Ball Screw Torque Calculation Formula
Estimate the torque required to drive a ball screw from axial load, lead, efficiency, and added drag torque. The calculator also estimates motor power at speed and visualizes how torque scales with load.
Force acting along the screw axis.
Linear travel per screw revolution.
Typical ball screw efficiency is often 85% to 98%.
Enter any extra torque losses in N·m.
Used to estimate mechanical power.
Apply a practical design margin to the final torque.
For most sizing work, choose Driving a load. Ball screws can often backdrive because efficiency is high.
Enter values and click Calculate Torque.
Expert Guide to the Ball Screw Torque Calculation Formula
Ball screws are among the most efficient and precise ways to convert rotary motion into linear motion. In CNC machinery, automated assembly equipment, semiconductor platforms, medical positioning systems, and packaging lines, they are often selected when designers need high repeatability, low friction, smooth travel, and a long service life. Even though the hardware looks simple from the outside, selecting the correct drive torque is one of the most important steps in the design process. If the torque is undersized, the axis may stall, overheat, or fail to meet acceleration targets. If it is oversized, the motion system may become unnecessarily expensive, heavy, and inefficient.
The ball screw torque calculation formula gives engineers a direct way to estimate how much rotational torque is needed at the screw to move a known axial load through a given lead. The reason the equation works is straightforward: each full revolution advances the nut by the screw lead, so the screw is doing linear work every turn. Torque is the rotary equivalent of that work. Efficiency must also be included because no real mechanical transmission is friction free. While ball screws are highly efficient compared with trapezoidal or Acme lead screws, they still lose some energy in rolling contact, recirculation, seals, support bearings, and preload.
What the Formula Means
The most common ball screw drive torque formula is:
T = (F × L) / (2π × η) + Textra
This equation is compact, but every term matters:
- F, axial load: This is the linear force the screw must overcome. It may come from gravity, cutting loads, process force, friction in guideways, or inertial loads reflected as linear force.
- L, lead: Lead is the linear distance traveled in one screw revolution. A higher lead gives more travel per turn, which improves linear speed at a given RPM, but it also increases torque demand.
- η, efficiency: Ball screw efficiency is usually high, often in the range of 0.85 to 0.98 depending on design, preload, lubrication, and operating condition.
- Textra: Many real systems have a baseline torque contribution from seal drag, bearing friction, coupling losses, preload, or contamination seals. This term helps bridge the gap between theory and field performance.
After the base torque is found, many engineers multiply by a safety factor. This is not because the equation is wrong, but because machines operate in the real world. Shock loads, transient acceleration, temperature changes, lubrication condition, contamination, and tolerances all influence actual running torque.
Why Ball Screw Torque Is Usually Lower Than Lead Screw Torque
A major reason ball screws are so popular is their excellent mechanical efficiency. Instead of sliding contact, they rely on recirculating balls that roll between the screw and nut raceways. Rolling contact dramatically reduces friction losses. That means more of the motor torque becomes useful linear force. In practical machine design, this often translates into smaller motors, lower heat generation, and better dynamic response.
| Drive Type | Typical Mechanical Efficiency | Backdrivable | Typical Use Case |
|---|---|---|---|
| Ball screw | 85% to 98% | Often yes | Servo axes, CNC, precision automation |
| Acme or trapezoidal lead screw | 20% to 70% | Often no or limited | Holding loads, low duty manual systems |
| Rack and pinion | 90% to 98% | Yes | Long travel linear axes |
The data above reflects common engineering ranges used in catalogs and machine design references. Notice that ball screws sit near the top of the efficiency spectrum for screw-driven linear motion. That is why they can deliver excellent force transmission with relatively modest torque.
How to Use the Formula Correctly
- Determine the axial load. Include process force, gravity if vertical, friction in linear guides, and any external forces that oppose motion.
- Convert to consistent units. The formula expects force in newtons and lead in meters per revolution if torque is desired in N·m.
- Choose a realistic efficiency. If the catalog gives a value, use it. If not, many designers start around 0.90 for a practical estimate and then refine from supplier data.
- Add extra torque losses. Preload, end bearing drag, seals, couplings, or contamination-resistant wipers can increase torque above ideal theory.
- Apply a safety factor. A design margin of 1.15 to 1.5 is common depending on duty, uncertainty, and consequence of failure.
- Check power at operating speed. Torque alone is not enough. The motor must provide the required torque at the planned RPM continuously or intermittently as the duty cycle demands.
Worked Example
Suppose a vertical axis must lift a 5,000 N load using a 10 mm lead ball screw with 90% efficiency. Assume an additional 0.2 N·m of drag torque from preload and bearings.
- Force: F = 5000 N
- Lead: L = 10 mm/rev = 0.01 m/rev
- Efficiency: η = 0.90
- Extra torque: 0.2 N·m
Base torque = (5000 × 0.01) / (2π × 0.90) = 8.84 N·m approximately.
Total torque with drag = 8.84 + 0.2 = 9.04 N·m.
If a 1.25 safety factor is applied, design torque becomes approximately 11.30 N·m. At 1500 RPM, mechanical power is about 1.77 kW. This example shows how quickly torque and power can grow when force and speed are both significant.
Lead Selection and Its Impact on Torque
Lead is a central design decision because it directly affects both speed and torque. A larger lead means the axis travels farther per revolution, which reduces the RPM needed to reach a target linear speed. However, the tradeoff is higher torque demand. A smaller lead lowers torque demand for the same force, but motor speed must increase to achieve the same linear velocity. That is why ball screw sizing is always a system-level optimization rather than a single equation exercise.
| Ball Screw Accuracy Grade | Typical Lead Error per 300 mm | Typical Application | Relative Cost |
|---|---|---|---|
| C0 | About ±5 µm | Ultra precision metrology and grinding | Very high |
| C3 | About ±7 µm | High-end CNC and precision automation | High |
| C5 | About ±18 µm | General CNC and industrial machinery | Medium |
| C7 | About ±52 µm | Positioning systems with moderate precision | Moderate |
| C10 | About ±210 µm | Transfer and utility motion | Lower |
These common catalog-grade figures are useful because they remind designers that torque is only one dimension of selection. The screw must also satisfy positional accuracy, stiffness, critical speed, and buckling requirements.
Important Factors the Basic Formula Does Not Fully Capture
The basic torque equation is excellent for first-pass sizing, but it should not be treated as the entire design model. A high-performance axis often requires additional checks:
- Acceleration torque: Rotating inertia from the screw, coupling, gearbox, and motor rotor can add substantial dynamic torque during starts and stops.
- Reflected inertia: The moving mass reflected through screw lead affects responsiveness and servo tuning.
- Critical speed: Long screws can whip at high RPM, limiting speed regardless of motor power.
- Buckling load: Compression-loaded screws on vertical or push applications must be checked against column buckling.
- Duty cycle: Continuous thermal loading may require a lower continuous torque rating than the peak value suggests.
- Lubrication condition: Poor lubrication can increase friction and shorten life.
Backdriving and Holding Considerations
Unlike many low-efficiency lead screws, ball screws are commonly backdrivable. This is a direct consequence of their high efficiency. If an axis is vertical, gravity can cause the load to drive the screw backward when power is removed. That is why vertical ball screw systems often use a motor brake, counterbalance, gas spring, or holding mechanism. The calculator includes a backdriving estimate mode because many designers want to compare the mechanical drive torque with the torque that may appear when the load tends to reverse-drive the system.
Best Practices for Motor Sizing With Ball Screws
- Use the torque formula for a clean baseline.
- Add preload and bearing losses rather than assuming zero drag.
- Apply a realistic safety factor based on application severity.
- Verify continuous and peak torque against the motor speed-torque curve.
- Check critical speed, buckling, and life from the screw supplier catalog.
- For servo systems, include inertia matching and acceleration profile analysis.
- For vertical axes, consider a holding brake because a ball screw may backdrive.
Authoritative References
For deeper engineering context, units, and motion-system fundamentals, review these authoritative sources:
- NIST: SI units and unit conversion guidance
- MIT OpenCourseWare: machine design and mechanics resources
- NASA Glenn Research Center: torque fundamentals
Final Takeaway
The ball screw torque calculation formula is one of the most useful first principles in precision motion design. It translates a linear load and screw lead into the rotary torque required at the drive. By accounting for efficiency, preload, and a practical safety factor, you can quickly create a realistic estimate for motor and drive sizing. The most important intuition is this: torque rises directly with load and directly with lead. If you double the load, torque doubles. If you double the lead, torque also doubles. Once that relationship is understood, the rest of the sizing process becomes far more systematic and predictable.
Engineering note: This calculator is ideal for preliminary sizing. For final design, confirm supplier efficiency data, preload torque, support-bearing drag, acceleration profile, screw critical speed, buckling limit, and life calculations.