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I was reading up on World War I fighter aircraft, and saw that rotary piston engines, which powered many of those planes, produced extreme gyroscopic effects due to the large, heavy chunk of metal spinning at several hundred RPM; as a result, single-engine aircraft using these engines had extreme difficulty turning to one side but could make exceptionally sharp and quick turns to the other side.

Why don't modern single-engine fighter jets experience even worse gyroscopic precession than their World War I counterparts, seeing how the compressor/shaft/turbine assembly in a turbojet engine is far heavier than a WWI rotary engine and spins far faster?

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Well, jet engines do have gyroscopic effects. It is a major concern in the design of the turbomachinery. When the plane pitches/yaws, the resulting gyroscopic moment causes the compressor and turbine blades to move closer to the case. If excessive, this can cause the blades to rub into the case, causing loss of performance.

As to why the plane is not affected by the gyroscopic moment of the engines, let's compare a few numbers.

Take a typical WWI era plane such as the Vickers FB19 powered by a Le Rhone 9J. The engine has a mass of 146 kg, the majority of which is rotating and produces 110 HP. The plane has a maximum takeoff mass of 675 kg, which includes the engine. The rotating mass is a fairly sizable fraction (~20%) of the mass of the plane. Therefore the gyroscopic effect of the engine has a big impact on manueverability.

Now compare that with an F16 powered by an GE F110 engine. The high pressure spool of the engine has a mass on the order of 200 kg (the whole engine is about 2000 kg and the high speed rotating spool is about 10% of that... sorry I don't have a specific reference). The max takeoff mass of the plane is about 19,200 kg. The rotating mass is only 1% of the total mass of the plane. So the gyroscopic effect is not so important. i.e. compared to the moment required just to turn the plane itself, the gyroscopic moment is not large.

Further, to be more precise, the gyroscopic moment is not actually proportional to the mass, but to the mass moment of inertia, which is proportional to radius squared. A lot of the rotating mass of an older rotary engine was at a fairly high radius, whereas a lot of the rotating mass of a jet engine rotor is at a fairly low radius. Although I have not done the calculations, my guess is that a modern fighter jet engine spool probably has a lower moment of inertia than a WWI rotary engine, despite the higher mass.

Edit:

In response to the comment by J Walters, let me try to make this a little more precise. The moment required to perform an angular acceleration is $M=I\alpha$, where $I$ is the moment of inertia. Let's assume that the plane is a rod (length >> width or height). That's not totally correct, but it's good for an order of magnitude approximation. Then from this formula, $I=(1/12)mL^2$. So for the FB19, the moment of inertia is $(1/12)(675)(5.54^2)=1726 kg-m^2$. For the F16, $(1/12)(19200)(15.06^2)=362885 kg-m^2$. So, comparing these two, the moment that the control surface have to apply to turn the plane (at a given angular acceleration) is 200x higher for the F16.

Now, let's look at the moments due to the gyroscopic effect. The gyroscopic moment is $M=J \Phi \times \Omega$, where $\Phi$ is the angular velocity of the aircraft maneuver and $\Omega$ is the spin angular velocity of the engine, and $J$ is the polar moment of inertia. I said before that I expected $J$ for the F16 to actually be lower due to lower radius, but let's just assume they are the same. The Le Rhone spins at 1,350 RPM. I don't know exactly, but I know the F110 high pressure spool top speed is between 15,000 - 20,000 rpm. So the gyroscopic moment is about 10 - 15x higher for the F16.

And as Peter Kampf pointed out, the aerodynamic forces go up too. The FB19 had a top speed of about 100 mph whereas the F16 can hit 900 mph at sea level. From NASA's excellent series, both lift and drag scale with velocity squared. So the aerodynamic forces are 81x higher for the F16.

So, in summary, yes, the gyroscopic moment is one order of magnitude larger on the F16. But everything else that the plane has to deal with during maneuvers (the moment of inertia of the plane itself, aerodynamic forces), are two orders of magnitude higher. So the gyroscopic forces are less relevant in comparison.

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    $\begingroup$ +1, and add to that the flight speed is much higher, so the aerodynamic forces acting on the jet aircraft are much higher relative to the inertial loads than they were on WW I aircraft. When aerodynamic loads are low, the engine needs to be balanced by counter-rotation of two spools. This was key to making the Harrier controllable in hover. $\endgroup$ – Peter Kämpf Mar 15 '18 at 12:12
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    $\begingroup$ This answer fails to take into account the angular momentum of a turbine engine. While these may possess a relatively low spool mass, the rotational velocity of is measured in tens-of-thousands of RPM, an order of magnitude above the comparative piston engine. The typical counter rotating spools design is also a significant omission. $\endgroup$ – J Walters Mar 17 '18 at 22:29
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The torque produced by piston engines was also an issue for military aircraft in the first and second world wars, particularly when engines were operating at full power, as during take-off when it could result in a tendency for the aircraft to yaw. In twin engined aircraft, designers often countered the problem by having the two engines and propellers turning in opposite directions (counterrotation).

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Rotary piston engines have a substantial diameter, and most of the mass (the cylinder heads in particular) is concentrated in the periphery. Hence, the moment of inertia of that rotating mass is quite high, and the gyroscopic effects are also much higher than in the case of a mass with less diameter and no so extreme a mass distribution. That's the case of the rotating compressor and turbine blades. It's true that the angular speed is much higher here, but it's not as important as the distance of the element of mass from the center of rotation, that is squared...

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The torque effect doesn't come from the spinning mass of metal, it comes from the propeller biting into the air to produce thrust, producing an opposite and equal reaction as Sir Issac Newton observed... same reason a helicopter needs a tail rotor to counteract the torque of the main rotor, which also works by deflecting air while producing torque in the opposite direction. It's the air resistance to the propeller/rotor that produces the bulk of the torque, not the gyroscopic effect of spinning metal.

All prop planes experience this, although the effect is not nearly as pronounced in a Cessna 152 with 110 hp as it is with a FG1D Corsair or Hawker Tempest with 3000 hp. With larger WW2 fighters, careless application of power could flip the plane over from the sudden application of torque, while any torque effect in a 152 is pretty much unnoticable due to the low power.

There was one WW2 fighter that didn't have this torque effect... the P38, with twin engines and props that rotated in opposite directions, canceling out the torque effect. A common tactic of P38 pilots attacked from the rear was to roll left. Single engine fighters couldn't match it's roll rate in that direction due to the torque effect of their single engine and prop. The P38's poor high altitude performance led to it being sidelined in the European theater, while it was quite popular in the Pacific theater, where the combat tended to occur at much lower altitudes.

The Dornier 335, with it's fore and aft engines, would have also been free of torque effect as the props rotated in opposite directions, only it never made full production and saw only very brief combat use.

Turbines in modern fighters don't produce nearly the torque effect because most of their thrust comes from hot gas being expelled to the rear from the turbine or afterburner which produces little (turbine) to no (afterburner) rotational torque. Most modern fighters have low bypass turbofans for better efficiency, but they still derive the bulk of their thrust from the exhaust, not the front turbine.

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    $\begingroup$ The question is specifically about gyroscopic forces, not torque. Maybe you wish to modify your answer? $\endgroup$ – Peter Kämpf Mar 15 '18 at 15:48

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