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 disp lacements are coupled, even in the linearized equations of motion. The flexural displacements in the rotating frame of reference are subject ed to a forcing (external) excitation due to gravity, while the combin ation of unbalance and gravity causes both a forcing excitation and a parametric excitation to act on the torsional displacement. The genera l linearized equations governing coupled flexural and torsional oscill ations are presented, in which the angular velocity is allowed to be a function of time. Then, Galerkin's method is applied to a simply supp orted 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 invest igated: (a) the angular velocity is near twice a torsional frequency; (b) the angular velocity is near a torsional frequency; and (c) the an gular velocity is near the difference or sum of a torsional frequency and a flexural frequency.