Noghrehabadi, A.R.; Yildirim, A.; 2012. Asymptotic Investigation of Buckled Beam Nonlinear Vibration. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 36(M2): 107-116.
[8] Sedighi, H.M.; Shirazi, K.H.; Zare, J.; 2012. Novel Equivalent Function for Deadzone Nonlinearity: Applied to Analytical Solution of Beam Vibration Using He’s Parameter Expanding Method. Latin American Journal of Solids and Structures, 9(4), 443-451.
[9] Sedighi, H.M.; Shirazi, K.H.; Reza, A.; Zare, J.; 2012. Accurate modeling of preload discontinuity in the analytical approach of the nonlinear free vibration of beams. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 226(10), 2474–2484, DOI: 10.1177/0954406211435196.
[10] Nikkhah Bahrami, M.; Khoshbayani Arani, M.; Rasekh Saleh, N.; 2011. Modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform beams. Scientia Iranica B, 18(5):1088–1094.
[11] Arvin, H.; Bakhtiari-Nejad, F.; 2011. Non-linear modal analysis of a rotating beam. International Journal of Non-Linear Mechanics, 46: 877–897.
[12] Zohoor, H.; Kakavand, F.; Vibration of Euler–Bernoulli and Timoshenko beams in large overall motion on flying support using finite element method, Scientia Iranica B, in press, doi:10.1016/j.scient.2012.06.019.
[13] Freno, B.A.: Cizmas, P.G.A.; 2011. A computationally efficient non-linear beam model. International Journal of Non-Linear Mechanics, 46: 854-869.
[14] Awrejcewicz, J.; Krysko, A.V.; Soldatov, V.; Krysko, V.A.; 2012. Analysis of the Nonlinear Dynamics of the Timoshenko Flexible Beams Using Wavelets. ASME Journal of Computational and Nonlinear Dynamics, 7(1): 011005.
[15] Andreaus, U.; Placidi, L.; Rega, G.; 2011. Soft impact dynamics of a cantilever beam: equivalent SDOF model versus infinite-dimensional system. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(10): 2444-2456. doi: 10.1177/0954406211414484.
[16] Chen, J.S.; Chen, Y.K.; 2011. Steady state and stability of a beam on a damped tensionless foundation under a moving load. International Journal of Non-Linear Mechanics, 46: 180–185.
[17] Sapountzakis, E.J.; Dikaros, I.C.; 2011. Non-linear flexural-torsional dynamic analysis of beams of arbitrary cross section by BEM. International Journal of Non-Linear Mechanics, 46: 782–794.
[18] Jang, T.S.; Baek, H.S.; Paik, J.K.; 2011. A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation. International Journal of Non-Linear Mechanics, 46: 339–346.
[19] Campanile, L.F.; Jähne, R.; Hasse, H.; 2011. Exact analysis of the bending of wide beams by a modified elastica approach, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(11): 2759-2764. Doi: 10.1177/0954406211417753.
[20] He, J.H., 2008. Max-Min Approach to Nonlinear Oscillators, International Journal of Nonlinear Sciences and Numerical Simulation, 9(2), 207-210.
[21] Liao, S.J. 2004. An analytic approximate approach for free oscillations of self-excited systems, International Journal of Non-Linear Mechanics, 39, 271-280.
[22] Sedighi, H.M.; Shirazi, K.H.; Zare, J.; 2012. An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method. International Journal of Non-Linear Mechanics, 47: 777- 784, DOI: 10.1016/j.ijnonlinmec.2012.04.008.
[23] Ghaffarzadeh, H; Nikkar, A; 2013. Explicit solution to the large deformation of a cantilever beam under point load at the free tip using the Variational Iteration Method-II, Journal of Mechanical Science and Technology, 27(11) 3433-3438.
[24] Bagheri, S.; Nikkar, A; Ghaffarzadeh, H; 2014. Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques, Latin American Journal of Solids and Structures, 11: 157-168.
[25] Shadloo, M.S.; Kimiaeifar, A.; 2011. Application of homotopy perturbation method to find an analytical solution for magneto hydrodynamic flows of viscoelastic fluids in converging/diverging channels. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225: 347-353.
[26] Soroush, R.; Koochi, A.; Kazemi, A.S.; Noghrehabadi, A.; Haddadpour, H.; Abadyan, M.; 2010. Investigating the effect of Casimir and van der Waals attractions on the electrostatic pull-in instability of nano-actuators, Phys. Scr., 82: 045801. doi:10.1088/0031-8949/82/04/045801.
[27] Bayat, M.; Shahidi, M.; Barari, A.; Domairry, G.; 2011. Analytical evaluation of the nonlinear vibration of coupled oscillator systems. Zeitschrift fur Naturforschung A-A Journal of Physical Sciences, 66(1-2): 67–74.
[28] He, J.H., 2002. Preliminary report on the energy balance for nonlinear oscillations. Mech. Res. Commun., 29, 107-111.
[29] Evirgen, F.; Özdemir, N.; 2011. Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System. ASME Journal of Computational and Nonlinear Dynamics, 6(2): 021003. Doi:10.1115/1.4002393.
[30] He, J.H; 2011. A short remark on fractional variational iteration method , PHYSICS LETTERS A, 375(38), 3362-3364, dio: 10.1016/j.physleta.2011.07.033.
[31] Khosrozadeh, A.; Hajabasi, M.A.; Fahham, H.R.; 2013. Analytical Approximations to Conservative Oscillators With Odd Nonlinearity Using the Variational Iteration Method, Journal of Computational and Nonlinear Dynamics, 8, 014502, DOI: 10.1115/1.4006789.
[32] Hasanov, A.; 2011. Some new classes of inverse coefficient problems in non-linear mechanics and computational material science, International Journal of Non-Linear Mechanics, 46(5): 667-684.
[33] He, J.H.; 2010. Hamiltonian approach to nonlinear oscillators, Physics Letters A, 374(23): 2312-2314.
[34] Baferani, A.H.; Saidi, A.R.; Jomehzadeh, E.; 2011. An exact solution for free vibration of thin functionally graded rectangular plates. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(3): 526-536. doi: 10.1243/09544062JMES2171.
[35] Naderi, A.; Saidi, A.R.; 2011. Buckling analysis of functionally graded annular sector plates resting on elastic foundations. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(2): 312-325.
[36] He, J.H.; 2002. Modified Lindstedt-Poincare methods for some strongly non-linear oscillations Part I: expansion of a constant, International Journal of Non-linear Mechanics, 37(2): 309-314. DOI: 10.1016/S0020-7462(00)00116-5.
[37] He, J.H., Homotopy Perturbation Method with an Auxiliary Term, Abstract and Applied Analysis, 2012, Article ID 857612, doi:10.1155/2012/857612.
[38] M.N. Hamden and N.H. Shabaneh, On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass, Journal of Sound and Vibration, 199(5) (1997), 711–736.
[39] M. R. M. Crespo da Silva & C. C. Glynn, Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. I. Equations of Motion, Journal of Structural Mechanics, Volume 6, Issue 4, 1978, DOI: 10.1080/03601217808907348.
[40] Lacarbonara, W. (1997). A theoretical and experimental investigation of nonlinear vibrations of buckled beams. Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg Virginia.
)
