The Theory of Meccano Gears Part 1 : Spur Gears

Appendix A

40DP Meccano gears (All dimensions in inches)

There is a school of thought that Meccano gears may have been cut with 40DP cutters (Reference 1). The argument is that the Meccano worm is in fact a 9/16-inch Whitworth thread (BSW), this being 12 teeth per inch (TPI) as is the worm. Meccano racks are also 12TPI. 38DP is 0.08267 circular pitch, 40DP is 0.07854 circular pitch and 12TPI is 0.08333 pitch. The pitch difference between 38DP and 12TPI is: 0.00066 (very small). The pitch difference between 40DP and 12TPI is: 0.00479 (significant).

Reference 1 proposes that 38 teeth were cut on a 1.00 PCD using 40DP geometry to minimise the pitch difference between 40DP and 12TPI because, by doing so, the circular pitch is 0.08267 (as with the 38DP cutter), thereby leading to the claim that Meccano gears are a thread based, not a gear based system.

The 'standard' pitch diameter for 38 teeth cut on a 1.00 PCD with a 40DP cutter is 0.95. In consequence, the cutter has to be 'withdrawn' so as to cut the 38 teeth on an 1.00 PCD, thereby facilitating correct meshing conditions for two such gears operating at 1.00 centre distance. This cutter withdrawal displaces the involute profile radially outwards and is termed "profile shift". Using the 38 tooth gears (P/N 31) as an example, I am trying to prove mathematically and by measuring my gears, that they may have been made to 40DP proportions.

Profile shifting (outward) increases pressure angle, which can have the visible effect of making the teeth more pointed (less crest width), but since the 40DP cutter is a finer pitch than 38DP, this effect may be less noticeable. Also, a 40DP tooth is less deep than a 38DP tooth, but the difference is only: 0.003 - not obvious, difficult to measure and lost within normal manufacturing tolerances anyway! The outside diameter differs also by only 0.003, which again, would be lost within the normal manufacturing tolerance. In any case, the OD is relatively unimportant and variable because it is merely the initial blank size (ignoring the effect of blank diameter on contact ratio). Thus, for two 38t gears, one cut with 38DP, the other cut with 40DP cutters, the difference would be very difficult to differentiate visually or even by measuring their features. This leaves determining the actual position of the involute tooth flanks as the most likely indicator of differing manufacturing methods.

One method of determining the involute position is to calculate the measurement over pins of known size, placed between teeth on opposite sides of the gear, and to compare this with actual measurements. Hence the mathematical/measured comparisons in chart 1, which assume 20° pressure angle and 0.4/DP root clearance (an industry standard). "Theory" denotes calculated values. "Actual" denotes measurements taken from a sample of six gears. From this evidence, the measurement over pins and crest width figures indicate that these gears were made using 38DP rather than 40DP cutters! Further study of 20 tooth pinions assuming 38 & 40DP generation is summarized in chart 2. This evidence supports the 40DP theory.

It would be easy to accept that early 20/25/40 & 50 tooth gears were cut to 40DP standard, as the pitch diameters calculate exactly to the ratios (standard gears). However, the 56-tooth gear does not conform? 40DP either produces 60 teeth on a 1.5 PCD, or a PCD of 1.4 for 56 teeth! So the 56 teeth are profile shifted to 1.5 PCD (at 40DP), but then the crest width would be only 0.006! Whereas my samples measure at least 0.015? Pin measurements merely indicate the radial position of the tooth flank at their point of contact. To determine the shape of the gear tooth profile by this means would require a number of pin measurements in order to plot the polar and radial co-ordinates of the curve. This would be very tedious and time consuming, which is why specialised equipment exists for this purpose such as shadowgraphs and involute checking machines. Other factors that detract from this kind of analysis are the (unknown) design intent and manufacturing tolerances, also wear of the sample gears being measured?

Despite these limitations, I conducted the same analysis of 19 & 57 tooth gears, taking three sets of measurements each from three of each gear. The data are shown in chart 3. These results are clearly in favour of 38DP proportions. Could it be that the early gears approximated to 40DP but were not involute in profile?

Conclusions:

1) The original (obsolete) 20/25/40/50 & 56 tooth gears could have been made to 40DP proportions, however...

2) It is more likely that they were cut to 0.65 module (39.07DP) of double circular arc form similar to BS 978 : Part 2 - Gears for Instruments & Clockwork Mechanisms. Although these gears (1903-1920) pre-date the British Standard, the standard would have incorporated the practice of clockmakers up to the time of compilation/publication. It is known that Hornby used clockmakers to produce early gears.

3) According to my measurements and calculations, the 19/38/57 tooth series conform to 38DP.

4) One would hope that replica; reproduction and compatible gears would conform similarly.

5) Meccano gears conform to a gear system rather than a thread standard.

6) It is possible to mix and mesh the obsolete 40DP gears with the later 38DP gears because the size and pitch differences are small, compared to the backlash and clearance allowances adopted in their manufacture.

7) The Meccano worm does indeed appear to conform to 9/16" BSW proportions at 12TPI, also by having an included angle of 55 degrees. The root diameter at 0.44 (0.4558-BSW) and outside diameter of 0554 (0.5625-BSW) are slightly undersize, probably to reduce the effective diameter of 0.509(BSW) to the 0.5 Meccano standard. Therefore it appears that the worm and racks are based on the 12TPI BSW thread system and the 0.00066 pitch difference is insignificant.

References:

1) http://neil.fraser.name/news/2004/04/03/

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)