This paper employs graph theory to analyze kinematic relationships, static force, and power flow for planetary gear systems. We start from the graphs of these trains to determine all the structurally distinct kinematic inversions. We then obtain all the constructive solutions resulting from every possible combination of gear configuration. Based on the kinematic and static force analysis model, related matrices are acquired. Hence, a kinematic and static force analysis of the planetary gear mechanism is achieved. The link power can be determined by the link speed and torque. Power flow diagrams of each gear are acquired. The efficiency is calculated by the transmission ratio method. This analysis is useful in the design of power transmission applications. A planetary gear system of a 6HP26 automatic transmission is further illustrated for the motion, static force, and efficiency calculations by using graph theory technique and matrix equations. This paper provides a guide for engineers in gear transmission fields to easily finish mechanical transmission design.

The lever method facilitates a straightforward analysis of the motion dynamics of an automatic transmission under various gear settings, offering a simple and intuitive approach to examining planetary gear transmissions. Specifically, the equivalent lever diagram for a parallel-double planetary gear transmission comprises four component points. From a control perspective, a transmission system with two planetary rows can accommodate up to two reduction gears, one direct gear, one overdrive gear, and one reverse gear. In contrast, the equivalent lever diagram for a parallel three-planetary gear transmission includes five component points, with such systems typically supporting up to three reduction gears, one direct gear, two overdrive gears, and two reverse gears. This methodological approach underscores the lever diagram’s utility in delineating the functional capabilities of complex transmission systems.

Using graph theory, computer aided motion, static force and power flow analysis of planetary gear mechanism is realized. The motion, static force and efficiency calculations of 6HP26 automatic transmission are carried out. Based on kinematic and static force analysis model, related matrices are acquired. Hence, kinematic and static force analysis of planetary gear mechanism is achieved. The link power can be determined by the link speed and torque. Power flow diagrams of each gear are acquired. The efficiency is calculated by transmission ratio method.

Dynamic analysis can provide a theoretical basis for controlling vibration and noise in gear systems, as well as calculating strength under dynamic conditions. This paper takes 6HP26 automatic transmission as the research object, takes the fifth-gear transmission as an example, adopts the centralized mass method to establish its translation-torsion coupling dynamic model, and analyzes the displacement response, torsional linear acceleration time-domain response and spectrogram of each component. This provides a reference for reducing vibration and noise in the system.