# Sun-Planet Worm Gear

Planetary gear set of carrier, worm planet, and sun wheels with adjustable gear ratio, worm thread type, and friction losses

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

## Description

The Sun-Planet Worm Gear block represents a
two-degree-of-freedom planetary gear built from a carrier and sun and planet gears. By
type, the sun and planet gears are crossed helical spur gears arranged as a worm drive,
in which the planet gear is the worm. Such transmissions are used in the Torsen^{®} T-1 differential. When transmitting power, the
sun gear can be independently rotated by the worm (planet) gear, by the carrier, or by
both.

You specify a fixed gear ratio, which signifies the worm angular velocity divided by the sun gear angular velocity. You control the direction by setting the worm thread type to left-hand or right-hand. Rotation of the right-hand worm in the positive direction causes the sun gear to rotate in the positive direction. The positive directions of the sun gear and the carrier are the same.

### 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
```

.

### Equations

**Variables**

Equation variables are:

R_{WG} | Gear ratio that signifies the worm angular velocity divided by the sun gear angular velocity: The ratio is positive for the right-hand worm and negative for the left-hand worm |

ω_{S} | Angular velocity of the sun gear |

ω_{P} | Angular velocity of the worm gear |

ω_{C} | Angular velocity of the carrier |

ω_{SC} | Angular velocity of the sun gear with respect to the carrier |

α | Normal pressure angle |

λ | Worm lead angle |

L | Worm lead |

d | Worm pitch diameter |

τ_{S} | Torque applied to sun shaft |

τ_{P} | Torque applied to planet shaft |

τ_{C} | Torque applied to carrier shaft |

τ | Torque due to meshing friction: The loss depends on the
device efficiency and the power flow direction. To avoid an
abrupt change of the friction torque at
ω = 0,
the friction torque is introduced via the hyperbolic
function._{S} |

τ_{instfr} | Instantaneous value of the friction torque used to simulate friction losses |

τ_{fr} | Torque due to friction in steady-state |

k | Friction coefficient |

η_{WG} | Efficiency of worm to gear power transfer |

η_{GW} | Efficiency of gear to worm power transfer |

p_{th} | Power threshold |

μ_{SC} | Viscous friction coefficient for the sun-carrier interface |

μ_{WC} | Viscous friction coefficient for the worm-carrier interface |

**Ideal Gear Constraints and Gear Ratio**

The Sun-Planet Worm Gear block imposes one kinematic constraint on the three connected axes:

$${\omega}_{\text{S}}=\frac{{\omega}_{\text{P}}}{{R}_{\text{WG}}}+{\omega}_{\text{C}}.$$

The gear has two independent degrees of freedom. The gear pair is (1,2) =
(*S*,*P*).

The torque transfer is:

$${R}_{\text{WG}}{\tau}_{\text{P}}+{\tau}_{\text{S}}\u2013{\tau}_{\text{loss}}=\text{}0$$

$${\tau}_{\text{C}}=\text{}\u2013{\tau}_{\text{S}}$$

In the ideal
case where there is no torque loss, *τ _{loss}* = 0.

**Nonideal Gear Constraints**

In the nonideal case, *τ _{loss}* ≠ 0. For more information, see Model Gears with Losses.

In a nonideal gear, the angular velocity and geometric constraints are unchanged, but the transferred torque and power are reduced by:

Coulomb friction due to worm-sun gear meshing, which is characterized by the friction coefficient

*k*or constant efficiencies [*η*,_{WG}*η*]_{GW}Viscous couplings of driveshafts with bearings, which are parametrized by viscous friction coefficients

*μ*and_{SC}*μ*_{WC}

Because the transmission incorporates a worm gear, the efficiencies are different for the direct and reverse power transfer. The table shows the value of the efficiency for all combinations of the power transfer.

Driving
Shaft | Driven
Shaft | ||

Planet | Sun | Carrier | |

Planet | N/A | η_{WG} | η_{WG} |

Sun | η_{GW} | N/A | No loss |

Carrier | η_{GW} | No loss | N/A |

**Geometric Surface Contact Friction**

When you set **Friction model** to ```
Constant
efficiency
```

and leave **Friction
parameterization** set to ```
Friction coefficient and
geometrical parameters
```

, the model considers geometric surface
contact friction. In this case, *η _{WG}*
and

*η*rely on:

_{GW}The worm-gear threading geometry, specified by lead angle

*λ*and normal pressure angle*α*.The surface contact friction coefficient

*k*.

$${\eta}_{\text{WG}}=\text{}\frac{\left(\text{cos}\alpha \u2013k\xb7\text{tan}\lambda \right)}{\left(\text{cos}\alpha +\frac{k}{\text{tan}\lambda}\right)}$$

$${\eta}_{\text{GW}}=\text{}\frac{\left(\text{cos}\alpha \u2013\frac{k}{\text{tan}\lambda}\right)}{\left(\text{cos}\alpha +k\xb7\text{tan}\alpha \right)}$$

**Constant Efficiencies**

When you set **Friction model** to ```
Constant
efficiency
```

and set **Friction
parameterization** to `Efficiencies`

, or
when you set **Friction model** to
`Temperature-dependent efficiency`

, the model
treats the efficiencies as constant. In this case, you specify
*η _{WG}* and

*η*independently of geometric details.

_{GW}**Self-Locking and Negative Efficiency**

You can enable self-locking behavior by making the efficiency negative. Power cannot be transmitted from the sun gear to the worm or from the carrier to the worm unless some torque is applied to the worm to release the train. In this case, the absolute value of the efficiency specifies the ratio at which the train is released. The smaller the train lead angle, the smaller the reverse efficiency.

**Meshing Efficiency**

The efficiencies, *η*, of meshing between the worm gear and
planet gear are fully active only if the transmitted power is greater than the
power threshold.

If the power is less than the threshold, the actual efficiency is automatically regularized to unity at zero velocity.

**Viscous Friction Force**

The viscous friction coefficients of the worm-carrier and sun-carrier bearings control the viscous friction torque experienced by the carrier from lubricated, nonideal gear threads. For details, see Nonideal Gear Constraints.

### 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.

### 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

## Parameters

## More About

## Extended Capabilities

## See Also

### Simscape Blocks

**Introduced in R2011a**