Analysis of Strongly Nonlinear Systems by Using HBM-AFT Method and Its Comparison with the Five-Order Runge–Kutta Method: Application to Duffing Oscillator and Disc Brake Model
Tóm tắt
Non-linear dynamic problems are difficult to analyze since no general methods exist to deal with them, which namely depend on several factors such as the nature of the problem, the type of non-linearity or the type of solution to be sought. While some methods are used to study weak nonlinearities, others can more easily solve strong nonlinearities. Therefore, the aim of the present paper is to find a method to solve the strong nonlinearities and to present the theoretical aspects of the harmonic balance method (HBM) in the form of algorithms to facilitate programming. Indeed, the first objective is to present the two continuation techniques based on the Newton–Raphson algorithm and the Alternating Frequency Time method (AFT). As for the second goal, it pertains to the proposition of a method for calculating the response curve by combining HBM and AFT with arc length continuation to solve the systems with strong non-linearities. For illustration, two applications were investigated, namely a Duffing model and a developed model describing a disc brake with non-linear friction. A comparison of the performances of the two continuation techniques was analyzed to demonstrate their advantages and disadvantages. Finally, the HBM–AFT was applied to study the Duffing oscillator and the disk brake model. The obtained results have shown a good agreement with those found in the literature for the Duffing model and with the Runge–Kutta numerical method of the order 5 for the brake model.
Tài liệu tham khảo
Kerschen, G.: Computation of nonlinear normal modes through shooting and pseudo-arclength computation. In: Modal Analysis of Nonlinear Mechanical Systems, vol. 555 of CISM International Centre for Mechanical Sciences, Springer, Vienna, pp. 215–250 (2014)
Ghorbel, A., Zghal, B., Abdennadher, M., Walha, L., Haddar, M.: Effect of the gear local damage and profile error of the gear on the drivetrain dynamic response. J. Theor. Appl. Mech. 56(3), 765–779 (2018)
Low, K.H.: Comments on: An assessment of time integration schemes for non-linear dynamic equations. J. Sound Vib. 201(2), 256–257 (1997)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, London (1979)
Hermann, M., Saravi, M.: Nonlinear Ordinary Differential Equations: Analytical Approximation and Numerical Methods. Springer, India (2016)
Saranen, J., Vainikko, G.: Trigonometric collocation methods with product integration for boundary integral equations on closed curves. SIAM J. Numer. Anal. 33(4), 1577–1596 (1996)
Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)
Nayfeh, A.H., Lacarbonara, W., Chin, C.M.: Nonlinear normal modes of buckled beams: three-to-one and one-to-one internal resonances. Nonlinear Dyn. 18, 253–273 (2004)
Zhu, F.R., Parker, R.G.: Perturbation analysis of a clearance-type nonlinear system. J. Sound Vib. 292, 969–979 (2006)
Yu, P., Leung, A.Y.T.: A perturbation method for computing the simplest normal forms of dynamical systems. J. Sound Vib. 261, 123–151 (2003)
Thomsen, J.J., Fidlin, A.: Analytical approximations for stick–slip vibration amplitudes. Int. J. Non-Linear Mech. 38, 389–403 (2003)
Murdock, J.: Perturbation methods for engineers and scientists (Alan W. Bush). SIAM Rev. 36, 136–137 (1994)
Cartmell, M.P., Ziegler, S.W., Khanin, R., Forehand, D.I.M.: Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems. Appl. Mech. Rev. 56, 455–492 (2003)
Khanin, R., Cartmell, M., Gilbert, A.: A computerised implementation of the multiple scales perturbation method using Mathematica. Comput. Struct. 76, 565–575 (2000)
Rong, H.W., Wang, X.D., Xu, W., Fang, T.: Saturation and resonance of nonlinear system under bounded noise excitation. J. Sound Vib. 291, 48–59 (2006)
Das, S.K., Ray, P.C., Pohit, G.: Free vibration analysis of a rotating beam with nonlinear spring and mass system. J. Sound Vib. 301, 165–188 (2007)
Okuizumi, Nobukatsu, Kimura, Koji: Multiple time scale analysis of hysteretic systems subjected to harmonic excitation. J. Sound Vib. 272, 675–701 (2004)
Marathe, Amol, Chatterjee, Anindya: Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales. J. Sound Vib. 289, 871–888 (2006)
Devoret, M.H., Martinis, J.M., Esteve, D., Clarke, J.: Resonant activation from the zero-voltage state of a current-biased Josephson junction. Phys. Rev. Lett. 53(13), 1260 (1984)
Saha, A.: Bifurcation, periodic and chaotic motions of the modified equal width-Burgers (MEW-Burgers) equation with external periodic perturbation. Nonlinear Dyn. 87(4), 2193–2201 (2017)
Saha, A., Pal, N., Chatterjee, P.: Dynamic behavior of ion acoustic waves in electron-positron-ion magnetoplasmas with superthermal electrons and positrons. Phys. Plasmas 21(10), 102101 (2014)
Saha, A., Pal, N., Chatterjee, P.: Bifurcation and quasiperiodic behaviors of ion acoustic waves in magnetoplasmas with nonthermal electrons featuring tsallis distribution. Braz. J. Phys. 45(3), 325–333 (2015)
Tamang, J., Sarkar, K., Saha, A.: Solitary wave solution and dynamic transition of dust ion acoustic waves in a collisional nonextensive dusty plasma with ionization effect. Phys. A 505, 18–34 (2018)
Beléndez, A., Pascual, C., Ortuño, M., Beléndez, T., Gallego, S.: Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities. Nonlinear Anal. Real World Appl. 10, 601–610 (2009)
Khan, Y., Wu, Q.: Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput. Math Appl. 61, 1963–1967 (2011)
He, J.H., Wu, G.C., Austin, F.: The variational iteration method which should be followed. Nonlinear Sci. Lett. A 1, 1–30 (2010)
Chen, Y.M., Meng, G., Liu, J.K.: Variational iteration method for conservative oscillators with complicated nonlinearities. Math. Comput. A 15, 802–809 (2010)
Leung, A.Y.T., Guo, Z.: Residue harmonic balance for discontinuous nonlinear oscillator with fractional power restoring force. Int. J. Nonlinear Sci. Numer. Simul. 11, 705–723 (2010)
Guo, Z., Leung, A.Y.T., Yang, H.X.: Iterative homotopy harmonic balance approach for conservative oscillator with strong odd-nonlinearity. Appl. Math. Model. 35, 1717–1728 (2011)
Ozis, T., Yildirim, A.: A comparative study of He’s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities. Int. J. Nonlinear Sci. Numer. Simul. 8, 243–248 (2007)
Beléndez, A., Gimono, E., Alvarez, M.L., Méndez, D.I.: Nonlinear oscillator with discontinuity by generalized harmonic balance method. Comput. Math Appl. 58, 2117–2123 (2009)
Eriksson, A., Pacoste, C., Zdunek, A.: Numerical analysis of complex instability behaviour using incremental-iterative strategies. Comput. Methods Appl. Mech. Eng. 179, 265–305 (1999)
Seydel, R.: Practical Bifurcation and Stability Analysis, vol. 5 of Interdisciplinary Applied Mathematics, 3rd edn. Springer, New York (2010)
Krauskopf, B., Osinga, H.M., Galan-Vioque, J. (eds.): Numerical Continuation Methods for Dynamical Systems. Springer, The Netherlands (2007)
Cameron, T.M., Griffin, J.H.: An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems. J. Appl. Mech. 56, 149–154 (1989)
Seiler, M.C., Seiler, F.A.: Numerical recipes in C: the art of scientific computing. Risk Anal. 9(3), 415–416 (1989)
Cesari, L.: Functional analysis and Galerkin’s method. In: Michigan Math. J. (1964). Cameron et al. 1989
Crisfield, M.: A fast incremental/iterative solution procedure that handles snap-through. Comput. Struct. 13, 55–62 (1981)
Peletan, L., Baguet, S., Torkhani, M., Jacquet-Richardet, G.: Quasi-periodic harmonic balance method for rubbing self-induced vibrations in rotor-stator dynamics. Nonlinear Dyn. 78(4), 2501–2515 (2014)
Peletan, L. : Stratégie de modélisation simplifiée et de résolution accélérée en dynamique non linéaire des machines tournantes: application au contact rotor-stator (Doctoral dissertation, INSA de Lyon) (2012)
Demailly, D.: Etude du comportement non-linéaire dans le domaine fréquentiel- Application à la dynamique rotor. 2003. Thèse de doctorat. Ecole Centrale de Lyon
Fonseca, T., Grigolini, P.: Theory of resonantly activated rate processes. Phys. Rev. A 33(2), 1122 (1986)
Liu, L., et al.: A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator. J. Comput. Phys. 215(1), 298–320 (2006)
Peng, Z.K., et al.: Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis. J. Sound Vib. 311(1), 56–73 (2008)
Lazarus, A., Olivier, T.: A harmonic-based method for computing the stability of periodic solutions of dynamical systems. Comptes Rendus Méc. 338(9), 510–517 (2010)