Gyroscope, Gimbals Frame, lack of precession

[fog37]

Hello,

A simple gyroscope, which is nothing but a spinning top, will exhibit precessional motion if a) it spins and b) a nonzero gravitational torque is applied to it.

The tip of the gyroscope rotates while the tip rotates on itself. If the gyroscope is mounted on gimbals, which are rings, the tip of the gyroscope continues to point in the same direction (say north) as if the gravitational torques disappeared. Precession is not occurring thanks to the 3 gimbals, correct? I am not sure why though. Is it because, as the tip precesses the gimbals frame rotates the other way cancelling the precession rotation?

For example, a compass could be placed inside the gimbals frame and the compass would keep pointing in the same direction regardless of any movement of the gimbal frame itself. Compasses are magnetic and are negatively affected by metals around while gyroscopes are not.

Thanks!

 

[A.T.]

...gravitational torque ...

Uniform gravity doesn't exert any torque around the center of mass (gyro in free fall doesn't precess). Only the support can exert a torque, that leads to precession. The gimbal support is designed to provide support without exerting a torque around the center of mass, thus avoiding precession.

 

[fog37]

OH! How did I miss that! Thank you A.T.!

 

[fog37]

I read how ships would wreck using compasses and the advent of gyroscopes completely changed the game.

 

[wrobel]

Free body (Euler's top) can also move with precession.

 

[wrobel]

gyro in free fall doesn't precess

it does

 

[A.T.]

Uniform gravity doesn't exert any torque around the center of mass (gyro in free fall doesn't precess).

it does

In uniform gravity?

 

[wrobel]

In uniform gravity?

without any gravity

 

[fog37]

without any gravity

What force generates the torque that will lead to precession of gravity is not acting?

 

[etotheipi]

You can have torque-free precession!

 

[wrobel]

What force generates the torque that will lead to precession of gravity is not acting?

V. Arnold Mathematical Methods of Classical Mechanics
section: Euler's equations. Poinsot's description of the motion

 

[vanhees71]

The problem is that "precession" and "nutation" have different meanings in different textbooks. I'm always confused by the naming. I'm not sure, which is most common. I learned that for a free top (i.e., a rigid body free to rotate around its center of mass) the rotation of the figure axis around the constant angular-momentum is called "nutation", while the motion of the figure axis of a heavy top around the direction of the constant acceleration of gravity $\vec{g}$ is called "precession".

As I said, that nomenclature is not unique in the literature, and e.g., Wikipedia also calls the "nutation" of a torquefree top "precession".

 

[wrobel]

if the body has axial symmetry then we can introduce the Euler angles such that

here x0y0z0 is an inertial frame;
θ the angle of nutation
ψ the angle of precession

 

[A.T.]

As I said, that nomenclature is not unique in the literature, and e.g., Wikipedia also calls the "nutation" of a torquefree top "precession".

OK, based on the question, I assumed the OP is asking about torque-induced-precession, so that's what "precession" refers to in my post #2.

 

[fog37]

Yes, torque induced processions was my question.
Nutation, for what I know, is just a gentle nodding of the axis while it precesses.

so the gyroscope inside the gimbals frame does not precess because there is no torque producing precession. However, there is torque-free precession, as wrobel states. After some quick research, it seems that torque free precession is possible when the moment of inertia is time-varying...That would imply a redistribution of the mass of the object... If that does not happen, there is no torque-free precession.

 

[vanhees71]

Well, given the confusion, perhaps it's better to avoid the words precession and nutation but describe it simply as follows.

For the force-free symmetric top as seen from the space-fixed system (i.e., us as observers toying around with a gyro in a gimbal with the fixed point of rotation adjusted such that it coincides with center of mass of the spinning body) you have a space-fixed axis given by the angular momentum of the top. Around this axis the figure axis (the symmetry axis of the top) moves with constant angular velocity around the angular momentum. The angle between the figure axis and the angular-momentum axis stays constant. The same holds for the momentaneous angular-momentum axis, i.e., it moves with the same angular velocity around the angular-momentum axis as the figure axis and also makes a constant angle to the angular-momentum axis.

In the body-fixed frame both the angular-momentum axis and the momentaneous angular-velocity axis rotate with the same constant angular velocity around the constant figure axis (making constant angles with it).

Which of these "precessions" you call nutation and which precession seems to me a great mess in the textbook literature. I know both German and English/American textbooks, where the nomenclature changes with time and also with position in an unsystematic manner. There are indeed also sources, where they only call the nodding of the figure axis wrt. the gravitational acceleration $\vec{g}$ of the heavy symmetric top "nutation". That's obviously the convention in @wrobel 's #13.

What makes the things worse is that even the naming of the Euler angles are changing. I once used two textbooks in parallel, where $\psi$ and $\phi$ where mutually interchanged. The naming of $\vartheta$ seems to be more stable over the literature. There's only one way to lift the confusion: Just choose one convention of Euler angles and solve the equations for yourself. It's puzzling but also fun!

 

[mpresic3]

After some quick research, it seems that torque free precession is possible when the moment of inertia is time-varying...That would imply a redistribution of the mass of the object... If that does not happen, there is no torque-free precession.

You would have to be careful, here. The moment of inertia (tensor) in the body fixed frame characterizes the distribution of mass in the rigid body. In the free-falling (i.e. torque free; force free) gyro; which we assume has axial symmetry, precession may occur based on the initial conditions. Precession, (the steady movement in psi in the figure that wrobel supplied) occurs , for general initial conditions. You can convince yourself of this precession by solving the two coupled first order differential equations and the third is easy, as it involves two equal moments of inertia (in the body frame) due to axial symmetry.

The mass distribution in the body frame has not changed. The moment of inertia tensor in the body frame characterizes that. However if you look at the moment of inertia (tensor) in a space fixed frame, it is changing in time, because the coordinate system is time-varying.

 

[wrobel]

For the force-free symmetric top as seen from the space-fixed system (i.e., us as observers toying around with a gyro in a gimbal with the fixed point of rotation adjusted such that it coincides with center of mass of the spinning body) you have a space-fixed axis given by the angular momentum of the top. Around this axis the figure axis (the symmetry axis of the top) moves with constant angular velocity around the angular momentum. The angle between the figure axis and the angular-momentum axis stays constant.

exactly!

 

[Ganesh Mahadevan]

Hello,

A simple gyroscope, which is nothing but a spinning top, will exhibit precessional motion if a) it spins and b) a nonzero gravitational torque is applied to it.

The tip of the gyroscope rotates while the tip rotates on itself. If the gyroscope is mounted on gimbals, which are rings, the tip of the gyroscope continues to point in the same direction (say north) as if the gravitational torques disappeared. Precession is not occurring thanks to the 3 gimbals, correct? I am not sure why though. Is it because, as the tip precesses the gimbals frame rotates the other way cancelling the precession rotation?

For example, a compass could be placed inside the gimbals frame and the compass would keep pointing in the same direction regardless of any movement of the gimbal frame itself. Compasses are magnetic and are negatively affected by metals around while gyroscopes are not.

Thanks!

Gimbals ensure that a free gyroscope can continue to point at an arbitrary point in space regardless of the movements of the structure that houses the gyro. Remember, it is only a gyroscope at this stage and not a gyrocompass. A gyrocompass is a device very specific to orientation on a spherical planet. A gyrocompass is normally located on the surface of planet earth, and has to be made to be north seeking and north pointing, for it to becomo become a compass. It has to be forcibly made to point along the local meridian towards terrestrial North and not at an arbitrary point in space like sirius or alpha centauri.
Precession is a property used for this purpose. The gyrocompass rotor is deliberately made a bit bottom heavy, such that whenever its axis strays from the meridian or points away from the meridian or horizontality, a gravitational force acts on the rotor in a direction so as to bring it back in line using precession. This straying from the meridian is a continuous phenomenon because the gyeocompass structure is rotating all the time, with the earth. Dont mix gimbals with precession eithet as a cause or effect. A well engineered set of gimbals should not affect gyro behaviour at all.

 

[Ganesh Mahadevan]

Hello,

A simple gyroscope, which is nothing but a spinning top, will exhibit precessional motion if a) it spins and b) a nonzero gravitational torque is applied to it.

The tip of the gyroscope rotates while the tip rotates on itself. If the gyroscope is mounted on gimbals, which are rings, the tip of the gyroscope continues to point in the same direction (say north) as if the gravitational torques disappeared. Precession is not occurring thanks to the 3 gimbals, correct? I am not sure why though. Is it because, as the tip precesses the gimbals frame rotates the other way cancelling the precession rotation?

For example, a compass could be placed inside the gimbals frame and the compass would keep pointing in the same direction regardless of any movement of the gimbal frame itself. Compasses are magnetic and are negatively affected by metals around while gyroscopes are not.

Thanks!

Gimbals ensure that a free gyroscope can continue to point at an arbitrary point in space regardless of the movements of the structure that houses the gyro. It is l a gyroscope at this stage and not a gyrocompass. A gyrocompass is normally located on the surface of a rotating planet earth, and has to be made to be north seeking and north pointing, for it to becomo become a compass. It has to be forcibly made to point along the local meridian towards terrestrial North and not at an arbitrary point in space like sirius or alpha centauri.
Precession is a property used for this purpose. The gyrocompass rotor is deliberately made a bit bottom heavy, such that whenever its axis strays from the meridian or points away from the meridian or horizontality, a gravitational force acts on the rotor in a direction so as to bring it back in line using precession. This straying from the meridian is a continuous phenomenon because the gyeocompass structure is rotating all the time, with the earth. Dont mix gimbals with precession eithet as a cause or effect.