Maximum efficiency and ball screws

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.