Based on the torsional vibration lumped-parameter model of the coaxial contra-rotating encased differential gear train

the natural frequencies and vibration modes of the system were solved. The results showed that all vibration modes of the planetary gear train can be categorized into three classes: rotational mode

stepped planet mode and planet mode

corresponding to the distinct frequency

L (Number of planet gear)-1 and N (Number of star gear)-1 multiple frequency respectively. The number of planets had no effect on the magnitude of the multiple frequency. When the change rate of stiffness was positive

the change rate of natural frequency was positive

and the change rate of natural frequency was negative when the change rate of moment of inertia was positive. Frequency loci veering occurred when two frequencies in the same vibration mode were approaching. The correctness of the theoretical analysis was verified by an example

providing a basis for the analysis and optimization of the natural characteristics design of the coaxial contra-rotating encased differential gear train.