The discussion started when it was argued that an undersquare engine had a geometric/mechanical advantage to produce torque due to a “longer torque arm” (post #22). As was demonstrated mathematically, for a given displacement, the bore/stroke ratio has no geometrical impact on the absolute torque produced by an engine. Instantaneous torque is only dependent on cylinder pressure, displacement and crank angle.
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Originally Posted by Boss330
However, I suspect "cylinder pressure" might be a keyword here... And that's where the complex calculations are coming into play.
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You are correct, the key lies in the cylinder pressure. The discussion has now evolved a bit, so here are my thoughts. An oversquare engine will have better ability to produce cylinder pressure higher in the RPM range and an undersquare engine will in the lower RPM range. As anything in life, there is no free lunch, everything is a compromise. I will spare you the formulas, because as you stated, it becomes mathematically very complex.
A bigger bore allows for bigger valves which in turn improves flow and volumetric efficiency resulting in greater cylinder pressures, especially so in the higher RPM range. A shorter stroke also has benefits in reducing piston speed for higher RPM (not citing anything new here).
However, a large bore also come with disadvantages. Because the engine relies on a greater force produced by the cylinder pressure (due to the big bore) rather than a longer arm (larger stroke) to generate a given amount of torque, the forces that all the components and bearings need to carry are that much greater. So stronger components and bearings are needed everywhere (piston, rod, crank) adding more complexity and cost. Further, these greater forces also generate more friction losses at the piston/cylinder wall interface. Another disadvantage of the bigger bore is the shape of the combustion chamber. For a given displacement and compression ratio, the combustion chambers of an oversquare and an undersquare engine are the same volume, hence the chamber of the oversquare engine is comparatively very broad in diameter but very short in height. Because of this, oversquare engines cannot tolerate as much ignition advance at lower RPMs since the flame will collide the piston and cause detonation, hence lower cylinder pressure. Much less of a problem at higher RPM since the piston speed moving away from the flame is sufficient to avoid the collision. Another disadvantage is the greater blowby losses caused by the greater circumference of the piston (worse at lower RPMs), which lowers effective cylinder pressure. There is more, but I will stop here .
In short, an oversquare engine will have better ability to produce torque higher in the RPM range and an undersquare engine will in the lower RPM range. It is a question of where torque is produced in the RPM range, not how much of it (Note however that producing torque higher in the RPM range will yield more power).
As swamp2 mentions, the bore/stroke relationship is selected by trying to find the best compromise for a given application and/or engine characteristics. Low cost, durability, reliability, overall efficiency favor long stroke low RPM engines. High specific output, power to weight favor bigger bore high RPM engines. On an industrial or marine engine that runs non-stop for days, reliability, durability and overall efficiency are very important. On an F1 car, where engine displacement is regulation constrained, high specific output, maximum power are very important; fuel economy (efficiency), durability and cost, much less so. And then there is everything in between.