Simple Gear with Variable Efficiency

Gear set with parallel-axis rotation and variable meshing efficiency

Library

Gears

Description

The block represents a simple gear train with variable meshing efficiency. The gear train transmits torque at a specified ratio between base and follower shafts arranged in a parallel configuration. Shaft rotation can occur in equal or opposite directions. Gear losses are optional. They include meshing and viscous bearing losses. To specify the variable meshing efficiency, the block contains a physical signal port that you can use to input a general time-varying signal. Inertia and compliance effects are ignored.

Dialog Box and Parameters

Main

Follower (F) to base (B) teeth ratio (NF/NB)

Enter the gear ratio. This is the fraction of follower over base gear teeth numbers, NF/NB. The ratio must be positive. The default value is 2.

Output shaft rotates

Select the relative rotation between shafts. This is the rotation direction of the output shaft with respect to the input shaft. Options include equal or opposite directions. The default setting is In opposite direction to input shaft.

Meshing Losses

Minimum Efficiency

Enter the smallest efficiency value allowed for the gear. The efficiency is the power ratio between output and input shafts. The physical signal input saturates for values below the minimum efficiency or above 1. The minimum efficiency must be positive. The default value is 0.01.

Follower angular velocity threshold

Enter the relative angular velocity above which full efficiency losses are included. Values below this threshold mark an efficiency transition region where the driving shaft becomes the driven shaft and vice-versa. The follower angular velocity threshold must be positive. Select a physical unit.

The default value is 0.01. The default unit is rad/s.

Viscous Losses

Viscous friction coefficients at base (B) and follower (F)

Enter a two-element vector with the viscous friction coefficients of the base and follower gears. Coefficients must be positive. The default vector is [0 0]. The default unit is N*m/(rad/s).

Simple Gear Model

Ideal Gear Constraint and Gear Ratio

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 + τFτloss = 0 ,

with τloss = 0 in the ideal case.

Nonideal Gear Constraint and Losses

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 FlowPower Loss ConditionOutput Driveshaft ωout
ForwardωBτB > ωFτFFollower, ωF
ReverseωBτB < ωFτFBase, ωB

Limitations

  • Gear inertia is negligible. It does not impact gear dynamics.

  • Gears are rigid. They do not deform.

  • Coulomb friction slows down simulation. See Adjust Model Fidelity.

Ports

PortDescription
BRotational conserving port representing the base shaft
FRotational Conserving port representing the follower shaft

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