I saw a youtube video of an ultralight airplane with a 5.5hp engine that has been modified to produce 30hp. So how do I modify a 5.5hp engine to make 15hp+ of output?
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to triple the output of a 5.5HP engine, you must either triple the displacement or triple the rotating speed. there is no way to triple the displacement by boring out the block and fitting a bigger piston and/or by installing a crank with a longer throw to increase the stroke n that engine. Now, tripling the rotating speed requires a long list of modifications to achieve which do not make economic sense for an engine this small, compared to simply buying a different engine with triple the cylinder displacement. Here is why.
The engine you picture has a "comfortable" design speed of about 3600RPM. It has valves operated by pushrods. Inertia in the valve-actuating mechanism limits its rotating speed to no more than about 2X 3600RPM or 7200RPM. To go faster than that requires re-engineering the engine for an overhead cam system (with reduced inertia) for a start, but there are other issues that must be dealt with.
For example, the induction system which supplies fuel and air mixture to the cylinder is proportioned to support a flow rate which matches the design speed of the engine. Tripling the design speed requires re-engineering this with a much-shortened manifold tube and a carburetor which can supply 3X the flow rate so the engine does not choke out at the new, higher RPM.
The "3X" engine will be generating heat at 3X the rate of the original, which must be gotten rid of. Since the fan that cools this engine runs at the same speed as the crankshaft, one would think that heat rejection is not a problem, but the finning pattern in the engine must be modified so as to pull heat out of the head (where the valves and valve seats reside) much faster than that required to support a 3600RPM engine, to prevent the valves and seats from overheating and getting burnt.
Similar comments apply to the lubrication system in the engine as well.
Now we come to final drive and reliability issues. Tripling the engine speed requires a redesign of the reduction system in the final drive, to maintain the prop tips subsonic. It also requires a prop redesign, to absorb 3X the power of the original engine.
Your "3X" engine will wear itself out in 1/3rd the time as the original design since its parts are moving 3X faster- but even if that were acceptable, what won't be is the overall reduction of reliability which accompanies an engine that fundamentally runs hot. So while racing motorcycles can use engines that run at 16,000 RPM, it's OK for them to blow up after one season's worth of races since they get torn down and completely rebuilt after ~tens of hours at 16,000RPM. For high-reliability applications like aircraft, you want a two-thousand-hour engine to statistically ensure an acceptably low failure rate in 1,000 hours of operation.
The bottom line is this: it is possible to re-engineer a 5.5HP lawnmower engine to run at 10,800RPM and it is possible to bore out the cylinder to get 1.2X the displacement and combine that with 1.2X the stroke with a new crank and it is possible to put a carb on that engine pulled from a 15HP engine, but it generally makes little sense.