×

On nonlinear motions of two-degree-of-freedom nonlinear systems with repeated linearized natural frequencies. (English) Zbl 1432.70038

Summary: In this paper, we investigate systematically the vibration of a typical 2DOF nonlinear system with repeated linearized natural frequencies. By application of Descartes’ rule of signs, we demonstrate that there are 14 types of roots describing different modal motions for varying nonlinear parameters. The method of multiple scales is used to obtain the amplitude-phase portraits by introducing the energy ratios and phase differences. The typical nonlinear in-unison and elliptic out-of-unison modal motions are located for the 14 types of roots and then validated by numerical simulations. It is found that the normal in-unison modal motions, elliptic out-of-unison modal motions are analogous to the polarization of classical optic theory. Further, some kinds of periodic and chaotic motions under out-of-unison and in-unison excitations are investigated numerically. The result of this study offers a detailed discussion of nonlinear modal motions and responses of 2DOF systems with cubic nonlinear terms.

MSC:

70H05 Hamilton’s equations
70K40 Forced motions for nonlinear problems in mechanics
70K42 Equilibria and periodic trajectories for nonlinear problems in mechanics
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] An, F. X. & Chen, F. Q. [2017] “ Multipulse orbits and chaotic dynamics of an aero-elastic FGP plate under parametric and primary excitations,” Int. J. Bifurcation and Chaos27, 1750050-1-18. · Zbl 1366.74038
[2] Cao, D. & Gao, Y. [2019] “ Free vibration of non-uniform axially functionally graded beams using the asymptotic development method,” Appl. Math. Mech.40, 85-96.
[3] Gendelman, O. V. [2004] “ Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment,” Nonlin. Dyn.37, 115-128. · Zbl 1081.70012
[4] Goldstein, D. [2003] Polarized Light (Marcel Dekker, Inc).
[5] Hill, T. L., Cammarano, A., Neild, S. A. & Wagg, D. J. [2015] “ Out-of-unison resonance in weakly nonlinear coupled oscillators,” Proc. Roy. Soc. A471, 20140659. · Zbl 1371.70056
[6] Kerschen, G., Peeters, M. & Golinval, J. C. [2009] “ Nonlinear normal modes, part I: A useful framework for the structural dynamic,” Mech. Syst. Sign. Process.1, 170-194.
[7] Krack, M. [2015] “ Nonlinear modal analysis of non-conservative systems: Extension of the periodic motion concept,” Comput. Struct.154, 59-71.
[8] Kuether, R. J., Renson, L. & Detroux, T. [2015] “ Nonlinear normal modes, modal interactions and isolated resonance curves,” J. Sound Vibr.351, 299-310.
[9] Lacarbonara, W. & Camillacci, R. [2004] “ Nonlinear normal modes of structural systems via asymptotic approach,” Int. J. Solids Struct.41, 5565-5594. · Zbl 1073.74032
[10] Liu, F., Yan, H. & Zhang, W. [2019] “ Nonlinear dynamic analysis of a photonic crystal nanocavity resonator,” Appl. Math. Mech.40, 139-152.
[11] Lu, Z., Li, K., Ding, H. & Chen, L. [2019] “ Nonlinear energy harvesting based on a modified snap-through mechanism,” Appl. Math. Mech.40, 167-180. · Zbl 1416.74071
[12] Luo, S., Li, S. & Tajaddodianfar, F. [2017] “ Chaos and adaptive control of the fractional-order magnetic-field electromechanical transducer,” Int. J. Bifurcation and Chaos27, 1750203-1-9. · Zbl 1379.93063
[13] Manevich, A. I. & Manevitch, L. I. [2003] “ Free oscillations in conservative and dissipative symmetric cubic two-degree-of-freedom systems with closed natural frequencies,” Meccanica38, 335-348. · Zbl 1062.70042
[14] Manevich, A. I. & Manevitch, L. I. [2005] The Mechanics of Nonlinear Systems with Internal Resonances (Imperial College Press, London). · Zbl 1082.70001
[15] Minorsky, N. [1962] Nonlinear Oscillations (Van Nostrand). · Zbl 0102.30402
[16] Peeters, M., Kerschen, G. & Golinval, J. C. [2011a] “ Modal testing of nonlinear vibrating structures based on nonlinear normal modes: Experimental demonstration,” Mech. Syst. Sign. Process.25, 1227-1247.
[17] Peeters, M., Kerschen, G. & Golinval, J. C. [2011b] “ Dynamic testing of nonlinear vibrating structures using nonlinear normal modes,” J. Sound Vibr.330, 486-509.
[18] Rand, R. & Vito, R. [1972] “ Nonlinear vibrations of two-degree-of-freedom systems with repeated linearized natural frequencies,” J. Appl. Mech.39, 296-297.
[19] Rosenberg, R. M. [1962] “ The normal modes of nonlinear \(<mml:math display=''inline`` overflow=''scroll``>\)-degree-of-freedom systems,” J. Appl. Mech.29, 7-14. · Zbl 0113.07502
[20] Shaw, S. W. & Pierre, C. [1993] “ Normal modes for non-linear vibratory systems,” J. Sound Vibr.164, 85-124. · Zbl 0925.70235
[21] Vakakis, A. F. [1997] “ Non-linear normal mode (NNMs) and their applications in vibration theory: An overview,” Mech. Syst. Sign. Process.11, 3-22.
[22] Yang, S. W., Hao, Y. X., Zhang, W. & Li, S. B. [2015] “ Nonlinear dynamic behavior of functionally graded truncated conical shell under complex loads,” Int. J. Bifurcation and Chaos25, 1550025-1-33. · Zbl 1309.74047
[23] Yang, X., An, H., Qian, Y., Zhang, W. & Melnik, R. V. N. [2016a] “ A comparative analysis of modal motions for the gyroscopic and non-gyroscopic two degree-of-freedom conservative systems,” J. Sound Vibr.385, 300-309.
[24] Yang, X., An, H., Qian, Y., Zhang, W. & Yao, M. [2016b] “ Elliptic motions and control of rotors suspending in active magnetic bearings,” J. Comput. Nonlin. Dyn.11, 054503.
[25] Ye, M., Sun, Y., Zhang, W., Zhan, X. & Ding, Q. [2005] “ Nonlinear oscillations and chaotic dynamics of an antisymmetric cross-ply laminated composite rectangular thin plate under parametric excitation,” J. Sound Vibr.287, 723-758. · Zbl 1243.74110
[26] Yin, S., Wen, G. & Wu, X. [2019] “ Suppression of grazing-induced instability in single degree-of-freedom impact oscillators,” Appl. Math. Mech.40, 97-110. · Zbl 1416.34047
[27] Zhang, W. & Song, C. [2006] “ Further studies on nonlinear oscillations and chaos of symmetric cross-ply laminated thin plate under parametric excitation,” Int. J. Bifurcation and Chaos2, 325-347. · Zbl 1111.37064
[28] Zhang, W. & Guo, X. Y. [2012] “ Periodic and chaotic oscillations of a composite laminated plate using the third-order shear deformation plate theory,” Int. J. Bifurcation and Chaos22, 1250103-1-25. · Zbl 1258.34120
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.