Double-Pinion Planetary Gear

Planetary gear train with two meshed planet gear sets

Libraries:
Simscape / Driveline / Gears

Description

The Double-Pinion Planetary Gear 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. In the Double-Pinion Planetary Gear block, you can specify the gear teeth ratios between the ring and sun gears and the outer planet and inner planet gears. A geometric constraint fixes the remaining teeth ratios—the ring gear to the outer planet gear and the inner planet gear to the sun gear. This constrains the ring gear radius to the sum of the sun gear radius and 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.

The ring to outer planet gear 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 Double-Pinion Planetary Gear 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 enabling the optional thermal port. To enable the port, set Friction model to Temperature-dependent efficiency.

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

expand all

C — Planet gear carrier
rotational mechanical

Rotational mechanical conserving port associated with the planet gear carrier.

R — Ring gear
rotational mechanical

Rotational mechanical conserving port associated with the ring gear.

S — Sun gear
rotational mechanical

Rotational mechanical conserving port associated with the sun gear.

H — Heat flow
thermal

Thermal conserving port associated with heat flow. Heat flow affects the power transmission efficiency by altering the gear temperatures.

Dependencies

To enable this port, set Friction model to Temperature-dependent efficiency.

Parameters

expand all

Main

Ring (R) to sun (S) teeth ratio (NR/NS) — Ring to sun gear rotation ratio
`2` (default) | positive scalar

Fixed ratio of the ring gear to the sun gear rotations as defined by the number of planet gear teeth divided by the number of sun gear teeth.

Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi) — Outer planet to inner planet gear rotation ratio
`1` (default) | positive scalar

Fixed ratio of the outer planet gear to the inner planet gear rotations as defined by the number of planet gear teeth divided by the number of sun gear teeth.

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 the gear wheel pairs is reduced by a constant efficiency, η, such that 0 < η ≤ 1.
Temperature-dependent efficiency — Transfer of torque between the gear wheel pairs is defined by the table lookup based on the temperature.

Sun-planet, ring-planet and planet-planet ordinary efficiencies — Torque transfer efficiencies for the
`[.98, .98, .99]` (default) | vector

Vector 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 vector elements must be in the interval (0,1].

Dependencies

To enable this parameter, set Friction model to Constant efficiency.

Temperature — Temperature
`[280, 300, 320]` `K` (default) | vector

Vector of temperatures used to construct a 1-D temperature-efficiency lookup table. The vector elements must increase from left to right.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Sun-planet efficiency — Torque transfer efficiency from the sun gear to the inner planet gears
`[.75, .65, .6]` (default) | vector

Vector of output-to-input power ratios that describe the power flow from the sun gear to the inner planet gears, η_SP. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Each element is an efficiency that relates to a temperature in the Temperature vector. The length of the vector must be equal to the length of the Temperature vector. Each element in the vector must be in the range (0,1].

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Ring-planet efficiency — Torque transfer efficiency from the ring gear to the outer planet gears
`[.5, .45, .4]` (default) | vector

Vector of output-to-input power ratios that describe the power flow from the ring gear to the outer planet gears, η_RP. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Planet-planet efficiency — Torque transfer efficiency from the inner planet gears to the outer planet gears
`[.5, .45, .4]` (default) | vector

Vector of output-to-input power ratios that describe 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

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Sun-carrier, ring-carrier, and planet-carrier power thresholds — Minimum efficiency power threshold for the sun-carrier, ring-carrier, and planet-carrier gear couplings
`[.001, .001, .001]` `W` (default) | vector

Vector of power thresholds above which the full efficiency factors apply. Enter the thresholds in the order sun-carrier, ring-carrier, planet-carrier.

When you set Friction model to Constant efficiency, the block lowers the efficiency losses to zero when no power is transmitted. When you set Friction model to Temperature-dependent efficiency, the block smooths the efficiency factors between zero when at rest and the values provided by the temperature-efficiency lookup tables at the power thresholds.

The power threshold should be lower than the expected power transmitted during simulation. Higher values may cause the block to underestimate efficiency losses. Very low values may raise the computational cost.

Dependencies

To enable the parameter, set Friction model 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) | vector

Vector 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.

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

Sun gear inertia.

Dependencies

To enable this parameter, set Inertia to On.

Sun gear radius — Sun gear radius
`0.01` `m` (default) | positive scalar

Sun gear radius.

Dependencies

To enable this parameter, set Inertia to On.

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

Ring gear inertia.

Dependencies

To enable this parameter, set Inertia to On.

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

Carrier gear inertia.

Dependencies

To enable this parameter, set Inertia to On.

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

Inertia of the inner planet gear about its own axis.

Dependencies

To enable this parameter, set Inertia to On.

Inner planet gear mass — Inner planet gear mass
`0.001` `kg` (default) | positive scalar

Mass of the inner planet gear.

Dependencies

To enable this parameter, set Inertia to On.

Inner planet gear radius — Inner planet gear radius
`0.01` `m` (default) | positive scalar

Radius of the inner planet gear.

Dependencies

To enable this parameter, set Inertia to On.

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

Outer planet gear inertia about its own axis.

Dependencies

Tho enable this parameter, set Inertia to On.

Outer planet gear mass — Inner planet gear mass
`0.001` `kg` (default) | positive scalar

Mass of the outer planet gear.

Dependencies

To enable this parameter, set Inertia to On.

Outer planet gear radius — Inner planet gear radius
`0.02` `m` (default) | positive scalar

Radius of the outer planet gear.

Dependencies

To enable this parameter, set Inertia to On.

Thermal Port

To enable these settings, set Friction model to Temperature-dependent efficiency.

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

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

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

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

Block temperature at the start of simulation. The initial temperature sets the initial component efficiencies according to their respective efficiency vectors.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

More About

expand all

Hardware-in-the-Loop Simulation

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

Extended Capabilities

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

Version History

Introduced in R2013b

expand all

R2024b: Inertia updates

You now can specify the inertia of all gear subcomponents including the sun, ring, and carrier gears. You also now can include the effects of planet procession in the planet inertia behavior.

Double-Pinion Planetary Gear

Description

Thermal Model

Assumptions and Limitations

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 rotation ratio 2 (default) | positive scalar

Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi) — Outer planet to inner planet gear rotation ratio 1 (default) | positive scalar

Meshing Losses

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 — Torque transfer efficiencies for the [.98, .98, .99] (default) | vector

Dependencies

Temperature — Temperature [280, 300, 320] K (default) | vector

Dependencies

Sun-planet efficiency — Torque transfer efficiency from the sun gear to the inner planet gears [.75, .65, .6] (default) | vector

Dependencies

Ring-planet efficiency — Torque transfer efficiency from the ring gear to the outer planet gears [.5, .45, .4] (default) | vector

Dependencies

Planet-planet efficiency — Torque transfer efficiency from the inner planet gears to the outer planet gears [.5, .45, .4] (default) | vector

Dependencies

Sun-carrier, ring-carrier, and planet-carrier power thresholds — Minimum efficiency power threshold for the sun-carrier, ring-carrier, and planet-carrier gear couplings [.001, .001, .001] W (default) | vector

Dependencies

Viscous Losses

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

Inertia

Inertia — Inertia model off (default) | on

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

Dependencies

Sun gear radius — Sun gear radius 0.01 m (default) | positive scalar

Dependencies

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

Dependencies

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

Dependencies

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

Dependencies

Inner planet gear mass — Inner planet gear mass 0.001 kg (default) | positive scalar

Dependencies

Inner planet gear radius — Inner planet gear radius 0.01 m (default) | positive scalar

Dependencies

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

Dependencies

Outer planet gear mass — Inner planet gear mass 0.001 kg (default) | positive scalar

Dependencies

Outer planet gear radius — Inner planet gear radius 0.02 m (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™.

Version History

R2024b: Inertia updates

See Also

Simscape Blocks

Topics

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 rotation ratio
`2` (default) | positive scalar

Outer planet (Po) to inner planet (Pi) teeth ratio (NPo/NPi) — Outer planet to inner planet gear rotation ratio
`1` (default) | positive scalar

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 — Torque transfer efficiencies for the
`[.98, .98, .99]` (default) | vector

Temperature — Temperature
`[280, 300, 320]` `K` (default) | vector

Sun-planet efficiency — Torque transfer efficiency from the sun gear to the inner planet gears
`[.75, .65, .6]` (default) | vector

Ring-planet efficiency — Torque transfer efficiency from the ring gear to the outer planet gears
`[.5, .45, .4]` (default) | vector

Planet-planet efficiency — Torque transfer efficiency from the inner planet gears to the outer planet gears
`[.5, .45, .4]` (default) | vector

Sun-carrier, ring-carrier, and planet-carrier power thresholds — Minimum efficiency power threshold for the sun-carrier, ring-carrier, and planet-carrier gear couplings
`[.001, .001, .001]` `W` (default) | vector

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

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

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

Sun gear radius — Sun gear radius
`0.01` `m` (default) | positive scalar

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

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

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

Inner planet gear mass — Inner planet gear mass
`0.001` `kg` (default) | positive scalar

Inner planet gear radius — Inner planet gear radius
`0.01` `m` (default) | positive scalar

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

Outer planet gear mass — Inner planet gear mass
`0.001` `kg` (default) | positive scalar

Outer planet gear radius — Inner planet gear radius
`0.02` `m` (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™.