How does the velocity of a turboprop engine relative to the airplane change when a plane is banking?

I saw this YouTube video recently:

And in it, a turboprop engine propeller and gearbox comes out and slashes across the underbelly of the plane, crippling the aircraft. My question is, what aerodynamic force is involved with the turboprop + gearbox being able to fall out specifically during the turn? I can imagine that for whatever reason, when the pilot decided to quickly turn around and head back, that the outermost turboprop must have been moving slower than the plane, and that those forces caused by the turn ultimately tugged the propeller out, and that the turning of the plane brought the propeller under the plane causing the massive damage.

• Why not ask instead, "What is the formula for the difference in airspeed between the inboard and outboard wingtips, as a function of airspeed (as measured at the fuselage) and bank angle?" You'll find the answer is rather small value during high-speed flight. Jan 15, 2022 at 7:46
• Well I don't really know much about how airplanes work and I just wasn't sure of how a propeller can lift out of a turboprop so I thought I'd ask here alongside my best assumption as to what was going on. So I'm assuming then that the turboprop didn't lift out because of the turn. Jan 15, 2022 at 8:00
• This question is "low hanging fruit" ripe for someone to pick-- but everyone seems to be sleeping-- but the very brief answer (I would post it as such if the question weren't so full of misconceptions) is a) there's no hint in the video that any pilot action caused the problem, and b) you can be sure that no roll rate (rate of change in bank angle), and no bank angle (steepness of "tilt") that this aircraft was capable of in normal non-violent-nausea-inducing operations ,could have remotely possibly have been the cause of this structural failure, absent some pre-existing flaw or failure. Jan 15, 2022 at 23:02
• Note: The video is not available in all locations worldwide. Jan 16, 2022 at 13:55

In a coordinated turn, this is purely a geometry question. Let's call the distance from centerline to engine $$d$$. The plane must bank to allow for centripetal acceleration $$a=v^2/r$$ with $$v$$ true airspeed and $$r$$ turn radius. This places the engines at a lateral distance of $$d^*=\frac{d}{\sqrt{1+\frac{a^2}{g^2}}}$$

from the centerline with $$g$$ acceleration due to gravity. Finally, the speed of the inner engine is $$v_1=v(1-d^*/r)$$ and the outer engine $$v_2=v(1+d^*/r)$$. The speed difference is thus $$\delta=\frac{2vd^*}{r}.$$ If you want you can simplify this to $$\delta=\frac{2gvd}{\sqrt{g^2r^2+v^4}}$$

You may play around with some real world values, but it's safe to say that no turn will rip off a turboprop engine just from the speed difference - these things are designed to apply many kilonewtons of thrust, also in reverse.

• Or course we could also consider the (vertical) velocity component imposed by rolling (changing bank angle) -- but we all know this is all a moot point in relation to this particular in-flight incident-- Jan 16, 2022 at 15:02

If the turn forces did not dislodge the prop, what did?

After that great crew brought their plane to a stop, the maintenance records for #4 engine may have been first on their minds, after safely deplaning the passengers.

When one considers the props pull the plane along, then what ever bearing that holds that (turbo) prop to its gear case comes into focus. If that bearing fails, the prop will literally pull itself ahead and out of the nacelle.

The rest is seen in the video. Normal turning forces, with passengers comfort in mind, cannot be expected to cause that type of failure.

With the huge propellers that drove that plane, any imbalance in the prop assembly could cause them to "wobble", weakening surrounding structures to the point of failure. This type of accident was all too common in the early age of passenger travel. In the video, one can see an unusual shudder in the co-pilot's control column and the captain making a decision to abort the flight, turn around, and get the plane back on the ground (before all hell literally broke loose).

Stringent quality control and maintenance procedures, along with proper design, are key factors in prevention of this type of accident.