Parametrically Excited Lateral Vibrations of an Asymmetric Slender Shaft in Asymmetrically Flexible Bearings

The lateral motion of a multi-degree-of-freedom shaft with cross-sectional asymmetry having common principal planes, mounted on asymmetrically flexible bearings having fixed common principal planes, is analysed in terms of a generating system consisting of a symmetric shaft with the mean cross-sectional rigidity mounted on the asymmetric bearings. Equations of motion are developed in terms of the principal co-ordinates and modes of the generating system, the shaft asymmetry producing coupling terms with periodic coefficients. The equations are solved by the perturbation-variation method of Hsu, leading to simple explicit formulae for instability bands in many practically important cases. The solutions are limited to small shaft asymmetry; any degree of bearing asymmetry is admissible.