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Worm Gear

Worm gear with adjustable gear ratio and friction losses




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, ωW and ωG have the same sign. If the worm thread is left-handed, ωW and ωG have opposite signs.

You can model the effects of heat flow and temperature change through an optional thermal conserving port. By default, the thermal port is hidden. To expose the thermal port, right-click the block in your model and, from the context menu, select Simscape > Block choices. Specify the associated thermal parameters for the component.



Gear ratio

Gear or transmission ratio RWG 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.

Friction Losses

Parameters for friction losses vary with the block variant chosen—that with a thermal port for thermal modeling or that without a thermal port.

 Without Thermal Port

 With Thermal Port

Viscous Losses

Viscous friction coefficients at worm (W) and gear (G)

Vector of viscous friction coefficients [μW μG], 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 Port

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 alters the component efficiency according to an efficiency vector that you specify, affecting the starting meshing or friction losses. The default value is 300 K.

Worm Gear Model

Model Variables

RWGGear ratio
ωWWorm angular velocity
ωGGear angular velocity
αNormal pressure angle
λWorm lead angle
LWorm lead
dWorm pitch diameter
τGGear torque
τWTorque on the worm
τlossTorque 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 ωG = 0, the friction torque is introduced via the hyperbolic function.
τfrSteady-state value of the friction torque at ωG → ∞.
kFriction coefficient
ηWGTorque transfer efficiency from worm to gear
ηGWTorque transfer efficiency from gear to worm
pthPower threshold

[μW μG]

Vector of viscous friction coefficients for the worm and gear

Ideal Gear Constraint and Gear Ratio

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

ωW = RWGωG .

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:

RWGτWτGτloss = 0 ,

with τloss = 0 in the ideal case.

Nonideal Gear Constraint

In the nonideal case, τloss ≠ 0. For general considerations on nonideal gear modeling, see Model Gears with Losses.

Geometric Surface Contact Friction

In the contact friction case, ηWG and ηGW are determined by:

  • The worm-gear threading geometry, specified by lead angle λ and normal pressure angle α.

  • The surface contact friction coefficient k.

ηWG = (cosαk·tanλ)/(cosα + k/tanλ) ,

ηGW = (cosαk/tanλ)/(cosα + k·tanλ) .

Constant Efficiencies

In the constant friction case, you specify ηWG and ηGW, independently of geometric details.

Self-Locking and Negative Efficiency

ηGW has two distinct regimes, depending on lead angle λ, separated by the self-locking point at which ηGW = 0 and cosα = k/tanλ.

  • In the overhauling regime, ηGW > 0, and the force acting on the nut can rotate the screw.

  • In the self-locking regime, ηGW < 0, and an external torque must be applied to the screw to release an otherwise locked mechanism. The more negative is ηGW, the larger the torque must be to release the mechanism.

ηWG is conventionally positive.

Meshing 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.

Viscous Friction Force

The viscous friction coefficient μW 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 –μWωW. ωW is the angular velocity of the worm with respect to its mounting.

The viscous friction coefficient μG controls the viscous friction torque experienced by the gear, mainly from viscous bearing losses. The viscous friction torque on a gear driveline axis is –μGωG. ωG 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.


WRotational conserving port representing the worm component
GRotational conserving port representing the gear component
HThermal conserving port for thermal modeling

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