Abstract: Based on the block structure of the assembled system matrix by gyroscopic mode synthesis technique, it is proved constructively that there is a series of the ( n -3) multiple eigenvalues in the spectrum of the gyroscopic eigenvalue problem for the rotarywing type structures, corresponding to the n repeated substructures mounted on the structure. The associated complete eigenvectors or modes are also obtained. The obtained analytical results are extended to the nonrotating, ...