# Sun-Planet Bevel

Planetary gear set of carrier, beveled planet, and sun wheels with adjustable gear ratio, assembly orientation, and friction losses

• Library:
• Simscape / Driveline / Gears / Planetary Subcomponents

## Description

The Sun-Planet Bevel gear block represents a set of carrier, planet, and sun gear wheels. The planet is connected to and rotates with respect to the carrier. The planet and sun corotate with a fixed gear ratio. You control the direction of rotation by setting the assembly orientation, left or right. A sun-planet and a ring-planet gear are basic elements of a planetary gear set. For model details, see Equations.

### 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```.

### Equations

#### Ideal Gear Constraints and Gear Ratios

The Sun-Planet Bevel block imposes one kinematic and one geometric constraint on the three connected axes:

`${r}_{C}{\omega }_{C}={r}_{S}{\omega }_{S}±{r}_{P}{\omega }_{P}$`

`${r}_{C}={r}_{S}±{r}_{P}$`

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.

• rP is the radius of the planet gear.

• ωP is the angular velocity of the planet gear.

The planet-sun gear ratio is defined as

`${g}_{PS}=\frac{{r}_{P}}{{r}_{S}}=\frac{{N}_{P}}{{N}_{S}},$`

where:

• gPS is the planet-sun gear ratio. As ${r}_{P}>{r}_{S}$, ${g}_{PS}>1$.

• NP is the number of teeth in the planet gear.

• NS is the angular velocity of the sun gear.

In terms of this ratio, the key kinematic constraint is:

• ${\omega }_{S}={g}_{PS}{\omega }_{P}-{\omega }_{C}$ for a left-oriented bevel assembly

• ${\omega }_{S}={g}_{PS}{\omega }_{P}+{\omega }_{C}$ for a right-oriented bevel assembly

The three degrees of freedom reduce to two independent degrees of freedom. The gear pair is (1,2) = (S,P).

### Warning

The planet-sun gear ratio, gPS, must be strictly greater than one.

The torque transfer is defined as

`${\tau }_{P}={\tau }_{loss}-{g}_{PS}{\tau }_{S},$`

where:

• τloss is the torque loss.

• τs is the torque for the sun gear.

• τp is the torque for the planet gear.

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

Then the torque transfer equation is ${\tau }_{P}={g}_{PS}{\tau }_{S}$.

#### Nonideal Gear Constraints and Losses

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

### Variables

Use the Variables settings to set the priority and initial target values for the block variables before simulating. For more information, see Set Priority and Initial Target for Block Variables (Simscape).

#### Dependencies

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

## Limitations and Assumptions

• Gear inertia is assumed negligible.

• Gears are treated as rigid components.

## 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 planet 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

Ratio gPS of the planet gear wheel radius to the sun gear wheel radius. This gear ratio must be strictly greater than 1.

Relative orientation of sun and planet gears, controlling their corotation direction. Left or right orientation imply, respectively, that the gears corotate in the same or opposite direction.

### 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.

Torque transfer efficiency, ηPS, for the planet gear to the sun gear pair meshing. This value 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 outer planet gear to the inner planet gear, ηPS. 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```.

Power threshold, pth, above which full efficiency is in effect. Below this 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

Viscous friction coefficient, μS, for the sun-carrier gear motion.

### 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`.

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