# Analysis of the Dynamic Characteristics of Coaxial Counter-Rotating Planetary Transmission System
[23] studied the inherent vibration modes of Ravigneaux compound planetary gear trains (PGT), proposing a two degrees of freedom simplified dynamic model of planet gears considering nonlinearities, such as time-varying mesh stiffness and backlash, and obtaining numerical solutions for the bifurcation and dynamic instability of two groups of planetary gear systems. In summary, the dynamics models of PGTs are mainly divided into concentrated mass analysis models, finite element analysis models, and rigid-flexible coupling analysis models. In the concentrated mass analysis model, the masses of all components (sun gear, planet gears, ring gear, carrier) in the PGTs are concentrated at single points, and these concentrated mass points are interconnected by springs with stiffness and damping. The finite element model utilizes finite element software to conduct dynamic analysis of gear transmission systems to obtain contact stresses, bending stresses, etc., while the rigid-flexible coupling model is a modeling method that treats components with large deformations as flexible based on the concentrated mass method. The aforementioned studies on the dynamics of planet gears have mostly focused on single-stage transmissions or simple two-stage serial planetary transmissions with a single input and single output. There has been relatively limited research on coaxial counter-rotating planetary transmission systems composed of fixed-axes PGT and differential PGT. Although finite element analysis models and rigid-flexible coupling analysis models can extract more dynamic information, there are issues such as low efficiency when solving complex systems. Therefore, for complex mechanisms like coaxial counter-rotating planetary transmission systems, employing concentrated mass analysis models can most quickly extract dynamic information that is closer to reality. When using finite element analysis models and rigid-flexible coupling analysis models for calculations, it often takes several hours or even days. However, using the model proposed in this paper for numerical calculations only takes a few minutes. Moreover, when the parameters of the transmission system change, simply replacing the parameters in the program is sufficient, avoiding the complex process of repetitive modeling and grid partitioning required by the finite element method. From the results, which are provided in Section 5 of this paper, through the comparison between ADAMS dynamic simulation and the results of this paper, also proves the correctness of the model. Therefore, this paper investigates the coaxial counter-rotating planetary transmission system, establishing a dynamic model of the transmission system that considers time-varying mesh stiffness and transmission error excitations. The dynamic characteristics of the system are computed and analyzed using the Runge-Kutta algorithm. This includes studying dynamic meshing forces, planet gear displacements, and load-sharing coefficients (LSCs) under both internal and external excitations, as well as different input torques. The main original findings in this study can be directly employed as preliminary principles for the design of coaxial counter-rotating planetary transmission systems.