The pitch P is the distance between two neighbouring thread flanks. The lead Ph is how far the nut travels in one turn of the screw. On a single-start screw the two are equal, on a multi-start screw the lead is a multiple of the pitch. Mixing them up gives the wrong speed, the wrong motor torque and, on vertical axes, a screw that no longer holds its load.
TorqueMath · checked against the standards listed below and the calculator’s test cases · Updated
Pitch, lead and number of starts
A screw can have one or more threads wound side by side, like the stripes of a barber's pole. Each one is a start. Between neighbouring flanks you always measure the pitch; the same thread comes round again only after all the other starts have passed.
Ph lead (mm per turn), n number of starts, P pitch (mm).
Linear speed in mm/min from the lead in mm and the screw speed in rpm. Example: Ph 4 mm at 300 rpm gives 1200 mm/min = 20 mm/s.
Use the speed of the screw. If a belt drive or gearbox sits between motor and screw, screw speed = motor speed / ratio. Efficiency does not change the travel per turn: the lead is pure geometry, efficiency only affects the torque.
Reading screw designations
Trapezoidal threads to ISO 2904 / DIN 103 give the nominal diameter and the lead, with the pitch added after P when they differ. Ball screws are usually named by diameter and lead in two two-digit groups.
| Designation | Type | Pitch P | Starts | Lead Ph |
|---|---|---|---|---|
| Tr 16×4 | trapezoidal, 30° | 4 mm | 1 | 4 mm |
| Tr 16×8 P4 | trapezoidal, 30° | 4 mm | 2 | 8 mm |
| Tr 8×8 P2 | trapezoidal (3D printers) | 2 mm | 4 | 8 mm |
| SFU 1605 | ball screw Ø16 | – | see maker | 5 mm |
| SFU 1610 | ball screw Ø16 | – | see maker | 10 mm |
| SFU 2005 | ball screw Ø20 | – | see maker | 5 mm |
For ball screws only the lead matters for speed and torque; the number of ball circuits is a maker's detail. For trapezoidal screws always check whether the second number is the lead or the pitch.
How to measure the lead
- Count the starts at the end of the screw: the number of thread ends that begin around the circumference is the number of starts.
- Measure the pitch with a caliper over several flanks, for example over 10 pitches, and divide by 10. A thread gauge also works for common pitches.
- Multiply: lead = starts × pitch.
- Check by turning: mark the nut, turn the screw exactly one revolution and measure the travel. This gives the lead directly and catches mistakes in counting starts.
Do not measure from one flank to the next one of the same colour on a photo or drawing without knowing the number of starts: that is the lead on a single-start screw, but it can be a fraction of the lead on a multi-start screw.
What the lead changes
For the same motor speed, a larger lead gives proportionally more linear speed, but the motor must deliver proportionally more torque and the screw is less likely to hold the load by itself.
T drive torque, F axial load, d₂ pitch diameter, λ lead angle, ρ' friction angle = atan(μ / cos 15°) for a 30° trapezoidal thread. With the running friction μ = 0.12 this gives ρ' = 7.08°. For the self-locking check the calculator uses static friction μs = 1.25 × μ = 0.15, giving ρ's = 8.83°. A screw meets the calculated self-locking condition when λ is smaller than ρ's. That describes the condition under the assumed friction values, not a guaranteed holding function: shock, vibration and changed friction can make a screw creep.
Example: 200 steps × 16 microsteps / 4 mm = 800 steps/mm. That is a commanded resolution of 1.25 µm per microstep, not a guaranteed positioning accuracy: missed steps, elastic deformation, backlash and microstep non-linearity are not included. With a belt or gearbox between motor and screw, include its ratio.
| Screw | Linear speed | Drive torque | Efficiency | Calculated self-locking? |
|---|---|---|---|---|
| Tr 16×4 | 1200 mm/min | 3.05 N·m | 42 % | yes: λ 5.2° < ρ's 8.8° |
| Tr 16×8 P4 | 2400 mm/min | 4.39 N·m | 58 % | no: λ 10.3°, the load needs 0.79 N·m holding torque |
| Ball screw 1605 | 1500 mm/min | 1.67 N·m | 95 % | no: ball screws are generally not self-locking |
Calculated self-locking is not a safety function. On vertical or safety-relevant axes a separately designed brake or holding function is required.
Worked example: Tr 16×4 or Tr 16×8 P4 for a 2 kN lifting axis
- Task: lift 2 kN at 20 mm/s or more, with a motor that runs at 300 rpm. The axis is vertical.
- Tr 16×4: one start, P = 4 mm, Ph = 4 mm. Speed = 4 × 300 = 1200 mm/min = 20 mm/s. The calculator gives 3.05 N·m drive torque and 42 % efficiency. The lead angle 5.20° is below the static friction angle 8.83° (μs 0.15), so in the simplified static model Tr 16×4 meets the self-locking condition. That is no guaranteed holding: on a vertical axis an independently designed holding or safety function is still required.
- Tr 16×8 P4: two starts, P = 4 mm, Ph = 2 × 4 = 8 mm. Speed doubles to 40 mm/s and efficiency rises to 58 %, but torque rises to 4.39 N·m and the lead angle 10.31° is above the friction angle: the load can turn the screw back, so a brake or self-locking gearbox is needed.
- With a stepper at 200 steps and 16 microsteps: Tr 16×4 gives 800 steps/mm, Tr 16×8 P4 gives 400 steps/mm.
Tr 16×4 reaches 20 mm/s and meets the calculated self-locking condition, Tr 16×8 P4 is twice as fast and more efficient but back-drives. On this vertical axis both need a separately designed brake or holding function; self-locking is never the safety function.
Common mistakes
- Reading Tr 16×8 P4 as pitch 8: the 8 is the lead, the pitch is 4 mm.
- Setting steps per mm from the pitch instead of the lead on a multi-start screw: every move is too short or too long by the number of starts.
- Relying on self-locking on a vertical axis. Ball screws are generally not self-locking because of their high efficiency, and even a calculated self-locking trapezoidal screw can creep under vibration: use a proper brake or holding function.
- Measuring one flank to the next and calling it the lead without counting the starts.
- Ordering a nut by pitch only. Nut and screw must match in diameter, pitch, number of starts and hand (right or left).
Sources and standards
- DIN 103 – metric trapezoidal threads, including the designation of multi-start threads (Tr d × Ph P). ISO 2901–2904 – metric trapezoidal threads: basic profile, general plan, tolerances and basic dimensions.
- ISO 3408 – Ball screws (nominal diameter and lead).
- Torque, efficiency and self-locking: thread friction formulas as in Roloff/Matek Maschinenelemente and Shigley's Mechanical Engineering Design, implemented in our lead screw calculator, how it was checked.