"Darwin123" escreveu na mensagem
...
On May 8, 9:03 pm, "El Enrrabadore-mor"
wrote:
"Darwin123" escreveu na
...

On May 8, 1:34 pm, "El Enrrabadore-mor"
wrote:
Do you think you know top/gyroscopes?
Do you know eigenvalues and eignevectors? The moment of inertia
is a tensor matrix. If you perform an eigenanalysis on the moment of
inertia, which is called diagonalization, you will get the the
symmetric axes of the object and the moment of inertia around each
body.
For a spherical body, the solution is three orthogonal axis with an
equal moment of inertia. However, this solution is degenerate. You can
chose any three orthogonal axis and it will be an axis of symmetry for
the sphere. I suspect that may be related to your problem. The
equations you are using are ill-conditioned due to the degeneracy of
the eigenvector solution. This is a math problem, not a physics
problem.

My equations don't have any trouble to predict the sphere
behaviour, for instant (the usual torque/precession for a
gyroscope moment I3w3):
T1 = I1 dw1/dt + I3 w3 w2 sin(theta) +
+ (I3-I2) w2^2 sin(theta) cos(theta)
Assuming that you have a "perfect sphere," you probably don't
have a "perfect torque." I mean, this formula seems to imply that only
the torque around the first axis is significant. There are no other
torques. You seem to be implying that:
T2=T3=0.

Well, the torque T3=0, theoretically. T3 is the torque around the
spinning axis, assumed to be zero. It is assumed zero friction
around gyro main bearings. Or else, I've a small DC motor
feed by batteries whose voltage/current can be controlled, so
that I can keep a constant angular speed w3 against friction.

Every text book I know considers torque T2 = 0.
T2 = 0 is the only theoretical approach known for gyroscopes.
In real world we have mechanical systems with friction and
torque T2 cannot be zero, ever. Only Gravity Probe B running
on magnetic fields can eventually be considered T2 = 0.
Such torque T2 will cause nutation and, for friction, it causes
a negative torque that causes a positive "falling" of the gyro.
That's why all gyros fall, not because the main gyroscopic
mass I3w3 is running slow with time.

What happens is that all torques T1, T2, T3 and all axes
I1, I2, I3 are orthogonal to each other. Therefore, they are
all independent from each other.
Due to said orthogonality, we go independent equations
for each of every torque. That's what Euler's equation of
motion and all text books say. That's logic too, and I agree.
That's a basic stone of the theory and a hope to be right.

The equation for T2, that causes the nutation w1, is much
more complicated and not covered on text books.
All text books consider that the gyroscope never falls,
hence T2 = 0, and the entire World lives happy with that.

The major challenge required was to define the potential
field V that gives rise to the nutation effect do to torque T:
F = - grad V
T = - grad V


Furthermore, I have no idea of how you determined T1.

T1 and T2 equations have been derived based on 3 methods:
1 - Newton: T = dL/dt
2 - Lagrange L = E - V
3 - Vectorial: T = w x L (cross product)


What if the direction of the torque is changing? Usually, the
assumption is that the torque is constant while the moment of inertia
matrix is constant. I am convinced (?) by your certainty (?) that you
know the moment of inertia. However, I am not sure you know where the
torque is coming from.

Torque changes all the time.
T1 = m g r sin(theta) (for gravity action = most usual)
T2 = user defined


So, I guess I could fortunately get out of that degeneracy
you've been talking about.
So, you agree that Euler's equations of motion don't
qualify for a sphere?

Plugging in numbers into an equation where the solution is
degenerate causes problems. When this happens to me, I go back to the
physics. I look for degeneracy, a place where there is too much
symmetry. In your case, I would look at the spherical body more
closely. Try to make a spherical top, or simulate a spherical top
using your hand. You may find a physical restriction that you weren't
aware of.

The best gyroscopes are spherical.
Gravity Probe B gyroscope is the most perfect sphere ever made.
Really? How is this gyroscope initially spun? It doesn't have an
axle. Not if its "the most perfect sphere ever made." I was trying to
figure out how an electromotive force could give it a spin. I figure
you must be running electric current through it, and placing it in an
magnetic field.

Yes, magnetic field are the source for Gravity Probe B.
No torques are applied over the gyroscope of Gravity Probe.
The aim of Gravity Probe B was to find if there was any possible
torque applied. That is, no precession/nutation should be measured
unless something external exists that causes precession/nutation.


Note: If you are using electromagnetic induction to spin the
sphere, the torque isn't just caused the mechanical force of your arm
on the axle. It also has a component due to the mechanical force of
the magnet doing the spinning.
I think you may have a weird torque, not a deviation from Euler's
equation. The torque probably depends on time and maybe is a function
of angular velocity. You found an empirical formula that sort of fits
on the assumption that only the mechanical force on the axle causes
torque. However, I'll bet you have a magnet somewhere around that is
spinning the sphere.

I've no problem with applied external torques.
Nevertheless, you comment is very important, indeed.
The axis 2 and T2 are not good to work with.
I'm working on the vertical FIXED axis z and a torque Tz and an
angular speed wz have been defined.
Tz = T2 sin(theta)
wz = w2
(simple trigonometry plus a fact seen and explained at least in
one text book - Harvard / Cambridge 2008).


You didn't say that your gyroscope is running down. So you are
providing a "constant" flow of energy somewhere. Is your instrument
drawing on electric current? If so, look to the wires. You may find a
wire stressed where you least suspect it.

Many thanks.
Can you point out any solutions besides Euler's equation
of motion? They look like this:
T1 = I1 dw1/dt + (I3-I2) W2 W3
T2 = I2 dw2/dt + (I1-I3) W3 W1
T3 = I3 dw3/dt + (I2-I1) W1 W2
No. I don't know of any. However, I know various transformations
on the Eulers equations. You haven't exhausted the Euler equation yet.

And what transformations on the Euler's equations are there?
Can you explain please?


I would suggest going into a deeper analysis of torque. List all
sources of external torque, not just the torque the gyroscope is
resisting. Write a function for T1, T2, and T3 as a function of time
and angular velocity. I don't know your apparatus. However, I really
doubt you fully understand the source of torque.
What if the torque in your apparatus was some matrix times the
angular velocity vector. What matrix would give your empirical result?
I know this would be a fit, not physics. However, your spin inducing
apparatus may be producing a strange torque for reasons that have
nothing to do with Eulers equations.
T1(w1,w2,w3), T2(w1,w2,w3), T3(w1,w2,w3) ?