Power Screw Efficiency⚠ unverified

Understanding the Parameters

• Useful work $F l$ — per revolution the load $F$ rises by the lead $l$, so the output work is $F l$. This is the numerator: what the screw actually accomplishes each turn.

• Input work $2\pi T$ — turning the screw one revolution against torque $T$ takes work $2\pi T$. The ratio of the two is the efficiency.

• Friction $\mu$ — buried in $T$: higher friction inflates the required torque and drives efficiency down. As $\mu \to 0$, $T \to F l/(2\pi)$ and $\eta \to 1$ (an ideal, frictionless — and non-self-locking — screw).

• Lead angle dependence — efficiency depends only on the lead angle $\lambda$ and friction: $\eta = \dfrac{\tan\lambda\,(1 - \mu\tan\lambda)}{\tan\lambda + \mu}$. It is zero at $\lambda = 0$, rises to a maximum near $\lambda \approx 45^\circ - \phi/2$ (where $\phi = \arctan\mu$), then falls — so there is an optimal lead angle, though self-locking screws operate on the steep low-angle side.

Derivation (Approaching a Proof)

Efficiency is (ideal input)/(actual input) — equivalently (output work)/(input work). Consider one revolution:

1. Output work. The load $F$ is raised by one lead $l$, doing useful work $W_{\text{out}} = F l$.
2. Input work. The applied torque $T$ turns through $2\pi$ radians, doing work $W_{\text{in}} = 2\pi T$.
3. Ratio. $$\eta = \frac{W_{\text{out}}}{W_{\text{in}}} = \frac{F l}{2\pi T}.$$

Substituting the frictionless (ideal) raising torque $T_0 = F l/(2\pi)$ shows the same thing as $\eta = T_0/T$: efficiency is the ideal torque divided by the actual torque. Plugging in the square-thread raising torque $T = \dfrac{F d_m}{2}\dfrac{l + \pi\mu d_m}{\pi d_m - \mu l}$ and using $\tan\lambda = l/(\pi d_m)$ reduces it to the lead-angle form

$$\eta = \frac{\tan\lambda\,(1 - \mu\tan\lambda)}{\tan\lambda + \mu},$$

which makes the friction/self-locking trade-off explicit: driving $\mu$ toward zero raises $\eta$ toward 1 but simultaneously destroys self-locking ($\mu > \tan\lambda$ fails).

Dimensional check. $\dfrac{F l}{2\pi T} = \dfrac{\text{N}\cdot\text{m}}{\text{N}\cdot\text{m}}$ = dimensionless. ✓

History and Development

Screw efficiency and its ~50 % ceiling for self-locking square threads are classical machine-design results (Shigley). The insight that efficiency and self-locking oppose each other drove the development of ball screws and roller screws (rolling contact, $\eta \approx 90$ %+) for applications like CNC feeds and aircraft actuators that need efficiency and accept back-driving with a brake.

Related Concepts: Power Screw Torque raise, Screw Lead Angle, Torque Lower Load, Ball Screw Efficiency, Total Torque Raise, Acme Thread Torque

Notes: Thread-only, square-thread efficiency (excludes collar friction; $\alpha$ not an input — see registry note). Peaks ~50 % for self-locking screws. Ball screws reach ~90 % but are not self-locking.