Ravigneaux planetary gearset

The Ravigneaux gearset is a double planetary gear set commonly used in automatic transmissions. This planetary gear set is constructed from two gear pairs, ring-planet and planet-planet. The Ravigneaux set has two sun gear wheels, a large sun and a small sun, and a single carrier gear with two independent planetary gear wheels connected to it, an inner planet and an outer planet. The carrier is one wheel but has two radii to couple with the inner and outer planets, respectively. The two planet gears rotate independently of the carrier but co-rotate with a fixed gear ratio with respect to each other. The inner planet couples with the small sun gear and co-rotates at a fixed gear ratio with respect to it. The outer planet couples with the large sun gear and co-rotates with a fixed gear ratio with respect to it. Finally, the ring gear also couples and co-rotates with the outer planet in a fixed gear ratio with respect to it.

Axis Motions and Constraints
The Ravigneaux block imposes four kinematic and four geometric constraints on the four connected axes and the two internal wheels (inner and outer planets):

rCiωC = rSsωSs + rPiωPi, rCi = rSs + rPi

rCoωC = rSlωSl + rPoωPo, rCo = rSl + rPo

(rCo - rCi)ωC = rPiωPi + rPoωPo, rCo - rCi= rPo + rPi

rRωR = rCoωC + rPoωPo, rR = rCo + rPo

In terms of the ring-to-small sun good ratio gRSs = rR/rSs and the ring-to-large sun gear ratio gRSl = rR/rSl, the key kinematic constraints are

(gRSs – 1)ωC = gRSs·ωR - ωSs

(gRSl + 1)ωC = gRSl·ωR + ωSl

The six degrees of freedom are reduced to two independent degrees of freedom.

The gear ratios are also the ratios of the number of teeth on each gear and the ratios of torques in each axis, gRSl = NR/NSl = τR/τSl and gRSs = NR/NSs = τR/τSs.