The widespread use of distribution transformers in power grids has significantly increased the maintenance workload. Traditionally, the operational condition of distribution transformers has been assessed by maintenance personnel through direct auditory judgment, which is prone to interference from external noise and subjective human factors. To address this issue, the paper introduces a method that takes advantage of the unique vibration characteristics of various components in distribution transformers. It employs a blind source separation algorithm to extract the vibration signals specific to the on-load tap changer (OLTC). Considering the nonlinear and chaotic nature of OLTC vibration signals, the paper introduces a phase space reconstruction technique for nonlinear chaotic signal analysis, based on the extraction of geometric features from the multi-dimensional space trajectory. This method extracts characteristic maps and geometric parameters of distribution transformers under different mechanical states. To enable fault state identification, A two-dimensional membership function is constructed by applying the phase space reconstruction technique. Fuzzy inference is then performed using predefined fuzzy rules, and characteristic parameters are calculated to build a comprehensive parameter library for state recognition. The proposed method is validated using field test data, achieving an average classification success rate exceeding 91.25%, thereby demonstrating its effectiveness in accurately identifying the operational state of OLTC.