# Angle Measurement Using Roller Gages

Quote of the Day

It is a universal truth that the loss of liberty at home is to be charged to the provisions against danger, real or pretended, from abroad.

## Introduction

Figure 1: Angle Measurement Example.

I am continuing to work through some basic metrology examples – today's example uses roller gages to measure the angle of a drilled hole (Figure 1). The technique discussed here uses two roller gages and a plug. The plug must fit the hole snugly (i.e. no backlash) as it will provide the surface that we will be measuring.  Using this approach assumes that you need a very accurate measurement of a hole's angle as rough measurements can be made using a protractor.

## Background

This example is based on the material found on this web page. I will derive the angle relationship presented there (Equation 1) and present a worked example that is confirmed using a scale drawing (Figure 1).

 Eq. 1 $\displaystyle \theta \left( {{{L}_{1}},{{L}_{2}},{{D}_{1}},{{D}_{2}}} \right)=2\cdot \text{arctan}\left( {\frac{1}{2}\cdot \frac{{{{D}_{1}}-{{D}_{2}}}}{{{{L}_{1}}-\frac{{{{D}_{1}}}}{2}-\left( {{{L}_{2}}-\frac{{{{D}_{2}}}}{2}} \right)}}} \right)$

where

• L1 is the distance from reference to outside edge of roller gage.
• L2 distance from reference to outside edge of roller gage.
• D1 diameter of the first roller gage.
• D2 diameter of the second roller gage.
• θ is the angle of the drill hole relative to the surface that is drilled.

These variables are all indicated in Figure 2.

Figure 2: Reference Drawing Showing Critical Variables.

## Analysis

### Derivation

Figure 3 shows how to derive Equation 1. The basic derivation process is simple:

• The center of each roller gage is on a line that is makes an angle of θ/2 with the plug.
• The slope of line connecting the roller gage centers has the value tan(θ/2).
• The line's slope is computed using the rise ($\frac{{{{D}_{1}}}}{2}\cdot \left( {1+\tan \left( {\frac{\theta }{2}} \right)} \right)-\frac{{{{D}_{2}}}}{2}\cdot \left( {1+\tan \left( {\frac{\theta }{2}} \right)} \right)$) and run (L1L2) values shown in Figure 2.

Figure 3: Derivation of Angle Relationship.

### Example

Figure 4 shows works through the angle calculation example of Figure 1.

Figure 4: Worked Example Using Values From Figure 1.

## Conclusion

I have some designs I plan to build that have angled holes. This procedure will give me a way to accurately measure the angle of these holes.

Save

Save

Save

Save

Save

Save

Save

Save

Save

Save

Save

Save

Save

Save

This entry was posted in Metrology. Bookmark the permalink.