What the calculation gives you
A lead screw turns a motor's rotation into a straight push. Given the load to move, the screw's lead and its efficiency, the calculation gives the torque the motor must deliver; with the speed you want, it gives the power. That is the number you take to a motor catalogue.
Torque to move the load
T = F · p / (2 π η)
T torque at the screw · F axial load to push (including friction of the guides and the weight if vertical) · p lead, the distance the nut travels per turn · η efficiency of screw and nut
The formula is work balance: one turn of the screw moves the load one lead, so the work out is F p, the work in is 2πT, and the efficiency is the ratio. Typical efficiencies: 0.2 to 0.4 for a plain (Acme or trapezoidal) screw with a bronze or plastic nut, 0.85 to 0.95 for a ball screw. The difference is why a ball screw needs a third of the motor for the same job.
If you prefer to work from friction rather than an efficiency, the same torque for a square-ish thread is T = (F dm/2) · (π μ dm + p) / (π dm − μ p), with dm the mean thread diameter and μ the friction coefficient, plus the collar or bearing friction. The efficiency form is the same equation with the friction folded into η.
Speed and power
n = v / p P = 2 π n T = F v / η
v linear speed of the nut · n screw speed in revolutions per unit time · P mechanical power at the motor shaft
The second form of P is a useful check: the power is the useful power Fv divided by the efficiency, whatever the lead. A finer lead lowers the torque and raises the speed; the power stays the same.
What else the motor must do
- Accelerate. Fast moves need extra torque Ta = J α, where J is the inertia of screw, coupling and rotor and α the angular acceleration. For a long screw at high speed this can exceed the load torque.
- Hold. A plain screw with efficiency below about 0.5 is usually self-locking: the load cannot back-drive it. A ball screw is not; a vertical axis on a ball screw needs a brake or a holding current.
- Keep torque at speed. A stepper's torque falls steeply with speed; read the torque-speed curve at the n you computed, not the holding torque.
- Margin. Friction grows with wear and dirt; a 25 to 50% margin over the computed torque is normal.
Worked example
The worksheet below pushes 1200 N with a 5 mm lead at an efficiency of 0.4, a typical plain screw. The torque comes out at 2.39 N·m; at 720 rpm the power is 180 W. Drag the load slider and watch both follow; change η to 0.9 to see what a ball screw would save.
Common mistakes
- Pitch instead of lead. On a single-start thread they are equal. On a two-start thread the lead is twice the pitch, and the torque and speed are both off by two if you use the pitch.
- Forgetting the guide friction. F is everything the nut has to push: the process load plus the friction of the linear bearings or ways, plus the weight on a vertical axis.
- Catalogue efficiency for a worn screw. Use the low end of the range for a plain screw, especially with a plastic nut.
- rpm and rad/s. P = 2πnT wants n in revolutions per second; with rpm, divide by 60. The worksheet carries the units and does this for you.
Questions
- Ball screw or lead screw?
- A ball screw is two to three times as efficient, more precise and lasts longer, but costs more, needs lubrication and does not hold a vertical load by itself. Plain screws win on cost, quiet running and self-locking.
- What about the critical speed?
- A long thin screw whips above a critical rotational speed set by its diameter, length and end supports. Check it for screws over about 40 diameters long; it is a separate calculation from the torque.
- Can I use this for a jack or a vise?
- Yes: it is the same screw-thread mechanics. For a jack, add the collar friction, which is often as large as the thread friction.
