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  • A Math and Mechanism problem

    For a project I am working on I need some "Timing " between components. If I have a disc with 15 evenly spaced holes arrayed around a circle and a second disc with 16 holes arrayed around the same diameter circle "How many combinations are possible?" If I am correct, there are 240 possibilities (15 x 16) or 1 degree 30 minutes (360/240). If that is correct, then 20 and 18 would yield 1 degree spacing.

    Is there a flaw in my reasoning?

    Thanks in advance
    Pete

  • #2
    I smell a can of worms here
    M

    Comment


    • #3
      He is talking about the Vernier concept.
      I don't smell anything.

      -Doozer
      DZER

      Comment


      • #4
        I am going to guess that 20 and 18 will give 2degree spacing.

        Igor

        Comment


        • #5
          This depends on a few things. First, is one disc staying stationary, or do both move? Also, you need to make sure that, in order to get the number of combinations you want, that the two numbers don't have common factors. 15 and 16 don't have common factors, but 18 and 20 do. This is probably the reasoning behind Igor's response. If you can find the right ratio of prime numbers to get what you want, that is your surest bet.

          Comment


          • #6
            One or both disc can move as well as the connecting pin.

            Comment


            • #7
              The trick with verniers of this type is deciding which holes to align to obtain a particular angle. On my website is a program (VERNIER) that generates hole counts and alignment cheat sheets given the desired number of divisions. I ran it for 240 divisions and the results are below.



              DATA FOR 240 DIVISION TWO PLATE VERNIER

              Total number of holes = 31

              Assume holes in plate with 15 holes are labeled with LETTERS.
              Assume holes in plate with 16 holes are labeled with NUMBERS.
              Either plate can be the movable plate.
              Assume 'A' and '1' holes are aligned initially.
              To reverse the direction of rotation, read the list backwards.

              Division Angle(deg) Holes to Align

              0 0.0000 A1
              1 1.5000 B2
              2 3.0000 C3
              3 4.5000 D4
              4 6.0000 E5
              5 7.5000 F6
              6 9.0000 G7
              7 10.5000 H8
              8 12.0000 I9
              9 13.5000 J10
              10 15.0000 K11
              11 16.5000 L12
              12 18.0000 M13
              13 19.5000 N14
              14 21.0000 O15
              15 22.5000 A16
              16 24.0000 B1
              17 25.5000 C2
              18 27.0000 D3
              19 28.5000 E4
              20 30.0000 F5
              21 31.5000 G6
              22 33.0000 H7
              23 34.5000 I8
              24 36.0000 J9
              25 37.5000 K10
              26 39.0000 L11
              27 40.5000 M12
              28 42.0000 N13
              29 43.5000 O14
              30 45.0000 A15
              31 46.5000 B16
              32 48.0000 C1
              33 49.5000 D2
              34 51.0000 E3
              35 52.5000 F4
              36 54.0000 G5
              37 55.5000 H6
              38 57.0000 I7
              39 58.5000 J8
              40 60.0000 K9
              41 61.5000 L10
              42 63.0000 M11
              43 64.5000 N12
              44 66.0000 O13
              45 67.5000 A14
              46 69.0000 B15
              47 70.5000 C16
              48 72.0000 D1
              49 73.5000 E2
              50 75.0000 F3
              51 76.5000 G4
              52 78.0000 H5
              53 79.5000 I6
              54 81.0000 J7
              55 82.5000 K8
              56 84.0000 L9
              57 85.5000 M10
              58 87.0000 N11
              59 88.5000 O12
              60 90.0000 A13
              61 91.5000 B14
              62 93.0000 C15
              63 94.5000 D16
              64 96.0000 E1
              65 97.5000 F2
              66 99.0000 G3
              67 100.5000 H4
              68 102.0000 I5
              69 103.5000 J6
              70 105.0000 K7
              71 106.5000 L8
              72 108.0000 M9
              73 109.5000 N10
              74 111.0000 O11
              75 112.5000 A12
              76 114.0000 B13
              77 115.5000 C14
              78 117.0000 D15
              79 118.5000 E16
              80 120.0000 F1
              81 121.5000 G2
              82 123.0000 H3
              83 124.5000 I4
              84 126.0000 J5
              85 127.5000 K6
              86 129.0000 L7
              87 130.5000 M8
              88 132.0000 N9
              89 133.5000 O10
              90 135.0000 A11
              91 136.5000 B12
              92 138.0000 C13
              93 139.5000 D14
              94 141.0000 E15
              95 142.5000 F16
              96 144.0000 G1
              97 145.5000 H2
              98 147.0000 I3
              99 148.5000 J4
              100 150.0000 K5
              101 151.5000 L6
              102 153.0000 M7
              103 154.5000 N8
              104 156.0000 O9
              105 157.5000 A10
              106 159.0000 B11
              107 160.5000 C12
              108 162.0000 D13
              109 163.5000 E14
              110 165.0000 F15
              111 166.5000 G16
              112 168.0000 H1
              113 169.5000 I2
              114 171.0000 J3
              115 172.5000 K4
              116 174.0000 L5
              117 175.5000 M6
              118 177.0000 N7
              119 178.5000 O8
              120 180.0000 A9
              121 181.5000 B10
              122 183.0000 C11
              123 184.5000 D12
              124 186.0000 E13
              125 187.5000 F14
              126 189.0000 G15
              127 190.5000 H16
              128 192.0000 I1
              129 193.5000 J2
              130 195.0000 K3
              131 196.5000 L4
              132 198.0000 M5
              133 199.5000 N6
              134 201.0000 O7
              135 202.5000 A8
              136 204.0000 B9
              137 205.5000 C10
              138 207.0000 D11
              139 208.5000 E12
              140 210.0000 F13
              141 211.5000 G14
              142 213.0000 H15
              143 214.5000 I16
              144 216.0000 J1
              145 217.5000 K2
              146 219.0000 L3
              147 220.5000 M4
              148 222.0000 N5
              149 223.5000 O6
              150 225.0000 A7
              151 226.5000 B8
              152 228.0000 C9
              153 229.5000 D10
              154 231.0000 E11
              155 232.5000 F12
              156 234.0000 G13
              157 235.5000 H14
              158 237.0000 I15
              159 238.5000 J16
              160 240.0000 K1
              161 241.5000 L2
              162 243.0000 M3
              163 244.5000 N4
              164 246.0000 O5
              165 247.5000 A6
              166 249.0000 B7
              167 250.5000 C8
              168 252.0000 D9
              169 253.5000 E10
              170 255.0000 F11
              171 256.5000 G12
              172 258.0000 H13
              173 259.5000 I14
              174 261.0000 J15
              175 262.5000 K16
              176 264.0000 L1
              177 265.5000 M2
              178 267.0000 N3
              179 268.5000 O4
              180 270.0000 A5
              181 271.5000 B6
              182 273.0000 C7
              183 274.5000 D8
              184 276.0000 E9
              185 277.5000 F10
              186 279.0000 G11
              187 280.5000 H12
              188 282.0000 I13
              189 283.5000 J14
              190 285.0000 K15
              191 286.5000 L16
              192 288.0000 M1
              193 289.5000 N2
              194 291.0000 O3
              195 292.5000 A4
              196 294.0000 B5
              197 295.5000 C6
              198 297.0000 D7
              199 298.5000 E8
              200 300.0000 F9
              201 301.5000 G10
              202 303.0000 H11
              203 304.5000 I12
              204 306.0000 J13
              205 307.5000 K14
              206 309.0000 L15
              207 310.5000 M16
              208 312.0000 N1
              209 313.5000 O2
              210 315.0000 A3
              211 316.5000 B4
              212 318.0000 C5
              213 319.5000 D6
              214 321.0000 E7
              215 322.5000 F8
              216 324.0000 G9
              217 325.5000 H10
              218 327.0000 I11
              219 328.5000 J12
              220 330.0000 K13
              221 331.5000 L14
              222 333.0000 M15
              223 334.5000 N16
              224 336.0000 O1
              225 337.5000 A2
              226 339.0000 B3
              227 340.5000 C4
              228 342.0000 D5
              229 343.5000 E6
              230 345.0000 F7
              231 346.5000 G8
              232 348.0000 H9
              233 349.5000 I10
              234 351.0000 J11
              235 352.5000 K12
              236 354.0000 L13
              237 355.5000 M14
              238 357.0000 N15
              239 358.5000 O16
              240 360.0000 A1

              Total holes to drill on letter plate = 15
              Total holes to drill on number plate = 16
              Regards, Marv

              Home Shop Freeware - Tools for People Who Build Things
              http://www.myvirtualnetwork.com/mklotz

              Location: LA, CA, USA

              Comment


              • #8
                Marv

                That is perfect. I think the 1 degree 30 minutes will work for terhis application. If I understand what others have said, to get 1 degree would take two numbers without common factors.

                Thanks for the chart.

                Pete

                Comment


                • #9
                  I think that for a 1 degree resolution I think you would need a 36 hole (10°) and a 40 hole (9°) (i.e. a 1 degree difference.)

                  Each degree position would however have four possible hole combinations giving the same result.

                  Cheers

                  .

                  Comment


                  • #10
                    If you have the time, read the thread, Mother of invention, written by RJ Newbould over on PM.

                    But only if you have time. ...... ....... It caused some late nights for me.

                    Dave

                    Comment


                    • #11
                      Barrrington

                      The 36 and 40 works perfect. As I do not need the entire 360 degrees I will just drill the holes lie in one quadrant.

                      Thanks to all who helped.

                      Pete

                      Comment


                      • #12
                        You are correct with your first example using 15 and 16 holes. This is because these two numbers do not have any common factors so when you align each line of the 16 hole circle with a hole of the 15 hole circle, no other lines will align. 15 = 3 X 5 and 16 = 2 X 2 X 2 X 2.

                        Unfortunately, in your second example the two numbers you choose DO have a common factor. 20 = 2 X 2 X 5 and 18 = 2 X 3 X 3. The common factor of 2 is the problem and it will pop up below.

                        Now, if you line up the first hole on the 20 hole circle with the first hole on the 18 hole circle and look down the line, you will see that the 11th and 10th holes also are in alignment. And as you rotate the 20 hole circle, every succeeding hole from 2 to 10 will have a hole on the 18 hole circle which it is in alignment with.

                        So instead of a 20 part Vernier, what you have is TWO 10 part Verniers in a row. (There is that 2 again, as I predicted.) And you will have only 18 X 10 or 180 distinct positions. That gives a 2 degree spacing.

                        Most Verniers are constructed with only ONE extra hole over the SAME distance on the primary scale. This ensures that there are no common factors. (That can be proven mathematically and I leave it as an exercise for the reader.) There are variations on this general method, but they also ensure that there are no common factors.

                        If you want 1 degree spacing you are going to have to use different numbers or a different scheme. Now, notice that I said "... over the SAME distance on the primary scale." So, we can make the the 18 and 20 hole idea work if you use the 20 hole circle as the primary scale and space the 18 holes over the space covered by 17 or 19 of those 20 holes. This is not convenient for a couple of reasons. First, using 20 holes for your primary gives 360/20 = 18 degrees and that is not a round number like 10 or 15 or 20 so counting around the complete circle is not so nice. Second, there will be a small space at the end of the Vernier scale which may also be confusing. And using the 18 hole circle will not allow the 20 holes to be spaced over 19 or 21 of those holes as you will run out of holes on the 18 hole primary.

                        So, you will need more holes on the primary circle. I would suggest 30 holes which would have a 12 degree spacing. Then a 12 part Vernier would provide one degree spacing. That would be the smallest number with convenient numbers in use. The 12 holes would be spaced over 13 holes on the primary circle or at 13 holes X 12 degrees /12 holes = 13 degree spacing.

                        The next good set of numbers would be 36 holes on the primary circle (10 degree spacing) and 10 on the Vernier. The Vernier holes would be at 11 holes X 10 degrees / 10 holes = 11 degree spacing.

                        Of course if a different final spacing, like 1.5 degrees, is allowed, then different, possibly smaller numbers can be used.



                        Originally posted by Stepside View Post
                        For a project I am working on I need some "Timing " between components. If I have a disc with 15 evenly spaced holes arrayed around a circle and a second disc with 16 holes arrayed around the same diameter circle "How many combinations are possible?" If I am correct, there are 240 possibilities (15 x 16) or 1 degree 30 minutes (360/240). If that is correct, then 20 and 18 would yield 1 degree spacing.

                        Is there a flaw in my reasoning?

                        Thanks in advance
                        Pete
                        Last edited by Paul Alciatore; 04-10-2015, 01:37 PM.
                        Paul A.
                        SE Texas

                        Make it fit.
                        You can't win and there is a penalty for trying!

                        Comment


                        • #13
                          Originally posted by becksmachine View Post
                          If you have the time, read the thread, Mother of invention, written by RJ Newbould over on PM.

                          But only if you have time. ...... ....... It caused some late nights for me.

                          Dave
                          Quite possibly the most interesting thread on PM, and I too lost a lot of time to it. Absolutely fascinating.

                          Comment


                          • #14
                            Paul, I loved your "exercise for the reader" comment. I am a physicist and my favorite thing to read in text books in school was something along the lines of "the astute reader will see" or something because it just assumed so much about the reader. My absolute favorite incarnation of that line was "even the most casual of observers will note...", which was science speak for "you're a total idiot if you don't see...", which of course the conclusion was not obvious.

                            Anyway, back to your regularly scheduled programming.

                            Comment


                            • #15
                              Originally posted by Ripthorn View Post
                              Paul, I loved your "exercise for the reader" comment. I am a physicist and my favorite thing to read in text books in school was something along the lines of "the astute reader will see" or something because it just assumed so much about the reader. My absolute favorite incarnation of that line was "even the most casual of observers will note...", which was science speak for "you're a total idiot if you don't see...", which of course the conclusion was not obvious.

                              Anyway, back to your regularly scheduled programming.
                              When I went to college, the line was,

                              It is intuitively obvious to the most casual observer that...
                              Regards, Marv

                              Home Shop Freeware - Tools for People Who Build Things
                              http://www.myvirtualnetwork.com/mklotz

                              Location: LA, CA, USA

                              Comment

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