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Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances. (English) Zbl 1430.37098
Summary: In this paper, a frequency-domain characterization of the period doubling bifurcation is proposed. This allows an efficient detection and localization of such points along frequency response curves computed through continuation and the harmonic balance method. A simple strategy for branch switching to sub-harmonic regimes is presented as well. Furthermore, these bifurcations are tracked in a two-dimensional parameter space, and extremum points with respect to the tracking parameter are characterized and linked to sub-harmonic isola formation. As a test case for these methods, a forced Duffing oscillator with asymmetric clearances is studied numerically. The results, which include the prediction of period doubling cascades and sub-harmonic isolas, are then compared to experimental results, yielding an excellent agreement.
37M20 Computational methods for bifurcation problems in dynamical systems
70K50 Bifurcations and instability for nonlinear problems in mechanics
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[1] Allgower, E.; Georg, K., Numerical Continuation Methods (1990), Berlin: Springer, Berlin · Zbl 0717.65030
[2] Bentvelsen, B.; Lazarus, A., Modal and stability analysis of structures in periodic elastic states: application to the ziegler column, Nonlinear Dyn., 91, 2, 1349-1370 (2017)
[3] Cameron, T.; Griffin, J., An alternating frequenct/time domain method for calculating the steady-state response of nonlinear dynamic systems, J. Appl. Mech. ASME, 56, 1, 149-154 (1989) · Zbl 0685.73036
[4] Cochelin, B.; Vergez, C., A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, J. Sound Vib., 324, 243-262 (2009)
[5] Dankowicz, H.; Schilder, F., Recipes for Continuation (2013), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1277.65037
[6] Dellwo, D.; Keller, H.; Matkowsky, B.; Reiss, E., On the birth of isolas, SIAM J. Appl. Math., 42, 5, 956-963 (1982) · Zbl 0503.35014
[7] Detroux, T.; Renson, L.; Kerschen, G.; Kerschen, G., The harmonic balance method for advanced analysis and design of nonlinear mechanical systems, Nonlinear Dynamics, 19-34 (2014), New York: Springer, New York
[8] Detroux, T.; Habib, G.; Masset, L.; Kerschen, G., Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber, Mech. Syst. Signal Process, 60-61, 799-809 (2015)
[9] Detroux, T.; Renson, L.; Masset, L.; Kerschen, G., The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems, Comput. Methods Appl. Mech. Eng., 296, 18-38 (2015) · Zbl 1423.70052
[10] Dhooge, A.; Govaerts, W.; Kuznetsov, Y., MATCONT: a MATLAB package for numerical bifurcation analysis of odes, SIGSAM Bull, 38, 1, 21-22 (2004)
[11] Doedel, E.J., Fairgrieve, T.F., Sandstede, B., Champneys, A.R., Kuznetsov, Y.A., Wang, X.: AUTO-07P: Continuation and bifurcation software for ordinary differential equations (2007)
[12] Duan, C.; Singh, R., Isolated sub-harmonic resonance branch in the frequency response of an oscillator with slight asymmetry in the clearance, J. Sound Vib., 314, 1-2, 12-18 (2008)
[13] Ferri, A., Leamy, M.: Error estimates for harmonic-balance solutions of nonlinear dynamical systems. In: 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics (2009)
[14] Gatti, G., Uncovering inner detached resonance curves in coupled oscillators with nonlinearity, J. Sound Vib., 372, 239-254 (2016)
[15] Grolet, A.; Thouverez, F., On a new harmonic selection technique for harmonic balance method, Mech. Syst. Signal Process., 30, 43-60 (2012)
[16] Groll, Gv; Ewins, D., The harmonic balance method with arc-length continuation in rotor/stator contact problems, J. Sound Vib., 241, 2, 223-233 (2001)
[17] Habib, G.; Cirillo, G.; Kerschen, G., Uncovering detached resonance curves in single-degree-of-freedom systems, Procedia Eng., 199, 649-656 (2017)
[18] Inayat-Hussain, J.; Kanki, H.; Mureithi, N., On the bifurcations of a rigid response in squeeze-film dampers, J. Fluids Struct., 17, 3, 433-459 (2003)
[19] Ing, J.; Pavlovskaia, E.; Wiercigroch, M.; Banerjee, S., Experimental study of impact oscillator with one-sided elastic constraint, Philos. Trans. R. Soc., 366, 679-704 (2008) · Zbl 1153.74302
[20] Keller, H.; Rabinowitz, P., Numerical solution of bifurcation and nonlinear eigenvalue problems, Applications of Bifurcation Theory, 384-395 (1977), Cambridge: Academic Press, Cambridge
[21] Kernevez, J.; Liu, Y.; Seoane, M.; Doedel, E.; Roose, D.; De Dier, B.; Spence, A., Optimization by continuation, Continuation and Bifurcations: Numerical Techniques and Applications, 349-362 (1990), Dordrecht: Springer, Dordrecht
[22] Kim, T.; Rook, T.; Singh, R., Effect of smoothening functions on the frequency response of an oscillator with clearance non-linearity, J. Sound Vib., 263, 3, 665-678 (2003) · Zbl 1237.70108
[23] Kim, T.; Rook, T.; Singh, R., Super- and sub-harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balance method, J. Sound Vib., 281, 3-5, 965-993 (2005)
[24] Kim, Wj; Perkins, N., Harmonic balance/galerkin method for non-smooth dynamic systems, J. Sound Vib., 261, 2, 213-224 (2003) · Zbl 1237.70080
[25] Krack, M., Nonlinear modal analysis of nonconservative systems: extension of the periodic motion concept, Comput. Struct., 154, 59-71 (2015)
[26] Kuether, R.; Renson, L.; Detroux, T.; Grappasonni, C.; Kerschen, G.; Allen, M., Nonlinear normal modes, modal interactions and isolated resonance curves, J. Sound Vib., 351, 299-310 (2016)
[27] Kuznetsov, Ya, Elements of Applied Bifurcation Theory (2004), New York: Springer, New York · Zbl 1082.37002
[28] De Langre, E.; Lebreton, G.; Pettigrew, M., An experimental and numerical analysis of chaotic motion in vibration with impact, Flow-Induced Vibration, 317-325 (1996), New York: American Society of Mechanical Engineers, New York
[29] Leine, R.; Van Campen, D.; Van De Vrande, B., Bifurcations in nonlinear discontinuous systems, Nonlinear Dyn., 23, 2, 105-164 (2000) · Zbl 0980.70018
[30] Natsiavas, S.; Gonzalez, H., Vibration of harmonically excited oscillators with asymmetric constraints, J. Appl. Mech., 59, 2, S284 (1992)
[31] Nayfeh, A.; Balachandran, B., Applied Nonlinear Dynamics (2004), New York: Wiley, New York
[32] Seydel, R., Practical Bifurcation and Stability Analysis (2010), Berlin: Springer, Berlin · Zbl 1195.34004
[33] Shaw, S.; Holmes, P., A periodically forced piecewise linear oscillator, J. Sound Vib., 90, 1, 129-155 (1983) · Zbl 0561.70022
[34] Strogatz, S., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (2015), Boulder: Westview Press, Boulder
[35] Van Til, J.; Alijani, F.; Voormeeren, S.; Lacarbonara, W., Frequency domain modeling of nonlinear end stop behavior in tuned mass damper systems under single- and multi-harmonic excitations, J. Sound Vib., 438, 139-152 (2018)
[36] Witte, Vd; Rossa, Fd; Govaerts, W.; Kuznetsov, Y., Numerical periodic normalization for codim 2 bifurcations of limit cycles: computational formulas, numerical implementation, and examples, SIAM J. Appl. Dyn. Syst., 12, 2, 722-788 (2013) · Zbl 1284.34049
[37] Xie, L.; Baguet, S.; Prabel, B.; Dufour, R., Bifurcation tracking by harmonic balance method for performance tuning of nonlinear dynamical systems, Mech. Syst. Signal Process., 88, 445-461 (2017)
[38] Yoon, Jy; Kim, B., Effect and feasibility analysis of the smoothening functions for clearance-type nonlinearity in a practical driveline system, Nonlinear Dyn., 85, 3, 1651-1664 (2016)
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