In-plane dynamics of high-speed rotating rings on elastic foundation

Rotating ring-like structures are very commonly used in civil, mechanical and aerospace engineering. Typical examples of such structures are components in turbomachinery, compliant gears, rolling tyres and flexible train wheels. At the micro-scale, rotating ring models find their applications in the field of ring gyroscopes, in which high accuracy of modelling is required. The in-plane vibrations of rotating rings are of particular interest since such structural components are usually subject to in-plane loads. The focus in this thesis is therefore placed on the in-plane dynamics of rotating rings. While the radial and circumferential motions of a stationary ring are coupled due to curvature, a steadily rotating ring, as any gyroscopic system, is subject to two additional fictitious forces induced by the gyroscopic coupling due to rotation, i.e. the Coriolis and centrifugal forces. Among them the centrifugal force associated with the steady rotation of the ring (quasi-static force) introduces an axi-symmetric radial expansion and a hoop stress; the latter has the tendency to stiffen the ring. In contrast, the dynamic part of the centrifugal force has the tendency to soften the system. Next to that, the Coriolis force bifurcates the natural frequencies of the ring. The proper consideration of the rotation effects is essential to determine the dynamic behaviour of rotating rings, such as stability of free vibrations and resonance of rotating rings under stationary loads. Although various models exist, the considerations of rotation effects are not always in agreement, resulting in distinct theoretical predictions of critical speeds associated with instability and resonance of rotating rings. In addition, in all the existing rotating ring models, the equations of motion were derived assuming the inner and outer surfaces of the ring to be traction-free. However, when one considers a ring whose inner surface is elastically restrained by distributed springs, this assumption is violated. The traction at the inner surface can significantly influence the stress distribution along the thickness of the ring and this effect has to be properly accounted for since the internal stresses may show a strong gradient from the inner surface to the outer surface, especially in the case of rings rotating at high speeds or when the latter are supported by stiff foundation. The primary aim of this thesis is to develop a highly accurate rotating ring model that properly accounts for both the rotation and boundary effects with rigorous mathematical derivation to fill the gap regarding the modelling and prediction of the dynamic behaviour of high-speed rotating rings. To achieve this aim, the following four objectives are set: (i) identify the reasons of disagreements between various existing rotating ring models and clarify the mathematically sound derivations of governing equations; (ii) develop a high-order rotating ring model which properly accounts for the rotation effects, as well as the non-zero tractions at boundaries; (iii) close the debate on the prediction of critical speeds associated with instability of free vibrations and resonance of forced vibrations; and (iv) apply the developed high-order model to predict the steadystate response of rotating rings under stationary loads and the stability of rotating ringstationary oscillator system.