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

Gears/Planetary Subcomponents

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

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

The block models the effects of heat flow and temperature change
through an optional thermal port. To expose the thermal port, right-click
the block and select **Simscape** > **Block choices** > **Show thermal
port**. Exposing the thermal port causes
new parameters specific to thermal modeling to appear in the block
dialog box.

**Gear ratio**Gear or transmission ratio

*R*_{WG}determined as the ratio of the worm angular velocity to the gear angular velocity. The default is`25`

.**Worm thread type**Choose the directional sense of gear rotation corresponding to positive worm rotation. The default is

`Right-handed`

. If you select`Left-handed`

, rotation of the worm in the generally-assigned positive direction results in the gear rotation in negative direction.

Parameters for meshing and friction losses vary with the block variant chosen—one with a thermal port for thermal modeling and one without it.

**Worm-carrier and sun-carrier viscous friction coefficients**Vector of viscous friction coefficients [

*μ*_{WC}*μ*_{SC}], for the worm-carrier and sun-carrier shafts, respectively. The default is`[0 0]`

.From the drop-down list, choose units. The default is newton-meters/(radians/second) (

`N*m/(rad/s)`

).

**Thermal mass**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. The default value is

`50`

J/K.**Initial temperature**Component temperature at the start of simulation. The initial temperature influences the starting meshing or friction losses by altering the component efficiency according to an efficiency vector that you specify. The default value is

`300`

K.

R_{WG} | Gear, or transmission, ratio determined as the ratio of the
worm angular velocity to the 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} | Planet (that is, worm) angular velocity |

ω_{C} | Carrier angular velocity |

ω_{SC} | Angular velocity of the sun 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 the sun shaft |

τ_{P} | Torque applied to the planet shaft |

τ_{C} | Torque applied to the carrier shaft |

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

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

τ_{fr} | Steady-state value of the friction torque |

k | Friction coefficient |

η_{WG} | Efficiency for worm-gear power transfer |

η_{GW} | Efficiency for gear-worm power transfer |

ω_{th} | Absolute angular velocity threshold |

μ_{SC} | Sun-carrier viscous friction coefficient |

μ_{WC} | Worm-carrier viscous friction coefficient |

Sun-planet worm gear imposes one kinematic constraint on the three connected axes:

*ω*_{S} = *ω*_{P}/*R*_{WG} + *ω*_{C} .

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

The torque transfer is:

*R*_{WG}*τ*_{P} + *τ*_{S} – *τ*_{loss} =
0 ,

*τ*_{C} =
– *τ*_{S},

with *τ*_{loss} =
0 in the ideal case.

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

Coulomb friction between thread surfaces on W and G, characterized by friction coefficient

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

*μ*_{SC}and*μ*_{WC}

The torque transfer for nonideal gear has the general form:

*τ*_{S} =
– *R*_{WG}(*τ*_{P} – *μ*_{WC}*ω*_{P})
+ *τ*_{instfr} ,

*τ*_{instfr} = *τ*_{fr}·tanh(4*ω*_{SC}/*ω*_{th})
+ *μ*_{SC}*ω*_{SC} .

The hyperbolic tangent regularizes the sign change in the friction torque when the sun gear velocity changes sign.

Condition | Friction
Torque τ_{fr} |
---|---|

ω_{SC}τ_{S} >
0 | |τ_{S}|·(1
– η_{GW}) |

ω_{SC}τ_{C} ≤
0 | |τ_{S}|·(1
– η_{WG})/η_{WG} |

Because the transmission incorporates a worm gear, the efficiencies are different for the direct and reverse power transfer. The following 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 |

In the contact friction case, *η*_{WG} and *η*_{GW} are
determined by:

The worm-gear threading geometry, specified by lead angle

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

*k*.

*η*_{WG} =
(cos*α* – *k*·tan*λ*)/(cos*α* + *k*/tan*λ*)
,

*η*_{GW} =
(cos*α* – *k*/tan*λ*)/(cos*α* + *k*·tan*α*)
.

In the constant efficiency case, you specify *η*_{WG} and *η*_{GW},
independently of geometric details.

If you set efficiency for the reverse power flow to a negative
value, the train exhibits *self-locking*. Power
can not be transmitted from sun gear to worm and from carrier to 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.

The efficiencies *η* of meshing between
worm and gear are fully active only if the absolute value of the gear
angular velocity is greater than the velocity tolerance.

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

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 the Nonideal Gear Constraints section.

Gear inertia is assumed negligible.

Gears are treated as rigid components.

Coulomb friction slows down simulation. See Adjust Model Fidelity.

Port | Description |
---|---|

C | Rotational conserving port representing the gear carrier |

W | Rotational conserving port representing the worm gear |

S | Rotational conserving port representing the sun gear |

H | Thermal conserving port for thermal modeling |

The sdl_torsen_differentialsdl_torsen_differential example model uses a Sun-Planet Worm gear to model a Torsen differential gear set.

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