Journal of Nonlinear Science

, Volume 5, Issue 3, pp 257–283

# Mode-locking in nonlinear rotordynamics

• G. H. M. van der Heijden
Article

## Summary

We present a computer-assisted study of the dynamics of two nonlinearly coupled driven oscillators with rotational symmetry which arise in rotordynamics (the nonlinearity coming from bearing clearance). The nonlinearity causes a splitting of the twofold degenerate natural frequency of the associated linear model, leading to three interacting frequencies in the system. Partial mode-locking then yields a biinfinite series of attracting invariant 2-tori carrying (quasi-) periodic motion.

Due to the resonance nature, the (quasi-) periodic solutions become periodic in a corotating coordinate system. They can be viewed as entrainments of periodic solutions of the associated linear problem. One presumably infinite family is generated by (scaled) driving frequencies ω = 1+2/n,n = 1,2,3,...; another one is generated by frequencies ω =m,m = 4,5,6,... Both integersn andm can be related to discrete symmetry properties of the particular periodic solutions.

Under a perturbation that breaks the rotational symmetry, more complicated behavior is possible. In particular, a second rational relation between the frequencies can be established, resulting in fully mode-locked periodic motion.

### Key words

rotor dynamics bearing clearance mode-locking resonance

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### References

1. [1]
G. H. M. van der Heijden, Bifurcation and chaos in drillstring dynamics,Chaos, Solitons & Fractals 3, 219–247 (1993).Google Scholar
2. [2]
J. D. Jansen, Nonlinear rotor dynamics as applied to oilwell drillstring vibrations,J. Sound and Vibration 147, 115–135 (1991).Google Scholar
3. [3]
W. B. Day, Asymptotic expansions in nonlinear rotordynamics,Quart. Appl. Math. 44, 779–792 (1987).Google Scholar
4. [4]
R. A. Zalik, The Jeffcott equations in nonlinear rotordynamics,Quart. Appl. Math. 47, 585–599 (1989).Google Scholar
5. [5]
Y. B. Kim, S. T. Noah, Bifurcation analysis for a modified Jeffcott rotor with bearing clearances,Nonlinear Dynam. 1, 221–241 (1990).Google Scholar
6. [6]
F. F. Ehrich, Some observations of chaotic vibration phenomena in high-speed rotordynamics,J. Vibration and Acoustics 113, 50–57 (1991).Google Scholar
7. [7]
G. Genta, C. Delprete, A. Tonoli, R. Vadori, Conditions for noncircular whirling of nonlinear isotropic rotors,Nonlinear Dynam. 4, 153–181 (1993).Google Scholar
8. [8]
S. K. Choi, S. T. Noah, Mode-locking and chaos in a Jeffcott rotor with bearing clearances,J. Appl Mech. 61, 131–138 (1994).Google Scholar
9. [9]
D. Bently, Forced subrotative speed dynamic action of rotating machinery, ASME Paper No. 74-PET-16, Petroleum Mechanical Engineering Conference, Dallas, Texas (1974).Google Scholar
10. [10]
R. F. Beatty, M. J. Hine, Improved rotor response of the updated high pressure oxygen turbo pump for the space shuttle main engine,J. Vibration, Acoustics, Stress and Reliability in Design 111, 163–169 (1989).Google Scholar
11. [11]
D. W. Childs, Rotordynamic characteristics of the HPOTP (high pressure oxygen turbopump) of the SSME (space shuttle main engine), Report FD-1-84, Turbomachinery Laboratories, Texas A & M University, 1984.Google Scholar
12. [12]
C. Kaas-Petersen, Computation, continuation, and bifurcation of torus solutions for dissipative maps and ordinary differential equations,Physica 25D, 288–306 (1987).Google Scholar
13. [13]
H. L. Swinney, Observations of order and chaos in nonlinear systems,Physica 7D, 3–15 (1983).Google Scholar
14. [14]
M. H. Jensen, P. Bak, T. Bohr, Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps,Phys. Rev. A 30, 1960–1969 (1984); Transition to chaos by interaction of resonances in dissipative systems. II. Josephson junctions, charge-density waves, and standard maps,Phys. Rev. A 30, 1970–1981 (1984).Google Scholar
15. [15]
D. G. Aronson, R. P. McGehee, I. G. Kevrekidis, R. Aris, Entrainment regions for periodically forced oscillators,Phys. Rev. A 33, 2190–2192 (1986).Google Scholar
16. [16]
R. S. MacKay, C. Tresser, Transition to topological chaos for circle maps,Physica 19D, 206–237 (1986).Google Scholar
17. [17]
J. A. Glazier, A. Libchaber, Quasi-periodicity and dynamical systems: an experimentalist's view,IEEE Trans. Circuits and Systems 35(7), 790–809 (1988); reprinted in H. Bai-Lin,Chaos II, World Scientific, Singapore, 1990.Google Scholar
18. [18]
D. Ruelle, F. Takens, On the nature of turbulence,Commun. Math. Phys. 20, 167–192 (1971)23, 343–344 (1971).Google Scholar
19. [19]
S. E. Newhouse, D. Ruelle, F. Takens, Occurrence of strange Axiom A attractors near quasi periodic flows onT m,m ≤ 3,Commun. Math. Phys. 64, 35–40 (1978).Google Scholar
20. [20]
P. S. Linsay, A. W. Gumming, Three-frequency quasiperiodicity, phase locking, and the onset of chaos,Physica 40D, 196–217 (1989).Google Scholar
21. [21]
C. Baesens, J. Guckenheimer, S. Kim, R. S. MacKay, Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos,Physica 49D, 387–475 (1991).Google Scholar
22. [22]
H. W. Broer, C. Simó, J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Preprint University of Groningen, The Netherlands (1993).Google Scholar
23. [23]
V. I. Arnol;d, Small denominators, I. Mappings of the circumference onto itself,Transl. Amer. Math. Soc., Ser. 246, 213–284 (1965).Google Scholar
24. [24]
E. J. Doedel, J. P. Kernevez, AUTO: Software for continuation and bifurcation problems in ordinary differential equations, Applied Mathematics Report, California Institute of Technology, 1986.Google Scholar
25. [25]
B. L. J. Braaksma, H. W. Broer, G. B. Huitema, Toward a quasi-periodic bifurcation theory,Mem. AMS 83(421), 83–175 (1990).Google Scholar
26. [26]
W. Szemplinska-Stupnicka,The Behaviour of Nonlinear Vibrating Systems, Vol. I and II, Kluwer Academic Publishers, Dordrecht, 1990.Google Scholar
27. [27]
W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling,Numerical Recipes. The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.Google Scholar
28. [28]
G. R. Qin, D. C. Gong, R. Li, X. D. Wen, Rich bifurcation behaviors of the driven Van der Pol oscillator,Phys. Lett. A 141, 412–416 (1989).Google Scholar
29. [29]
G. R. Qin, R. Li, D. C. Gong, L. Jiang, Equal periodic bifurcation in a real dissipative system,Phys. Lett. A 137, 255–258 (1989).Google Scholar
30. [30]
L. Glass, R. Ferez, Fine structure of phase locking,Phys. Rev. Lett. 48, 1772–1775 (1982).Google Scholar
31. [31]
D. G. Aronson, M. A. Chory, G. R. Hall, R. P. McGehee, Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study,Commun. Math. Phys. 83, 303–354 (1982).Google Scholar
32. [32]
D. L. Gonzalez, O. Piro, Chaos in a nonlinear driven oscillator with exact solution,Phys. Rev. Lett. 50, 870–872 (1983).Google Scholar
33. [33]
J. A. Sanders, F. Verhulst,Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985.Google Scholar
34. [34]
J. Guckenheimer, P. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.Google Scholar
35. [35]
F. Verhulst, Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies,Phil. Trans. Roy. Soc. London Ser. A 290, 435–465 (1979).Google Scholar
36. [36]
S. Wiggins,Global Bifurcations and Chaos, Springer-Verlag, New York, 1988.Google Scholar