Modelling of mechanical gears with losses
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The Gears library blocks contain additional built-in friction loss models to represent non-ideal gear couplings. In a non-ideal gear pair (1,2), the angular velocity, tooth radii, tooth constraints and gear ratio remain unchanged. The transmitted torque and power are reduced by:
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Coulomb friction between imperfect tooth meshing surfaces on gears 1 and 2, parameterised by the efficiency . This coefficient depends on the torque load on the teeth. But it is often assumed to be approximately constant.
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Viscous coupling of cardan shafts with bearings, parameterised by the viscous friction coefficients .
Constant efficiency
In the simplest non-ideal transmission loss model, the efficiency of meshing in the gear pair (1,2) is constant and independent of the load (torque or transmitted power). * The friction losses represented by , are only fully utilised efficiently if the transmitted power exceeds a threshold power value . Below this value, the hyperbolic tangent function flattens the efficiency. When no power is transferred, Engee uses the frictional torque equation.
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For carrier gears, represents the normal efficiency determined when the carrier is not moving.
For gears with different efficiencies for forward and reverse power flow:
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, where is the efficiency of torque transfer from the driven shaft to the main shaft.
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, where is the efficiency of torque transfer from the basic shaft to the driven shaft.
The friction torque is calculated as:
where:
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- transmitted torque;
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- transmitted power;
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- threshold power at the basic shaft, above which the total efficiency loss occurs.
For some gear models, such as Simple Gear, efficiency is assumed to be equal for both forward and reverse power flow . For units such as Leadscrew, where is replaced by , the same expression for power is applicable.
Efficiency as a function of load
Load-dependent efficiency is a way to make the loss model more accurate. An example of load-dependent efficiency is shown on the Simple Gear block page.
Geometry dependent efficiency
Dependence of on gear meshing geometry is another way to make the loss model more accurate. An example of geometry dependent efficiency is given on the Leadscrew block page.
Viscous friction
On a drive shaft attached to a gear wheel using lubricated non-ideal bearings, the viscous friction experienced by the axle is controlled by the coefficient of viscous friction . The viscous friction torque on the drive shaft a is equal to , where is the angular velocity of the drive shaft relative to its mount or holder (if a holder is present).