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Gear train with sun, planet, and ring gears

**Library:**Simscape / Driveline / Gears

This block models a gear train with sun, planet, and ring gears. Planetary gears are
common in transmission systems, where they provide high gear ratios in compact
geometries. A carrier connected to a drive shaft holds the planet gears. Ports
**C**, **R**, and **S** represent
the shafts connected to the planet gear carrier, ring gear, and sun gear.

The block models the planetary gear as a structural component based on Sun-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, sun, and ring gears, connect
Simscape
Inertia blocks to ports
**C**, **S**, 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** tab, set the **Friction
model** parameter to ```
Temperature-dependent
efficiency
```

.

The Planetary Gear block imposes two kinematic and two geometric constraints on the three connected axes and the fourth, internal gear (planet):

$${r}_{C}{\omega}_{C}={r}_{S}{\omega}_{S}+{r}_{P}{\omega}_{P}$$

$${r}_{C}={r}_{S}+{r}_{P}$$

$${r}_{R}{\omega}_{R}={r}_{C}{\omega}_{C}+{r}_{P}{\omega}_{P}$$

$${r}_{R}={r}_{C}+{r}_{P}$$

where:

*r*is the radius of the carrier gear._{C}*ω*is the angular velocity of the carrier gear._{C}*r*is the radius of the sun gear._{S}*ω*is the angular velocity of the sun gear._{S}*r*is the radius of planet gear._{P}*ω*is the angular velocity of the planet gears._{p}*r*is the radius of the ring gear._{R}

The ring-sun gear ratio is

$${g}_{RS}={r}_{R}/{r}_{S}={N}_{R}/{N}_{S}$$

Where *N* is the number of teeth on each
gear.

In terms of this ratio, the key kinematic constraint is:

$$(\text{1}+{g}_{RS}){\omega}_{C}={\omega}_{S}+{g}_{RS}{\omega}_{R}$$

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

**Warning**

The gear ratio *g*_{RS} must be
strictly greater than one.

The torque transfer is

$${g}_{RS}{\tau}_{S}+{\tau}_{R}\u2013{\tau}_{loss}=\text{}0$$

Where:

*τ*is torque transfer for the sun gear._{S}*τ*is torque transfer for the ring gear._{R}*τ*is torque transfer loss._{loss}

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.