Abstract
Nonlinear rotors are often considered as potential sources of chaotic vibrations. The aim of the present paper is that of studying in detail the behaviour of a nonlinear isotropic Jeffcott rotor, representing the simplest nonlinear rotor. The restoring and damping forces have been expanded in Taylor series obtaining a ‘Duffing-type’ equation. The isotropic nature of the system allows circular whirling to be a solution at all rotational speeds. However there are ranges of rotational speed in which this solution is unstable and other, more complicated, solutions exist.
The conditions for stability of circular whirling are first studied from closed form solutions of the mathematical model and then the conditions for the existence of solutions of other type are studied by numerical experimentation. Although attractors of the limit cycle type are often found, chaotic attractors were identified only in few very particular cases. An attractor supposedly of the last type reported in the literature was found, after a detailed analysis, to be related to a nonchaotic polyharmonic solution.
As the typical unbalance response of isotropic nonlinear rotors has been shown to be a synchronous circular whirling motion, the convergence characteristics of Newton-Raphson algorithm applied to the solution of the set of nonlinear algebraic equations obtained from the differential equations of motion are studied in some detail.
c damping coefficient
i imaginaty unit (i=% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbqfgBHr% xAU9gimLMBVrxEWvgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvA% Tv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9% vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea% 0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabe% aadaabauaaaOqaamaakaaabaGaeyOeI0IaaGymaaWcbeaaaaa!3E66!\[\sqrt { - 1}\])
k stiffness
m mass
t time
x istate variables i=1, 4
z complex co-ordinate (z=x+iy)
[J] Jacobian matrix
Oxyz inertial co-ordinate frame
Oξηz rotating co-ordinate frame
δ perturbation term
ε eccentricity
ζ complex co-ordinate (ζ=ξ+iη)
λ system eigenvalues
μ nonlinearity parameter
τ nondimensional time
ϕ phase
ω spin speed
u nonrotating
t rotating
0 amplitude
t nondimensional terms
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Genta, G., Delprete, C., Tonoli, A. et al. Conditions for noncircular whirling of nonlinear isotropic rotors. Nonlinear Dyn 4, 153–181 (1993). https://doi.org/10.1007/BF00045252
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DOI: https://doi.org/10.1007/BF00045252