Synchronous whirling of spinning homogeneous elastic cylinders: linear and weakly nonlinear analyses

Stationary whirling of slender and homogeneous (continuous) elastic shafts rotating around their axis, with pin-pin boundary condition at the ends, is revisited by considering the complete deformations in the cross section of the shaft. The stability against a synchronous sinusoidal disturbance of any wavelength is investigated and the analytic expression of the buckling amplitude is derived in the weakly nonlinear regime by considering both geometric and material (hyper-elastic) nonlinearities. The bifurcation is supercritical in the long wavelength domain for any elastic constitutive law, and subcritical in the short wavelength limit for a limited range of nonlinear material parameters.