Ball and Acme Leadscrews

§1From thread to axis

A leadscrew is the power thread promoted to a machine element — and the promotion changes what matters: from static capacity to continuous duty, positional truth and life.

The Acme page sized a jack that lifts occasionally; a leadscrew strokes for a living, and three new ledgers open. Duty: the friction heat priced per lift now arrives per metre travelled, all shift long, so efficiency stops being a curiosity and becomes the electricity bill and the temperature of the nut. Accuracy: the screw is now the machine’s ruler — every micrometre of lead error and every hundredth of backlash prints directly into the work — so lead precision (the ground-thread inheritance of the grinding page) and §5’s backlash management become first-class specifications. Life: a sliding nut wears toward the tolerance limit; a rolling nut fatigues on the bearing page’s L₁₀ clock; either way the axis has a computable lifetime the jack never needed. And one member joins the assembly that the threads section could ignore: the shaft itself, now long, slender, spinning and squeezed between its end bearings — §6’s subject, and the limit that surprises more axis designs than any nut ever has.

§2Acme in service

The sliding trapezoid remains the right axis wherever its two gifts — the built-in brake and the split-nut’s wear recovery — are worth its one recurring bill.

Everything the Acme page established transfers whole: the 29°/30° form, the T = F(d₂/2)tan(λ ± ρ′) sizing, the self-locking test, and the bronze-nut-and-oil economy in which the nut is the sacrificial, replaceable partner. What service adds is the discipline of living with η around a third. The heat ledger — roughly 1.78 units into the flanks per unit of lift — now runs continuously, so an Acme axis is sized against nut temperature and PV exactly as the plain-bearings page sized a bush, and duty cycle is a rating, not a footnote. The wear ledger arrives as growing backlash, and the trapezoid answers with its own patent virtue: the split or adjustable nut, closed periodically to chase the wear — the lathe cross-slide’s ritual, institutionalised. In exchange the axis keeps the two properties no ball screw offers free: it holds position unpowered, a vertical slide or clamp parked forever on its own friction; and it is cheap, quiet and dirt-tolerant, a rolled thread and a bronze nut against a precision-ground raceway system. Manual machines, jacks, vices, clamps, adjusters and modest positioning axes remain, quite rationally, Acme country.

§3The ball screw

Fill the space between screw and nut with balls and a return path, and the leadscrew becomes a rolling bearing wrapped on a helix — with everything that family promised and everything it charged.

The hero shows the anatomy. Screw and nut carry matching gothic-arch grooves — an ogival profile giving each ball four-point-capable contact and a defined running angle — and the balls roll along the loaded turns, exit through a pick-up into the return (a tube over the nut, or internal deflectors turn-to-turn), and re-enter at the start: an endless circuit, so the nut can travel indefinitely on a finite crowd of balls. The bearing page’s physics arrives intact. Friction falls to rolling values — efficiency around 90%, from the first degree of rotation — so drive torque, motor size and heat collapse (§4 prices it); load passes through point contacts of hardened steel, so capacity and life are catalogue numbers on the L₁₀ clock, lubrication is elastohydrodynamic and non-negotiable, and dirt is the enemy that bearing-grade sealing and wipers exist to exclude. And the family’s statics come too: the ball nut, like any rolling assembly, has internal clearance unless §5 removes it, and it back-drives — the load can spin the screw, which is not a defect but the flip side of the friction it no longer wastes, and the single most consequential line in §4’s table.

§4One job, two bills

Put the same load on the same lead through both nuts and the whole choice is four numbers: what the motor pays, and what holds the load when the motor stops.

§5Backlash and preload

An axis reverses; whatever clearance lives between flank and ball is lost motion printed into every reversal — so precision screws remove it the bearing page’s way, with preload, and pay the bearing page’s price.

The Acme’s answer was §2’s split nut, closed onto the worn flanks. The ball screw’s answers are the angular-contact pair’s, translated. A double-nut assembly loads two nuts against each other through a ground spacer or spring, so one nut’s balls bear each flank direction and the reversal crosses no gap at all; a single nut achieves the same with oversize balls pressed into the gothic arch’s four-point contact, or with a lead-shifted nut body doing the double-nut’s job internally. The result is an axis stiff and true through zero — and a familiar invoice. Preload is permanent internal load: drag torque the motor pays every revolution, heat that grows with speed, and consumption of a slice of the catalogue’s capacity before the work applies any — the same thermal-spiral caution the spindle bearings carried, and the reason preload is specified as a class (a percentage of dynamic capacity), not maximised. Field diagnosis closes the loop: growing reversal error on a preloaded axis is the preload wearing off — the L₁₀ clock made audible — and the servo compensating in software is a bookmark, not a repair.

§6The shaft between the bearings

Long before a good nut runs out of capacity, the slender spinning shaft it rides on runs out of physics — twice: once in whirl, once in buckling.

§7Quick reference

The working core of the page on one card rack.