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.

Constant Efficiency

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 p_th. 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 – η_FB), η_FB is the torque transfer efficiency from the follower shaft to the base shaft.
BackwardLoss = (1/η_BF – 1), where η_BF is the torque transfer efficiency from the base shaft to the follower shaft.

The frictional torque is calculated as:

T_f = T / 2((ForwardLoss + BackwardLoss)tanh(4p / p_th) + ForwardLoss – BackwardLoss)

where:

T is the transferred torque.
p is the transferred power.
p_th 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, η_BF = η_FB. For blocks like the Leadscrew block that use F_loss instead of T_f, the same expression applies in terms of force.

Load-Dependent Efficiency

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.

Geometry-Dependent Efficiency

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.

Viscous Friction

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 –μ_a·ω_a, where ω_a is the angular velocity of the driveshaft with respect to its mounting or carrier (if a carrier is present).