## 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 *g*_{12} =
*r*_{2}/*r*_{1}
=
*ω*_{1}/*ω*_{2} 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
*η*_{12} of meshing in gear pair (1,2) is
constant, independent of load (torque or power transferred).

The friction loss represented by

*η*_{12}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,

*η*_{12}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(4*p* /*
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*, the same expression applies in terms of force.

_{f}### 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).