Power Screw Efficiency

The amount of work on the power screw can be determined by comparing the power for the input or output.

For a square thread, we have:

Input:

$$W_{\text{in}}=2\pi T=2\pi \frac{d_p}{2}F$$

where $F=P\cdot \frac{\mu \cos \lambda +\sin \lambda }{\cos \lambda -\mu \sin \lambda }$ for lifting-up situation (while neglecting collar friction). With collar friction, this changes to $T+T_B$ where $T_B={\mu }_B\frac{P{d}_c}{2}$.

Output:

$$W_{\text{out}}=PL$$

Then, the efficiency is given by

$$\begin{array}{rl}\eta & =\frac{W_{\text{out}}}{W_{\text{in}}}=\frac{PL}{2\pi \cdot \frac{d_p}{2}\cdot F}=\frac{PL}{2\pi \cdot \frac{d_p}{2}\cdot P\cdot \frac{\mu \cos \lambda +\sin \lambda }{\cos \lambda -\mu \sin \lambda }}\\ & =\frac{L}{\pi d_p}\cdot \frac{\cos \lambda -\mu \sin \lambda }{\mu \cos \lambda +\sin \lambda }=\frac{\sin \lambda }{\cos \lambda }\cdot \frac{\cos \lambda -\mu \sin \lambda }{\mu \cos \lambda +\sin \lambda }=\frac{1-\mu \tan \lambda }{1+\mu \cot \lambda }\end{array}$$

For ACME thread,

$$\eta =\frac{\cos \alpha -\mu \tan \lambda }{\cos \alpha +\mu \cot \lambda }$$

For worm gear:

$$\eta =\frac{\cos {\varphi }_n-\mu \tan \lambda }{\cos {\varphi }_n+\mu \cot \lambda }$$