Details of the rotation of the rigid body

[czarodziej_snow]

   I am interested in the mechanics of rigid body rotation. I entered the subject very deeply, and from my calculations and conclusions it follows that, not everything is exactly that haw it is written in books and few details have hidden in present science . Of course, these are my conclusions and should be verified but no one did not get interested, and physicists are afraid of the subject, like the devil of holy water.

   I thought I might find people here with whom I could discuss the issue, but because the questions for I am looking answers can not be found in books, they only hear excuses. Unfortunately, the local community is just waiting for easy questions which answers can be found in books, and he has not been interested in searching for answers that have not yet been found.

   To better introduce you to the topic I prepared simulations showing rigid body rotation based on Poisnot solutions using the ellipsoid concept. Solutions in the sources which I have, it was very chaotically written but I finally managed to understand and sort it.

What is matter?

   During the rotation of the rigid body, the moment of inertia is the most important thing. Each rigid body has three constant moments of inertia which is permanently attached to the solid frame. It means that in the body configuration, the moment of inertia is constant over time. Poinsot presented it in the form of an ellipsoid of inertia in the body configuration. It is interesting that, if we have different main moments of inertia and the body rotates near the middle moment of inertia, comes to the Dzibibekov effect that is, for spontaneous change of the instantaneous axis of rotation. There is a spontaneous change of the angular velocity and changes in the moment of inertia. These changes are not chaotic and predicted that angular velocity vector moves at the intersection of ellipsoid of angular momentum and energy ellipsoid. That's was written but how to understand ellipsoid of angular momentum when the angular momentum is constant over time?

My solution is quite simple. Because angular momentum is constans he creates a sphere.

L²=I_x²ω_x²+ I_y²ω_y²+ I_z²ω_z² =L_x²+ L_y²+L²_z

L=constans

 

Energy is also constant over time and if we used it directly, we would get another sphere. Energy is

E=Iω²/2

Angular momentum is

L=Iω

we can now write energy as

E=L²/2I

It gives us an ellipsoid 2EI

L²=2EI_x+2EI_y+2EI_z

 

Angular momentum is constans in the inertial frame, but is not must be constans relative to the rigid body. This sphere of angular momentum and ellipsoid 2EI are solid relative to the rigid body, so they will rotate with this body in the inertial system. Angular momentum vector is constans in inertial and always it's at the intersection this sphere and the ellipsoid. In rigid body frame this sphere and the ellipsoid they do not move but the angular momentum vector moves along their intersection.

Analogously, we can determine the ellipsoid for the angular velocity vector

Ω²=2E/I_x+2E/I_y+2E/I_z

Ω²=(L/I_x)²+(L/I_y)²+(L/I_z)²
Angular velocity vector moves along their intersection these ellipsoids. Everything is showing my animation.

https://youtu.be/p8pwQ39Tx9A

 

The rotation of the rigid body describes Euler's equations

I_x(ω_x/dt)= (I_y - I_z)ω_zω_y

I_y(ω_y/dt)= (I_z - I_x)ω_zω_x

I_z(ω_z/dt)= (I_x - I_y)ω_xω_y

   I do not know why no one did not see of the relations that resulted from them, ω/dt is definitely the vector of angular accelerations ɛ and Iɛ it is a moment of force. Important, I is the tensor of the moment of inertia, which very complicates calculations but I have pass through it and now I can use it effectively in simulations.

   There is a superstition that isolated bodies can not have inner moments of forces and internal angular accelerations, but Euler's equations deny it. I have heard a lot of opinion that such a moment of froce would have to be incompatible with the laws of physics, but I examined this moment of force and dont found nothing in it that would contradict any of the laws of Physics. These opinions is result from ignorance, lack of scientific approach and intuitive estimates, dont have nothing to do with the truth.

I have proof though it is not yet complete.

Conservation of linear angular momentum

dL/dt=0                 (1)

Angular momentum is

L= Iω                  (2)

use (1) and (2)

d(Iω)/dt= ω(dI/dt) + Iɛ =0     (3)

moment of force is

M= Iɛ                     (4)

We know that during the rigit body rotation mechanics, the moment of inertia changes over time I showed it in my simulation above . Changing (3)

dI/dt=-M/ω              (5)

we have a pattern for the inner moment of strength

M=-ω(dI/dt)              (6)

we count the derivative with dI/dt

M=-ω(dmr^2)/dt             (7)

the mass is constant, so

M=-ωm(2r(dr/dt))=-2ωmrv         (8)

momentum is

p=mv                  (9)

We have now

M=-2ωrp        (10)

Second law of Kepler

dA/dt=r(dr/dt)/2= (r x v) /2 = (r x p)/2m             (11)

S=dA/dt= constans                             (12)

Angular momentum is

L = r x p = r x mv                         (13)

Use (11) and (12)

r x v = 2S                                 (14)

Angular momentum is

L=m(r x v)=m2S=2mvr(sina)             (15)

a - angle between r,v

Use (15) to (8)

M=-ωL              (16)

and I found my guess

M=-ω x L                 (17)

Using the equation (17) I can effectively simulate the mechanics of rotation. At the end I sending the code in Vpython. How does this moment of force look shows my simulation.

https://www.youtube.com/edit?o=U&video_id=8OnWhW1-15s

It is only a fragment of my conclusions, there are still some details that still elude me but I'm on the best way to find them. If anyone would like to help or cooperate, I am open to any suggestions. You will find more simulations and details in simulations on my Youtube profile.

[Bender]

On 5/5/2018 at 7:04 PM, czarodziej_snow said:

d(Iω)/dt= ω(dI/dt) + Iɛ =0     (3)

moment of force is

M= Iɛ                     (4)

We know that during the rigit body rotation mechanics, the moment of inertia changes over time I showed it in my simulation above . Changing (3)

dI/dt=-M/ω              (5)

I didn't go through all the details, but you made an error in this step. On the left hand side, you assume omega is constant, which makes the right hand side 0. 

[czarodziej_snow]

49 minutes ago, Bender said:

I didn't go through all the details, but you made an error in this step. On the left hand side, you assume omega is constant, which makes the right hand side 0. 

which exactly step?

During the mechanics of rotation of the rigid body  angular velocity is constant only in three special cases only when it rotates exactly around one of the main axes of the moment of inertia in other cases, the vector is not stable in the body frame and inertial frame.  It shows my animation

 

another simulation shows the same

There was an error in my Euler patterns, I will try to fix it quickly

How to edit a note?

correction of Euler's equations

I_x(dω_x/dt)= (I_y - I_z)ω_zω_y

I_y(dω_y/dt)= (I_z - I_x)ω_zω_x

I_z(dω_z/dt)= (I_x - I_y)ω_xω_y

 

[Bender]

Never mind, I misread. It is getting late.

So in summary : what's new? The simulation looks nice, but you aren't exactly the first to make one of this well known and well documented effect.

[czarodziej_snow]

22 hours ago, Bender said:

Never mind, I misread. It is getting late.

So in summary : what's new? The simulation looks nice, but you aren't exactly the first to make one of this well known and well documented effect.

 

My simulations show a lot more of details which you will not see elsewhere, and angular acceleration or moment of force is a completely new approach to this issue