In order to analyze the fault characteristics of the cracked multi-stage gear transmission system， the fault components are analyzed. Therefore，the three-dimensional contact model of spur gears under fault conditions was established by finite element method，and the time-varying stiffness of gear meshing was calculated. Sequentially， the bearing stiffness as well as the flexibility of the transmission shaft were comprehensively considered， and a secondary gear with gear cracks was established. The dynamic model of the transmission system is used to solve the vibration response of different crack positions. Meanwhile， the variational mode decomposition （VMD） is used for the problems of the test signal to noise ratio and the uneven energy distribution. Combined with fast spectral kurtosis （FSK） to extract the fault components，the correlation between the crack location and the fault frequency was analyzed. Results show that due to the existence of cracks， the system vibration response appears periodic shocks in the time domain history. Frequency domain appears as side frequency modulation near the meshing frequency of the gear pair of this stage. The interval of the side spectrum lines is the frequency of the axis of the fault tooth. VMD-FSK method can effectively extract its fault characteristics in a noisy environment and has good robustness，providing a theoretical basis for the identification and location of gear failures.