# Compound Planetary Gear

Planetary gear train with stepped planet gear set

**Library:**Simscape / Driveline / Gears

## Description

The Compound Planetary Gear block represents a planetary gear train with composite planet gears. Each composite planet gear is a pair of rigidly connected and longitudinally arranged gears of different radii. One of the two gears engages the centrally located sun gear while the other engages the outer ring gear.

**Compound Planetary Gear**

The block models the compound planetary gear as a structural component based on the Simscape™ Driveline™ Sun-Planet and Ring-Planet blocks. The figure demonstrates the equivalent block diagram for the Compound planetary gear block.

To increase the fidelity of the gear model, 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 Compound Planetary Gear block imposes two kinematic and two geometric constraints.

$${r}_{C}{\omega}_{C}={r}_{S}{\omega}_{S}+{r}_{P1}{\omega}_{P},$$

$${r}_{R}{\omega}_{R}={r}_{C}{\omega}_{C}+{r}_{P2}{\omega}_{P},$$

$${r}_{C}={r}_{S}+{r}_{P1},$$

$${r}_{R}={r}_{C}+{r}_{P2},$$

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 1._{P1}*ω*is the angular velocity of the planet gears._{P}*r*is the radius of planet gear 2._{P2}*r*is the radius of the ring gear._{R}

The ring-planet and planet-sun gear ratios are:

$${g}_{RP}={r}_{R}/{r}_{P2}={N}_{R}/{N}_{P2}$$

and

$${g}_{PS}={r}_{P1}/{r}_{S}={N}_{P1}/{N}_{S},$$

where:

*g*is the ring-planet gear ratio._{RP}*N*is the number of teeth on the ring gear._{R}*N*is the number of teeth on planet gear 2._{P2}*g*is the planet-sun gear ratio._{PS}*N*is the number of teeth on planet gear 1._{P1}*N*is the number of teeth on the sun gear._{S}

In terms of the gear ratios, the key kinematic constraint is:

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

The four degrees of freedom reduce to two independent degrees of freedom. The
gear pairs are (1, 2) = (*P2*, *R*) and
(*S*, *P1*).

**Warning**

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

The torque transfers are:

$${g}_{RP}{\tau}_{P2}+{\tau}_{R}\u2013{\tau}_{loss}\left(P2,R\right)\text{}=\text{}0$$

and

$${g}_{PS}{\tau}_{S}+{\tau}_{P1}\u2013{\tau}_{loss}\left(S,P1\right)=0,$$

where:

*τ*is torque transfer for planet gear 2._{P2}*τ*is torque transfer for the ring gear._{R}*τ*is torque transfer loss._{loss}*τ*is torque transfer for the sun gear._{S}*τ*is torque transfer for planet gear 1._{P1}

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**