Dynamics of a beam with both axial moving and spinning motion: An example of bi-gyroscopic continua

Classical gyroscopic continua include axially moving materials and spinning structures. The gyroscopic effect of the axially moving material is pronounced via the gyroscopic coupling among the basis functions in the same motional direction. On the other hand, the gyroscopic coupling of the spinning structure acts in the two different directions of motion. In this paper, we study the dynamics of a beam with both axial moving motion and spinning motion as a prototype of bi-gyroscopic continua. The influence of bi-gyroscopic effects on the natural frequencies, modes, and stability is investigated by an analytical method applied to the discretized equations of the axially moving and spinning beam. Distinct bifurcation series of the eigenvalues and corresponding physical interpretations are discussed by numerical display of the modal motions. The complex modes describing both whirling motions and traveling waves are investigated in detail for such bi-gyroscopic system. New interesting phenomena have been analyzed numerically and important conclusions have been drawn for such bi-gyroscopic system.