Primary and parametric resonances of asymmetrical rotating shafts with stretching nonlinearity

In this paper, primary and parametric resonances of a simply supported nonlinear rotating asymmetrical shaft with unequal mass moments of inertia and bending stiffness in the direction of principal axes are simultaneously considered. The nonlinearities are due to the stretching and large amplitude. The method of multiple scales is applied to the ordinary and partial differential equations of motion. The achieved results are in a very good agreement. The influences of inequality of mass moments of inertia and bending stiffness 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 asymmetrical rotating shaft are investigated. The loci of bifurcation points are plotted as functions of damping coefficient. The numerical simulation is utilized to verify the multiple scales method results. The results of multiple scales method are in a good agreement with those of numerical simulation.