In Situ Identification of Natural Frequency Branches in Gyroscopic Systems via Autoresonance and Phase-Locked Loop

The methods described allow one to directly measure the individual branches of the Campbell diagram of a physical gyroscopic system at any rotation speed. Typically, such data are acquired by exciting the vibration modes through naturally occurring unbalance forces. During run-up, these forces expose some of the Campbell diagram, but mainly the forward whirling branches, leaving the backward whirling branches mostly hidden. Furthermore, good modal frequency data are only acquired at critical speeds. The methods proposed here allow one to excite either a forward or backward whirling mode at any rotation speed in a precisely controlled manner, greatly improving the quality of an acquired Campbell diagram. The technique employs an external excitation device that automatically produces oscillating forces at a chosen modal frequency. Control is based on the autoresonance feedback algorithm, which can excite a mechanical system at resonance effectively. It will also be shown that with two actuators and two sensors, one can choose which bending mode to excite at resonance in either the forward or backward whirling direction. As the rotation speed is then gradually increased, one can measure the speed dependence of the resonance frequency. Furthermore, when combining autoresonance with a phase-locked loop, one can acquire very clean measurements by removing most of the noise generated by the unbalance and other sources. The technique is demonstrated analytically, numerically, and experimentally.