Rotating, Axially Loaded Timoshenko Shaft: Modeling and Stability

For studying the stability of a rotating shaft subject to axial load, the derivation of correct stability equations is the essential preliminary problem. Here, the model of a uniform non-circular Timoshenko shaft under a compressive end load of constant magnitude is dealt with. Starting point is the nonlinear boundary value problem for coupled extensional-bending-torsional oscillations where a finite strain beam theory in a floating reference frame following the rigid body rotation is applied. First, the equation set describing the stationary shaft configuration is deduced. Next, the variational equations for small superimposed perturbations are derived. The only interesting stability problem for usual properties of the shaft cross section is constituted by a linear boundary value problem describing the bending vibrations. The corresponding characteristic equation is evaluated finally to find the critical buckling load also for the case of an oval shaft not considered before.