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Worm Gear

Worm gear with adjustable gear ratio and friction losses

Library

Gears

Description

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.

Ports

W and G are rotational conserving ports. The ports represent the worm and the gear, respectively.

Dialog Box and Parameters

Main

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 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

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)).

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
ωthAbsolute angular velocity 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 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 [ηWG ηGW]

  • Viscous coupling of driveshafts with bearings, parametrized by viscous friction coefficients μ

The loss torque has the general form:

τloss = τfr·tanh(4ωG/ωth) + μGωG + μWωW.

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

Power FlowPower Loss ConditionOutput DriveshaftFriction Torque τfr
ForwardωWτW > ωGτGGear, ωGRWG|τW|·(1 – ηWG)
ReverseωWτWωGτGWorm, ωW|τG|·(1 – ηGW)

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 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 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.

Limitations

  • 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.

See Also

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