Stability of Gyroscopic Systems Near Vanishing Eigenvalues

Renshaw and Mote (1996) proposed a conjecture concerning the growth of vibrating eigensolutions of gyroscopic systems in the neighborhood of a vanishing eigenvalue when the system operators depend on an independent system parameter. Although the conjecture was not proved, it was supported by several examples drawn from well-known continuous physical systems. Lancaster and Kliem (1997), however, recently presented three two-degree-of-freedom counter examples. Unlike the examples tested by Renshaw and Mote (1996), these counter examples lack a definiteness property that is usually found in models derived from physical systems which appears to be essential to the conjecture. This Brief Note revises the original conjecture to include this definiteness criterion and proves the conjecture for general two-degree-of-freedom systems.