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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 293-311 
    ISSN: 1573-269X
    Keywords: Oil whirl ; bifurcation ; unbalance ; instability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear behavior of an unbalanced rotor supported in a fluid film bearing is analyzed. A simplified two dimensional model is adopted which uses the long-bearing approximation with a π-film to account for cavitation. This model has been thoroughly studied by Myers [1] in the balanced case, where it is shown that the whirl instability is the result of a Hopf bifurcation. The implications of imbalance are studied in this paper. This leads to the study of a periodically perturbed Hopf bifurcation. It is shown that the dynamics in this situation can, especially under certain nonlinear resonance conditions, have an extremely complicated dependence on the system parameters and the rotor speed. Complete bifurcation diagrams are presented for a particular rotor model which demonstrate this dependence.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 4 (1993), S. 251-268 
    ISSN: 1573-269X
    Keywords: Singular perturbation problems ; nonlinear effects ; turbulence via chaos techniques ; bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Flow-induced oscillations of rigid cylinders exposed to fully developed turbulent flow can be described by a fourth order autonomous system. Among the pertinent constants, the mass ratio is the control parameter governing the transition from limit cycle oscillations to chaotic vibrations. Particular attention is paid to the stability of the limit cycles: it has been found that they lose their stability at the point of appearance of quasi-periodic motion. The documentation of this transition is performed in terms of Lyapunov exponents, phase plots, Fourier spectra, bifurcation diagrams, and Poincaré maps. As opposed to the calculation of the Lyapunov exponents where remarkable numerical difficulties were encountered, the investigation of the remaining quantities shows clearly the passage of cylinder motions from limit cycle oscillations to more and more irregular vibrations, leading finally to chaos.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 4 (1993), S. 389-408 
    ISSN: 1573-269X
    Keywords: Plate ; bifurcation ; chaos ; homoclinic orbit
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we investigate global bifurcations in the motion of parametrically excited, damped thin plates. Using new mathematical results by Kovačič and Wiggins in finding homoclinic and heteroclinic orbits to fixed points that are created in a resonance resulting from perturbation, we are able to obtain explicit conditions under which Silnikov homoclinic orbits occur. Furthermore, we confirm our theoretical predictions by numerical simulations.
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  • 4
    ISSN: 1573-269X
    Keywords: Collocation ; stability ; bifurcation ; modal interaction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A methodology is first presented for analyzing long time response of periodically exited nonlinear oscillators. Namely, a systematic procedure is employed for determining periodic steady state response, including harmonic and superharmonic components. The stability analysis of the located periodic motions is also performed, utilizing results of Froquet theory. This methodology is then applied to a special class of two degree of freedom nonlinear oscillators, subjected to harmonic excitation. The numberical results presented in the second part of this study illustrate effects caused by the interaction of the modes as well as effects of the nonlinearities on the steady state response of these oscillators. In addition, sequences of bifurcations are analyzed for softening systems, leading to unbounded response of the model examined. Finally, the importance of higher harmonics on the response of systems with strongly nonlinear characteristics is investigated.
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  • 5
    ISSN: 1573-269X
    Keywords: Shaft ; stability ; bifurcation ; analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The dynamic stability and self-excited posteritical whirling of rotating transversally loaded shaft made of a standard material with elastic and viscous nonlinearities are analyzed in this paper using the theory of bifurcations as a mathematical tool. Partial differential equations of motion are derived under assumption that von Karman's nonlinearity is absent but geometric curvature nonlinearity is included. Galerkin's first-mode discretization procedure is then applied and the equations of motion are transformed to two third-order nonlinear equations that are analyzed using the theory of bifurcation. Condition for nontrivial equilibrium stability is determined and a bifurcating periodic solution of the second-order approximation is derived. The effects of dimensionless stress relaxation time and cubic elastic and viscous nonlinearities as well as the role of the transverse load are studied in the exemplary numerical calculations. A strongly stabilizing influence of the relaxation time is found that may eliminate self-excited vibration at all. Transition from super- to subcritical bifurcation is observed as a result of interaction between system nonlinearities and the transverse load.
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