On the quality of stability and bifurcation sets in rotors with permanent shaft bow on nonlinear supports

Abstract

The paper investigates the dynamics of simple and complex geometry rotors of various configurations with permanent bow, mounted on nonlinear bearings, in regards to nonlinear stability, quality of motion, and bifurcation set. Permanent shaft bow is a fault existing in most slender rotors in thermal machines, e.g. gas turbines, steam turbines, and turbopumps, and other applications of rotating machinery like transmission shafts and geared shafts. In this work, finite element models of multi-segment rotors are implemented, including permanent bow, and a generic rotor system of a uniform rotor with three masses is studied first on its stability characteristics for various bow values, and for different slenderness. A realistic rotor of complex geometry is used as the second application. The rotors are mounted on two plain short bearings to include the nonlinearity of impedance forces with analytical formulas; this is the unique source of nonlinearity in the systems.The rotor-bearing systems are found to produce the well-known oil whirl instability through a Hopf, or a secondary Hopf bifurcation, depending the unbalance grade, in speeds higher than the first critical speed, when bow is not included. Supercritical and subcritical Neimark-Sacker and period doubling bifurcations are produced in speeds lower than critical speed when bow is included above some threshold values. Stable and unstable limit cycles are generated both for unbalanced or balanced rotors with bow, and their periodicity is investigated. A bifurcation set of the rotor-bearing systems is evaluated utilizing the pseudo-arc length continuation method with embedded collocation method, for the evaluation of periodic limit cycles (where these appear). Neimark-Sacker bifurcations are found to collide and vanish as the bow magnitude changes from high to low values. Bow phase angle with respect to unbalance force is found not to influence the respective bifurcation sets.