Aiming at the problem of quantitatively identifying and tracing the source of machining chatter related to insufficient dynamic stiffness caused by weak faults of main components of complex nonlinear machine tools，a research method of dynamic stiffness characteristics based on rough identification of fault components based on PCA/D-S method（PCA：principal component analysis；D-S：evidence theory）and fine-grained identification based on FUKL fusion algorithm（fusion of fuzzy set and relative entropy）is proposed. In this method，the vibration characteristics of multi parts in the processing state are collected，and the time-frequency domain eigenvalues are separated，then the PCA is used to reduce the dimension to obtain the low-dimensional features with strong correlation. By calculating the synthetic evidence probability through D-S，the fault parts are roughly located，and then the fault identification results are further accurately calculated through FUKL fusion algorithm. The proposed method is applied to the fault traceability research of the actual chatter machine tool. From the four main components，the insufficient dynamic stiffness of the spindle system is identified with the synthetic evidence probability of 78.69%. The fault essence of the insufficient axial dynamic stiffness of the spindle system is identified by FUKL. The correctness of the operation results of the proposed algorithm is analyzed and verified by disassembling the faulty components and testing the actual load.