Worm gear with adjustable gear ratio and friction losses

Gears

The block represents a rotational gear that constrains the two
connected driveline axes, worm (W) and gear (G), to rotate together
in a fixed ratio that you specify. You can choose whether the gear
rotates in a positive or negative direction. Right-handed rotation
is the positive direction. If the worm thread is right-handed, *ω*_{W} and *ω*_{G} have
the same sign. If the worm thread is left-handed, *ω*_{W} and *ω*_{G} have
opposite signs.

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*determined as the ratio of the worm angular velocity to the gear angular velocity. The default is_{WG}`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 friction losses vary with the block variant chosen—that with a thermal port for thermal modeling or that without a thermal port.

**Viscous friction coefficients at worm (W) and gear (G)**Vector of viscous friction coefficients [

*μ*_{W}*μ*_{G}], for the worm and gear, 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 ratio |

ω_{W} | Worm angular velocity |

ω_{G} | Gear angular velocity |

α | Normal pressure angle |

λ | Worm lead angle |

L | Worm lead |

d | Worm pitch diameter |

τ_{G} | Gear torque |

τ_{W} | Torque on the worm |

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_{G} =
0, the friction torque is introduced via the hyperbolic function. |

τ_{fr} | Steady-state value of the friction torque at ω_{G} →
∞. |

k | Friction coefficient |

η_{WG} | Torque transfer efficiency from worm to gear |

η_{GW} | Torque transfer efficiency from gear to worm |

ω_{th} | Absolute angular velocity threshold |

[ | Vector of viscous friction coefficients for the worm and gear |

Worm gear imposes one kinematic constraint on the two connected axes:

*ω*_{W} = *R*_{WG}*ω*_{G} .

The two degrees of freedom are reduced to one independent degree of freedom. The forward-transfer gear pair convention is (1,2) = (W,G).

The torque transfer is:

*R*_{WG}*τ*_{W} – *τ*_{G} – *τ*_{loss} =
0 ,

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

In a nonideal worm-gear pair (W,G), 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*

The loss torque has the general form:

*τ*_{loss} = *τ*_{fr}·tanh(4*ω*_{G}/*ω*_{th})
+ *μ*_{G}*ω*_{G} + *μ*_{W}*ω*_{W}.

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

Power Flow | Power Loss Condition | Output Driveshaft | Friction Torque τ_{fr} |
---|---|---|---|

Forward | ω_{W}τ_{W} > ω_{G}τ_{G} | Gear, ω_{G} | R_{WG}|τ_{W}|·(1
– η_{WG}) |

Reverse | ω_{W}τ_{W} ≤ ω_{G}τ_{G} | Worm, ω_{W} | |τ_{G}|·(1
– η_{GW}) |

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*λ*)
.

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

*η*_{GW} has two
distinct regimes, depending on lead angle *λ*,
separated by the *self-locking point* at which *η*_{GW} =
0 and cos*α* = *k*/tan*λ*.

In the

*overhauling regime*,*η*_{GW}> 0, and the force acting on the nut can rotate the screw.In the

*self-locking regime*,*η*_{GW}< 0, and an external torque must be applied to the screw to release an otherwise locked mechanism. The more negative is*η*_{GW}, the larger the torque must be to release the mechanism.

*η*_{WG} is conventionally
positive.

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 coefficient *μ*_{W} controls
the viscous friction torque experienced by the worm from lubricated,
nonideal gear threads and viscous bearing losses. The viscous friction
torque on a worm driveline axis is –*μ*_{W}*ω*_{W}. *ω*_{W} is
the angular velocity of the worm with respect to its mounting.

The viscous friction coefficient *μ*_{G} controls
the viscous friction torque experienced by the gear, mainly from viscous
bearing losses. The viscous friction torque on a gear driveline axis
is –*μ*_{G}*ω*_{G}. *ω*_{G} is
the angular velocity of the gear with respect to its mounting.

Gear inertia is assumed negligible.

Gears are treated as rigid components.

Coulomb friction slows down simulation. See Adjust Model Fidelity.

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

W | Rotational conserving port representing the worm component |

G | Rotational conserving port representing the gear component |

H | Thermal conserving port for thermal modeling |

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