Gear Force Calculator: How to Find Gear Load

A gear force calculator works out the tangential force (also called tangential load) on a gear tooth from the torque you put in and the gear’s pitch radius. You enter the input torque and the gear’s pitch diameter (or radius), and it returns the tangential force in newtons or pounds-force, plus the radial and axial components if you add the pressure angle. This page gives you the standard AGMA-style formulas, a worked example, and the common mistakes that make gear calculations wrong.

Formula

The basic gear force formula comes from the definition of torque: torque equals force times the lever arm. For a gear, the lever arm is the pitch radius (half the pitch diameter). So:

F_t = T / r

Where:

• F_t = tangential force (N or lbf)
• T = torque (N·m or lbf·in)
• r = pitch radius (m or in) = pitch diameter / 2

Once you have the tangential force, the radial force (which pushes the gears apart) is:

F_r = F_t · tan(φ)

Where φ is the pressure angle (typically 20° or 14.5°). For helical gears, there is also an axial force component, but that requires the helix angle, which is not included in the simple calculator above.

Worked Example

Using the default values from the calculator: torque = 50 N·m, pitch diameter = 100 mm, pressure angle = 20°.

1. Convert pitch diameter to radius: r = 100 mm / 2 = 50 mm = 0.05 m
2. Tangential force: F_t = 50 N·m / 0.05 m = 1000 N
3. Radial force: F_r = 1000 N · tan(20°) = 1000 N · 0.36397 = 363.97 N ≈ 364.0 N

The calculator shows F_t = 1000.0 N and F_r = 364.0 N, matching the hand calculation.

When to Use a Gear Force Calculation

You need gear forces whenever you are:

• Selecting bearings for a gearbox – bearings must handle both radial and axial loads.
• Checking gear tooth bending strength (Lewis equation) or surface durability (Hertzian contact).
• Designing shafts – the tangential force creates bending moment and torsion.
• Calculating housing loads – the radial force pushes the bearings outward.
• Estimating efficiency losses – friction force is proportional to normal load.

For a simple spur gear pair, the tangential force is the same on both gears (Newton’s third law). The radial forces are also equal and opposite. So you only need to calculate once per mesh.

How to Read the Results

The tangential force is the useful force that transmits power. It acts along the line of action at the pitch point, perpendicular to the radial line. The radial force pushes the gears apart and loads the bearings. For a 20° pressure angle, the radial force is about 36% of the tangential force. For a 14.5° pressure angle, it’s about 26%.

If you are designing a shaft, remember that the tangential force also creates torque on the shaft. The bending moment from the radial force is often the limiting factor for shaft diameter.

Common Mistakes and Unit Mix-ups

The most frequent error is mixing units. If you input torque in N·m and pitch diameter in mm, you must convert mm to meters first. The calculator above does that for you (it multiplies torque by 1000 and divides by diameter in mm). If you use imperial units, keep everything consistent: torque in lbf·in, radius in inches, force in lbf.

Another common mistake is using the outside diameter instead of the pitch diameter. The pitch diameter is the theoretical circle where the teeth mesh – it is not the outer tip diameter. For a standard gear, pitch diameter equals number of teeth times module (metric) or pitch diameter equals number of teeth divided by diametral pitch (imperial).

Also, do not confuse pressure angle with helix angle. The pressure angle is the angle between the line of action and the tangent to the pitch circle. The helix angle is the angle of the teeth relative to the gear axis. The calculator above only uses the pressure angle.

Limits of the Simple Formula

The formula F_t = T/r is exact for ideal gears with no friction. In reality, there is sliding friction between teeth, which adds a small tangential component. However, for design purposes, the friction force is usually negligible (less than 1% of the tangential force) for well-lubricated gears.

The formula also assumes the force acts at the pitch point. In practice, the contact point moves along the line of action, and the force distribution is not uniform. But for bearing loads and shaft design, the pitch-point approximation is standard and conservative enough.

For helical gears, the axial force component is significant and must be calculated separately. The axial force is F_a = F_t · tan(ψ), where ψ is the helix angle. This calculator does not include that because it requires an additional input.

How to Choose the Right Pressure Angle

Most standard gears use a 20° pressure angle. It gives a good balance between tooth strength and bearing load. Older gears (pre-1960s) often used 14.5°, which has lower radial loads but weaker teeth. If you are replacing a gear, use the same pressure angle as the original – mixing pressure angles will cause interference and rapid wear.

For high-load applications, 25° pressure angles are sometimes used, but they increase bearing loads significantly. The calculator lets you compare: try 14.5° and 25° to see how the radial force changes.

Frequently Asked Questions

What is the difference between tangential force and radial force on a gear?

Tangential force is the force that actually transmits power – it acts along the direction of motion at the pitch point. Radial force pushes the gears apart and loads the bearings. For a 20° pressure angle, radial force is about 36% of tangential force.

How do I convert torque to force on a gear?

Divide the torque by the pitch radius. For example, 50 N·m of torque on a gear with a 50 mm pitch radius gives 1000 N of tangential force. Make sure to use consistent units – torque in N·m and radius in meters.

Does the number of teeth affect the gear force?

Not directly. The force depends on torque and pitch radius, not tooth count. However, for a given module, more teeth means larger pitch diameter, which reduces the force for the same torque. So indirectly, tooth count affects force through the radius.

What is a typical pressure angle for spur gears?

The most common is 20°. Some older or specialized gears use 14.5° or 25°. The pressure angle affects the radial force and tooth strength. Always match the pressure angle of the mating gear.

How do I calculate gear force in imperial units?

Use torque in lbf·in and pitch radius in inches. The result is in pounds-force. For example, 100 lbf·in of torque on a 2-inch pitch diameter gear (1-inch radius) gives 100 lbf of tangential force. Then multiply by tan(20°) = 0.364 for the radial force.

Why does the calculator show axial force as zero?

Because the axial force only exists on helical or bevel gears. For a spur gear (straight teeth), the axial force is zero. The calculator includes the axial output for completeness, but you need to enter a helix angle to get a non-zero value – which this simple version does not support.

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in t | Input torque | N·m | 50 | 0
in d | Pitch diameter | mm | 100 | 0
in pa | Pressure angle | deg | 20 | 0
out ft | Tangential force | N | (t * 1000) / (d / 2) | 1
out fr | Radial force | N | ft * tand(pa) | 1
out fa | Axial force (helical) | N | ft * tand(0) | 1