A Math and Mechanism problem

[Stepside]

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

[boslab]

I smell a can of worms here
M

[Doozer]

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

-Doozer

[ikdor]

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

Igor

[Ripthorn]

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.

[Stepside]

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

[mklotz]

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

[Stepside]

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

[Barrington]

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

.

[becksmachine]

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

[Stepside]

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

[Paul Alciatore]

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.

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

[adatesman]

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.

[Ripthorn]

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.

[mklotz]

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...