The Sommerfeld effect of second kind: passage through parametric instability in a rotor with non-circular shaft and anisotropic flexible supports

Abstract

A rotor system with asymmetry in its bending flexibility as well as in its support bearings has many stable and unstable speed ranges. The dynamics of such a system is governed by differential equations with time-varying parametric coefficients which lead to parametric instability in certain rotor speed ranges. In this study, an asymmetric rotor shaft having a rotor disc placed firmly on its centre is considered. The rotor shaft is mounted on two bearings at its two ends having stiffness asymmetry. A permanent magnet-type DC motor is used to drive the rotor. When rotor approaches the lower limit of any unstable speed range, the rotor spin speed is captured at that speed with increasing whirl amplitudes and the rotor speed does not respond to an increase in the motor power, unless there is sufficient surplus power to accelerate the rotor through the corresponding unstable speed range. With this surplus power, the escape from lower instability limit speed to a much higher stable speed takes place as a nonlinear jump phenomenon. In certain situations, escape from one unstable speed range may lead to capture at another unstable speed range or simultaneous escape from the next unstable range. This specifically occurs due to the presence of the non-ideal drive and is termed here as the Sommerfeld effect of second kind. Unlike regular Sommerfeld effect (of first kind) where the power scarcity at the resonance is the cause of speed capture; there is, ideally, no need for a residual rotor unbalance in the Sommerfeld effect of second kind. In both the cases of speed capture, the excess motor power is spent to increase the whirl amplitudes. The Sommerfeld effect of the first kind relates to the resonance at the synchronous rotor whirl (critical speeds), whereas that of the second kind relates to instability of the rotor whirl. Due to such nonlinear jumps, some of the speed ranges where the rotor is theoretically stable under ideal or mathematical conditions may not be reached in practice, i.e. with a real drive which is naturally non-ideal. The dynamics of this rotor system is analytically and numerically studied in this article. The numerical simulations are performed with a multi-energy domain bond graph model which guarantees energetic consistency of the model.