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:

$r_{r} = r_{s} + 2 \cdot r_{p i} + 2 \cdot r_{p o},$

where:

r_r is the ring gear radius
r_s is the sun gear radius
r_pi is the inner planet gear radius
r_po 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

$\frac{r_{r}}{r_{p o}} = 2 \cdot \frac{\frac{r_{r}}{r_{s}}}{(\frac{r_{r}}{r_{s}} - 1)} \cdot \frac{(\frac{r_{p o}}{r_{p i}} + 1)}{\frac{r_{p o}}{r_{p i}}},$

The inner planet-sun teeth ratio is

$\frac{r_{p i}}{r_{s}} = \frac{(\frac{r_{r}}{r_{s}} - 1)}{2 (\frac{r_{p o}}{r_{p i}} + 1)},$

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 tab, set the Friction model parameter to Temperature-dependent efficiency.

Ports

Conserving

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`C` — Planet gear carrier
rotational mechanical

Rotational conserving port associated with the planet gear carrier.

`R` — Ring gear
rotational mechanical

Rotational conserving port associated with the ring gear.

`S` — Sun gear
rotational mechanical

Rotational conserving port associated with the sun gear.

`H` — Heat flow
thermal

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

`Ring (R) to sun (S) teeth ratio (NR/NS)` — Ring-to-sun gear ratio
`2` (default)

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.

`Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi)` — Outer planet-to-inner planet gear ratio
`1` (default)

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` — Friction model
`No meshing losses - Suitable for HIL simulation` (default) | `Constant efficiency` | `Temperature-dependent efficiency`

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.

`Sun-planet, ring-planet and planet-planet ordinary efficiencies` — Sun-planet, ring-planet and planet-planet Torque transfer efficiencies
`[.98, .98, .99]` (default) | array | `0` < η ≤ `1`

Array of torque transfer efficiencies, [η_SP, η_RP, η_RPP], 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.

`Temperature` — Temperature
`[280, 300, 320]` `K` (default) | array of increasing values

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.

`Sun-planet efficiency` — Sun-planet efficiency
`[.95, .9, .85]` (default) | array | `0` < η_SP ≤ `1`

Array of mechanical efficiencies, ratios of output power to input power, for the power flow from the sun gear to the planet gear, η_SP. 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.

`Ring-planet efficiency` — Ring-planet efficiency
`[.95, .9, .85]` (default) | array | `0` < η_RP ≤ `1`

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

Dependencies

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

`Planet-planet efficiency` — Planet-planet efficiency
`[.5, .45, .4]` (default) | array | `0` < η_SP ≤ `1`

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, η_PP. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Dependencies

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

`Sun-carrier, ring-carrier and planet-carrier power thresholds` — Sun-carrier, ring-carrier and planet-carrier power thresholds
`[.001, .001, .001]` `W` (default) | array

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

`Sun-carrier, ring-carrier, and planet-carrier viscous friction coefficients` — Gear viscous friction
`[0, 0, 0]` `N*m/(rad/s)` (default) | array

Array of viscous friction coefficients, [μ_S, μ_R, μ_P], for the sun-carrier, ring-carrier, and planet-carrier gear motions, respectively.

Inertia

`Inertia` — Inertia model
`off` (default) | `on`

Inertia model for the block:

Off — Model gear inertia.
On — Neglect gear inertia.

Dependencies

When this parameter is set to On exposes related parameters.

`Inner planet gear inertia` — Inner planet gear inertia
`0.001` `kg*m^2` (default) | positive scalar

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.

`Outer planet gear inertia` — Outer planet gear inertia
`0.001` `kg*m^2` (default) | positive scalar

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 mass` — Thermal mass
`50` `J/K` (default) | scalar

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.

`Initial temperature` — Initial temperature
`300` `K` (default) | scalar

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.

More About

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Hardware-in-the-Loop Simulation

For optimal simulation performance, in the Meshing Losses settings, set the Friction model parameter to the default value, No meshing losses - Suitable for HIL simulation.

Documentation

Double-Pinion Planetary Gear

Description

Thermal Model

Ports

Conserving

C — Planet gear carrier rotational mechanical

R — Ring gear rotational mechanical

S — Sun gear rotational mechanical

H — Heat flow thermal

Dependencies

Parameters

Main

Ring (R) to sun (S) teeth ratio (NR/NS) — Ring-to-sun gear ratio 2 (default)

Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi) — Outer planet-to-inner planet gear ratio 1 (default)

Meshing Losses

Friction model — Friction model No meshing losses - Suitable for HIL simulation (default) | Constant efficiency | Temperature-dependent efficiency

Dependencies

Sun-planet, ring-planet and planet-planet ordinary efficiencies — Sun-planet, ring-planet and planet-planet Torque transfer efficiencies [.98, .98, .99] (default) | array | 0 < η ≤ 1

Dependencies

Temperature — Temperature [280, 300, 320] K (default) | array of increasing values

Dependencies

Sun-planet efficiency — Sun-planet efficiency [.95, .9, .85] (default) | array | 0 < ηSP ≤ 1

Dependencies

Ring-planet efficiency — Ring-planet efficiency [.95, .9, .85] (default) | array | 0 < ηRP ≤ 1

Dependencies

Planet-planet efficiency — Planet-planet efficiency [.5, .45, .4] (default) | array | 0 < ηSP ≤ 1

Dependencies

Sun-carrier, ring-carrier and planet-carrier power thresholds — Sun-carrier, ring-carrier and planet-carrier power thresholds [.001, .001, .001] W (default) | array

Dependencies

Viscous Losses

Sun-carrier, ring-carrier, and planet-carrier viscous friction coefficients — Gear viscous friction [0, 0, 0] N*m/(rad/s) (default) | array

Inertia

Inertia — Inertia model off (default) | on

Dependencies

Inner planet gear inertia — Inner planet gear inertia 0.001 kg*m^2 (default) | positive scalar

Dependencies

Outer planet gear inertia — Outer planet gear inertia 0.001 kg*m^2 (default) | positive scalar

Dependencies

Thermal Port

Thermal mass — Thermal mass 50 J/K (default) | scalar

Dependencies

Initial temperature — Initial temperature 300 K (default) | scalar

Dependencies

More About

Hardware-in-the-Loop Simulation

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

See Also

Simscape Blocks

Topics

Simscape Driveline Documentation

Support

`C` — Planet gear carrier
rotational mechanical

`R` — Ring gear
rotational mechanical

`S` — Sun gear
rotational mechanical

`H` — Heat flow
thermal

`Ring (R) to sun (S) teeth ratio (NR/NS)` — Ring-to-sun gear ratio
`2` (default)

`Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi)` — Outer planet-to-inner planet gear ratio
`1` (default)

`Friction model` — Friction model
`No meshing losses - Suitable for HIL simulation` (default) | `Constant efficiency` | `Temperature-dependent efficiency`

`Sun-planet, ring-planet and planet-planet ordinary efficiencies` — Sun-planet, ring-planet and planet-planet Torque transfer efficiencies
`[.98, .98, .99]` (default) | array | `0` < η ≤ `1`

`Temperature` — Temperature
`[280, 300, 320]` `K` (default) | array of increasing values

`Sun-planet efficiency` — Sun-planet efficiency
`[.95, .9, .85]` (default) | array | `0` < η_SP ≤ `1`

`Ring-planet efficiency` — Ring-planet efficiency
`[.95, .9, .85]` (default) | array | `0` < η_RP ≤ `1`

`Planet-planet efficiency` — Planet-planet efficiency
`[.5, .45, .4]` (default) | array | `0` < η_SP ≤ `1`

`Sun-carrier, ring-carrier and planet-carrier power thresholds` — Sun-carrier, ring-carrier and planet-carrier power thresholds
`[.001, .001, .001]` `W` (default) | array

`Sun-carrier, ring-carrier, and planet-carrier viscous friction coefficients` — Gear viscous friction
`[0, 0, 0]` `N*m/(rad/s)` (default) | array

`Inertia` — Inertia model
`off` (default) | `on`

`Inner planet gear inertia` — Inner planet gear inertia
`0.001` `kg*m^2` (default) | positive scalar

`Outer planet gear inertia` — Outer planet gear inertia
`0.001` `kg*m^2` (default) | positive scalar

`Thermal mass` — Thermal mass
`50` `J/K` (default) | scalar

`Initial temperature` — Initial temperature
`300` `K` (default) | scalar

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.