Quote of the Day
Preparation is the be-all of good trial work. Everything else – felicity of expression, improvisational brilliance – is a satellite around the sun. Thorough preparation is that sun.
— Louis Nizer
I am still working through some examples of using gage balls for machine shop work. The following reference on Google Books has great information on using gage balls (Figure 1) in measuring the characteristics of a countersink and I will be working through the presentations there. These are good, practical applications of high-school geometry.
I will be working two examples:
- Measuring the diameter of a sharp-edged countersink
This is the simplest case and assumes that the edge is so sharp that the countersink edge touches the gage ball at a single point.
- Measuring the theoretical diameter of rounded or burred edge countersink
The theoretical countersink diameter is the diameter prior to the rounding or burring occurring.
Diameter of a Sharp-Edged Countersink
Equation 1 is the formula for the diameter of countersink in terms of the parameters shown in Figure 2(a).
- B is the diameter of the ball.
- H is the height of the ball above the surface with the countersink.
- E is the edge diameter of the countersink.
Figure 2(b) shows a construction that I use to derive Equation 1.
Figure 3(a) shows a sharp-edged countersink example that I have contrived. Figure 3(b) shows my derivation using the Pythagorean theorem and the calculation for the example of Figure 3(a).
Diameter of a Rounded-Edge Countersink
A countersink with a rounded or burred edge represents a measurement problem. If we know the diameter of the countersink taper, we can use the depth that a gage ball that fits down into the countersink to compute the theoretical (unrounded) diameter of the countersink.
Equation 2 is the formula for the diameter of countersink in terms of the parameters shown in Figure 4(a).
- D is the theoretical diameter of the countersink (i.e. with no rounding or burring).
Figure 4(a) shows the construction from the reference. Figure 4(b) shows a construction that I made that is just a bit less busy and I used it to derive Equation 2.
Figure 5(a) shows a rounded-edged countersink example that I have contrived. Figure 5(b) shows my derivation using the Pythagorean theorem and the calculation for the example of Figure 5(a).