Spur Gear Forces

In this article we'll look at spur gear forces that result from gear mesh.  These forces are important to determine the loads on shafts and bearings.

Gear Mesh

When two spur gears mesh, their pitch circles touch as shown below.

Looking at the free body diagrams for each gear, there is a normal force acting on each gear.  The normal force acts at an angle from tangent called the pressure angle, Φ.  In the case of the two gear set-up, there are opposing torques as well.  For idler gears, there is no torque, but the normal forces from all gear meshes balance out.

Spur Gear Forces Example

Consider the gear train below.  There is a driving gear, an idler, and a driven gear.  For this example we'll analyze the idler gear.  We want to know the load on the idler shaft and bearings so they can be sized.

Forces of interest are shown on the the diagram below (not a free body diagram).

The normal force for each gear mesh can be broken into two components: tangential force, Ft and radial force, Fr.  The radial force is also known as separating force since it tends to push the gears apart.  The equations to determine these forces are:

Ft = Fn cos Φ

Fr = Fn sin Φ

This isn't enough information to determine forces.  If we know the torque on one of the other gears the tangential forces can be found.  Let's assume the driver gear has a torque of 100 in-lb and a pitch diameter of 1.5 inches.  The tangential forces in a gear mesh are equal and opposite.  Therefore if we find the tangential force on the driver, we know the tangential force on the idler.

Ft = Tdriver ÷ rpdriver = 100 ÷ .75 = 133.33 lb

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

Using substitution of the equations for Ft and Fr, we can find Fr by:

Fr = Ft tan Φ

Knowing these forces, we can now find the resultant force that the shaft exerts on the gear and its opposing force on the shaft.

The MEboost Gear Forces Tool

MEboost has a spur 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 spur 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 0 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.

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

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