Parametric, external and combination resonances in coupled flexural and torsional oscillations of an unbalanced rotating shaft

If a rotating elastic shaft is unbalanced, flexural and torsional displacements are coupled, even in the linearized equations of motion. The flexural displacements in the rotating frame of reference are subjected to a forcing (external) excitation due to gravity, while the combination of unbalance and gravity causes both a forcing excitation and a parametric excitation to act on the torsional displacement. The general linearized equations governing coupled flexural and torsional oscillations are presented, in which the angular velocity is allowed to be a function of time. Then, Galerkin’s method is applied to a simply supported shaft. For the case of constant angular velocity, the method of multiple scales is utilized to obtain approximate motions of the shaft. The following resonances involving torsional oscillations are investigated: (a) the angular velocity is near twice a torsional frequency; (b) the angular velocity is near a torsional frequency; and (c) the angular velocity is near the difference or sum of a torsional frequency and a flexural frequency.