# Off center torque applied to a rotating body

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• Chenkel
Chenkel
TL;DR Summary
I give the example of an airplane, where the rudder causes a torque about the center of mass of the airplane, how does this effect the momentum vectors for the engines?
Hello everyone!

So I've been studying gyroscopes, and see that a torque about the shaft alters the momentum, we can find the new momentum vector by finding the torque, multiplying by a small amount of time, and finally adding that vector to the momentum vector. This will create a precession for torques not colinear with the momentum vector.

This makes me wonder what happens to the momentum vector about A when the torque is not applied about the point A, but is instead applied off center at some point B. An example is an airplane, the engines have momentum vectors pointing in front of the plane (this might be a simplification of what really happens, if you want to enlightenment me, feel free to reply.) As the plane turns the rudder applies a torque to the center of mass. I'm wondering what happens to the momentum vectors of the engines, there seems to be a change in direction of the momentum vectors, and I'm thinking the torque vectors causing the change in momentum of the engines must not be colinear with the momentum vector of the engine, because the momentum vectors for the engines are changing direction.

How does the rudder create these torques to the engine, and does the housing of the engine have to be strong to resist these torques.

Hopefully this question isn't too difficult to answer, I'm looking forward to replies, if you have some wisdom you can shed on this issue it will be greatly appreciated, thank you!

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How does the rudder create these torques to the engine, and does the housing of the engine have to be strong to resist these torques.

When you talk about the engines having "momentum vectors", you presumably mean that they have "angular momentum vectors". Or, in particular, that the spinning turbine shaft and blades has a non-trivial amount of angular momentum.

So we have the rudder applying a torque to the plane. The plane yaws in response. A turn to the right or left within the plane of the craft-relative horizontal is called a "yaw". As opposed to a "roll" or a "pitch".

The yaw rate (so many degrees per second) will result in a rate of change in the angular momentum of the spinning turbines. This would tend to cause the spinning turbines to precess. They will tend to either pitch up or pitch down depending on the direction of the turn and the turbine's direction of rotation.

The engine bearings that hold the turbines in place will naturally deal with this pitching effect, applying a counter-torque that prevents the turbine shaft from pitching up or pitching down. The effect of that counter-torque is to cause the turbines to precess so that they continue to match the orientation of the aircraft as a whole. It is not that much torque. The bearings can deal with it. Obviously they can. Because they do.

If the pilot has to crank in a tiny bit of elevator (the horizontal control surfaces on the tail) to compensate for the up or down pitching torque from the engines, he will do that automatically and without even thinking about it. "Plane pitching a bit down -- I'd better pull back slightly on the stick"

If the angular momentum in the engines were all that big of a deal, one would expect to see aircraft doing cartwheels on the tarmac every time they throttled up their engines. We do not see that. We do not see the aircraft tilt at all when throttling up. At least I sure do not.

Edit: I have not been able to Google up very much with respect to engine specifications for moment of inertia and rotation rate. However, there is some interesting reading out there.

Your typical "jet engine" is actually a turbofan. With the turbine engine running a shrouded fan at a lower rotation rate, lower exhaust velocity and higher mass flow rate compared to the engine itself. This is a big win for fuel efficiency.

Here, the concern about moment of inertia is apparently centered on how fast one can spool the engine up, rather than on how rapidly the engine bearings can force it to precess...

https://theflyingengineer.com/flightdeck/pw1100g-gtf/ said:
"With a gear ratio of 3:1, the equivalent moment of inertia of the fan, as seen by the low pressure spool, is only 1/9th of the actual moment of inertia. The radius of the fan is 27% more than the IAEV2527-A5’s fan, making the volume of the fan approximately 2 times that of the IAE’s, implying very crudely that the mass is double that of the IAE’s, assuming the same material is used to make the blades. With double the mass and 27% greater radius, the moment of inertia of the PW1100G’s fan is about 3.3 times that of the IAE’s. This makes the equivalent moment of inertia of the fan, as seen by the low pressure spool, just 36% of the IAE’s, despite the larger mass and radius. Considering that the PW1100G’s low pressure spool has only 3 compressors and 3 turbines, as opposed to larger and heavier 4 compressors and 5 turbines on the IAEV2527-A5, the overall equivalent moment of inertia of the low pressure spool of the PW1100G’s is atleast around 25% that of the IAEV25257’s. With the PW1100G’s low pressure spool estimated to spin at twice the angular speed of the IAEV2527’s, and the overall moment of inertia around 25%, the spool up time may be reduced to around 50% of that in an IAEV2527-A5. This allows thrust to be made available faster, increasing safety margins through enhanced engine response times."

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Chenkel
Chenkel

When you talk about the engines having "momentum vectors", you presumably mean that they have "angular momentum vectors". Or, in particular, that the spinning turbine shaft and blades has a non-trivial amount of angular momentum.

So we have the rudder applying a torque to the plane. The plane yaws in response. A turn to the right or left within the plane of the craft-relative horizontal is called a "yaw". As opposed to a "roll" or a "pitch".

The yaw rate (so many degrees per second) will result in a rate of change in the angular momentum of the spinning turbines. This would tend to cause the spinning turbines to precess. They will tend to either pitch up or pitch down depending on the direction of the turn and the turbine's direction of rotation.

The engine bearings that hold the turbines in place will naturally deal with this pitching effect, applying a counter-torque that prevents the turbine shaft from pitching up or pitching down. The effect of that counter-torque is to cause the turbines to precess so that they continue to match the orientation of the aircraft as a whole. It is not that much torque. The bearings can deal with it. Obviously they can. Because they do.

If the pilot has to crank in a tiny bit of elevator (the horizontal control surfaces on the tail) to compensate for the up or down pitching torque from the engines, he will do that automatically and without even thinking about it. "Plane pitching a bit down -- I'd better pull back slightly on the stick"

If the angular momentum in the engines were all that big of a deal, one would expect to see aircraft doing cartwheels on the tarmac every time they throttled up their engines. We do not see that. We do not see the aircraft tilt at all when throttling up. At least I sure do not.
The pitching torque from the engine onto the plane should have a counter pitch from the plane onto the engine, I'm wondering if this torque is applied through the center of mass of the airplane.

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The pitching torque from the engine onto the plane should have a counter pitch from the plane onto the engine, I'm wondering if this torque is applied through the center of mass of the airplane.
I do not understand the question. Torques are not applied "through" particular points. Forces are applied at points. Torques, not so. They are relative to a chosen axis rather than needing a point of application.

In the case of the engine bearings acting on a the shaft+blade assembly in a turbine, it is convenient to represent the net effect as:

1. A linear force applied at the shaft+blade center of mass plus...
2. A pure torque (or "couple") applied to the shaft+blade as a whole.

A "pure torque" is a set of forces that sum to zero but have a non-zero net torque. The simplest example would be a pair of equal and opposite forces applied at opposite ends of a rod. Hence the name "couple".

A "pure torque" has no associated point of application and needs no reference axis. Regardless of where you put the reference axis, the torque will be the same.

If you are concerned about the turbine's effect on the plane, you could have:

1. A linear force applied at a point on the plane's frame corresponding to the turbine's center of mass plus...
2. A pure torque (or "couple") applied to the aircraft as a whole.

Edit: Let me state this a bit more compactly.

The pitching torque arising from the precession associated with yaw is a pure torque. It does not matter where the engine is. It does not matter where the aircraft's center of mass is. It does not even matter where you choose to place a reference axis.

The pitching torque will be the same regardless.

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Chenkel
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