The reaction of a radial flexible ring with flanges in the supports of machine rotors to radial displacement has been studied. Two models of deformation and contact interaction of an elastic element with two rigid bodies positioned inside and outside of the ring have been developed. These models are based on variational statements with respect to the unknown forces and displacements. The options of gap and offset on both sides of the contact between the ring and the rigid bodies have been considered. A substantially nonlinear static reaction of the ring to radial displacement has been established. Radial reaction curves have been obtained across a wide range of design parameters, enabling a well-justified choice of an elastic damper with the desired stiffness properties.