Angle Measurement Using Roller Gages

Posted on 23-November-2016 by mathscinotes

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.

— James Madison

Introduction

Figure 1: Angle Measurement Example.

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.

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 (L1–L2) values shown in Figure 2.
Figure 3: Derivation of Angle Relationship.

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.

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.

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