Frequency split and vibration localization in imperfect rings

The dynamics of linearly elastic, imperfect rings vibrating in their own plane is considered in this paper. Imperfections are modeled as perturbations of the uniform linear mass density and bending stiffness of a perfect ring. A perturbation expansion and a spectral representation are employed, and a variational formulation of the vibration problem is obtained. A linear theory is deduced by retaining only the leading-order terms in the variational formulation. The linear theory yields simple, closed-form expressions for the eigenfrequencies and the modal shapes, which are accurate when the imperfections are sufficiently small. An enhanced, nonlinear theory is also derived, which is accurate even when the ring imperfections are not small: in this case, an iterative solution procedure is developed. The proposed theories are validated by considering some case-study problems and using the Ritz-Rayleigh solution as a benchmark. Finally, the linear theory is applied to the frequency trimming problem of an imperfect ring. A simple, closed-form expression for the trimming masses is presented, valid for trimming any selected number of eigenmodes.