Resonance analysis of gyroscopic nonlinear spinning shafts with parametric excitations and speed fluctuations

This study investigates simultaneous primary resonances due to imbalance periodic force and parametric resonances due to two sources of parametric excitations for a simply supported nonlinear spinning shaft with unequal mass moments of inertia and flexural rigidities in the direction of principal axes. The sources of parametric excitations are asymmetry of shaft and speed fluctuations. The sources of nonlinearity are the stretching and large amplitude. The method of multiple scales is applied for the analysis of the ordinary and partial differential equations of motion. Finally, the results are compared, out of which a reasonable agreement is concluded. The influences of speed fluctuations, inequality of mass moments of inertia and flexural rigidities in the direction of principal axes, inequality between two eccentricities corresponding to the principal axes and external damping on the steady state response of the spinning asymmetrical and symmetrical shafts are investigated. In addition, the effect of speed fluctuations on the stability and jump phenomenon are considered. The locus of bifurcation points for asymmetrical and symmetrical shafts once considering and once neglecting the effects of speed fluctuations are plotted as functions of damping coefficient. In order to verify the solutions the numerical simulations are utilized. The results of multiple scales method are in a good agreement with those of numerical simulation.