Screws

Introduction

Screws are used to transmit power or motion to another part of a machine. The most common type of screw is a square threaded screw.

lead angle: $\theta = \tan^{- 1}(\frac{l}{2\pi r})$

where $l$ = lead of screw

Three Main Types of Screws:

Upward Impending Motion
$\rightarrow \Sigma F_{x} = 0;\$$\frac{M}{r} - Rsin(\theta + \Phi_{s}) = 0$

$\uparrow \Sigma F_{y} = 0;$ $Rcos(\theta + \Phi_{s}) - W = 0$

Solve to get: $M = rWtan(\theta + \Phi_{s})$

Downward Impending Motion
When $\theta > \Phi_{s}$

$\rightarrow \Sigma F_{x} = 0;\$$\frac{M'}{r} - Rsin(\theta - \Phi_{s}) = 0$

$\uparrow \Sigma F_{y} = 0;$ $Rcos(\theta - \Phi_{s}) - W = 0$

Solve to get: $M' = rWtan(\theta - \Phi_{s})$

Downward Impending Motion

When $\theta < \Phi_{s}$

This is a type of self locking screw.

$\rightarrow \Sigma F_{x} = 0;\$$\frac{M''}{r} - Rsin(\Phi_{s} - \theta) = 0$

$\uparrow \Sigma F_{y} = 0;$ $Rcos(\Phi_{s} - \theta) - W = 0$

Solve to get: $M'' = rWtan(\Phi_{s} - \theta)$

Example 1: Problem 8-74

The square-threaded bolt is used to join two plates together. If the bolt has a mean diameter of d=20mm and a lead of l=3mm determine the smallest torque M required to loosen the bolt if the tension in the bolt is T=40kN. The coefficient of static friction between the threads and the bolt is $\mu_{s} =$0.15.

We know: $\Phi = \tan^{- 1}(\mu_{s})$, so $\Phi = \tan^{- 1}(0.15) \Rightarrow \Phi = 8.53{^\circ}$

$\theta = \tan^{- 1}(\frac{l}{2\pi r}) \Rightarrow \theta = \tan^{- 1}(\frac{3mm}{20\pi mm}) \Rightarrow \theta = 2.73{^\circ}$

Using the downward impending motion when $\Phi > \theta$; $M = rWtan(\Phi - \theta)$

$M = 10mm \cdot 40kN \cdot tan(8.53{^\circ} - 2.73{^\circ}) \Rightarrow M = 40.63Nm$