# Worm Gear Forces

Calculating worm gear forces is necessary to determine bearing reactions and shaft loads.  Because of the sliding action between the worm and gear, friction plays a major role in a worm gear set's efficiency.

For this discussion, we assume the worm and gear shafts are 90 degrees apart and that the worm is driving the gear.

## Worm Gear Force Notation

The diagram below shows the worm gear forces that need to be determined in order to analyze shaft and bearing loads.

For shafts at 90 degrees:

Worm axial force, Faw = Gear tangential force, Ftg

Worm tangential force, Ftw = Gear axial force, Fag

Worm radial force, Frw = Gear radial force, Frg

## Mesh Forces

When a worm and gear mesh, there are two forces present.  The normal force, Fn is the force normal to the worm (and gear) tooth.  There is also a friction force, Ff due to the sliding action of the worm against the gear.

The diagram below shows the forces on the worm.  Fn is also decomposed into its x, y, and z components which helps to determine tangential, radial, and axial forces later.

λ is the worm lead angle.  Φn is the normal pressure angle.

The normal force, Fn is decomposed by the following equations:

Fx = Fn (cos Φn sin λ)

Fy = Fn sin Φn

Fz = Fn (cos Φn cos λ)

The friction force, Ff acts in the x-z plane.  It is decomposed into x and z components by the following:

Ffx = μ Fn cos λ

Ffz = μ Fn sin λ

Where μ is the coefficient of friction.

## Calculating Final Forces

Now that we've got the forces from the gear mesh, we can calculate tangential, radial, and axial forces which are necessary to determine shaft and bearing loads.  To start we either have to assume an input torque to the worm or an output torque from the gear.

### Worm Tangential and Gear Axial Forces

We will assume we know the input torque, and the worm tangential force is determined by:

Ftw = Ti ÷ rpw

Where Ti is input torque, and rpw is the pitch radius of the worm.  NOTE: Appropriate unit conversion must also be incorporated.

From earlier, we know that Ftw = Fag.  Also Ftw = Fx + Ffx therefore:

Ftw = Fn [(cos Φn sin λ) + μ cos λ]     {Equation 1}

### Worm Axial and Gear Tangential Forces

The worm axial force is given by:

Faw = Fz - Ffz = Fn [(cos Φn cos λ) - μ sin λ]     {Equation 2}

Solving equation 1 for Fn and substituting into equation 2:

Faw = [Ftw ((cos Φn cos λ) - μ sin λ)] ÷ [(cos Φn sin λ) + μ cos λ]

From earlier, Faw = Ftg.

The gear output torque is:

To = Ftg rpg

Where To is output torque, and rpg is the pitch radius of the gear.  NOTE: Appropriate unit conversion must also be incorporated.

### Worm Radial and Gear Radial Forces

The worm radial force is given by:

Frw = Fy = Fn sin Φn     {Equation 3}

Solving equation 1 for Fn and substituting into equation 3:

Frw = [Ftw sin Φn] ÷ [(cos Φn sin λ) + μ cos λ]

From earlier, Frw = Frg.

## References

Budynas, Richard; Nisbett, Keith, Shigley's Mechanical Engineering Design, 8th Edition, McGraw-Hill, 2006, pp. 694-700.

Wilson, Charles; Sadler, Peter; Michels, Walter Kinematics and Dynamics of Machinery, Harper Collins, 1983, pp. 436-438.

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