Three kinds of resonant modes of a single layered circular elastic cylinder embedded in the elastic medium are analysed by considering the oscillation of the scatter’s core, based on the fact that the core moves as a rigid body when the shell material is very compliant. The resonant frequencies of the single resonator acquired by our method are in good agreement with those calculated by the local interaction simulation approach (LISA) for the local resonant phononic crystal. Therefore, the local resonant characteristics of a single layered circular elastic cylinder can be used to evaluate the resonant frequencies of the phononic crystal. The effects of the geometrical and physical parameters of the shell and the core are also studied in details. This work is significant for designing the locally resonant phononic crystal based on the local resonant characteristics of the single resonator, and the resonant frequencies can be tuned by selecting the geometrical sizes and the materials.