An interesting question with a complicated answer.
For a start, not all synchronisation methods were the same. Some were mechanical, the RAF one was hydrosonic, and some (like the German one) electrical.
Secondly, it made a big difference what type of prop was used: fixed pitch, two-speed (etc) or constant speed.
Thirdly, not all guns were equally suited to synchronisation.
Put these three variables together and you get very different results.
The Italian fighters in use at the start of WW2 had fixed-pitch props which varied in rpm with the engine speed. So the RoF of the guns varied as well - quite a lot. This is how I described similar WW1 systems in
Flying Guns – World War 1: Development of Aircraft Guns, Ammunition and Installations 1914-32:
"The effect of synchronisation on the rate of fire can best be explained by describing a simple system like that introduced by Fokker, in which one firing signal was sent to the gun for each rotation of the propeller. If the gun was capable of firing at 500 rounds per minute, then for propeller speeds of up to 500 revolutions per minute the RoF would be the same as the propeller rpm. However, as soon as the propeller exceeded 500 rpm, the gun mechanism could no longer keep up and could then only fire on every other rotation, so the RoF would drop to 250 rpm. It would then accelerate again with increasing propeller speed but at half the rate, so when the propeller was spinning at 1,000 rpm, the gun would be back to firing at 500 rpm again. Once more, propeller revs faster than this would cause the RoF to drop, but this time only to two-thirds of the full RoF, as it would fire on every third rotation, so it would be achieving 330 rpm. As the propeller continued to accelerate to 1,500 rpm, the gun would be back up to 500 rpm again, and so on. Any quoted figure for synchronised rates of fire could therefore only be an average."and:
"More advanced systems like the CC and the later German types sent two firing signals per propeller revolution (logical with a two-bladed propeller, in which there would be two firing opportunities per revolution), although possibly at the expense of some reliability in these primitive early systems, as they would have to work twice as fast. In this case, the maximum RoF for our 500 rpm gun would be reached twice as often, at 250, 500, 750, 1,000, 1,250 and 1,500 rpm. A still more sophisticated variation was to use a “critical sector cam”, which instead of just sending a single firing impulse sent a continuous one during the “safe” period when the propeller blades were out of the way. The effect of this was much less regular, with the gun firing in erratic bursts, but the average RoF was the highest of all."Clearly, a constant-speed prop (such as became standard in WW2) avoided this sort of problem, but it was helpful if the prop rpm was compatible with the natural firing cycle of the gun. For instance, the Germans stuck with three-bladed props (possibly because of the synchronisation issue, at least in part), which gave three opportunities per prop rev for each gun to fire. With a "free" gun RoF of 1,200 rpm, then a prop rev of 1,200 rpm would be fine; so would 1,600 rpm (firing once every fourth blade - if my back-of-the-envelope is working this morning!). The Germans sorted this quite well; you will never get an exact match between prop revs and gun rpm, but they only lost about 10% of the free RoF in their synch guns.
The average loss in RoF overall was probably around 20-30% (e.g. Russian 12.7mm UB = 24%)
Some guns just couldn't be synchronised for technical reasons, others suffered a lot more when they were. For example, the .50 BMG lost around 40% of its RoF, possibly because it had a heavy firing pin assembly with a lot of inertia. The Japanese Army's Browning-based guns suffered to a similar extent.
The Germans achieved the best solution later in the war, by adopting electric ignition of the ammunition. So instead of their usual practice of linking the position of the prop to a solenoid which released the sear at the appropriate moment, they just had the synch system sending an electrical impulse which fired the primer directly. This made the MG 131 and MG 151 by far the best large-calibre guns for synchronisation.
Tony Williams: Military gun and ammunition
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