# 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 shows the equivalent circuit for the structural component.

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 only. 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 exposing an optional thermal port. To expose the port, in the Meshing Losses tab, set the Friction model parameter to ```Temperature-dependent efficiency```.

### Equations

Ideal Gear Constraints and Gear Ratios

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

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

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

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

and

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

where:

• rC is the radius of the carrier gear.

• ωC is the angular velocity of the carrier gear.

• rS is the radius of the sun gear.

• ωS is the angular velocity of the sun gear.

• rP1 is the radius of planet gear 1.

• ωP is the angular velocity of the planet gears.

• rP2 is the radius of planet gear 2.

• rR is the radius of the ring gear.

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:

• gRP is the ring-planet gear ratio.

• NR is the number of teeth on the ring gear.

• NP2 is the number of teeth on planet gear 2.

• gPS is the planet-sun gear ratio.

• NP1 is the number of teeth on planet gear 1.

• NS is the number of teeth on the sun gear.

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

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 gRP must be strictly greater than one.

The torque transfers are:

and

where:

• τP2 is torque transfer for planet gear 2.

• τR is torque transfer for the ring gear.

• τloss is torque transfer loss.

• τS is torque transfer for the sun gear.

• τP1 is torque transfer for planet gear 1.

In the ideal case, there is no torque loss, that is τloss = 0.

Nonideal Gear Constraints and Losses

In the nonideal case, τloss ≠ 0. For more information, see Model Gears with Losses.

## Limitations and Assumptions

• Gears are assumed rigid.

## Ports

### Conserving

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Rotational conserving port associated with the planet gear carrier.

Rotational conserving port associated with the ring gear.

Rotational conserving port associated with the sun gear.

Thermal conserving port associated with heat flow. Heat flow affects gear temperature, and therefore, power transmission efficiency.

#### Dependencies

This port is exposed when, in the Meshing Losses settings, the Friction parameter is set to `Temperature-dependent efficiency`.

Exposing this port also exposes related parameters.

## Parameters

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### Main

Fixed ratio, gRP, of the ring gear to the planet gear. This gear ratio must be strictly greater than `1`.

Fixed ratio, gPS, of the planet gear to the sun gear. This gear ratio must be strictly greater than `0`.

### Meshing Losses

Friction model for the block:

• `No meshing losses - Suitable for HIL simulation` — Gear meshing is ideal.

• `Constant efficiency` — Transfer of torque between gear wheel pairs is reduced by a constant efficiency, η, such that 0 < η ≤ 1.

• `Temperature-dependent efficiency` — Transfer of torque between gear wheel pairs is defined by table lookup based on the temperature.

#### Dependencies

If this parameter is set to:

• `Constant efficiency` — Related parameters are exposed.

• `Temperature-dependent meshing losses` — A thermal port and related parameters are exposed.

Array of torque transfer efficiencies, [ηSP, ηRP], for sun-planet and ring-carrier gear wheel pair meshings, respectively. The array element values must be greater than `0` and less than or equal to `1`.

#### Dependencies

This parameter is exposed when the Friction model parameter is set to ```Constant efficiency```.

Array of temperatures used to construct a 1-D temperature-efficiency lookup table. The array values must increase from left to right.

#### Dependencies

This parameter is exposed when Friction model is set to `Temperature-dependent efficiency`.

Array of mechanical efficiencies, ratios of output power to input power, for the power flow from the sun gear to the planet gear, ηSP. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Each array element values is the efficiency at the temperature of the corresponding element in the Temperature array. The number of elements in the Efficiency array must be the same as the number of elements in the Temperature array. The value of each Efficiency array element must be greater than `0` and less than or equal to `1`.

#### Dependencies

This parameter is exposed when the Friction model parameter is set to ```Temperature-dependent efficiency```.

Array of component efficiencies, ηRP—the ratio of output power to input power, that the block uses to construct a 1-D temperature-efficiency lookup table.

Each array element values is the efficiency at the temperature of the corresponding element in the Temperature array. The number of elements in the Efficiency array must be the same as the number of elements in the Temperature array. The value of each Efficiency array element must be greater than `0` and less than or equal to `1`.

#### Dependencies

This parameter is exposed when the Friction model parameter is set to ```Temperature-dependent efficiency```.

Array of power thresholds, pth, above which full efficiency factors apply, for the sun-carrier and planet-carrier, respectively. Below these values, a hyperbolic tangent function smooths the efficiency factor. For a model without thermal losses, the function lowers the efficiency losses to zero when no power is transmitted. For a model that considers thermal losses, the function smooths the efficiency factors between zero at rest and the values provided by the temperature-efficiency lookup tables at the power thresholds.

#### Dependencies

This parameter is exposed when the Friction model parameter is set to ```Constant efficiency``` or ```Temperature-dependent efficiency```.

### Viscous Losses

Array of viscous friction coefficients, [μS, μP], for the sun-carrier and planet-carrier gear motions, respectively.

### Inertia

Inertia model for the block:

• `Off` — Model gear inertia.

• `On` — Neglect gear inertia.

#### Dependencies

When this parameter is set to `On` exposes related parameters.

Moment of inertia of the combined planet gears. This value must be positive.

#### Dependencies

This parameter is exposed when the Inertia parameter is set to `On`.

### Thermal Port

These settings are exposed when, in the Meshing Losses settings, the Friction model parameter is set to `Temperature-dependent efficiency`.

Thermal energy required to change the component temperature by a single degree. The greater the thermal mass, the more resistant the component is to temperature change.

#### Dependencies

This parameter is exposed when, in the Meshing Losses settings, the Friction model parameter is set to `Temperature-dependent efficiency`.

Component temperature at the start of simulation. The initial temperature alters the component efficiency according to an efficiency vector that you specify, affecting the starting meshing or friction losses.

#### Dependencies

This parameter is exposed only if, in the Meshing Losses settings, the Friction model parameter is set to `Temperature-dependent efficiency`.

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