Sun-Planet Worm Gear

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.


C, W, and S are rotational conserving ports. They represent the carrier, worm (planet), and sun shafts, respectively.

Dialog Box and Parameters


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

Friction model

Select how to implement friction losses from nonideal meshing of gear threads. The default is No friction losses.

  • No friction losses — Suitable for HIL simulation — Gear meshing is ideal.

  • Constant efficiency — Transfer of torque and force between worm and gear is reduced by friction. If you select this option, the panel expands.

     Constant Efficiency

Angular velocity threshold

Absolute angular velocity threshold above which full efficiency loss is applied. Must be greater than zero. The default is 0.01.

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

Viscous Losses

Worm-carrier and sun-carrier viscous friction coefficients

Vector of viscous friction coefficients [μWC μSC], 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)).

Sun-Planet Worm Gear Model

Model Variables

RWGGear, 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.
ωSAngular velocity of the sun gear
ωPPlanet (that is, worm) angular velocity
ωCCarrier angular velocity
ωSCAngular velocity of the sun with respect to the carrier
αNormal pressure angle
λWorm lead angle
LWorm lead
dWorm pitch diameter
τSTorque applied to the sun shaft
τPTorque applied to the planet shaft
τCTorque applied to the carrier shaft
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 ωS = 0, the friction torque is introduced via the hyperbolic function.
τinstfrInstantaneous value of the friction torque added to the model to simulate friction losses
τfrSteady-state value of the friction torque
kFriction coefficient
ηWGEfficiency for worm-gear power transfer
ηGWEfficiency for gear-worm power transfer
ωthAbsolute angular velocity threshold
μSCSun-carrier viscous friction coefficient
μWCWorm-carrier viscous friction coefficient

Ideal Gear Constraints and Gear Ratio

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

ωS = ωP/RWG + ωC .

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

The torque transfer is:

RWGτP + τSτloss = 0 ,

τC = – τS,

with τloss = 0 in the ideal case.

Nonideal Gear Constraints

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 [ηWG, ηGW]

  • Viscous coupling of driveshafts with bearings, parametrized by viscous friction coefficients μSC and μWC

The torque transfer for nonideal gear has the general form:

τS = – RWG(τPμWCωP) + τinstfr ,

τinstfr = τfr·tanh(4ωSC/ωth) + μSCωSC .

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

ConditionFriction Torque τfr
ωSCτS > 0|τS|·(1 – ηGW)
ωSCτC ≤ 0|τS|·(1 – ηWG)/ηWG

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 shaftDriven shaft
SunηGWn/aNo loss
CarrierηGWNo lossn/a

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·tanα) .

Constant Efficiencies

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

Self-Locking and Negative Efficiency

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.

Meshing 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.

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 the Nonideal Gear Constraints section.


  • Gear inertia is negligible. It does not impact gear dynamics.

  • Gears are rigid. They do not deform.

  • Coulomb friction slows down simulation. See Adjust Model Fidelity.


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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