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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

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 by 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-hand or right-hand. Rotation of the right-hand 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.

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
```

.

Equation variables are:

*R*is the 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._{WG}*ω*is the angular velocity of the sun gear._{S}*ω*is the planet (that is, worm) angular velocity._{P}*ω*is the carrier angular velocity._{C}*ω*is the angular velocity of the sun with respect to the carrier._{SC}*α*is the normal pressure angle.*λ*is the worm lead angle.*L*is the worm lead.*d*is the worm pitch diameter.*τ*is the torque applied to the sun shaft._{S}*τ*is the torque applied to the planet shaft._{P}*τ*is the torque applied to the carrier shaft._{C}*τ*is the 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*= 0, the friction torque is introduced via the hyperbolic function._{S}*τ*is the instantaneous value of the friction torque added to the model to simulate friction losses._{instfr}*τ*is the steady-state value of the friction torque._{fr}*k*is the friction coefficient.*η*is the efficiency for worm-gear power transfer._{WG}*η*is the efficiency for gear-worm power transfer._{GW}*p*is the power threshold._{th}*μ*is the sun-carrier viscous friction coefficient._{SC}*μ*is the worm-carrier viscous friction coefficient._{WC}

The sun-planet worm gear 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, there is no torque loss, that is *τ _{loss}* = 0.

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 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*and_{SC}*μ*_{WC}

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

The worm-gear threading geometry, specified by lead angle

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

*k*.

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

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

In the constant efficiency case, you specify
*η _{WG}* and

If you set efficiency for the reverse power flow to a negative value, the
train exhibits *self-locking*. Power cannot 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 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.

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.

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.

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

.

Gear inertia is assumed negligible.

Gears are treated as rigid components.

Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.