In this article we'll look at spiral bevel gear forces that result from gear mesh. These forces are important to determine the loads on shafts and bearings. Spiral bevel gears are useful when input and output shafts are not parallel. Most often, spiral 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 spiral bevel gear pair where the pinion drives the gear.

The forces on the teeth need to be broken into different planes to get radial and axial forces. We'll look at the pinion to show the derivation of these forces, but they apply to the gear as well.

The engagement of the teeth is also important. In one rotation direction the convex side of the pinion's teeth will engage with the concave side of the gear teeth. If rotation is reversed, the teeth engagement will be pinion concave with gear convex.

### Convex Engagement

When the convex side of the pinion teeth is engaged, we have a normal force, Fn assumed to be at the mid-point of the tooth face. β is the helix angle, Φn is the normal pressure angle, and γ is the pitch angle.

Starting with the normal force, Fn we can resolve it to forces F1 and F2. F1 is in a plane that is tangent to the pitch cone. F2 is perpendicular to the pitch cone.

F1 = Fn cos Φn

F2 = Fn sin Φn

In the view of the pinion face, we can resolve F1 into F3 and F4. These forces are in a plane that is tangent to the pitch cone. F4 is the tangential force on the pinion.

F3 = F1 sin β

F4 = Ft = F1 cos β

To find the radial and axial forces, we use F2 and F3.

Fr = F2 cos γ + F3 sin γ

Fa = F2 sin γ - F3 cos γ

We can rewrite Fr and Fa in terms of Ft, since Ft can be found with a known torque.

Ft = Fn cos Φn cos β → Fn = Ft ÷ cos Φn cos β

F1 = (Ft cos Φn) ÷ cos Φn sin β = Ft ÷ cos β

Substituting the equation above for F1:

F2 = (Ft sin Φn) ÷ cos Φn cos β = Ft tan Φn ÷ cos β

F3 = Ft sin β ÷ cos β = Ft tan β

Substituting the equations above for F2 and F3 and rearranging:

Fr = (Ft ÷ cos β)(tan Φn cos γ + sin β sin γ) **{Equation 1}**

Fa = (Ft ÷ cos β)(tan Φn sin γ - sin β cos γ) **{Equation 2}**

### Concave Engagement

When the concave side of the teeth is engaged, we have a normal force, Fn assumed to be at the mid-point of the tooth face. β is the helix angle, Φn is the normal pressure angle, and γ is the pitch angle.

Starting with the normal force, Fn we can resolve it to forces F1 and F2. F1 is in a plane that is tangent to the pitch cone. F2 is perpendicular to the pitch cone.

F1 = Fn cos Φn

F2 = Fn sin Φn

In the view of the pinion face, we can resolve F1 into F3 and F4. These forces are in a plane that is tangent to the pitch cone. F4 is the tangential force on the pinion.

F3 = F1 sin β

F4 = Ft = F1 cos β

To find the radial and axial forces, we use F2 and F3.

Fr = F2 cos γ - F3 sin γ

Fa = F2 sin γ + F3 cos γ

We can rewrite Fr and Fa in terms of Ft, since Ft can be found with a known torque.

Ft = Fn cos Φn cos β → Fn = Ft ÷ cos Φn cos β

F1 = (Ft cos Φn) ÷ cos Φn sin β = Ft ÷ cos β

Substituting the equation above for F1:

F2 = (Ft sin Φn) ÷ cos Φn cos β = Ft tan Φn ÷ cos β

F3 = Ft sin β ÷ cos β = Ft tan β

Substituting the equations above for F2 and F3 and rearranging:

Fr = (Ft ÷ cos β)(tan Φn cos γ - sin β sin γ) **{Equation 3}**

Fa = (Ft ÷ cos β)(tan Φn sin γ + sin β cos γ) **{Equation 4}**

## Calculating the Spiral Bevel Gear Forces

Now that we have formulas for the axial and radial forces, we can calculate them as well as tangential forces. Our example bevel gears have the following dimensions:

Φn = 20 degrees

β = 35 degrees

γpinion = 30.96 degrees

γgear = 59.036 degrees

Pinion pitch diameter = 3 inches

Gear pitch diameter = 5 inches

Face width = .75 inches

The first step is to calculate the pinion tangential force. The pinion shaft torque is 125 in-lb. 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(125) ÷2.614 = 95.63 lb

Where:

Tpinion is pinion torque

Dmean is mean diameter of pinion

### Pinion Convex to Gear Concave

First we'll calculate forces when the pinion rotates so that the convex side of its teeth engage with the concave side of the gear's teeth.

For the pinion we use equations 1 and 2 that were derived earlier.

Fr =70.89 lb

Fa = -35.562

For the gear, Ft is the same as the pinion. We use equations 3 and 4 that were derived earlier.

Ft = 95.63 lb

Fr = -35.56 lb

Fa = 70.887 lb

For radial and axial forces, a positive force separates the gears and a negative force pulls the gears together.

The forces are shown on the diagram below. Note that the pinion axial force and gear radial force are pulling the gears together.

### Pinion Concave to Gear Convex

Next we'll calculate forces when the pinion rotates so that the concave side of its teeth engage with the convex side of the gear's teeth.

For the pinion we use equations 3 and 4 that were derived earlier.

Fr = 1.98 lb

Fa = 79.28 lb

For the gear, Ft is the same as the pinion. We use equations 1 and 2 that were derived earlier.

Ft = 95.63 lb

Fr = 79.28 lb

Fa = 1.98 lb

The forces are shown on the diagram below. Note that all the radial and axial forces are pushing the gears apart.

## The MEboost Gear Forces Tool

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

#### Results

The pinion and gear tangential, radial, and axial forces are shown. The radial and axial forces are calculated for both rotation directions.

- Pinion convex side engages with gear concave side.
- Pinion concave side engages with gear convex side.

The tool also calculates the pitch angle for each gear.

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