LOCAL STABILITY OF GYROSCOPIC SYSTEMS NEAR VANISHING EIGENVALUES

Vanishing eigenvalues of a gyroscopic system are always repeated and a s a result of this degeneracy, their eigenfunctions represent a combin ation of constant displacements with zero velocity and the displacemen ts derived from constant, nonzero velocity. In a second-order formulat ion of the equations of motion, the assumption of harmonic vibration i s not sufficiently general to represent this motion as the displacemen ts derived from constant, nonzero velocity are not included. In a firs t order formulation, however, the assumption of harmonic vibration is sufficient. Solvability criteria are required to determine the complet e form of such eigenfunctions and in particular whether or not their v elocities are identically zero. A conjecture Soi gyroscopic systems is proposed that predicts whether the eigenvalue locus is imaginary or c omplex in the neighborhood of a vanishing eigenvalue. If the velocitie s of all eigenfunctions with vanishing eigenvalues are identically zer o, the eigenvalues are imaginary; if any eigenfunction exists whose ei gen value is zero but whose velocity is nonzero, the corresponding eig envalue locus is complex. The conjecture is shown to be true for many commonly studied gyroscopic systems; no counter examples have yet been found. The conjecture can be used to predict divergence instability i n many cases without extensive computation.