---
title: "Maximum efficiency and ball screws"
subject: "Machine Design II"
chapter: 8
key: "semester_7/machine_design_ii:self-locking-maximum-efficiency-ball-screw"
source: https://www.bctengineer.com/q/semester_7/machine_design_ii/self-locking-maximum-efficiency-ball-screw
site: BCT Engineering
---

# Maximum efficiency and ball screws

Machine Design II · Chapter 8: Power screw

- Asked in: year not given, 6 marks
- Video: [Power Screws - Torque to Force Relationships in Just Over 10 Minutes!](https://www.youtube.com/watch?v=BstZUC4tcOA) — Less Boring Lectures
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/machine_design_ii/self-locking-maximum-efficiency-ball-screw
- Whole chapter: https://www.bctengineer.com/md/semester_7/machine_design_ii/chapter-8

## Question

Show that the maximum efficiency of a square-threaded screw is (1 - sin phi)/(1 + sin phi) and find the helix angle at which it occurs. For a friction coefficient of 0.1 calculate both. Explain why such a helix angle is not used in practice, the effect of thread angle on the effective friction in an Acme thread (mu = 0.12, half angle 14.5 degrees), and the advantage of a ball screw.

## Short answer

For square threads, max efficiency (1-sin φ)/(1+sin φ) at α=45°-φ/2. For μ=0.1, α_opt=42.1°, η_max=81.9%. Such a large helix is not self-locking and gives low mechanical advantage. Acme: effective μ'=0.12/cos14.5°≈0.124, raising φ' and reducing efficiency. Ball screws (rolling elements, μ~0.005-0.01) achieve >90% efficiency, reversible, need brake.

## Answer

#### Maximum efficiency
$$
\eta = \frac{\tan\alpha}{\tan(\alpha+\varphi)}
$$

Writing the tangents as sines and cosines and using the product-to-sum identities:

$$
\eta = \frac{\sin\alpha\cos(\alpha+\varphi)}{\cos\alpha\sin(\alpha+\varphi)} = \frac{\sin(2\alpha+\varphi)-\sin\varphi}{\sin(2\alpha+\varphi)+\sin\varphi}
$$

The expression is largest when $\sin(2\alpha+\varphi)$ is largest, that is, when $2\alpha+\varphi = 90^\circ$:

$$
\alpha_{opt} = 45^\circ - \frac{\varphi}{2}, \qquad \eta_{max} = \frac{1-\sin\varphi}{1+\sin\varphi}
$$

#### Numerical ($\mu = 0.1$)
$\varphi = \tan^{-1}0.1 = 5.71^\circ$, $\sin\varphi = 0.0995$.

$$
\alpha_{opt} = 45 - 2.86 = 42.1^\circ, \qquad \eta_{max} = \frac{1-0.0995}{1+0.0995} = 0.819 = 81.9\%
$$

#### Why not used in practice
A helix angle of 42 degrees means a very large lead: the screw is **not self-locking** ($\alpha > \varphi$), has a small mechanical advantage and cannot hold a load. Practical power screws use $\alpha$ of 2 to 6 degrees (self-locking, large force) and accept efficiencies of 25 to 40%. Self-locking requires $\alpha \le \varphi$.

#### Acme thread
The thread angle increases the normal force on a thread flank, so the effective friction coefficient becomes

$$
\mu' = \frac{\mu}{\cos\theta} = \frac{0.12}{\cos14.5^\circ} = 0.124
$$

and $\varphi' = \tan^{-1}\mu'$ is used in the torque equation $T = W\dfrac{d_m}{2}\tan(\alpha+\varphi')$. The efficiency is slightly lower than for a square thread of the same lead.

#### Ball screw
The screw and nut have grooves with recirculating steel balls between them, which replace sliding friction by rolling friction ($\mu \approx 0.005$ to $0.01$).
- Efficiency of 90% and above, and low starting torque (no stick-slip).
- Reversible: it is not self-locking, so a brake is needed for vertical loads.
- Used in CNC machine feed drives, aircraft actuators and steering gear, where accuracy and efficiency matter more than cost.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Efficiency is maximum when
   A. alpha = phi
   B. 2 alpha + phi = 90 degrees, so alpha = 45 - phi/2
   C. alpha = 0
   D. alpha + phi = 45 degrees
   Answer: B. Efficiency (sin(2 alpha + phi) - sin phi)/(sin(2 alpha + phi) + sin phi) is largest when the sine is 1. (Key point: Condition for max efficiency)

2. For mu = 0.1, the maximum efficiency and optimum helix angle are about
   A. 81.9% at 42.1 degrees
   B. 81.9% at 5.7 degrees
   C. 50% at 42.1 degrees
   D. 90% at 45 degrees
   Answer: A. phi = 5.71 degrees, alpha = 45 - 2.86 = 42.1 degrees, eta = (1 - 0.0995)/(1 + 0.0995) = 0.819. (Key point: Numerical value)

3. Why is a helix angle of about 42 degrees not used in power screws?
   A. Efficiency would be too low
   B. The screw is not self-locking (alpha > phi) and cannot hold a load
   C. It cannot be machined
   D. It would raise friction
   Answer: B. Practical screws use 2 to 6 degrees to be self-locking and give large force. (Key point: Why not used)

4. What is the advantage of a ball screw?
   A. It is self-locking
   B. Rolling friction (mu about 0.005 to 0.01) gives efficiency of 90% or more
   C. It is cheaper than a plain screw
   D. It needs no brakes for vertical loads
   Answer: B. Recirculating balls replace sliding by rolling friction, but the screw is reversible and needs brakes for vertical loads. (Key point: Ball screw advantage)

## Related in Machine Design II

- [Screw jack with square thread](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Ascrew-jack-square-thread-numerical) — Machine Design II, chapter 8
- [Torque and efficiency of a power screw](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Apower-screw-torque-efficiency-derive) — Machine Design II, chapter 8
- [Power screw threads, standards and materials](https://www.bctengineer.com/question.md?key=semester_7%2Fmachine_design_ii%3Ascrew-thread-types-standards-stress) — Machine Design II, chapter 8

## Related in other subjects

- [Screw thread production and change gears](https://www.bctengineer.com/question.md?key=semester_4%2Fmanufacturing_and_production_processes%3Ascrew-thread-manufacturing-change-gears) — Manufacturing and Production Processes, chapter 11
- [Thread errors and effect of pitch error](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Ascrew-thread-errors-and-pitch-effect) — Metrology, chapter 12
- [Measuring thread pitch, angle and diameters](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Atoolmaker-microscope-thread-measurement) — Metrology, chapter 12
- [Maximum demand and need to calculate it](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_power_distribution_and_illumination%3Amaximum-demand-need) — Industrial Power Distribution and Illumination, chapter 1