Abstract This paper presents a methodology for the analysis of the free, in-plane, vibration of thin rings with profile variations in the circumferential direction. The methodology is suitable for any thin ring which is bounded by closed curves which are single valued functions of circumferential position. The inner and outer profiles are expressed as Fourier series, thus allowing any profile to be approximated with any degree of accuracy. An iterative numerical procedure for determining the true middle surface and the corresponding thickness at each cross-section around the circumference is established. A reduced (plane stress) form of Novozhilov's thin-shell theory is used to model the deformation mechanics of the ring. The eigenvalue problem is then formulated using the Rayleigh–Ritz method in conjunction with a harmonic series description of the displacements. General expressions are presented for the corresponding mass and stiffness matrices. A companion paper presents a comprehensive set of results which illustrates application of the theory.