Design the ball screw that converts a motor's rotary motion into the linear motion of a table. Adjust the lead, speed and driving torque to see the feed speed, thrust, lead angle, driving power and load acceleration update in real time, and study the feed mechanism of CNC machines and robot axes.
As the screw shaft rotates, the ball nut and carriage advance by one lead ℓ for every turn, carried by the recirculating balls. One revolution moves the table by one lead.
Linear feed speed V and thrust F. A larger lead raises the speed but lowers the thrust for the same torque (the speed-versus-thrust trade-off).
$$\lambda=\arctan\!\frac{\ell}{\pi d}$$
Lead angle λ. ℓ is the lead, n the speed, T the driving torque, η the mechanical efficiency and d the shaft diameter. A larger lead angle raises efficiency but makes the screw less self-locking.
What is a Ball Screw?
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A "ball screw" is that long threaded rod that moves a machine-tool table, right? How is it different from an ordinary screw?
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Exactly — the long screw that drives the X, Y and Z axes of a CNC machine. Its job is to convert the motor's rotary motion into the linear motion of the table. So far that is the same as an acme screw (an ordinary trapezoidal-section lead screw). But a ball screw sandwiches lots of small steel balls between the screw groove and the nut, and recirculates those balls. So the nut does not "slide" along the screw — the balls "roll".
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Does rolling really make that much difference?
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A dramatic difference. An acme screw with sliding friction has a mechanical efficiency of roughly 30-50% — so it throws away more than half the motor's power as friction heat. A ball screw uses rolling friction, so the efficiency is 90-95%. Look at the "Mechanical efficiency" slider on the left. For the same torque and lead, the higher the efficiency the more thrust F = 2π·η·T/ℓ you get out. You can move heavier loads faster with the same motor — that is exactly why CNC machines and robots use ball screws.
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The word "lead" keeps coming up — what is it?
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The lead is the distance the nut travels in one full turn. A 10 mm-lead screw advances 10 mm per revolution. Feed speed is V = ℓ·n/60, so a larger lead is faster. But it is not free. Look at the thrust F = 2π·η·T/ℓ — the lead ℓ is in the denominator. So a larger lead is faster but produces less thrust for the same torque. The "Thrust vs lead" chart below is exactly that falling curve. Use a large lead for fast transport and a small lead for heavy cutting.
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So a high-efficiency ball screw is all upside, then!
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Not quite — high efficiency has a hidden downside. An acme screw is so inefficient that it stays put even when you push on it: it is "self-locking". But a ball screw is too efficient to self-lock. Push the nut axially and the screw spins backward easily — "back-driving". So if you use a ball screw on a vertical Z axis, the moment you cut the motor power the spindle head drops under gravity. To prevent that, servo motors on vertical axes always get an electromagnetic brake. Convenience and a safety measure come as a set.
Frequently Asked Questions
A ball screw moves the nut by one lead ℓ for every full turn. The feed speed is therefore V = ℓ·n/60, where ℓ is the lead in mm, n is the speed in rpm and V is in mm/s. For a 10 mm lead at 1500 rpm, V = 10×1500/60 = 250 mm/s = 0.25 m/s. Feed speed is simply proportional to both lead and speed, so for a fast axis you choose a screw with a larger lead or run the servo at a higher speed.
The thrust generated from the driving torque T is F = 2π·η·T/ℓ, where ℓ is the lead converted to metres and η is the mechanical efficiency. A smaller lead gives more thrust for the same torque, but a slower feed speed — this is the speed-versus-thrust trade-off. For a 10 mm lead, 5 N·m torque and η = 0.90, the thrust is F = 2π×0.90×5/0.010 ≈ 2827 N. Choose a small lead to push heavy cutting loads and a large lead for fast transport.
An acme (sliding) lead screw has the external and internal threads rubbing face to face, so friction is high and the mechanical efficiency stays around 30-50%. A ball screw circulates many balls between the screw groove and the nut, replacing sliding friction with rolling friction. This raises the efficiency to 90-95%, giving more thrust and a faster feed from the same motor. That is why ball screws drive the feed axes of CNC machines, industrial robots and precision stages.
The flip side of a ball screw's high efficiency is that it is NOT self-locking. Push the nut axially and back-driving occurs easily, so the moment the motor power is cut the load drops under gravity. A vertical-axis ball screw (such as a Z axis) therefore always needs a holding brake to keep the load in place during a power loss or emergency stop. An acme screw is self-locking because its efficiency is low, but with a ball screw a holding brake is an essential part of a safe design.
Real-World Applications
Feed axes of CNC machine tools: The X, Y and Z feed axes of machining centres and lathes are the most representative use of ball screws. Because large cutting reaction forces act on the table during machining, a small-lead screw with thrust to spare is chosen, and preload (an initial tightening of the balls) brings the backlash close to zero for positioning accuracy. High-speed machines use large-lead screws or a "rotating-nut" design — turning the nut instead of the screw — to gain feed speed.
Industrial robots and transfer equipment: Cartesian robots (gantry loaders), electric cylinders and the stage drives of semiconductor equipment all use ball screws. The acceleration a = F/m from this tool is a quick check of whether the load can be accelerated and decelerated within the required cycle time. For heavy loads or fast cycles, thrust is consumed by acceleration and less remains for cutting or pressing, making the acceleration design critical.
Injection moulding machines and presses: Modern electric injection moulding machines perform clamping and injection with a servo motor and a ball screw. Compared with hydraulics they respond faster, are more energy-efficient and make position, speed and pressure easier to control. Because large clamping forces are needed, several ball screws may be used in parallel, or a very small-lead, high-thrust type is selected.
Precision positioning stages: The drive axes of measuring instruments, optical equipment, 3D printers and medical devices require micron-level positioning. With rolling contact, a ball screw has low friction and little stick-slip (jerkiness) at start-up, making it well suited to smooth, repeatable fine feed. Where even higher accuracy is needed, linear motors are considered as an alternative, but ball screws are often chosen for their thrust, stiffness and cost.
Common Misconceptions and Pitfalls
A common mistake is to assume that a ball screw is so close to 100% efficient that losses can be ignored. 90-95% is indeed high, but the remaining 5-10% heats the screw shaft as friction heat. Thermal expansion of the shaft enlarges the positioning error, and the longer the screw the worse it gets. To suppress this thermal displacement, precision machines run cooling oil through a hollow shaft, fix both ends to apply pretension, or apply temperature compensation. The efficiency η in this tool is a calculated value; on a real machine, always study heat generation and thermal displacement together.
Next, assuming that as long as the thrust is there, the speed can be raised without limit. Feed speed V is the product of lead and speed, but the speed has two upper limits: the "critical speed" (bending resonance of the shaft) and the "DN value" (the rolling limit of the balls). Spin a thin, long screw shaft too fast and the shaft itself whips like a skipping rope at its critical speed and vibrates violently. However attractive the V and F from this tool look, you must separately check the manufacturer's catalogue to confirm the speed is below the critical speed and the allowable DN value.
Finally, assuming that choosing a ball screw means you need not worry about life. A ball screw also has a rolling-fatigue life, and that life shrinks inversely with the cube of the axial load. In other words, double the thrust and the life drops to about one-eighth. This tool deals with the static relationship between torque, thrust and speed, but a real design computes the rated life (travel distance or hours) from the mean axial load and chooses a size that meets the required life. In addition, preload, lubrication and protection from contamination (bellows and wipers) all strongly affect life.
How to Use
Enter ball screw lead (mm/rev) in the leadNum field—typical values range 5–20 mm for CNC machines
Set screw diameter (mm) using diaNum; standard sizes include 16, 20, 25 mm for spindle applications
Input motor speed (rpm) in rpmNum; servo drives commonly operate 3000–6000 rpm
Specify driving torque (N·m) in torqueNum based on motor nameplate ratings
Click Calculate to compute linear feed speed (m/s), thrust force (N), lead angle, power consumption (W), mechanical efficiency (%), and load acceleration (m/s²)
Worked Example
A CNC milling machine uses a ball screw with lead=10 mm/rev, diameter=20 mm, motor speed=4500 rpm, driving torque=3.5 N·m. Linear feed speed V = (4500 rpm × 10 mm) / 60000 = 0.75 m/s. Thrust force F = (3.5 N·m × 2 × tan(λ)) / (π × 0.020 m), where lead angle λ ≈ 4.55°, yields F ≈ 1240 N. Driving power P = 3.5 × (4500 × 2π/60) ≈ 1648 W. For 3 kg carriage mass, acceleration a = 1240 / 3 ≈ 413 m/s² assuming 95% mechanical efficiency.
Practical Notes
Preload class (light, standard, heavy) affects no-load torque 5–15% in high-speed grinding spindles; account for thermal growth on extended runs
Lead angles below 3° require higher torque but generate thrust; angles above 10° risk self-locking degradation in vertical axis applications
Backlash compensation: standard ±0.05 mm, precision grade ±0.01 mm for 5-axis simultaneous interpolation
Lubrication duty cycles at >8000 rpm demand synthetic grease ISO VG 32 instead of mineral oil to prevent cavitation