Abstract
The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.
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Seyranian, A.P., Mailybaev, A.A.: Multiparameter Stability Theory with Mechanical Applications. Word Scientific, New Jersey (2003)
Meiss, J.D.: Differential Dynamical Systems. Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics, Philadelphia (2007)
Crandall, S.H., Dugundji, J.: Resonant whirling of aircraft propeller-engine systems. J. Appl. Mech. 48(4), 929–935 (1981)
Genta, G.: On the stability of rotating blade arrays. J. Sound Vib. 273(4–5), 805–836 (2004). doi:10.1016/s0022-460x(03)00784-3
Genta, G.: Dynamics of Rotating Systems, 1st edn. Springer, Berlin (2005)
Hartog, J.P.D.: Mechanical Vibrations, 4th edn. McGraw-Hill Book Company, New York (1956)
Sanches, L., Michon, G., Berlioz, A., Alazard, D.: Instability zones for isotropic and anisotropic multibladed rotor configurations. Mech. Mach. Theory 46(8), 1054–1065 (2011). doi:10.1016/j.mechmachtheory.2011.04.005
Shahgholi, M., Khadem, S.: Stability analysis of a nonlinear rotating asymmetrical shaft near the resonances. Nonlinear Dyn. 70(2), 1311–1325 (2012). doi:10.1007/s11071-012-0535-7
Crandall, S.H., Mroszczyk, J.W.: Conservative and Nonconservative Coupling in Dynamic Systems. Institution of Mechanical Engineers, London (2003)
Leissa, A.W.: On a curve veering aberration. ZAMP (J. Appl. Math. Phys.) 25, 99–111 (1974)
Afolabi, D.: Modal Interaction in Linear Dynamic Systems Near Degenerate Modes. NASA, Ohio (1991)
Afolabi, D., Mehmed, O.: On Curve Veering and Flutter of Rotating Blades, pp. 1–45. NASA, Ohio (1993)
Afolabi, D.: The cusp catastrophe and the stability problem of helicopter ground resonance. Proc. R. Soc. Lond. 441(1912), 399–406 (1993)
Seiranyan, A.P.: Collision of eigenvalues in linear oscillatory systems. J. Appl. Math. Mech. 58(5), 805–813 (1994). doi:10.1016/0021-8928(94)90005-1
Seyranian, A.P., Mailybaev, A.A.: Interaction of eigenvalues in multi-parameter problems. J. Sound Vib. 267(5), 1047–1064 (2003). doi:10.1016/s0022-460x(03)00360-2
Kirillov, O.N.: Unfolding the conical zones of the dissipation-induced subcritical flutter for the rotationally symmetrical gyroscopic systems. Phys. Lett. A 373(10), 940–945 (2009). doi:10.1016/j.physleta.2009.01.013
Kirillov, O.N., Seyranian, A.O.: The effect of small internal and external damping on the stability of distributed non-conservative systems. J. Appl. Math. Mech. 69(4), 529–552 (2005). doi:10.1016/j.jappmathmech.2005.07.004
Kirillov, O.N.: Campbell diagrams of weakly anisotropic flexible rotors. Proc. R Soc. A 465(2109), 2703–2723 (2009)
Cleghorn, W.L., Fenton, R.G., Tabarrok, B.: Steady-state vibrational response of high-speed flexible mechanisms. Mech. Mach. Theory 19(4—-5), 417–423 (1984). doi:10.1016/0094-114X(84)90100-9
Peletan, L., Baguet, S., Torkhani, M., Jacquet-Richardet, G.: A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics. Nonlinear Dyn. 72(3), 671–682 (2013). doi:10.1007/s11071-012-0744-0
Sadeghi, M., Ansari Hosseinzadeh, H.: Parametric excitation of a thin ring under time-varying initial stress: theoretical and numerical analysis. Nonlinear Dyn. 74(3), 733–743 (2013). doi:10.1007/s11071-013-1001-x
van der Meer, J.-C.: The Hamiltonian Hopf Bifurcation. Springer, Berlin (1985)
Nayfeh, A.H.: Introduction to perturbation techniques. Wiley, New York (1993)
Nayfeh, A.H., Balachandran, B.: Computational and Experimental Methods. Applied Nonlinear Dynamics: Analytical. Wiley, New York (2008)
Lahiri, A., Sinha Roy, M.: The Hamiltonian Hopf bifurcation: an elementary perturbative approach. Int. J. Non-Linear Mech. 36, 787–802 (2001). doi:10.1016/S0020-7462(00)00045-7
Gils, S., Krupa, M., Langford, W.F.: Hopf bifurcation with non-semisimple 1:1 resonance. Nonlinearity 3, 825–850 (1990)
Luongo, A., Egidio, A.D., Paolone, A.: Multiscale analysis of defective multiple-Hopf bifurcations. Comput. Struct. 82(31–32), 2705–2722 (2004). doi:10.1016/j.compstruc.2004.04.022
Luongo, A., Egidio, A.: Bifurcation equations through multiple-scales analysis for a continuous model of a planar beam. Nonlinear Dyn. 41(1–3), 171–190 (2005). doi:10.1007/s11071-005-2804-1
Luongo, A., Paolone, A.: On the reconstitution problem in the multiple time-scale method. Nonlinear Dyn. 19(2), 135–158 (1999). doi:10.1023/A:1008330423238
Anegawa, N., Fujiwara, H., Matsushita, O.: Vibration diagnosis featuring blade-shaft coupling effect of turbine rotor models. J. Eng. Gas Turbines Power 133(2), 022501–022508 (2011)
Chiu, Y.-J., Chen, D.-Z.: The coupled vibration in a rotating multi-disk rotor system. Int. J. Mech. Sci. 53(1), 1–10 (2011). doi:10.1016/j.ijmecsci.2010.10.001
Diken, H., Alnefaie, K.: Effect of unbalanced rotor whirl on blade vibrations. J. Sound Vib. 330(14), 3498–3506 (2011). doi:10.1016/j.jsv.2011.02.017
Wang, L., Cao, D.Q., Huang, W.: Nonlinear coupled dynamics of flexible blade-rotor-bearing systems. Tribol. Int. 43(4), 759–778 (2010). doi:10.1016/j.triboint.2009.10.016
Ghazavi, M.R., Najafi, A., Jafari, A.A.: Bifurcation and nonlinear analysis of nonconservative interaction between rotor and blade row. Mech. Mach. Theory 65(0), 29–45 (2013). doi:10.1016/j.mechmachtheory.2013.02.008
Younesian, D., Esmailzadeh, E.: Non-linear vibration of variable speed rotating viscoelastic beams. Nonlinear Dyn. 60(1–2), 193–205 (2010). doi:10.1007/s11071-009-9589-6
Bridges, T.J.: Bifurcation of periodic solutions near a collision of eigenvalues of opposite signature. Math. Proc. Cambridge Philos. Soc. 108, 575–601 (1990)
Bolotin, V.V.: Nonconservative Problems of the Theory of Elastic Stability. Pergamon, New York (1963)
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Najafi, A., Ghazavi, MR. & Jafari, AA. Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor. Nonlinear Dyn 78, 1049–1064 (2014). https://doi.org/10.1007/s11071-014-1495-x
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DOI: https://doi.org/10.1007/s11071-014-1495-x