# Planetary Gear

Gear train with sun, planet, and ring gears

**Library:**Simscape / Driveline / Gears

## Description

The Planetary Gear 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 to be negligible. The block enables you to specify the inertias of the
internal planet gears. To model the inertias of the carrier, sun, and ring gears,
connect Simscape
Inertia blocks to ports
**C**, **S**, and **R**.

### Thermal Model

You can model
the effects of heat flow and temperature change by enabling the optional thermal port. To enable
the port, set **Friction model** to ```
Temperature-dependent
efficiency
```

.

### Equations

**Ideal Gear Constraints and Gear Ratios**

The Planetary Gear block imposes two kinematic and two geometric constraints,

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

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

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

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

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 where there is no torque loss, *τ _{loss}* = 0.

**Nonideal Gear Constraints and Losses**

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

### Assumptions and Limitations

Gear inertia is assumed to be negligible.

Gears are treated as rigid components.

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

## Ports

### Conserving

## Parameters

## More About

## Extended Capabilities

## See Also

### Simscape Blocks

**Introduced in R2011a**