Simple gear of base and follower wheels with adjustable gear ratio and friction losses
The Simple Gear block represents a gearbox that constrains the two connected driveline axes, base (B) and follower (F), to corotate with a fixed ratio that you specify. You can choose whether the follower axis rotates in the same or opposite direction as the base axis. If they rotate in the same direction, ωF and ωB have the same sign. If they rotate in opposite directions, ωF and ωB have opposite signs. For model details, see Simple Gear Model.
B and F are rotational conserving ports representing, respectively, the base and follower gear wheels.
The dialog box has one active area, Parameters, with three tabs.
Fixed ratio gFB of the follower axis to the base axis. The gear ratio must be strictly positive. The default is 2.
Choose the direction of motion of the follower (output) driveshaft relative to the motion of the base (input) driveshaft. The default is In opposite direction to input shaft.
Select how to implement friction losses from nonideal meshing of gear teeth. The default is No meshing losses.
No meshing losses — Suitable for HIL simulation — Gear meshing is ideal.
Constant efficiency — Transfer of torque between gear wheel pairs is reduced by a constant efficiency η satisfying 0 < η ≤ 1. If you select this option, the panel changes from its default.
Load-dependent efficiency — Transfer of torque between gear wheel pairs is reduced by variable efficiency 0 < η < 1, a function of load across the gear. If you select this option, the panel changes from its default.
Simple Gear imposes one kinematic constraint on the two connected axes:
rFωF = rBωB .
The follower-base gear ratio gFB = rF/rB = NF/NB. N is the number of teeth on each gear. The two degrees of freedom reduce to one independent degree of freedom.
The torque transfer is:
gFBτB + τF – τloss = 0 ,
with τloss = 0 in the ideal case.
In the nonideal case, τloss ≠ 0. For general considerations on nonideal gear modeling, see Model Gears with Losses.
In a nonideal gear pair (B,F), the angular velocity, gear radii, and gear teeth constraints are unchanged. But the transferred torque and power are reduced by:
Coulomb friction between teeth surfaces on gears B and F, characterized by efficiency η
Viscous coupling of driveshafts with bearings, parametrized by viscous friction coefficients μ
τloss = τCoul·tanh(4ωout/ωth) + μωout , τCoul = |τF|·(1 – η) .
The hyperbolic tangent regularizes the sign change in the Coulomb friction torque when the angular velocity changes sign.
|Power Flow||Power Loss Condition||Output Driveshaft ωout|
|Forward||ωBτB > ωFτF||Follower, ωF|
|Reverse||ωBτB < ωFτF||Base, ωB|
In the constant efficiency case, η is constant, independent of load or power transferred.
In the load-dependent efficiency case, η depends on the load or power transferred across the gears. For either power flow, τCoul = gFBτidle + kτF. k is a proportionality constant. η is related to τCoul in the standard, preceding form but becomes dependent on load:
η = τF/[gFBτidle + (k + 1)τF] .
Gear ratios must be positive. Gear inertia and compliance are ignored. Coulomb friction reduces simulation performance. See Adjust Model Fidelity.