The presence of cracks in solid structures can be detected by measuring changes in the natural frequencies of appropriate vibration modes. In the present work a simple energy-based model of a vibrating ring is presented to estimate the effect of a crack on the frequencies of repeated mode pairs. Introduction of a crack breaks the symmetry of the structure, causing the frequencies of the degenerate modes to split to a measurable extent. A simple expression is derived that relates the split in frequency to the frequency of the modes in the un-cracked ring, the dimensions of the ring and the crack length. The treatment is extended to the case of a ring that is not perfectly symmetrical prior to the introduction of a crack. In the latter case, when the crack is small, the mode pairs have a preferred orientation determined by the initial asymmetry. As the crack grows the mode pairs rotate to become aligned with the axis defining the location of the crack.