# Double-Pinion Planetary Gear

Planetary gear train with two meshed planet gear sets

• Library:
• Simscape / Driveline / Gears

## Description

This block represents a planetary gear train with two meshed planet gear sets between the sun gear and the ring gear. A single carrier holds the two planet gear sets at different radii from the sun gear centerline, while allowing the individual gears to rotate with respect to each other. The gear model includes power losses due to friction between meshing gear teeth and viscous damping of the spinning gear shafts.

Structurally, the double-pinion planetary gear resembles a Ravigneaux gear without its second, large, sun gear. The inner planet gears mesh with the sun gear and the outer planet gears mesh with the ring gear. Because it contains two planet gear sets, the double-pinion planetary gear reverses the relative rotation directions of the ring and sun gears.

The teeth ratio of a meshed gear pair fixes the relative angular velocities of the two gears in that pair. The parameter settings provide two parameters to set the ring-sun and outer planet-inner planet gear teeth ratios. A geometric constraint fixes the remaining teeth ratios—ring-outer planet and inner planet-sun. This geometric constraint requires that the ring gear radius equal the sum of the sun gear radius with the inner and outer planet gear diameters:

`${r}_{r}={r}_{s}+2\cdot {r}_{pi}+2\cdot {r}_{po},$`

where:

• rr is the ring gear radius

• rs is the sun gear radius

• rpi is the inner planet gear radius

• rpo is the outer planet gear radius

In terms of the ring-sun and outer planet-inner planet teeth ratios, the ring-outer planet teeth ratio is

`$\frac{{r}_{r}}{{r}_{po}}=2\cdot \frac{\frac{{r}_{r}}{{r}_{s}}}{\left(\frac{{r}_{r}}{{r}_{s}}-1\right)}\cdot \frac{\left(\frac{{r}_{po}}{{r}_{pi}}+1\right)}{\frac{{r}_{po}}{{r}_{pi}}},$`

The inner planet-sun teeth ratio is

`$\frac{{r}_{pi}}{{r}_{s}}=\frac{\left(\frac{{r}_{r}}{{r}_{s}}-1\right)}{2\left(\frac{{r}_{po}}{{r}_{pi}}+1\right)},$`

The block is a composite component. It contains three underlying blocks—Ring-Planet, Planet-Planet, and Sun-Planet—connected as shown in the figure. Each block connects to a separate drive shaft through a rotational conserving port.

### 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 settings, set the Friction parameter to ```Temperature-dependent efficiency```.

## 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

Gear ratio between the ring and sun gears. This ratio is the number of teeth in the ring gear divided by the number of teeth in the sun gear.

Gear ratio between the outer-planet and inner-planet gears. This ratio is the number of teeth in the outer planet divided by the number of gear teeth in the inner planet.

### 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, ηRPP], for sun-planet, and ring-planet, and planet-planet 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 mechanical efficiencies, ratios of output power to input power, for the power flow from the inner planet gear to the outer planet gear, ηPP. 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 power thresholds above which the full efficiency factors apply. Enter the thresholds in the order sun-carrier, ring-carrier, and planet-carrier. A hyperbolic tangent function smooths the efficiency factors between zero when at rest and the values provided by the temperature-efficiency lookup tables when at the power threshold.

As a guideline, the power threshold should be lower than the expected power transmitted during simulation. Higher values might cause the block to underestimate efficiency losses. Very low values might, however, raise the computational cost of simulation.

#### 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, μR, μP], for the sun-carrier, ring-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 inner planet gears. This value must be positive.

#### Dependencies

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

Moment of inertia of the outer 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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