Dynamic modeling and stability analysis of a nonlinear system with primary resonance
DOI:
https://doi.org/10.31686/ijier.vol8.iss3.2245Keywords:
nonlinear mechanical systems, primary resonances, multiple scales method, Lyapunov exponent, Poincaré map, stability analysis and chaosAbstract
In recent years, there has been growing interest in the study of nonlinear phenomena. This is due to the modernization of structures related to the need of using lighter, more resistant and flexible materials. Thus, this work aims to study the behavior of a mechanical system with two degrees of freedom with nonlinear characteristics in primary resonance. The structure consists of the main system connected to a secondary system to act as a Nonlinear Dynamic Vibration Absorber, which partially or fully absorbs the vibrational energy of the system. The numerical solutions of the problem are obtained using the Runge-Kutta methods of the 4th order and approximate analytical solutions are obtained using the Multiple Scales Method. Then, the approximation error between the two solutions is analyzed.
Using the aforementioned perturbation method, the responses for the ordinary differential equations of the first order can be determined, which describe the modulation amplitudes and phases. Thus, the solution in steady state and the stability are studied using the frequency response. Furthermore, the behavior of the main system and the absorber are investigated through numerical simulations, such as responses in the time domain, phase planes and Poincaré map; which shows that the system displays periodic, quasi-periodic and chaotic movements. The dynamic behavior of the system is analyzed using the Lyapunov exponent and the bifurcation diagram is presented to better summarize all the possible behaviors as the force amplitude varies. In general, the main characteristics of a dynamic system that experiences the chaotic response will be identified.
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[2] Rade, D. A. and Steffen, V. Jr.: Dynamic Vibration Absorber. Encyclopedia of Vibration, Academic Press, ISBN 0-12-227085-1, 9-26 (2011).
[3] Koronev, B. G., Reznikov, L. M.: Dynamic Vibration Absorbers: Theory and Technical Applications. John Wiley and Sons Ltd., Chichester, UK, (1993).
[4] Borges, R. A., de Lima A.M.G., Steffen Jr, V.: Robust optimal design of a nonlinear dynamic vibration absorber combining sensitivity analysis, Shock and Vibration 17, 507-520, (2010).
[5] Awrejcewicz, J., Krysko, A. V., Zagniboroda, N. A., Dobriyan, V. V., Krysko, V. A.: On the general theory of chaotic dynamics of flexible curvilinear Euler–Bernoulli beams, Nonlinear Dynamics 79,11 – 79, (2015).
[6] Thiery, F., Aidanpää, J. O.: Nonlinear vibrations of a misaligned bladed Jeffcott rotor, Nonlinear Dynamics 86, 1807 – 1821, (2016).
[7] Thomsen, J. J. Vibration and Stability Advanced Theory, Analysis, and Tools. Springer-Verlag, 2nd Edition, (2003).
[8] Awrejcewicz, J., Krysko, V. A.: Chaos in Structural Mechanics, Springer-Verlag Berlin Heidelberg, (2008).
[9] Vadasz, P., Equivalent initial conditions for compatibility between analytical and computational solutions of convection in porous media, International Journal of Non-Linear Mechanics 36, 197-208 (2001).
[10] Nayfeh, A, H.: Perturbation Methods. John Wiley and Sons, New York, (2004).
[11] Rabelo M. Silva, L., Borges, R.A., Gonçalves, R. Henrique, M., Computational and Numerical Analysis of a Nonlinear Mechanical System with Bounded Delay, International Journal of Non-Linear Mechanics 91, (2018), 36-57.
[12] Lee, K. H., Han, H. S., Park, S., Bifurcation analysis of coupled lateral/torsional vibrations of rotor systems, Journal Sound and Vibration 386, (2017), 372 – 389.
[13] Srinil N and Zanganeh H 2012 Modelling of coupled cross-flow/in-line vortex-induced vibrations using double Duffing and van der Pol oscillators, Ocean Engineering., 53, 83-97
[14] Nayfeh, A. H., Balachandran, Experimental Investigation of Resonantly Forced Oscillations of a Two-Degree-of-Freedom Structure, International Journal of Non-Linear Mechanics 25, 199-209, (1990).
[15] Norouzi H., Younesian D., Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads, 69, 121-128, (2015).
[16] (zhang) Peng, Z. K., Meng, G., Lang, Z. Q., Zhang, W. M., Chu, F. L., Study of the efects of a cubic nonlinear damping on vibrations isolations using harmonic balance method. International Journal of Non-Linear Mechanics, 47, 1073-1080, (2012).
[17] Zhua, Zhengb and Fu (2004) - - ZHUA; ZHENGB and FU: Analysis of Non-Linear Dynamics of a Two-Degree-of-Freedom Vibration System with Non-Linear Damping and Non-Linear Spring. Journal of Sound and Vibration, Vol. 271, n. 1-2, 15 – 24 (2004).
[18] J.J. Lou, Q.W. He, S.J. Zhu, Chaos in the softening Duffing system under multi-frequency periodic forces, Applied Math. Mech. 25 (12) (2004), 1421–1427.
[19] M.H. Ghayesh, Stability characteristics of an axially accelerating string supported by an elastic foundation, Mech. Mach. Theory 44 (10) (2009) 1964–1979.
[20] Kevorkian, J. K., Cole, J. D., Multiple Scale and Singular Perturbation Methods, Springer, (1996).
[21] Burden, R. L., Faires, J. D., Numerical Analysis, Brooks/Cole, Cengage Learning, Ninth Edition, (2011).
[22] Borges, R. A.; Lobato, F S. ; Steffen, V. . Application of Three Bioinspired Optimization Methods for the Design of a Nonlinear Mechanical System. Mathematical Problems in Engineering (Print), v. 2013, p. 1-12, 2013.
[23] Nayfeh, A., Mook, D., Marshall, L., Non-linear Coupling of Pitch and Roll Modes in Ship Motions, J. Hydronautic 7,145-152, (1973).
[24] Kahn, P.B.,Mathematical Methods for Scientists and Engineers: Linear and Nonlinear Systems, Jphn Wiley & Sons, New York, (2004).
[25] Nayfeh, A.H., Balachandran, B., Applied Nonlinear Dynamics: Analytical, Computation and Experimental Methods. John Wiley and Sons, N.Y., (1995).
[26] Batista, M., On stability of elastic rod planar equilibrium configurations, International Jornal of Solids and Structures 72,144-152, (2015).
[27] Saberi, L., Nahvi, H., Vibration Analysis of a Nonlinear System with a Nonlinear Absorber under the Primary and Super-harmonic Resonances, International Journal of Engineering, vol. 27, no. 3, 499-508, (2014).
[28] Elnaggar, A.M.; and Khalil, K. M.: The Response of Nonlinear Controlled System under an External Excitation via Time Delay State Feedback. Journal of King Saud University-Engineering Sciences. Online publication, (2014).
[29] Chen G., Hill D. J., Yu X., Bifurcation Control – Theory and Applications, Springer, (2003).
[30] Feng, Z. C., Sethna P. R., Global bifurcation and chaos in parametrically forced system with one-one resonance, Dynamics and Stability of Systems, 5, 201-225, (1990).
[31] Yang, W. Y., CAO, W.; CHUNG, T. S. and MORRIS, J. (2005): Applied Numerical Methods Using Matlab®. John Wiley and Sons, Inc., Hoboken NJ.
[32] Lynch, S. Dynamical Systems with Applications using MATLAB. Manchester: Birkhäuser Science, 2014.
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