On the continuation of quasi-periodic solutions of rotor systems with fluid-film bearings

Abstract Fluid-film bearings have widespread use in many rotating machines, mainly due to their low friction, high load capacity and ease of manufacture. However, these bearings generally display a self-excited phenomenon, labeled oil-whirl or oil-whip, the latter being known to be more dangerous. In perfectly balanced rotors, this phenomenon can be seen as a Hopf bifurcation, where limit cycles emerge as a parameter is varied in the system, which in this case is the shaft speed. As unbalance is included in the system, the limit cycles turn to quasi-periodic or periodic orbits, depending on the relation between the rotating speed and frequency of the cycle. This quasi-periodic solution can be seen as occurring on a $N$-dimensional torus. This work presents a continuation method to obtain invariant tori of rotors with fluid-bearings subjected to unbalance. The basis of the approach is to use a multidimensional harmonic balance method (MHBM) adopting the shaft speed and the frequency of the limit cycle as harmonic components to estimate the torus function and continue the solution using the pseudo arc-length scheme. The method is valid for obtaining both stable and unstable quasi-periodic solutions, and it is evaluated by comparing the responses with transient simulations and an open-source numerical continuation package (MATCONT). The contribution of this work lies in presenting a new tool to analyze quasi-periodic motions of rotors with unbalance and fluid instability, as well as demonstrating the effect of external excitation on the limit cycles of the autonomous system.