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2020 | 140 | 26-58
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On the Analysis of Jump and Bifurcation Phenomena in a Forced Vibration of Geometrical Nonlinear Cantilever Beam: Application of Differential Transformation Method

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EN
One of the classical features exhibited in nonlinear dynamics of engineering systems is the jump phenomenon, which is the discontinuous change in the steady state response of a system as a parameter is slowly varied. Such phenomenon is characterized by large amplitude dynamic responses of systems to small amplitude disturbances. It is established that this phenomenon cannot be described by the standard asymptotic and perturbation methods because they are limited to the study of small amplitude responses to small disturbances. Therefore, this paper presents the application of differential transformation method-Padé approximant to the solution of jump and bifurcation phenomena for a geometrical nonlinear cantilever beam subjected to a harmonic excitation. The accuracy and validity of the analytical solutions obtained by the differential transformation method are shown through a comparison of the results of the analytical solution with the corresponding results of the numerical solution obtained by fourth-order Runge-Kutta method and also with the results in a past study using harmonic balancing method. With the aid of the differential transformation method-Padé approximant, the effects of the nonlinear parameters in the model equation on the dynamic response of the beam are investigated. Also, the sensitivity of the beam to the external excitation amplitude is analyzed. In the distributed forced vibration, the jump phenomenon appeared in the response amplitude by variation of the excitation frequency while in the resonance frequency, the beat phenomenon with harmonic motion is seen for low level of excitation amplitude. At a certain frequency, the jump and bifurcation phenomena are seen in the curves of responses versus excitation amplitude. Additionally, the plots of the phase plane and time history of the system response are shown. It is established that the differential transformation method is a very useful mathematical tool for dealing with the nonlinear problems.
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Year
Volume
140
Pages
26-58
Physical description
Contributors
  • Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria
author
  • Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria
author
  • Department of Biomedical Engineering, University of Lagos, Akoka, Lagos, Nigeria
author
  • Department of Civil and Environmental Engineering, University of Lagos, Akoka, Lagos, Nigeria
author
  • Department of Civil and Environmental Engineering, University of Lagos, Akoka, Lagos, Nigeria
References
  • [1] J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell, New York, 1968.
  • [2] Joshi N, Lustri CJ. Stokes phenomena in discrete Painlevé I. Proc Math Phys Eng Sci. 2015 May 8; 471(2177): 20140874. https://doi.org/10.1098/rspa.2014.0874
  • [3] R. Lyon, M. Heck, and C. B. Hazelgrove. Response of Hard-Spring oscillator to Narrow-Band Excitation. J. Acoust. Soc. Am. 33(1985), 1404-1411 (1961).
  • [4] [4] K. Sato, S, Yamamoto, O. Kamadaa and N. Takatsu. Jump phenomena Gear System to Random Excitation. Bull. JSME 28 (1985), 1271-1278.
  • [5] J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell, New York, 1968, 29-38.
  • [6] W. B. Bush. The hypersonic approximation for the shock structure of a perfect gas with the Sutherland viscosity law. J. De Macanique, 1 (1962) pp. 27-33.
  • [7] W. B. Bush and F. Fendell, Asymptotic analysis of laminar flame propagation for general Lewis numbers. Combustion Science Technology, 1 (1970) 421-428.
  • [8] D. R. Kassoy. Perturbation methods for mathematical models of explosion phenomena. Quart. J. Mech. Appl. Math. 28 (1975) 63-74.
  • [9] Richard Haberman. Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems. SIAM J. Appl. Math., 37(1), 69–106. https://doi.org/10.1137/0137006
  • [10] A. K. Kapila. Arrhenius Systems: Dynamics of Jump Due to Slow Passage Through Criticality. SIAM J. Appl. Math., 41(1), 29–42. https://doi.org/10.1137/0141004
  • [11] Do, D., & Weiland, R. (1982). Dynamics of rapid extinction in a lumped system with Arrhenius chemistry. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 23(3), 278-290. doi:10.1017/S0334270000000230
  • [12] Kapila, A. K., Arrhenius systems: dynamics of jump due to slow passage through criticality. SIAM J. Appl. Math. 41 (1981), 29–42.
  • [13] D. R. Kassoy. A note on Asymptotic Methods for Jump Phenomena. SIAM J. Appl. Math. 42(1982)(4), 1982
  • [14] D. Schaeffer and M. Golubitsky, Boundary conditions and mode jumping in the buckling of a rectangular plate. Commun. Math. Phys. 69 (1979) 209-236.
  • [15] H. Suchy. H. Troger and R. Weiss. A numerical study of mode jumping of rectangular plates. ZAMM Z. Angew. Math. u. Mech. 65(Z) (1985) 71-78
  • [16] W. Jia and T. Fang. Jump phenomena in coupled Duffing oscillators under random excitation. J. Acoust. Soc. Am. 81 (4) (1987) 961-965.
  • [17] P. B. Gomcjalves. Jump phenomena, bifurcations, and chaos in a pressure loaded spherical cap under harmonic excitation. ASME Appl Mech Rev. 46 (1993) (11) (2) 280-288.
  • [18] M. S. Soliman. Jump Phenomena Resulting in Unpredictable Dynamics in the Driven Damped Pendulum. Int. J Non-Lmear Mechnnm 31 (1996) (2) 167-174.
  • [19] E. Riks, C. C. Rankin, F. A. Brogan. On the solution of mode jumping phenomena in thin-walled shell structures. Comput. Methods Appl. Mech. Engrg. 136 (1996) 59-92
  • [20] M. Brennan, I. Kovacic, A. Carrella, T. Waters. On the jump up and jump-down frequencies of the Duffing oscillator. J Sound Vib 318 (2008) (4): 1250–1261
  • [21] D. Banerjee and J. K. Bhattacharjee Analyzing jump phenomena and stability in nonlinear oscillators using renormalization group arguments. American Journal of Physics 78 (2010) 142.
  • [22] J. Wang and T. C. Lim. On the Boundary between Nonlinear Jump Phenomenon and Linear Response of Hypoid Gear Dynamics. Advances in Acoustics and Vibration Volume 2011, Article ID 583678, 13 pages
  • [23] A. Allahverdizadeh R. Oftadeh M. J. Mahjoob A. Soleimani H. Tavassoli Analyzing the effects of jump phenomenon in nonlinear vibration of thin circular functionally graded plates. Arch Appl Mech 82 (2012) 907–918
  • [24] Walter V. Wedig. Velocity Jumps in Road-Vehicle Dynamics. Procedia Engineering Volume 144, 2016, Pages 1076-1085. https://doi.org/10.1016/j.proeng.2016.05.064
  • [25] A. Motallebi S. Irani S. Sazesh. Analysis on jump and bifurcation phenomena in the forced vibration of nonlinear cantilever beam using HBM. Journal of the Brazilian Society of Mechanical Sciences and Engineering 38(2) (2016) 515-524
  • [26] Y. Yu and Q. Wang and C. W. Lim. Multiple-S-Shaped Critical Manifold and Jump Phenomena in Low Frequency Forced Vibration with Amplitude Modulation. International Journal of Bifurcation and Chaos, Vol. 29, No. 5 (2019) (13 pages)
  • [27] J.K. Zhou, Differential Transformation and Its Applications for Electrical Circuits. Huarjung University Press, Wuuhahn, China, 1986, (in Chinese).
  • [28] M.G. Sobamowo. Singular perturbation and differential transform methods to two-dimensional flow of nanofluid in a porous channel with expanding/contracting walls subjected to a uniform transverse magnetic field. Thermal Science and Engineering Progress 4 (2017) 71–84.
  • [29] S. Ghafoori, M. Motevalli, M.G. Nejad, F. Shakeri, D.D. Ganji, M. Jalaal. Efficiency of differential transformation method for nonlinear oscillation: Comparison with HPM and VIM. Current Applied Physics 11 (2011) 965-971
  • [30] H. N. Arafat. Nonlinear response of cantilever beams. Ph.D. thesis, Virginia Polytechnic Institute and State University. Blacksburg. Virginia, 1999.
  • [31] M. C. da Silva, C. Glynn. Nonlinear flexural-flexural- torsional dynamics of inextensional beams I Equations of motion. J Struct Mechn 6 (1978) (4): 437–448
  • [32] M.G. Sobamowo. Nonlinear thermal and flow-induced vibration analysis of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations. Thermal Science and Engineering Progress 4 (2017) 133–149
  • [33] A. Barari, H. Kaliji, M. Ghadimi, G. Domairry. Non-linear vibration of Euler-Bernoulli beams. Lat Amour Solids Struct 8 (2011) (2): 139–148
  • [34] M. Belhaq, A. Bichri, J. Der Hogapian, J. Mahfoud. Effect of electromagnetic actuations on the dynamics of a harmonically excited cantilever beam. Int J Non-Linear Mech 46 (2011) (6): 828–833
  • [35] S.B. Shiki, V. Lopes Jr, S. da Silva. Identification of nonlinear structures using discrete-time Volterra series. J Braz Soc Mech Eng 36 (2014) (3):523–532
  • [36] M. Yaman. Direct and parametric excitation of a nonlinear cantilever beam of varying orientation with time-delay state feedback. J Sound Vib 324 (2009) (3): 892–902
  • [37] W. Zhang. Chaotic motion and its control for nonlinear nonplanar oscillations of a parametrically excited cantilever beam. Chaos Solitons Fractals 26 (2005) (3): 731–745.
  • [38] G.A. Baker, P. Graves-Morris, Pade Approximants, Cambridge U.P., 1996
Document Type
article
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YADDA identifier
bwmeta1.element.psjd-a4689f40-2ae8-4eb6-8ed4-4a617a7aa79c
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