Standard forms of the eigenvalue problems associated with gyroscopic systems

Abstract The eigenvalue problem arising in the free vibration and stability analysis of gyroscopic systems is associated with a λ-matrix in which λ as well as its square appears. The characteristic polynomial of a non-dissipative gyroscopic system however, is a function of λ 2 , and an equivalent standard eigenvalue formulation involving only λ 2 as an eigenvalue, therefore, seems to be a more appropriate representation of the physical system. Such forms and the related transformations are discussed herein. Similarly, when the loading parameter also appears explicitly in the λ-matrix, an equivalent double-eigenvalue problem involving λ 2 and the loading parameter as eigenvalues is generated. The extremum properties of the Rayleigh Quotient leads to a convenient proof of an upper bound theorem on λ 2 . The use of left and right eigenvectors and the new double-eigenvalue formulation allows for the establishment of a flutter condition similar to one obtained for circulatory systems earlier. An example illustrating some of the concepts is presented.