The equations of motion of planetary gear sets including pinion dynamics are derived using the Lagrange method. The Lagrange method provides a systematic procedure for derivation and yields a single set of equations that are valid for all gears and shifts for a given configuration. This procedure is applied to the coupled planetary gear set in GM Hydramatic 440 transmission. The planetary gear set equations along with a simplified engine, torque converter, friction elements and vehicle model are simulated for 2–3 and 3–4 shifts. The simulation results demonstrate that the equations derived for planetary gear sets can be used for studying shift quality.

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