Power screws and ball screws

Why it matters

Screws turn motor rotation into precise linear motion: screw jacks, presses and vices use sliding power screws for large forces, while CNC axes, 3D printers, injection-moulding machines and electric actuators use ball screws for efficient, accurate positioning. Choosing between them and sizing the motor depends on torque, efficiency and whether the screw must hold the load by itself (self-locking).

Key ideas

Thread forms for power screws. Square threads have the highest efficiency but are hard to cut and cannot compensate wear. Trapezoidal (ISO metric, 30° included angle, designated e.g. Tr 40 × 7) and Acme (29°) threads are easier to make and allow a split nut to take up wear, at the cost of more friction. Buttress threads carry heavy load in one direction only.

Terms. Pitch p is the axial distance between adjacent threads; lead L is the axial advance per revolution; L = n·p for an n-start thread. Mean diameter dm = d − p/2 (square and trapezoidal, approximately). Lead (helix) angle α: tan α = L/(π·dm).

Mechanics of a square thread. Unwrap one turn of the thread into an inclined plane of angle α. Raising the load W against friction (friction angle φ, tan φ = μ) needs a tangential effort P = W·tan(φ + α) at the mean radius, so the torque is T = W·(dm/2)·tan(φ + α). Lowering needs T = W·(dm/2)·tan(φ − α); if this is positive (φ > α), the load does not run down by itself - the screw is self-locking.

Trapezoidal/Acme threads. The thread flank is inclined at θ/2 (half the thread angle), which raises the normal force; use a virtual friction coefficient μ′ = μ/cos(θ/2) in the square-thread formulas.

Collar friction. In a screw jack the rotating screw or nut bears on a thrust collar. Collar torque Tc = μc·W·Rm with Rm = (Ro + Ri)/2 for uniform wear (Rm = (2/3)(Ro³ − Ri³)/(Ro² − Ri²) for uniform pressure). It often equals or exceeds the thread torque and must not be forgotten.

Efficiency. η = work output per revolution / work input = W·L/(2π·T). For the thread alone η = tan α / tan(α + φ). It is maximum at α = 45° − φ/2, where ηmax = (1 − sin φ)/(1 + sin φ). A self-locking square-thread screw always has η < 50%.

Design checks. Direct compression/tension and torsional shear in the screw core (combine with max shear theory), shear of the threads at the root, bearing pressure between screw and nut threads (limits nut length), and buckling for long slender screws in compression.

Ball screws replace sliding with rolling balls that recirculate through the nut. Efficiency is typically about 90% or more, so they are not self-locking: a vertical axis needs a brake or counterbalance. Preload (double nut, oversized balls or offset lead) removes backlash and increases stiffness for precision positioning. Life is rated like a rolling bearing (dynamic load rating), and long screws are limited by critical (whirling) speed and buckling - values from the manufacturer's catalogue.

Formulas

L = n·p · tan α = L / (π·dm) · tan φ = μ

• L: lead (m); n: number of starts; p: pitch (m); dm: mean diameter (m); α: lead angle; φ: friction angle; μ: coefficient of thread friction.

T_raise = W·(dm/2)·tan(φ + α) · T_lower = W·(dm/2)·tan(φ − α) (square thread)

• T: torque on the thread (N·m); W: axial load (N).

μ′ = μ / cos(θ/2) (trapezoidal θ = 30°, Acme θ = 29°)

Tc = μc·W·Rm, Rm = (Ro + Ri)/2 (uniform wear)

• Tc: collar friction torque (N·m); μc: collar friction coefficient; Ro, Ri: collar outer and inner radii (m).

η = W·L / (2π·T_total) · η_thread = tan α / tan(α + φ) · ηmax = (1 − sin φ)/(1 + sin φ)

Self-locking: φ > α

T_motor = F·L / (2π·η) (ball screw, driving) · T_back = F·L·η′ / (2π) (torque produced by back-driving load)

• F: axial force on the nut (N); η: forward efficiency; η′: reverse efficiency.

p_b = W / (π/4·(d² − dc²)·z)

• p_b: bearing pressure on threads (Pa); dc: core diameter (m); z: number of threads engaged.

Worked examples

Example 1 (standard). A screw jack has a single-start square thread, d = 50 mm, p = 8 mm, μ = 0.1. The load of 20 kN rests on a collar of mean radius 30 mm with μc = 0.12. Find the raising torque, overall efficiency and whether it is self-locking.

1. dm = 50 − 8/2 = 46 mm; tan α = 8/(π × 46) → α = 3.17°; φ = tan⁻¹ 0.1 = 5.71°.
2. Thread torque: T = W·(dm/2)·tan(φ + α) = 20,000 × 23 × tan 8.88° = 71.9 N·m.
3. Collar torque: Tc = μc·W·Rm = 0.12 × 20,000 × 0.030 = 72.0 N·m.
4. Total torque = 143.9 N·m.
5. Overall efficiency: η = W·L/(2π·T) = 20,000 × 0.008 / (2π × 143.9) = 17.7% (thread alone 35.4%).
6. φ (5.71°) > α (3.17°) → self-locking; lowering torque = 20,000 × 0.023 × tan 2.54° = 20.4 N·m.

Example 2 (GATE level). A Tr 40 × 7 single-start trapezoidal screw (θ = 30°, dm = 36.5 mm) lifts 15 kN. Thread μ = 0.15; collar mean radius 25 mm, μc = 0.15. Find the total torque and the overall efficiency.

1. Virtual friction: μ′ = 0.15 / cos 15° = 0.1553 → φ′ = 8.83°.
2. tan α = 7/(π × 36.5) → α = 3.49°.
3. Thread torque: 15,000 × 0.01825 × tan 12.32° = 59.8 N·m.
4. Collar torque: 0.15 × 15,000 × 0.025 = 56.25 N·m.
5. Total = 116.0 N·m; efficiency = 15,000 × 0.007 / (2π × 116.0) = 14.4%.

Example 3 (ball screw sizing). A ball screw with lead 10 mm and η = 0.9 must push 400 N at 0.5 m/s. Motor torque = 400 × 0.01/(2π × 0.9) = 0.707 N·m; speed = 0.5/0.01 = 50 rev/s = 3000 rpm; power = 0.707 × 2π × 50 = 222 W (plus acceleration torque).

Common mistakes

• Using pitch instead of lead for multi-start screws.
• Forgetting collar friction, which can double the torque.
• Using μ instead of μ/cos(θ/2) for trapezoidal and Acme threads.
• Calculating efficiency from the thread torque only when the question asks for overall efficiency.
• Assuming a ball screw holds a vertical load - it back-drives.
• Using the outside diameter instead of dm in tan α.

For GATE ME

Expect torque to raise or lower a load, efficiency of a square-thread screw (with and without collar), the self-locking condition, maximum efficiency, and lead from pitch and starts. Practise the inclined-plane derivation so you can handle raising versus lowering signs without memorising, and remember μ′ for trapezoidal threads.

Quick check

1. Pitch 4 mm, three starts. Lead?
2. Condition for self-locking?
3. Maximum efficiency of a square thread with φ = 6°?
4. Why does a vertical ball-screw axis need a brake? Answers: 1. 12 mm. 2. φ > α. 3. (1 − sin 6°)/(1 + sin 6°) = 0.811, about 81%. 4. Its high efficiency makes it back-drivable (not self-locking).