jj.palmer wrote: ...see attached calculation sheets...
Thank you very
much for catching my mistake. What a difference a '-' sign in the wrong place makes. In the hopes that the third time's the charm:
-- revised a second time with another math error corrected ------
A formula for calculating the Balance Factor can be written in the form:Balance Factor = balance weight + small end weight / piston weight + small end weight
As can be seen, to solve this requires determining three weights as well as having a fixture for holding the crankshaft so it can rotate freely when the balance weights are added. Small end weight:
With the 13/16" bushing to reduce it for a smaller gudgeon pin than the original 1" pin it weighs 267.5 +/-1 g. However, from this subtract 48.2 g for the "excess" weight of the bronze in this smaller ID bushing so it originally would have weighed 219.3 +/-1 g.-- Current small end weight = 267.5 +/-1 g-- Original small end weight = 219.3 +/-1 gPiston weight:
The "piston weight" is that of the complete assembly of piston, gudgeon pin, circlips and rings. Although it doesn't enter into the calculations shown here I'll note that the weight of the additional Al used in a, say, +30 piston is not negligible. It can be calculated from the annular volume of a piston of stock diameter and one 0.03" larger than that.-- weight of +30 piston assembly that was currently in my bike 467.5 +/-0.5g
I was lucky to find two people with original piston assemblies for the Ariel. The one in Australia is used and weighs 507.2 g and the one in Canada is new and weighs 503.5 g.-- weight of original piston assembly weight (average of above) = 505 +/-2 g-- weight of aftermarket +60 Gandini piston assembly 516.5 +/-0.5 g-- weight of aftermarket +60 Omega piston assembly 435.0 +/-0.5 gBalance weight:
I hung balance weights and washers from a wire attached to the small end until the crankshaft was in balance and weighed the final total mass. It took 196.59 g plus 10 g on the rim at 90 deg. Taking into account the off-axis imbalance I estimate the uncertainty in balancing the crank using only weights hanging from the connecting rod and none at 90 deg. is +/-3 g.
The weight of the heads of the two 1/4" cap screws pinning the big end is 2 x 2.74 grams = 5.48 g. Without the cap screws it would have required that much additional weight, plus the 48.2 g that the current bronze reducing bush weighs more than the original 1" bush. So, to originally balance the crank it would have required 196.6 + 5.5 + 48.2 = 250.3 g. -- weight to balance crankshaft in its current form = 196.6 +/-0.1 g-- weight to originally balance crankshaft = 250.3 +/-2 g
If the 5/16" holes were added sometime later the original weight required to balance it would have been 38.9 grams less.-- weight to originally balance crankshaft without the four 5/16" holes =211.4 +/-3 g Original factory balance factor:
If the crankshaft in its current form (less the cap screws) is how it left the factory, the original balance factor was:
250.3 + 219.3 / 505.0 + 219.3 = 469.6 / 724.3 = 64.8 +/-1%
If the four 5/16" holes were added later it would have required 38.9 grams less to balance it originally. In this case the original balance factor would have been
211.4 + 219.3 / 505 + 219.3 = 430.7 / 724.3 = 59.5 +/-1%
For comparison, a 1960 BSA Service Bulletin shows 60% for the 250cc 'C' series, 58% for Gold Stars, 55% for the essentially identical 'B' series singles in the same frame as the Gold Star, and 55% for both the 500cc and 650cc 'A' series twins, also in the same frame as the Gold Star. A 1930s Vincent Comet used 66% (claimed weight 390 lbs. vs. 290 for the Ariel) but this had to be reduced to 61% in a lightweight speedway frame. Current Balance Factor:
With current piston in it:
196.6 + 267.5 / 467.5 + 267.5 = 464.1 / 735 = 63.1% +/-0.3%
If Gandini piston were used in it without any other changes:
196.6 + 267.5 / 516.5 + 267.5 = 464.1 / 784 = 59.2 +/-0.3%
If Omega piston were used in it without any other changes:
196.6 + 267.5 / 435.0 + 267.5 = 464.1 / 702.5 = 66.1 +/-0.3%
To reduce the balance factor to 65% if using the Omega piston would require reducing the required balance weight by 7.5 g which in turn would mean removing roughly half that weight from the rim of the flywheel. This could be achieved by, for example, drilling two additional 5/16" holes approx. 3/16" deep each. If I did this I would carefully
make any such changes only after assembling the flywheel and checking the balance.
Again, thank you very much for catching this mistake. I would like to think I would have caught it myself when I went through the calculation a final time, but I certainly might not have.