Planetary gear with two sun gears and two planet gear sets

**Library:**Simscape / Driveline / Gears

The Ravigneaux Gear block represents a planetary gear train with dual sun and planet gear sets. The two sun gears are centrally located and separated longitudinally along a common rotation axis. The smaller of these gears engages an inner planet gear set, which in turn engages an outer planet gear set. The outer planet gear set, whose length spans the distance between the two sun gears, engages both the larger sun gear and the ring gear.

A carrier holds the planet gear sets in place at different radii. The carrier, which rigidly connects to a drive shaft, can spin as a unit with respect to the sun and ring gears. Revolute joints, each located between a planet gear and the carrier, enable the gears to spin about their individual longitudinal axes.

The relative angular velocities of the sun, planet, and ring gears follow from the kinematic constraints between them. For more information, see Equations.

The block models the Ravigneaux gear as a structural component based on Sun-Planet, Planet-Planet, and Ring-Planet Simscape™ Driveline™ blocks. The figure shows the block diagram of this structural component.

To increase the fidelity of the gear model, you can specify properties such as gear
inertia, meshing losses, and viscous losses. By default, gear inertia and viscous losses
are assumed negligible. The block enables you to specify the inertias of the internal
planet gears only. To model the inertias of the carrier, large sun, small sun, and ring
gears, connect Simscape
Inertia blocks to ports
**C**, **SL**, **SS**, and
**R**.

You can model
the effects of heat flow and temperature change by exposing an optional thermal port. To expose
the port, in the **Meshing Losses** settings, set the
**Friction** parameter to ```
Temperature-dependent
efficiency
```

.

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

$${r}_{Ci}{\omega}_{C}={r}_{SS}{\omega}_{SS}+{r}_{Pi}{\omega}_{Pi}$$

$${r}_{Ci}={r}_{SS}+{r}_{Pi}$$

$${r}_{Co}{\omega}_{C}={r}_{SL}{\omega}_{SL}+{r}_{Po}{\omega}_{Po}$$

$${r}_{Co}={r}_{SL}+{r}_{Po}$$

$$\left({r}_{Co}-{r}_{Ci}\right){\omega}_{C}={r}_{Pi}{\omega}_{Pi}+{r}_{Po}{\omega}_{Po}$$

$${r}_{Co}-{r}_{Ci}={r}_{Po}+{r}_{Pi}$$

$${r}_{R}\omega R={r}_{Co}{\omega}_{C}+{r}_{Po}{\omega}_{Po}$$

$${r}_{R}={r}_{Co}+{r}_{Po}$$

Where:

*r*is radius of the inner carrier gear._{Ci}*ω*is angular velocity of the carrier gears._{C}*r*is radius of the small sun gear._{SS}*ω*is angular velocity of the small sun gear._{SS}*r*is radius of the inner planet gear._{Pi}*ω*is angular velocity of the inner planet gear._{Pi}*r*is radius of the outer carrier gear._{Co}*r*is radius of the large sun gear._{SL}*ω*is angular velocity of the large sun gear._{SL}*r*is radius of the outer planet gear._{Po}*ω*is angular velocity of the outer planet gear._{Po}*ω*is angular velocity of the ring gear._{R}

The ring-to-sun ratios are:

$${g}_{RSS}={r}_{R}/{r}_{SS}={N}_{R}/{N}_{SS}$$

$${g}_{RSL}={r}_{R}/{r}_{SL}={N}_{R}/{N}_{SL}$$

Where:

*g*is the ring-to-small sun gear ratio._{RSS}*N*is the number of teeth in the ring gear._{R}*N*is the number of teeth in the small sun gear._{SS}*g*is the ring-to-large sun gear ratio._{RSS}*N*is the number of teeth in the large sun gear._{SL}

In terms of these gear ratios, the key kinematic constraints are:

$$\left({g}_{RSS}-1\right){\omega}_{C}={g}_{RSS}{\omega}_{R}-{\omega}_{SS}$$

$$\left({g}_{RSL}-1\right){\omega}_{C}={g}_{RSL}{\omega}_{R}-{\omega}_{SL}$$

The six degrees of freedom reduce to two independent degrees of freedom. The
gear pairs are (1,2) = (*LS*,*P*),
(*SS*,*P*),
(*P*,*R*), and
(*P*,*P*).

The gear ratio *g _{RSS}* must be
strictly greater than the gear ratio

The torque transfers are:

$${g}_{RSS}{\tau}_{SS}+{\tau}_{R}-{\tau}_{loss}\left(SS,R\right)=0$$

$${g}_{RSL}{\tau}_{SL}+{\tau}_{R}-{\tau}_{loss}\left(SL,R\right)=0$$

Where:

*τ*is torque transfer for the small sun gear._{SS}*τ*is torque transfer for the ring gear._{R}*τ*is torque transfer loss between the small sun gear and the ring gear._{loss}(SS,R)*τ*is torque transfer for the large sun gear._{SL}*τ*is torque transfer loss between the large sun gear and the ring gear._{loss}(SL,R)

In the ideal
case, there is no torque loss, that is *τ _{loss}* = 0.

In the nonideal case, *τ _{loss}* ≠ 0. For more information, see Model Gears with Losses.

Gears are assumed rigid.

Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.