Sizing a Lead-Screw or Ball-Screw Positioning Stage

On a horizontal stage the screw does not lift the carriage, it only overcomes the friction of the linear guides, so the axial thrust it must produce is F = µ·m·g, where µ is the guide (or bearing) friction coefficient — low for recirculating linear guides (≈0.003–0.05), higher for plain slides. On an inclined or vertical stage add the gravity component along the axis, m·g·sin(θ); on a fast move add the inertial force m·a. The cylinder/load builder assembles this force for you from the mass, the friction coefficient and the incline, so you never hand-total it.

Getting this force right matters because everything downstream scales from it: the screw torque is proportional to the thrust, and an over-stated friction load over-sizes the whole drive. For a horizontal stage on good linear guides the friction load is often small, and the drive is then sized more by acceleration (Step 3) than by the steady thrust.

The screw converts rotation into that axial thrust; the torque it needs depends on the lead, the thread geometry and the friction. For a lead screw the power-screw torque model (Shigley) gives the raise torque from the mean diameter, the lead angle and the modified friction coefficient of the thread form; a lower thread friction or a ball screw (rolling contact, efficiency ≈ 0.9) needs far less torque than a plain Acme screw for the same thrust. The same model returns the efficiency and whether the screw is self-locking — a valuable property for a vertical stage that must hold position with the motor off.

Use the power-screw calculator for a lead/Acme screw and the ball-screw calculator (ISO 3408) for a ball screw, entering the thrust from Step 1. The torque you get is the steady drive torque to move the load; the motor must add the acceleration torque on top, which is where the reflected inertia in Step 3 comes in.

The motor does not just hold torque against friction — it has to accelerate the whole moving mass, and to a rotating motor the linear carriage appears as an equivalent rotary inertia through the screw. A ball/lead screw reflects a linear mass m to the motor shaft as J = m·(lead/2π)², a very small number because the lead is small — which is exactly why screw drives suit precise positioning. Add the screw's own inertia and any coupling, and reflect anything on the far side of a reduction by dividing by the ratio squared.

The number that decides whether the servo is tunable is the inertia ratio: the reflected load inertia divided by the motor's rotor inertia. Servo practice keeps this below roughly 5–10; too high and the drive is sluggish and hard to settle, too low and you are over-motored. The motor drive-train reflection calculator reflects the mass (and any gear or belt stage) to the shaft, computes the motor speed from the linear speed and the lead, and returns the inertia ratio so you can pick a motor whose rotor inertia lands the ratio in a healthy band.

The guide friction is small but not zero, and it is the number you know least precisely — bound it. On a vertical stage, check the holding case separately: a ball screw is not self-locking and will back-drive under gravity, so it needs a brake, whereas a high-friction Acme lead screw may self-lock (the power-screw calculator tells you which). Keep the inertia ratio in a healthy band by choosing the lead and the motor together — a longer lead moves faster but reflects more inertia and needs more torque. And confirm the screw's own limits (critical speed and column buckling) against your travel and speed; the ball-screw calculator checks those to ISO 3408.