Insights into the gyroscopic behaviour of axially and torsionally loaded rotating shafts

A distributed parameter model of a straight uniform shaft rotating at constant angular velocity is analytically investigated including the effects of transverse shear, rotatory inertia, gyroscopic moments and considering the additional contribution of combined end thrust and twisting moment. The equations of motion are derived by applying Hamilton’s principle according to the Timoshenko beam theory, and cast in dimensionless form to highlight the influence of the main governing parameters (slenderness ratio, angular velocity, applied external end thrust and twisting moment) on natural frequencies and critical speeds of the rotor. The results of this study constitute the basis for further developments, including comparison with finite element models and rotor stability analysis under combined loads.