Abstract: In order to accurately analyze the magnetostriction phenomenon of transformer cores, a theoretical model of core vibration coupled with elastodynamics and thermodynamics is established, and a method for diagnosing loosening based on singularity decomposition is proposed. Firstly, by coupling the strain energy in elastodynamics and the elastic free energy in thermodynamics, a mathematical model for magnetostrictive vibrations characterising the multi-band core of a transformer is developed. Secondly, the effect of a non-linear force-magnetic coupling mechanism based on elastic dynamics is taken into consideration, a vibration multi-band multi-physical field simulation model is established, and multi-band vibration characteristics of the core from 0 Hz to 1000 Hz are obtained. Thirdly, a magnetostrictive vibration acquisition platform for transformer cores is built and vibration monitoring tests are carried out on 10 kV transformer cores. The experimental findings show that, besides the primary frequency of 100 Hz, the core vibration also contains vibration signals from 200 Hz to 600 Hz, with the measured results of 0.009 m/s² and 0.023 m/s² for the A and B phases, and the simulation results of 0.009 m/s² and 0.023 m/s². The simulations agree with the experimental results and demonstrate the correctness of the simulation model. Afterwards, a core loosening fault model is developed to obtain the vibration characteristics in the partially and fully loosened state. Finally, the vibration signals are used to diagnose faults based on different states of the iron core, such as in normal, partially loose, and completely loose states. A method which is based on the singular decomposition theory is proposed, and characteristic values of 0.0524, 0.2822, and 1.5059, respectively can be obtained. It is confirmed that, as the looseness worsens, the characteristic values increase, validating the feasibility of the diagnostic approach.