Helical Gear Forces

In this article we'll look at helical gear forces that result from gear mesh. These forces are important to determine the loads on shafts and bearings.  Unlike spur gears, single helical gears generate an axial force.  This thrust can be canceled by using a double helical gear (herringbone). This article focuses on single helical gears.

Helical Gear Forces Example

Consider the helical gear train below.  The bottom gear drives an idler gear which in turn, drives a driven gear in the same rotational direction as the driver gear.

Unlike a spur gear, the normal force on the teeth is not at a 90 degree angle from the gear shaft.  The diagram below shows the normal force on the driver teeth.  We can decompose the normal force into tangential, radial, and axial components.

Notation for force diagram:

Fn is normal force on tooth

Ft is tangential force

Fr is radial force

Fa is axial force

ψ is helix angle

Φn is normal pressure angle

Φt is tangential pressure angle

The relationship between helix angle, normal and tangential pressure angles is:

cos ψ = tan Φn ÷ tan Φt

The force components are determined by the following equations:

Ft = Fn cos Φn cos ψ

Fr = Fn sin Φn = Ft tan Φt

Fa = Fn cos Φn sin ψ = Ft tan ψ

If we know the torque on the driver gear we can determine Ft.  Let's assume the driver has a torque of 125 in-lb, and a pitch diameter of 1.63 inches.  Ft is then:

Ft = Tdriver ÷ rpdriver = 125 ÷ .815 = 153.37 lb

Where Tdriver is torque on the driver gear and rpdriver is the pitch radius of the driver gear.

Now that we've covered the force on gear teeth, let's look at forces in general that are of interest.  We want to know what forces are applied to the shaft and bearings.  Let's look at the idler gear forces (not a free body diagram).

Looking at the diagram above, there is a net zero axial force on the shaft.  Just because the net axial force is zero does not mean we should ignore the axial forces!  The forces create a moment that must be counter acted by the bearings and need to be included when calculating bearing loads.

The MEboost Gear Forces Tool

MEboost has a helical gear forces tool that can easily make these calculations. We'll use the same example to illustrate its use. To run the tool, click the Gear Forces button on the Excel ribbon.

The gear forces form will appear.  There are tabs for different gear types.  In our case we'll use the helical gear tab.  Information on the gear to analyze, in our example, the idler gear, is entered in the top pane.

Information on the gears that mesh with the idler are entered in the bottom pane.  We have a driver gear that is mounted at 270 degrees from the x axis.  We need to supply the tool with the driver gear tangential force that we calculated earlier.  There is also a driven gear mounted at 90 degrees.


The results pane shows the resultant force on the shaft as well as the angle from the x axis.  The shaft resultant force is also broken into its x and y components for easy calculation of bearing loads.  As we found earlier, net axial force is zero.

The tangential, radial (separating), and axial forces for each gear mesh are shown as well.

Excel is a registered trademark of Microsoft Corporation. Used with permission from Microsoft.