The Theory of Meccano Gears Part 1: Spur Gears
Appendix B
Meccano gears - working centres and ratios (All dimensions in inches)
Sometimes we need a 'special' ratio, which can be achieved by meshing gears not normally paired together. This invariably involves a non-standard centre distance to achieve the correct mesh conditions. There is always some scope for variation of the true centres due to backlash and running clearances, but chart 1 shows the theoretically correct working centre distances for all combinations of Meccano and compatible gears. If any pair of gears do not run efficiently at these centre distances, it could be due to manufacturing errors or excessive tolerances adopted in their production (e.g. I have some oversize 22t pinions and undersize 2-1/2 ring gears).
The virtual* numbers of
teeth are those used to calculate the working centre distance from the formula:
CD = (T + t)/2/DP, where CD is the centre distance, T & t are the number of
teeth in the gear and pinion respectively and DP the diametral
pitch (38). Chart 2 is a quick reference of the gear ratios for all
combinations of tooth numbers.
|
Chart 1: Working Centres for Meccano and Compatible gears (Inches) |
||||||||||
|
Actual No. of teeth |
- |
11 |
13 |
15 |
19 |
22 |
25 |
30 |
38 |
45 |
|
Virtual No. of teeth |
10.5 |
12 |
15.5 |
19 |
21.5 |
25.5 |
30.5 |
38 |
45.5 |
|
|
11 |
10.5 |
0.2763 |
0.2961 |
0.3421 |
0.3882 |
0.4211 |
0.4737 |
0.5395 |
0.6382 |
0.7368 |
|
13 |
12 |
0.2961 |
0.3158 |
0.3618 |
0.4079 |
0.4408 |
0.4934 |
0.5592 |
0.6579 |
0.7566 |
|
15 |
15.5 |
0.3421 |
0.3618 |
0.4079 |
0.4539 |
0.4868 |
0.5395 |
0.6053 |
0.7039 |
0.8026 |
|
19 |
19 |
0.3882 |
0.4079 |
0.4539 |
0.5000 |
0.5329 |
0.5855 |
0.6513 |
0.7500 |
0.8487 |
|
22 |
21.5 |
0.4211 |
0.4408 |
0.4868 |
0.5329 |
0.5658 |
0.6184 |
0.6842 |
0.7829 |
0.8816 |
|
25 |
25.5 |
0.4737 |
0.4934 |
0.5395 |
0.5855 |
0.6184 |
0.6711 |
0.7368 |
0.8355 |
0.9342 |
|
30 |
30.5 |
0.5395 |
0.5592 |
0.6053 |
0.6513 |
0.6842 |
0.7368 |
0.8026 |
0.9013 |
1.0000 |
|
38 |
38 |
0.6382 |
0.6579 |
0.7039 |
0.7500 |
0.7829 |
0.8355 |
0.9013 |
1.0000 |
1.0987 |
|
45 |
45.5 |
0.7368 |
0.7566 |
0.8026 |
0.8487 |
0.8816 |
0.9342 |
1.0000 |
1.0987 |
1.1974 |
|
50 |
50.5 |
0.8026 |
0.8224 |
0.8684 |
0.9145 |
0.9474 |
1.0000 |
1.0658 |
1.1645 |
1.2632 |
|
55 |
54.5 |
0.8553 |
0.8750 |
0.9211 |
0.9671 |
1.0000 |
1.0526 |
1.1184 |
1.2171 |
1.3158 |
|
57 |
57 |
0.8882 |
0.9079 |
0.9539 |
1.0000 |
1.0329 |
1.0855 |
1.1513 |
1.2500 |
1.3487 |
|
60 |
60.5 |
0.9342 |
0.9539 |
1.0000 |
1.0461 |
1.0789 |
1.1316 |
1.1974 |
1.2961 |
1.3947 |
|
65 |
64 |
0.9803 |
1.0000 |
1.0461 |
1.0921 |
1.1250 |
1.1776 |
1.2434 |
1.3421 |
1.4408 |
|
66 |
65.5 |
1.0000 |
1.0197 |
1.0658 |
1.1118 |
1.1447 |
1.1974 |
1.2632 |
1.3618 |
1.4605 |
|
95 |
95 |
1.3882 |
1.4079 |
1.4539 |
1.5000 |
1.5329 |
1.5855 |
1.6513 |
1.7500 |
1.8487 |
|
133 |
133 |
1.8882 |
1.9079 |
1.9539 |
2.0000 |
2.0329 |
2.0855 |
2.1513 |
2.2500 |
2.3487 |
|
Actual No. of teeth |
- |
45 |
50 |
55 |
57 |
60 |
65 |
66 |
95 |
133 |
|
Virtual No. of teeth |
45.5 |
50.5 |
54.5 |
57 |
60.5 |
64 |
65.5 |
95 |
133 |
|
|
11 |
10.5 |
0.7368 |
0.8026 |
0.8553 |
0.8882 |
0.9342 |
0.9803 |
1.0000 |
1.3882 |
1.8882 |
|
13 |
12 |
0.7566 |
0.8224 |
0.8750 |
0.9079 |
0.9539 |
1.0000 |
1.0197 |
1.4079 |
1.9079 |
|
15 |
15.5 |
0.8026 |
0.8684 |
0.9211 |
0.9539 |
1.0000 |
1.0461 |
1.0658 |
1.4539 |
1.9539 |
|
19 |
19 |
0.8487 |
0.9145 |
0.9671 |
1.0000 |
1.0461 |
1.0921 |
1.1118 |
1.5000 |
2.0000 |
|
22 |
21.5 |
0.8816 |
0.9474 |
1.0000 |
1.0329 |
1.0789 |
1.1250 |
1.1447 |
1.5329 |
2.0329 |
|
25 |
25.5 |
0.9342 |
1.0000 |
1.0526 |
1.0855 |
1.1316 |
1.1776 |
1.1974 |
1.5855 |
2.0855 |
|
30 |
30.5 |
1.0000 |
1.0658 |
1.1184 |
1.1513 |
1.1974 |
1.2434 |
1.2632 |
1.6513 |
2.1513 |
|
38 |
38 |
1.0987 |
1.1645 |
1.2171 |
1.2500 |
1.2961 |
1.3421 |
1.3618 |
1.7500 |
2.2500 |
|
45 |
45.5 |
1.1974 |
1.2632 |
1.3158 |
1.3487 |
1.3947 |
1.4408 |
1.4605 |
1.8487 |
2.3487 |
|
50 |
50.5 |
1.2632 |
1.3289 |
1.3816 |
1.4145 |
1.4605 |
1.5066 |
1.5263 |
1.9145 |
2.4145 |
|
55 |
54.5 |
1.3158 |
1.3816 |
1.4342 |
1.4671 |
1.5132 |
1.5592 |
1.5789 |
1.9671 |
2.4671 |
|
57 |
57 |
1.3487 |
1.4145 |
1.4671 |
1.5000 |
1.5461 |
1.5921 |
1.6118 |
2.0000 |
2.5000 |
|
60 |
60.5 |
1.3947 |
1.4605 |
1.5132 |
1.5461 |
1.5921 |
1.6382 |
1.6579 |
2.0461 |
2.5461 |
|
65 |
64 |
1.4408 |
1.5066 |
1.5592 |
1.5921 |
1.6382 |
1.6842 |
1.7039 |
2.0921 |
2.5921 |
|
66 |
65.5 |
1.4605 |
1.5263 |
1.5789 |
1.6118 |
1.6579 |
1.7039 |
1.7237 |
2.1118 |
2.6118 |
|
95 |
95 |
1.8487 |
1.9145 |
1.9671 |
2.0000 |
2.0461 |
2.0921 |
2.1118 |
2.5000 |
3.0000 |
|
133 |
133 |
2.3487 |
2.4145 |
2.4671 |
2.5000 |
2.5461 |
2.5921 |
2.6118 |
3.0000 |
3.5000 |
|
Chart 2: Gear ratios for Meccano gears : T/t |
|
|||||||||
|
# Teeth |
- |
|
||||||||
|
t\T |
11 |
13 |
15 |
19 |
22 |
25 |
30 |
38 |
45 |
|
|
11 |
1.0000 |
1.1818 |
1.3636 |
1.7273 |
2.0000 |
2.2727 |
2.7273 |
3.4545 |
4.0909 |
|
|
13 |
0.8462 |
1.0000 |
1.1538 |
1.4615 |
1.6923 |
1.9231 |
2.3077 |
2.9231 |
3.4615 |
|
|
15 |
0.7333 |
0.8667 |
1.0000 |
1.2667 |
1.4667 |
1.6667 |
2.0000 |
2.5333 |
3.0000 |
|
|
19 |
0.5789 |
0.6842 |
0.7895 |
1.0000 |
1.1579 |
1.3158 |
1.5789 |
2.0000 |
2.3684 |
|
|
22 |
0.5000 |
0.5909 |
0.6818 |
0.8636 |
1.0000 |
1.1364 |
1.3636 |
1.7273 |
2.0455 |
|
|
25 |
0.4400 |
0.5200 |
0.6000 |
0.7600 |
0.8800 |
1.0000 |
1.2000 |
1.5200 |
1.8000 |
|
|
30 |
0.3667 |
0.4333 |
0.5000 |
0.6333 |
0.7333 |
0.8333 |
1.0000 |
1.2667 |
1.5000 |
|
|
38 |
0.2895 |
0.3421 |
0.3947 |
0.5000 |
0.5789 |
0.6579 |
0.7895 |
1.0000 |
1.1842 |
|
|
45 |
0.2444 |
0.2889 |
0.3333 |
0.4222 |
0.4889 |
0.5556 |
0.6667 |
0.8444 |
1.0000 |
|
|
50 |
0.2200 |
0.2600 |
0.3000 |
0.3800 |
0.4400 |
0.5000 |
0.6000 |
0.7600 |
0.9000 |
|
|
55 |
0.2000 |
0.2364 |
0.2727 |
0.3455 |
0.4000 |
0.4545 |
0.5455 |
0.6909 |
0.8182 |
|
|
57 |
0.1930 |
0.2281 |
0.2632 |
0.3333 |
0.3860 |
0.4386 |
0.5263 |
0.6667 |
0.7895 |
|
|
60 |
0.1833 |
0.2167 |
0.2500 |
0.3167 |
0.3667 |
0.4167 |
0.5000 |
0.6333 |
0.7500 |
|
|
65 |
0.1692 |
0.2000 |
0.2308 |
0.2923 |
0.3385 |
0.3846 |
0.4615 |
0.5846 |
0.6923 |
|
|
66 |
0.1667 |
0.1970 |
0.2273 |
0.2879 |
0.3333 |
0.3788 |
0.4545 |
0.5758 |
0.6818 |
|
|
95 |
0.1158 |
0.1368 |
0.1579 |
0.2000 |
0.2316 |
0.2632 |
0.3158 |
0.4000 |
0.4737 |
|
|
133 |
0.0827 |
0.0977 |
0.1128 |
0.1429 |
0.1654 |
0.1880 |
0.2256 |
0.2857 |
0.3383 |
|
|
Overdrive Ratios |
Reduction Ratios |
|
||||||||
|
t\T |
45 |
50 |
55 |
57 |
60 |
65 |
66 |
95 |
133 |
|
|
11 |
4.0909 |
4.5455 |
5.0000 |
5.1818 |
5.4545 |
5.9091 |
6.0000 |
8.6364 |
12.0909 |
|
|
13 |
3.4615 |
3.8462 |
4.2308 |
4.3846 |
4.6154 |
5.0000 |
5.0769 |
7.3077 |
10.2308 |
|
|
15 |
3.0000 |
3.3333 |
3.6667 |
3.8000 |
4.0000 |
4.3333 |
4.4000 |
6.3333 |
8.8667 |
|
|
19 |
2.3684 |
2.6316 |
2.8947 |
3.0000 |
3.1579 |
3.4211 |
3.4737 |
5.0000 |
7.0000 |
|
|
22 |
2.0455 |
2.2727 |
2.5000 |
2.5909 |
2.7273 |
2.9545 |
3.0000 |
4.3182 |
6.0455 |
|
|
25 |
1.8000 |
2.0000 |
2.2000 |
2.2800 |
2.4000 |
2.6000 |
2.6400 |
3.8000 |
5.3200 |
|
|
30 |
1.5000 |
1.6667 |
1.8333 |
1.9000 |
2.0000 |
2.1667 |
2.2000 |
3.1667 |
4.4333 |
|
|
38 |
1.1842 |
1.3158 |
1.4474 |
1.5000 |
1.5789 |
1.7105 |
1.7368 |
2.5000 |
3.5000 |
|
|
45 |
1.0000 |
1.1111 |
1.2222 |
1.2667 |
1.3333 |
1.4444 |
1.4667 |
2.1111 |
2.9556 |
|
|
50 |
0.9000 |
1.0000 |
1.1000 |
1.1400 |
1.2000 |
1.3000 |
1.3200 |
1.9000 |
2.6600 |
|
|
55 |
0.8182 |
0.9091 |
1.0000 |
1.0364 |
1.0909 |
1.1818 |
1.2000 |
1.7273 |
2.4182 |
|
|
57 |
0.7895 |
0.8772 |
0.9649 |
1.0000 |
1.0526 |
1.1404 |
1.1579 |
1.6667 |
2.3333 |
|
|
60 |
0.7500 |
0.8333 |
0.9167 |
0.9500 |
1.0000 |
1.0833 |
1.1000 |
1.5833 |
2.2167 |
|
|
65 |
0.6923 |
0.7692 |
0.8462 |
0.8769 |
0.9231 |
1.0000 |
1.0154 |
1.4615 |
2.0462 |
|
|
66 |
0.6818 |
0.7576 |
0.8333 |
0.8636 |
0.9091 |
0.9848 |
1.0000 |
1.4394 |
2.0152 |
|
|
95 |
0.4737 |
0.5263 |
0.5789 |
0.6000 |
0.6316 |
0.6842 |
0.6947 |
1.0000 |
1.4000 |
|
|
133 |
0.3383 |
0.3759 |
0.4135 |
0.4286 |
0.4511 |
0.4887 |
0.4962 |
0.7143 |
1.0000 |
|
|
Overdrive Ratios |
Reduction ratios |
|
||||||||
* I use the term 'virtual' because some gear tooth numbers do not compute correctly for 38DP. For example, for the 60/15-tooth combination giving a ratio of 4:1, the operating pitch circle diameters are 1.6 & 0.4. However, at 38DP, these tooth numbers indicate standard PCD's of 0.3947 & 1.5789 respectively! Still 4:1 but the sum of the pitch circle radii is 0.9868, which is 0.0132 short of the required 1.000 working centre distance. Alternatively, the 0.4 & 1.6 PCD's at 38DP indicate tooth numbers of 15.2 & 60.8!
This situation is overcome by extending the standard centre distance from 0.9868 to 1.0000, and is achieved by shifting the involute profile of each gear radially outward. Then 15 & 60 teeth are cut on the extended PCD's (+0.01055 & 0.00265 profile shift per gear respectively). By this means, 15 & 60 tooth gears are made using 38DP tooling that mesh correctly at 1-inch centre distance.
Other apparently non-standard gears such as the 11/66, 22/55, 25/50 & 30/45 combinations are also produced using negative or positive profile shift. Another way to treat this situation is to consider that one tooth must be lost from the pair (15.2+60.8)-75=1), then by using a 'virtual' pitch of 37.5DP (half a tooth less per gear), the PCD's calculate correctly. Similarly, by adding half a tooth to each gear, giving 'virtual' numbers of teeth of 15.5 & 60.5, then at 38DP, the centre distance calculates correctly.
Note: This virtual analogy is an expedient for the above chart only and should be treated with caution. It does not apply to individual gears and the actual pitch of a gear is determined by the cutter or tooling.
Please direct any comments or queries to the 'Meccano / General Meccano Chat' section of the public FORUM. General engineering issues can also be addressed to the 'Other / General Resources / Prototype Information & Research' section, if you feel that is more appropriate.
© A Wenbourne (September 2007)