Straight Bevel Gear Forces

In this article we'll look at straight bevel gear forces that result from gear mesh. These forces are important to determine the loads on shafts and bearings.  Straight bevel gears are useful when input and output shafts are not parallel.  Most often, straight bevel  gear shafts are 90 degrees apart.

Anatomy of a Bevel Gear

The diagram below shows important factors in a bevel gear.

The broken lines represent the pitch line and together form the pitch cone.  The pitch diameter is measured at the outside of the pitch lines.  Pitch diameter is sometimes referred to as outside pitch diameter.

Using the assumption that all forces act at the center of each tooth at the pitch line we introduce the mean diameter.  This is the diameter at the center of the teeth at the pitch line.  Mean diameter is calculated by:

Mean diameter = Pitch diameter - Face width cos (90 - γ)

Example Bevel Gear Pair

Consider the bevel gear pair where the pinion drives the gear.

Now let's look at the force on the pinion's teeth.  The normal force, Fn can be broken into tangential, radial, and axial components for easier computation of shaft and bearing loads.  The axial load also tells how much thrust the bearings must handle.

The equations for tangential force, Ft, radial force, Fr, and axial force, Fa are:

Ft = Fn cos Φ

Fr = Fn sin Φ cos γ = Ft tan Φ cos γ

Fa = Fn sin Φ sin γ = Ft tan Φ sin γ


Fn is normal force on tooth

Φ is pressure angle

γ is pitch angle

Let's assume we know the torque on the pinion's shaft is 225 in-lb.  The pinion pitch diameter is 3 inches, face width is .75 inches, and pitch angle is 30.96 degrees.  First, we need to calculate mean diameter:

Dmean = 3 - .75 cos (90 - 30.96) = 2.614 inches

We can calculate Ft by the following:

Ft = 2Tpinion ÷ Dmean = 2(225) ÷2.61 = 172.14 lb


Tpinion is pinion torque

Dmean is mean diameter of pinion

Now let's look at the tangential, radial, and axial forces on both gears.

For shafts that are at 90 degrees:

Pinion radial force = Gear axial force

Pinion axial force = Gear radial force

For shafts that are not at 90 degrees, the relations above do not apply.

The tangential force for both are equal in value and opposite in direction.

The MEboost Gear Forces Tool

MEboost has a gear forces tool that can easily determine straight bevel gear forces. 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 Straight Bevel tab.  The pinion and gear data are entered and the pinion input torque must be supplied.


The pinion and gear tangential, radial, and axial forces are shown.  The tool also calculates the pitch angle for each gear.

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