A high-order model for in-plane vibrations of rotating rings on elastic foundation

A new high-order model for in-plane vibrations of rotating rings is developed in this paper. The inner surface of the ring is connected to an immovable axis through an elastic foundation (distributed springs), whereas the outer surface is traction free. The developed model enables the dynamic analysis of the rings on stiff elastic foundation that rotate with a high speed. The traction force at the inner surface of such rings is so high that it influences significantly the through-thickness stress distribution. This boundary effect cannot be captured by the classical low order theories while the model proposed in this paper can account for this effect. Nonlinear equations of motion are first derived, considering the geometrical nonlinearity of the system while assuming the linear elastic behaviour of the ring material. The formulation accounts for the stress caused by rotation and the significant normal and tangential traction forces at the inner surface of the ring. The displacement fields are assumed to be polynomials of the through-thickness coordinate in both the radial and circumferential directions. The derivation is generic and can yield ring theories of different order, i.e. of the Timoshenko-type and beyond, with proper consideration of both the internal state of the body and the boundary effects at the surfaces. Two types of critical speeds are investigated, namely the one at which the free vibrations become unstable and the one at which the forced vibration of a rotating ring subjected to a constant stationary point load experiences resonance. A comparison is presented of the predictions of the developed model to those of the lower order theories. It is shown that even for thin rings on elastic foundation, high order corrections, beyond the ones of the Timoshenko theory, need to be considered for an accurate estimation of the critical speeds of rotating rings. The new high-order model is superior to the existing ring models in predicting dynamic behaviour of either stationary or rotating rings. Without loss of generality, the model is applicable to both plane strain and plane stress configurations.