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Graduating a vernier scale - maths inside!

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  • Graduating a vernier scale - maths inside!

    This is going to make you guys laugh! Though I fear for the wrong reasons.

    As I am refurbing an Atlas 7B shaper, I decided to do something about the original p*ss poor 7/8ths diameter graduated collars for the cross feed and for the downfeed, ....them guys sure had good eyesight back in the '40's!

    So I turned up some bar and settled on a 1.800" diameter grad ring, a suitably stepped diameter with a knurl on to set it off and then transferred it to the mill with the dividing head on to scribe the 100 graduations on the diameter, with fullest lines at 0.010" intervals and intermediate lines at the 0.005" intervals, every 0.001" being about 0.1" long.

    Set the DH up with a 36 hole plate (well it was already on the DH!) and calculated that:-

    36*40/100=12 ~ therefore 12 hole indexing steps would give a 100 graduations over the periphery of the dial. No problem so far!

    I realised that because of the larger dial size the datum line scribed on the fixed crossfeed bracket would also need to be increased in size to match the new graduations. No problem, so I turned up a thin disc 0.2" thick with a 1/2 hole through it for the crossfeed shaft and attached it to the previous bracket with a couple of 8-32 c'sks.

    Rather than put a mark on the fixed datum with something as uncouth as a saw blade! I decided to mount the disc on a mandrel and scribe it off in the DH again (as the mill was still set-up with the DH and the scriber on centre line). Then the bonus of having a vernier scale to read in 0.0002" increments took hold and I thought I'll put the vernier markings on either side of the datum line, and I can then read it positive going, or negative going. What a bright little b*gger I am!

    Since that point the maths has left me confused as to which way to go with the indexing. So far as I can see, given the dividing head plates are as follows:-
    #1 plate = 22,24,29,36,37,49,57,63
    #2 plate = 16,27,30,33,41,47,53,61
    #3 plate = 23,25,28,31,39,43,51,59

    I calculated that what was needed was a 20% increase in the spacing of the original 100 divisions. So converting the original 100 divs to degrees gives 360/100=3.6* per div; 3.6* + 20% = 3.96* so that divs required will be 360/3.96 = 90.909, which will round out to 91 divs, 63 hole plate #1 ~ 28 holes, and use a 72T x 32T differential gearing, or 90 divs without using the diff. gearing.

    Then I looked at the possibility of using direct indexing by looking for a 100 division setup, with an even number of hole indexing steps that had a factor of 5 in them. This I thought I'd found in the following:-

    Plate #3 ~ 25 holes. Gives 25*40 = 1000, therefore 100 divs = 1000/100 = 10 step indexing. Alright so far! Now if I factor 10 by +20% I get 12. So 12 holes will give me a 1/100th div + the 20% for the 0.0002" vernier scaler.

    But on rechecking I now find that 25*40 = 1000, and 1000/12 = 83.3333. This looks to be the inverse of 90.909. Where have I gone wrong? (yes! I know - thinking that having a 0.0002" vernier scaler on a shaper was a bright idea!!)

    Please help, you maths whizz-kids.


  • #2
    Quote:"Set the DH up with a 36 hole plate (well it was already on the DH!) and calculated that:-

    36*40/100=12 ~ therefore 12 hole indexing steps would give a 100 graduations over the periphery of the dial. No problem so far!"

    Sorry that bits wrong for a start.
    Formula is 40/100 = 2/5 or any variation of this like 8 holes on a 20 circle.

    12 holes in a 36 plate equals 1/3 = 40/120 so 12 holes will get you a 120 division dial.

    To cut your verier scale you need to scribe 10 marks onto a line equal to 9 of the original marks. So you will be one divison short.
    The maths for a full rotation to cut a vernier scale will be 100 / 9 *10 = 111.11 divisions.

    Your main dial needs 100 divisions and the vernier needs 20 division out of 111.11 to give you the plus and minus you need.

    Splitting the difference and calling it 111 you can get that by differential indexing using 13 holes on a 39 circle and a 72 tooth wheel on the spindle drivng a 24 tooth on the input shaft.

    A good web site that explains about verniers is here:-

    John S.

    [This message has been edited by John Stevenson (edited 01-14-2004).]

    Sir John , Earl of Bligeport & Sudspumpwater. MBE [ Motor Bike Engineer ] Nottingham England.


    • #3
      Color me 'an interested spectator'. While seemingly simple on the surface, dividing head calculations give me a splitting headache.
      Lynn (Huntsville, AL)


      • #4
        "A good web site that explains about verniers is here:-"
        Very nice site.
        Do you have anymore sites on Engineering,etc?
        please visit my webpage:


        • #5
          I built a set of rear jacks for my fifth wheel trailers some years back. They are a 2" square tube that has a foot and slides inside a square sleeve bolted to the back bumper. I wanted to be able to adjust them every 1/2". I used 3/8 hardened pins that fit through a hole in the sleeve and the jack to hold position. The problem is that if I wanted 1/2 inch adjustment I couldn't drill the holes in the jack that close together without weakening it seriously. Besides, thats a lot of holes. Solution, drill holes in the jack every 1.5" and three holes in the sleeve spaced 1" apart vertically. Presto, vernier jack with 1/2" adjustment.

          [This message has been edited by Evan (edited 01-14-2004).]
          Free software for calculating bolt circles and similar: Click Here


          • #6
            Thanks for that John, you've rescued my sanity. An excellent site you guided me too, as well.

            I realised that after writing the topic last night that I had dropped two huge ones! The mistake in my original calculation was that I'd added only 10% to the 1/100 degree angle calculation (3.6* + 10% = 3.96* and 360/3.96 = 90.909 divisions). When that item is corrected to read 3.6* + 20% = 4.32*, then the equation correlates with the later one, i.e. 360/4.32 = 83.333 divisions = 25*40/12!

            I was transfixed by how to do the vernier scale by only 5 lines (6 including the zero datum) - to give a 0.0002" accuracy. I'd written about 2 sides of scribled working notes, and those notes were confusing to say the least!

            I did the 100 divisions correctly, NOT using the 36 hole plate, but using the 30 hole plate #2 - 12 hole indexing giving ~ 30*40/100=12, the 36 hole plate was an attempt at calculating the vernier!

            I can only blame advancing years, and accumulated 'dirty brown water' intake for the confused writing in my earlier post.

            Ah Well, it just goes to prove that even some one that's been 'at it' for as long as me can '%&*) up' occasionally.



            • #7
              Maybe John will adopt us both. I have tools and can learn to cuss in British! They probably wouldn't understand common Canadian cussin' - I could bring a dictionary to translate! Say...that just might work