Ball Screw Force Calculator & Formula

This calculator works out the axial force produced by a ball screw when a known torque is applied, or the torque needed to produce a desired axial force. You enter the applied torque (or force), the screw’s lead, and its efficiency, and you get the corresponding axial force (or torque) in newtons and newton-meters. The calculation uses the standard lead-screw force equation and accounts for friction losses through the efficiency term.

Formula

The axial force \(F\) generated by a ball screw from an applied torque \(T\) is given by:

\(F = \dfrac{2 \pi \cdot T \cdot \eta}{L}\)

Where:

• \(F\) = axial force (newtons, N)
• \(T\) = applied torque (newton-meters, N·m)
• \(\eta\) = efficiency of the ball screw (as a decimal, e.g., 0.9 for 90%)
• \(L\) = lead of the screw (meters, m)

If you start with a desired axial force and want the required torque, rearrange to:

\(T = \dfrac{F \cdot L}{2 \pi \cdot \eta}\)

Note that the lead must be in meters to match the torque units. If your lead is in millimeters, convert by dividing by 1000 (as done in the calculator).

Worked Example

Using the default values: torque \(T = 5\,\text{N·m}\), lead \(L = 10\,\text{mm} = 0.010\,\text{m}\), efficiency \(\eta = 90\% = 0.90\).

1. Convert lead to meters: \(10\,\text{mm} = 0.010\,\text{m}\).
2. Plug into the formula: \(F = \dfrac{2 \pi \times 5 \times 0.90}{0.010}\).
3. Calculate numerator: \(2 \pi \times 5 \times 0.90 = 28.2743\,\text{N·m}\).
4. Divide by lead: \(28.2743 / 0.010 = 2827.43\,\text{N}\).

The calculator rounds to 2827 N. So 5 N·m of torque on a 10 mm lead, 90% efficient ball screw produces about 2.83 kN of linear force.

How Ball Screws Work

A ball screw converts rotary motion into linear motion using recirculating ball bearings that roll between the screw shaft and the nut. Unlike a conventional lead screw (which slides with high friction), the balls roll, greatly reducing friction and increasing efficiency. Typical ball screw efficiency ranges from 85% to 95%, depending on lead angle, preload, and lubrication. The lead (also called pitch) is the linear distance the nut travels in one full revolution of the screw. A larger lead means more travel per turn but requires more torque to produce the same force.

How to Choose a Ball Screw

When selecting a ball screw for an application, consider these factors:

• Required axial force: Use the calculator to determine the torque needed from your motor.
• Lead: Choose a lead that balances speed and resolution. Small leads give finer positioning but lower speed for a given motor RPM.
• Efficiency: Higher efficiency reduces heat and energy loss, but may cost more.
• Critical speed: Long, thin screws can whip at high RPM; check the manufacturer’s critical speed chart.
• Buckling load: For compression loads, ensure the screw does not buckle – this depends on unsupported length and diameter.
• Accuracy grade: Ball screws come in accuracy classes (e.g., C3, C5) that define lead error; choose based on positioning tolerance.

Common Mistakes and Misconceptions

One frequent error is confusing lead with pitch. On a single-start screw, they are the same, but on a multi-start screw, lead = pitch × number of starts. Always use lead in the formula, not pitch.

Another mistake is forgetting to convert units. Torque in N·m and lead in mm will give a force that is off by a factor of 1000. Always convert lead to meters.

Efficiency is not constant – it varies with load, speed, and lubrication. The 90% default is typical, but check the datasheet for your specific screw. Also, the formula assumes a horizontal or vertical orientation without additional friction from guides or seals.

Finally, the calculated force is the theoretical ideal. In reality, the motor must also overcome inertia, friction in bearings, and any external loads like gravity or cutting forces. Add a safety factor (often 1.5 to 2) when sizing a motor.

Alternatives to Ball Screws

Ball screws are not the only way to convert rotary to linear motion. Depending on your application, consider:

• Lead screws (Acme or trapezoidal): Cheaper and self-locking (can hold position without a brake), but lower efficiency (20–50%) and more wear.
• Belt drives: Fast, cheap, and quiet, but lower precision and stiffness; prone to stretching.
• Rack and pinion: Good for very long travel, but has backlash and requires maintenance.
• Linear motors: Direct drive, high speed and precision, but expensive and require a controller.

Choose based on speed, precision, load, cost, and environmental factors.

Safety and Maintenance Considerations

Ball screws can generate high forces, so always design with proper guards and limit switches to prevent over-travel. Overloading a ball screw can cause permanent damage to the balls or raceways, leading to failure. Regular lubrication is essential – use the manufacturer-recommended grease or oil. Check for backlash and preload periodically; excessive backlash reduces positioning accuracy. When mounting, ensure the screw is aligned with the load to avoid side loads that reduce life. If you are unsure about a specific application, consult the screw manufacturer’s engineering guide.

Frequently Asked Questions

What is the difference between lead and pitch?

Pitch is the distance between adjacent threads, while lead is the distance the nut travels in one revolution. For a single-start screw, they are equal. For a multi-start screw (e.g., 2-start), lead = pitch × number of starts. Always use lead in force calculations.

How does efficiency affect the force?

Efficiency directly multiplies the ideal force. A 90% efficient screw produces 90% of the force you would get with a 100% efficient (frictionless) screw. Lower efficiency means more torque is lost to friction, so you need a larger motor to achieve the same force.

Can I use this calculator for a lead screw (Acme thread)?

Yes, the same formula applies, but the efficiency will be much lower (typically 20–50%). Use the actual efficiency of your lead screw, which depends on the thread angle and friction coefficient. The calculator will then give a reasonable estimate.

Why is my calculated force different from the actual force?

The formula is a static idealization. Real systems have friction in bearings, seals, and guides, plus inertia during acceleration. Also, the screw’s efficiency varies with load and speed. Always add a safety factor and verify with a load cell if precise force is critical.

How do I convert the force to pounds or torque to inch-pounds?

To convert newtons to pounds-force, multiply by 0.2248. To convert N·m to in-lb, multiply by 8.851. For example, 2827 N is about 635 lbf, and 5 N·m is about 44.3 in-lb.

What is a ball screw’s maximum speed?

Maximum speed is limited by critical speed (whirling) and the nut’s ball recirculation. For long screws, critical speed is the main limit. You can estimate it using the screw diameter, length, and end fixity, but always refer to the manufacturer’s data.

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in t | Applied torque | N·m | 5 | 0
in lead | Screw lead | mm | 10 | 0
in eff | Efficiency | % | 90 | 0
out force | Axial force | N | (2 * pi * t * (eff / 100)) / (lead / 1000) | 0