Calculations for the volume of a key

[Snelson]

I am trying to find some calculations for the volume of a key. Specifically a key that will complete the radius of a shaft. I have most of what is needed, the problem that I have comes with calculating the volume in the radius portion of the keyway. This type of key is typically used for balancing so it is used to make up the missing mass taken up by the machining of the keyway. The geometry of the upper portion of the key on the radius end gets very complicated. Does anyone have any suggestions?

 

[MintJulep]

Two suggestions:

Solid model.

Graduated beaker full of water.

 

[arunmrao]

Archimede'principle

 

[TheTick]

Key would e shaped like rectangle plus chord segment. Calculate area of rectangle plus area of chord. Use math to add them together.

 

[TheTick]

Chord segment calculator:
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http://www.1728.com/circsect.htm>

 

[Cockroach]

Yes, TheTick is correct. Volume is simply the cross sectional area of the key times it's height.

For the cord area, this is simply a fraction of a circle of equal radius. You can easily map out the angular displacement of the cord and then note that A = 2 * pi * r for a FULL circle of radius r, or more simply stated A = theta * r for angular displacement theta. In other words, the area of the chord is just (theta / [2 * pi]) that of full circle area.

As previously stated by TheTick, add the rectangular and chord areas together, multiply by key height and the result is the volume of the key.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[MintJulep]

I think the OP's problem is at the end of the key.

I am assuming that the keyway is cut with an end mill, not a slot cutter. This will leave a half-round end of the key.

Also, since this is really a balance weight, not a key in the normal sense I assume that the upper surface matches the OD of the shaft.

So, at the end we have a volume made up of the intersection of a half cylinder and the chord of a cylinder.

This little bit is more than my feeble mind can tackle on only one cup of coffee. Depending on the size of the rest of the key, I could get it down to a probably negligable error using the techniques mentioned above.

 

[TheTick]

Maybe cut two keyways on opposite sides and use two keys?