Force Analysis for Power Screws

Square Thread

Imagine that a single thread of a power screw is unrolled for a turn:

• Note that $\tan \lambda =\frac{\pi d_p}{L}$, where $L=p\cdot n_s$.

Then, one edge of the thread will form the hypotenuse of a right triangle whose base is the circumference of the mean-thread-diameter circle and whose height is the lead. The angle $\lambda$ is the lead angle of the thread. We represent the summation of all the axial forces acting upon the normal thread area by $P$.

To raise the load, a force $F$ acts to the right. To lower the load, a force $F$ acts to the left. The friction force $f$ is the product of the coefficient of friction $f$ with the normal force $N$, and acts to oppose the motion.

Force Analysis

We can conduct force analysis to find the required torque $T$ to resist linear load $P$.

$$\begin{array}{r}\sum F_x=F-\mu N\cos \lambda -N\sin \lambda =0\\ \sum F_y=N\cos \lambda -P-\mu N\sin \lambda =0\end{array}$$

The first equation above gives

$$F=N(\mu \cos \lambda +\sin \lambda )$$

The second equation gives:

$$N=\frac{P}{\cos \lambda -\mu \sin \lambda }$$

We can then write

$$F=P\frac{\mu \cos \lambda +\sin \lambda }{\cos \lambda -\mu \sin \lambda }=P\frac{\mu +\tan \lambda }{1-\mu \tan \lambda }$$

When lifting up, we are work against the load, which gives us:

$$T=F\cdot \frac{d_p}{2}=\frac{P{d}_p}{2}\cdot \frac{\mu +\cos \alpha \tan \lambda }{\cos \alpha -\mu \tan \lambda }$$

When lowering down, we work with the load, which gives us:

$$T^{\prime }=\frac{P{d}_p}{2}\cdot \frac{\mu -\cos \alpha \tan \lambda }{\cos \alpha +\mu \tan \lambda }$$

In practice, we must also deal with additional torque required to overcome the friction in the bearings of the system:

$$T_{\text{up}}=T+T_B,T_{\text{down}}=T^{\prime }+T_B$$

where $T_B={\mu }_B\cdot \frac{P{d}_c}{2}$. Here, ${\mu }_B$ is the coefficient of friction in the bearings, and $d_c$ is the diameter of the shaft.

If $T_{\text{down}}<0$, the load is large enough to make the screw spin with no external torque. If $T_{\text{down}}>0$, the screw is self-locking.

ACME Thread

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Relation to Worm Gears

From worm gear load analysis, we derived

$$W_{tW}=W_{xW}\frac{\cos {\varphi }_n\sin \lambda +\mu \cos \lambda }{\cos {\varphi }_n\cos \lambda -\mu \sin \lambda }$$

From power screws, we have just derived

$$F=P\frac{\cos \alpha \sin \lambda +\mu \cos \lambda }{\cos \alpha \cos \lambda -\mu \sin \lambda }$$

Examples

Example 1: Power Screw Jack

Example 2: Scissor Jack