Model Gears with Losses
The blocks of the Simscape™ Driveline™ Gears Library contain optional built-in models of frictional losses, allowing you to represent nonideal gear couplings. In a nonideal gear pair (1,2), the angular velocity, gear radii, gear teeth constraints, and gear ratio g12 = r2/r1 = ω1/ω2 are unchanged. The transferred torque and power are reduced by:
Coulomb friction between imperfectly meshing teeth surfaces on gears 1 and 2, parameterized by an efficiency η, 0 < η ≤ 1. This efficiency depends on the torque load on the teeth. But it is often approximated as constant.
Viscous coupling of driveshafts with bearings, parameterized by viscous friction coefficients μ.
In the simplest nonideal gear loss model, the efficiency η12 of meshing in gear pair (1,2) is constant, independent of load (torque or power transferred).
The friction loss represented by η12 is effectively applied in full only if the transmitted power is greater than the power threshold pth. Below this value, a hyperbolic tangent function smooths the efficiency factor. When no power is transmitted, Simscape uses the frictional torque equation.
For gear sets with a carrier, η12 represents the ordinary efficiency, defined when the carrier is not moving.
For gears with different efficiencies for the forward and reverse power flow:
ForwardLoss = (1 – ηFB), ηFB is the torque transfer efficiency from the follower shaft to the base shaft.
BackwardLoss = (1/ηBF – 1), where ηBF is the torque transfer efficiency from the base shaft to the follower shaft.
The frictional torque is calculated as:
Tf = T / 2((ForwardLoss + BackwardLoss)tanh(4p / pth) + ForwardLoss – BackwardLoss)
T is the transferred torque.
p is the transferred power.
pth is the power threshold at the base shaft above which full efficiency losses are in effect.
For certain gear models, such as the Simple Gear, efficiency is assumed equal for both the forward and reverse power flow, ηBF = ηFB. For blocks like the Leadscrew block that use Floss instead of Tf, the same expression applies in terms of force.
Making η dependent on the load is a way to make the loss model more accurate. For an example of load-dependent efficiency, see the Simple Gear block reference page.
Making η dependent on the geometry of gear meshing is another way to make the loss model more accurate. For an example of geometry-dependent efficiency, see the Leadscrew block reference page.
On a driveshaft mounted to a gear wheel by lubricated, nonideal bearings, the viscous friction experienced by the axis is controlled by the viscous friction coefficient μ. The viscous friction torque on a driveshaft “a” is –μa·ωa, where ωa is the angular velocity of the driveshaft with respect to its mounting or carrier (if a carrier is present).