Velocity, acceleration and forces for point in spinning rigid body

[czarodziej_snow]

Hi I would like to present my new simulation rotated rigid ball.
Brief introduction to the Dzanibek Effect



When I learned to simulate the effect and study the secrets of vector relationships with no problem to finde speed vectors for points.


Next step is finde acceleration. It is a=(v1-v0)/dt. To spread this vector I set the angle n of inclination of both vectors and line acceleration it is al=cosn*a. Centrifugal acceleration it is ad=a-al.


Now it is easy to determine the force acting on the point. It is F=am. Now I set the angle m between force vector and main axis. Central force is Fc=cosm*F. Remaining component is forces creating a moment of force.


Yellow- Velocity
Red, lighbrown - acceleration and forces
Green - line acceleration
Light blue - centrifugal acceleration
Pink - Central force
Orange – forces give moment of force.

My simulation is not perfect and gives errors of the order of 0.01.
Sum of moments of forces give result max 0.01 but most likely this is a method error.
https://m.salon24.pl/9236008c61194a8bfd80bc37832b8ae7,750,0,0,0.jpg

[studiot]

In a perfect world rotation about any of the three Euler axes is stable, an governed by three non linear coupled differential equations.

[math]A\frac{{d{\omega _1}}}{{dt}} = \left( {B - C} \right){\omega _2}{\omega _3}[/math]

[math]B\frac{{d{\omega _2}}}{{dt}} = \left( {C - A} \right){\omega _3}{\omega _1}[/math]

[math]C\frac{{d{\omega _3}}}{{dt}} = \left( {A - B} \right){\omega _1}{\omega _2}[/math]

Where constants A < B < C

Because of the couping the slightest perturbation of  [math]{\omega _2}[/math] will lead to regenerative instability of rotation about the intermediate axis as your vids show.

The equation set can be linearised and the linearised equations set still exhibits this instability.

 

Would you like to explain how your animations improve on this?

Also the above calculations show what happens if you introduce the perturbation into either [math]{\omega _1}[/math]  or [math]{\omega _3}[/math]

Can you animations show this?

 

Edited October 23, 2017 by studiot

[czarodziej_snow]

21 hours ago, studiot said:

In a perfect world rotation about any of the three Euler axes is stable, an governed by three non linear coupled differential equations.

(B−C)ω2ω3

Bdω2dt=(C−A)ω3ω1

Cdω3dt=(A−B)ω1ω2

Where constants A < B < C

Because of the couping the slightest perturbation of  ω2 will lead to regenerative instability of rotation about the intermediate axis as your vids show.

The equation set can be linearised and the linearised equations set still exhibits this instability.

My animation works on Euler's equations. In non-inertial frame I finded angular velocity vector and I rotate this frame in interial frame.  In this way, I finded the coordinates of the temporary axis of rotation.  And I repeat this step a lot time.  One step is a very small error but  Thousands of steps cause an error to occur. I know how to check those moments force, but I must find time.  I'm a very busy person.

 

21 hours ago, studiot said:

Would you like to explain how your animations improve on this?

Also the above calculations show what happens if you introduce the perturbation into either ω1   or ω3

Can you animations show this?

What improve?  This animation shows simple properties this vectors. a=(v1-v0)/dt and F=am.  That's all.

Sorry but I do not understand the rest.

 

[studiot]

25 minutes ago, czarodziej_snow said:

My animation works on Euler's equations. In non-inertial frame I finded angular velocity vector and I rotate this frame in interial frame.  In this way, I finded the coordinates of the temporary axis of rotation.  And I repeat this step a lot time.  One step is a very small error but  Thousands of steps cause an error to occur. I know how to check those moments force, but I must find time.  I'm a very busy person.

 

What improve?  This animation shows simple properties this vectors. a=(v1-v0)/dt and F=am.  That's all.

Sorry but I do not understand the rest.

 

Perhaps I should have said that omega 1, omega2 and omega 3 are the angular velocities about the 3 Euler axes.

A, B and C are the positive constant moments of inertia about these axes.

Without perturbation the system is stable.

If we introduce even the slightest perturbation to omega 2 the instability you are modelling results.

If we introduce that perturbation to omega 1 or 3 then something different happens.

I was asking if your animation can show what happens then?

What equations are you solving to generate the vectors?

Edited October 24, 2017 by studiot

[czarodziej_snow]

18 minutes ago, studiot said:

Perhaps I should have said that omega 1, omega2 and omega 3 are the angular velocities about the 3 Euler axes.

A, B and C are the positive constant moments of inertia about these axes.

Without perturbation the system is stable.

If we introduce even the slightest perturbation to omega 2 the instability you are modelling results.

If we introduce that perturbation to omega 1 or 3 then something different happens.

I was asking if your animation can show what happens then?

What equations are you solving to generate the vectors?

Euler's equations.

I_x(dω_x /dt) + (I_z - I_y)ω_zω_y   = 0

I_y(dω_y /dt)+ (I_x - I_z)ω_zω_x = 0

I_z(dω_z /dt)+ (I_y - I_x)ω_xω_y   = 0

Omega must have minimum two no zero elements. If that hapen equations show you dω.

Haw this vectors work in nointeria frame show difrent my animation.