An Old Problem Reconsidered: The Wahl–Fischer Torsional Instability Problem

The phenomenon of torsional instabilities found in rotating mechanical systems powered by induction motors was encountered in the late 1930s by Wahl and Fisher in a turnpike exhaust system. Subsequently, this phenomenon has been repeatedly found to affect a number of similar physical systems and has led to a number of studies and analysis that have clarified the reasons for these instabilities, associated with the torque characteristics of induction motors. Surprisingly, none of these studies have presented a mathematical qualitative analysis of the eigenvalues of the differential equations that describe such electromechanical systems. This is the central purpose of this paper. It depends on the identification and exploitation of a particular “central” solution to the differential equations that describe the system, and of its relationship to a simpler formulation of the same system. This turns out to be a rather modest mathematical endeavor which, however, yields result that illuminate the nature of the instabilities encountered and provides a designer of such systems with the tools to abate or avoid these instabilities.