Resistance and Precession Torque of a Gyroscope

[dishwasher95]

Hi,

Say there's a gyroscope with moments of inertia I_x, I_y and I_z spinning around a vertical z-axis (see attached illustration) with a given angular velocity ω_z. Notice that the gyroscope is floating in space as in that there's no gravity acting on the gyroscope.

Now I apply a torque τ_x around the x-axis.

The torque will introduce angular momentum L_x around the x-axis.

What I want to know is, how can I determine the resisting torque τ_res,x that resists the motion of the applied torque τ_x?

And what is the size of the torque τ_y that will occur due to the applied torque τ_x?

I'm look for an analytical approach to relate the applied torque τ_x to τ_y and τ_res,x.

I feel like I've searched the whole internet for a clear explanation but failed to find one.

If anyone would be willing to go through the theory step-by-step or just show me some literature that does I'd appreciate it immensely!

Attached is also some formulas I found online but I would love a derivation.

Thanks in advance!

 

[A.T.]

What I want to know is, how can I determine the resisting torque τ_res,x that resists the motion of the applied torque τ_x?

What to you mean by "resist the motion"? If you apply τ_x to the gyroscope, the gyroscope applies -τ_x to you.

 

[dishwasher95]

I'm not really sure how to describe the 'resisting torque'.

I'm not thinking about Newton's 3rd law.

I feel that whenever I try to apply a torque to my spinning gyroscope, it tries to resist the motion around the applied torque's axis. But I'm not 100% that this is what is happening. Maybe it just 'feels' as if there's a resisting torque because the gyroscope starts to rotate around the y-axis with the induced τ_y?

So if there is no τ_res,x (of course there's the equal but opposite reaction -τ_x that you mentioned) do you have any idea what the 'Resistance Torque' formula in the attached table is?

I still want to know how to relate τ_x to τ_y though.

Thanks!

 

[A.T.]

do you have any idea what the 'Resistance Torque' formula in the attached table is?

Maybe you should post a link to the source.

 

[dishwasher95]

Sure thing! Here you go:
https://scialert.net/fulltextmobile/?doi=ajsr.2017.380.386
Here's another article talking about the same formulas:
Gyroscope Mystery is Solved

 

[A.T.]

Sure thing! Here you go:
https://scialert.net/fulltextmobile/?doi=ajsr.2017.380.386
Here's another article talking about the same formulas:
Gyroscope Mystery is Solved

Both is from the same author. Neither shows any derivation, just the formulas. The torque predictions disagree with the standard approach based on on angular momentum conservation (2nd link, Eq 6 vs Eq 7). The explanation of this disagreement based on centrifugal and Coriolis forces doesn't make much sense to me.

 

[dishwasher95]

I'm glad I'm not the only one who didn't understand it - thought I was missing something.

But let's forget about the articles, τ_res,x and the formulas from the table.

I'm interested in finding a relation between τ_x and τ_y.

I found this website (see the bottom 'Mathematical Discussion'):
Gyroscope physics

It seems that the 'Mathematical Discussion' derives τ_y (which in the article is just τ) from ω_x (ω_s - the swivelling rate in the article).

I think this might be what I'm looking for - what do you think @A.T. ?

I have some trouble understanding how the 'tendency to pull ahead/lag behind overall swiveling' (F = -2mω_sv_r) is supposed to be understood. I think I need a more intuitive explanation of what is happening.

 

[A.T.]

I'm interested in finding a relation between τ_x and τ_y.

If you apply a torque around x, then τ_y = 0.

I think this might be what I'm looking for - what do you think @A.T. ?

I don't know what you are looking for. Here is how the applied torque relates to the procession rate:
https://en.wikipedia.org/wiki/Precession#Classical_(Newtonian)

I think I need a more intuitive explanation of what is happening.

For intuition it might help to look at it in terms of linear motion:

 

[dishwasher95]

I must apologise for any confusion. Let me try again:

The wikipedia article that you linked was very informative. I'll use its description to explain my problem.

Please read the text in the attached screenshot.

After you've read the text, notice the red box in the bottom with the marked word 'torque'. THIS torque (as it is described in the article) is the one I'm looking for. This torque is my τ_y. The pitching torque.

The article states that the pitching torque arises from the Coriolis Force (marked with blue squares in the screenshot).

The only question I have now (after newly gained information) is how the Coriolis Force can be calculated. From the article that I posted before:

'Gyroscope Physics' (at this moment the website is down?)

- the Coriolis Force is derived to be:

F = -2 * m * ω_s * v_r

- where m is the mass of a point on the rotating disc of the gyro
- ω_s is the spinning angular velocity
- v_r is the radial velocity

I understand what the Coriolis Effect is, but I have trouble understanding how the Coriolis Force is derived.

So here's my (updated) question/wish:

For someone to show me an intuitive and relatively simple derivation of the Coriolis Force F = -2 * m * ω_s * v_r

Thanks!

 

[jbriggs444]

I understand what the Coriolis Effect is, but I have trouble understanding how the Coriolis Force is derived.

Let us verify that we have common ground for discussion.

The Coriolis effect does not arise from looking at rotating objects from an inertial point of view. It arises from adopting a frame of reference that is itself rotating. This is often convenient because it means that we can look at a rotating object as if it were not rotating.

Let us bring that Wikipedia article in as text rather than as an attached image...

To distinguish between the two horizontal axes, rotation around the wheel hub will be called spinning, and rotation around the gimbal axis will be called pitching. Rotation around the vertical pivot axis is called rotation.

This is a nice choice of terminology.

First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a torque around the gimbal axis.

In order to invoke Coriolis, we need a rotating frame of reference. So we pick one which is locked to the uniform rotation of the device around the vertical axis. In this frame of reference the device is no longer rotating. It is just sitting there and spinning in place.

But this is a rotating frame of reference. In a rotating frame of reference, angular momentum is not necessarily conserved. We can have a changing angular momentum with no external torque. Or we can have constant angular momentum with a non-zero net external torque. The claim is that the latter is the case here -- we have a wheel that is rotating in place but only so long as a net external torque is supplied.

The task is to calculate the required external torque. Coriolis allows it to be calculated. It turns out to be a pitching torque.

 

[A.T.]

- the Coriolis Force is derived to be:

F = -2 * m * ωs * vr

- where m is the mass of a point on the rotating disc of the gyro
- ωs is the spinning angular velocity
- vr is the radial velocity

Note that ωs is the angular velocity of your reference frame (around the vertical pivot axis), not of the rotating disc. And vr is the velocity component perpendicular to the vertical pivot axis. The Coriolis force and its lever arm around the pitch axis are not constant along the disc, so to get the total Coriolis torque you would have to use integration.

 

[vanhees71]

I'd rather use a good textbook on classical mechanics than some strange online encyclopedia quoted in #5, which cannot even be downloaded in readable form as a pdf! Contrary to what's claimed in non-relativistic mechanics the theory of a top is a fully understood topic of mathematical physics. It goes back as early as to Euler's equations of the spinning top. It's not an easy topic though, but learning about it teaches you a lot about the rotation group, which is of great value for all further studies in physics. A good source is

A. Sommerfeld, Lectures on theoretical physics, vol. 1.