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- From: Lawrence London <lfljvenaura@gmail.com>
- To: machinist@lists.ibiblio.org
- Subject: [machinist] Trig Formula Help
- Date: Mon, 11 Nov 2013 21:25:02 -0500
http://www.practicalmachinist.com/vb/general/trig-formula-help-275007/
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Trig Formula Help
Need help with measuring the bore diameter of a tapered hole using a steel ball. In this case the bore of countersunk hole - calipers & a magnifing glass wont work & don"t have a chamfer gage. Balls will be like 3/16, 1/4, 5/16 ect. Miking the ball, lower taper + thickness of rest of the material. Or come down from the top with a depth mike & a known riser block or ring.
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How accurate do you need the measurement?
You could pack the hole with silly putty or some other semi solid. -
11-11-2013, 08:27 PM #3
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11-11-2013, 08:31 PM #4
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Sounds like you want diameter-at-depth of a known taper angle. I'll caution that if the actual taper angle is not exactly the "known" taper, calculations of diameter-at-depth will go to Hell in a hurry, especially for small included angle tapers. So check the actual angle on your countersink.
Start with two known nominal quantities, the radius of the ball (r) and the half angle of the taper (a). The radius of the ball meets the wall of the taper at a right angle, so now you have a triangle with all three angles (90°, a, 90°-a) and one side (r) known. Add an additional line from the point where the radius intersects the taper, perpendicular to the axis. This defines another, smaller right triangle geometrically similar to the first one. The additional line is the radius of the tapered hole, at a depth defined by the ball center and the triangle side along the axis. Measure the ball depth (or height) relative to the part surface, subtract the ball radius to get the ball center, subtract the length of that triangle side and you've got the depth of the circle where the ball meets the taper.
Let me know if that's not enough to get you going. -
I could probably figure out the trig, but if you only have one or two tapers to measure, it'd be faster for me to cad it out. Give me the ball diameters, and the distance of the top of each ball to some fixed point. (OR the diameter of each ball and the difference in height.) I'll do it right now if you have the info...
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If you can come down with a depth gauge from the top...this should be easy.
Use a 1/4" ball first, say you measure the ball to be a depth of 2.000". add the radius of the ball, and you end up with 2.125"
Use a 1/2" ball next, say you measure the ball to be a depth of 1.000". add the radius of the ball, and you end up with 1.250"
2.125-1.250 = .8750
The diameter has changed .250" over 7/8" depth.
To make it easier, lets make a right triangle with one leg being 7/8", the other leg being .125", and solve for the hypotenuse using the pathagorian theorem.
Hypot. = .8838"
Now to find the angle, but I gotta go...locking up the building...maybe someone will finish it before I get home. -
11-11-2013, 08:35 PM #7
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... "figure out"... what was I thinking? That is what google is for. see page 2
http://ledyardmachine.com/downloads/files/tapers.pdf -
I think this site might get you started:
Tangent lines to two circles calculator
It's math oriented, but if you assume that you will just go along the X axis, it's simpler. I don't have any real numbers, but let's try some made up numbers.
You drop a 1" ball in the hole and it goes in .5". That defines the first circle. It's center is at x=1, since it went in .5" and it's radius is .25" . Then you drop a 1/2" ball in the hole and it measures 2.5". The center of the second ball is at x=2.75, since it went in 2.5 inches and it's radius is .25 inches.
The application reports:
Outer line equation: 4.02y - 0.45x + 2.25 = 0
Outer line equation: -4.02y - 0.45x + 2.25 = 0
Now, we don't care about anything but the slope of the lines. Rearrange either of the equations by putting x and the constant on the right:
4.02y = .45x -2.25
Then, divide both sides by the y coefficient to get it in the form Y=MX+b
y = (.45/4.02)X - (2.25/4.02)
y= .1119 x - .5597
All we care about is the x coefficient, which is the slope of the line. The slope is the rise over the run, but it's only half the taper. The total taper would be .2238/inch.
I'm doing this in advanced mode, and I notice several faster answers. I don't think lbhsbz is quite right. He says "The diameter has changed .250" over 7/8" depth. " If you were using disks, rather than balls, it would work, but the ball doesn't contact the taper at it's diameter, it's a little bit off. When I run his numbers using what I've done above, I get:
-8.08y + 1.17x - 3.5 = 0
-8.08y + 1.17x = 3.5
-8.08y = 3.5-1.17x
y = .1448 x - .00433
This says the angle changes .2892"/inch, where his calculation is .2857". ( I really should be saying .289 for my work, as there are only 3 significant figures in the equation)
Here's an illustration:
As you can see, the contact points of the circles don't fall on a line through the centers of the circles, and the distance between the contact points is not the diameter of the ball. I think the Ledyard machine .pdf is making the same mistake. their diagram actually shows that a line through the diameter doesn't contact the taper on both sides.
I'm not 100% sure I'm right either. The equations that the site provides don't have enough significant figures to make me comfortable. The continuing discussion will be interesting.
Here's all the math that's hidden in the applet on the web site. This is why I used the applet...
Last edited by wheels17; 11-11-2013 at 09:10 PM. Reason: Just saw the Ledyard machine .pdf
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- [machinist] Trig Formula Help, Lawrence London, 11/11/2013
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