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.
W and G are rotational conserving ports. The ports represent the worm and the gear, respectively.
The dialog box has one active area, Parameters, with three tabs.
Gear or transmission ratio RWG determined as the ratio of the worm angular velocity to the gear angular velocity. The default is 25.
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.
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.
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).
|ωW||Worm angular velocity|
|ωG||Gear angular velocity|
|α||Normal pressure angle|
|λ||Worm lead angle|
|d||Worm pitch diameter|
|τW||Torque on the worm|
|τloss||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 ωG = 0, the friction torque is introduced via the hyperbolic function.|
|τfr||Steady-state value of the friction torque at ωG → ∞.|
|ηWG||Torque transfer efficiency from worm to gear|
|ηGW||Torque transfer efficiency from gear to worm|
|ωth||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:
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:
RWGτW – τG – τloss = 0 ,
with τloss = 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 [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 Flow||Power Loss Condition||Output Driveshaft||Friction Torque τfr|
|Forward||ωWτW > ωGτG||Gear, ωG||RWG|τW|·(1 – ηWG)|
|Reverse||ωWτW ≤ ωGτG||Worm, ωW|||τG|·(1 – ηGW)|
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 = (cosα – k·tanλ)/(cosα + k/tanλ) ,
ηGW = (cosα – k/tanλ)/(cosα + k·tanλ) .
In the constant friction case, you specify ηWG and ηGW, independently of geometric details.
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.
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 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 ratios must be positive. Gear inertia and compliance are ignored. Coulomb friction reduces simulation performance. See Adjust Model Fidelity.