Intermediate axis theorem (Tennis racket theorem)

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The discussion revolves around the Intermediate Axis Theorem, particularly focusing on the Euler equations for a rigid body with different moments of inertia. The user is struggling to solve the equations for angular velocities, specifically how to express ω1 and ω2 to satisfy the derived equations. There is confusion regarding the signs of the terms in the equations, particularly with the definition of Ω2 being negative. Suggestions are made to redefine Ω2 to avoid imaginary quantities in the solutions. The conversation emphasizes the importance of correctly interpreting the parameters to derive meaningful solutions for the angular motion.
Lambda96
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Homework Statement
see screenshot
Relevant Equations
Euler equation
Hi,

unfortunately, I am not getting anywhere with task b

Bildschirmfoto 2022-12-17 um 20.39.23.png


In the lecture we had the special case that ##\vec{M}=0## , ##I_x=I_y=I , I \neq I_z## and ##\omega_z=const.##

Then the Euler equation looks like this.

$$I_x\dot{\omega_x}+\omega_y \omega_z(I_z-i_y)=0$$
$$I_y\dot{\omega_y}+\omega_z \omega_x(I_x-i_z)=0$$
$$I_z\dot{\omega_z}=0$$

With this, we then set up the following equations, where ##\Omega=\frac{I_z-I}{I}##.

$$\dot{\omega_x}+\Omega \omega_y=0$$
$$\dot{\omega_y}-\Omega \omega_x=0$$

The solution for ##\dot{\omega_x}## and ##\dot{\omega_y}## are then as follows

$$\dot{\omega_x}(t)=Acos{\Omega t + \alpha}$$
$$\dot{\omega_y}(t)=Asin{\Omega t + \alpha}$$

I then wanted to use the approximation ##\omega_1 \gg \omega_2## , ##\omega_1 \gg \omega_3## and ##\omega_3 \omega_2=0## to solve the Euler equation also according to the same recipe

The Euler equation looks like this:

$$I_1\dot{\omega_1}=0$$
$$I_2\dot{\omega_2}+\omega_1 \omega_3(I_1-I_3)=0$$
$$I_3\dot{\omega_3}+\omega_2 \omega_1(I_2-I_1)=0$$

After that I got the following equation

$$\dot{\omega}_2+\frac{(I_1-I_3)\omega_1}{I_2}\omega_3=0$$
$$\dot{\omega}_3-\frac{(I_2-I_1)\omega_1}{I_3}\omega_2=0$$

With ##\frac{(I_1-I_3)\omega_1}{I_2}=\Omega_1## and ##\frac{(I_2-I_1)\omega_1}{I_3}=\Omega_2##, the above equation is

$$\dot{\omega}_2+\Omega_1 \omega_3=0$$
$$\dot{\omega}_3- \Omega_2 \omega_2=0$$

Unfortunately, I'm not getting anywhere now, because I don't know what ##\omega_1## and ##\omega_2## have to look like to satisfy the two equations, because unfortunately I don't have ##\Omega##, but two different ones with ##\Omega_1## and ##\Omega_2##.
 
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Lambda96 said:
After that I got the following equation$$\dot{\omega}_2+\frac{(I_1-I_3)\omega_1}{I_2}\omega_3=0$$$$\dot{\omega}_3-\frac{(I_2-I_1)\omega_1}{I_3}\omega_2=0$$
Shouldn't the second equation have a plus sign for the second term on the left side?

Lambda96 said:
With ##\frac{(I_1-I_3)\omega_1}{I_2}=\Omega_1## and ##\frac{(I_2-I_1)\omega_1}{I_3}=\Omega_2##,
With these definitions, note that ##\Omega_2## is a negative number. It might be less confusing if you define ##\Omega_2 = \frac{(I_1-I_2)\omega_1}{I_3}## so that both ##\Omega_1## and ##\Omega_2## are positive.

Lambda96 said:
$$\dot{\omega}_2+\Omega_1 \omega_3=0$$
$$\dot{\omega}_3- \Omega_2 \omega_2=0$$

Unfortunately, I'm not getting anywhere now, because I don't know what ##\omega_1## and ##\omega_2## have to look like to satisfy the two equations, because unfortunately I don't have ##\Omega##, but two different ones with ##\Omega_1## and ##\Omega_2##.

Take the time derivative of the first equation and then use the second equation to substitute for ##\dot \omega_3##.
 
Thank you TSny for your help 👍

Regarding the minus sign, I had made a mistake, I actually meant that ##\Omega_2## is negative, since ##I_1>I_2## and thus becomes negative. But I'll leave this interpretation out of the derivation for now.

The equation ##\dot{\omega_2}+\Omega_1 \omega_3=0## derived in time is as follows, actually I would have to use the product rule, but since ##\dot{\omega_1}=0## I can leave this term out directly and get

$$\ddot{\omega_2}+\Omega_1 \dot{\omega_3}=0$$

The second equation then solved for ##\dot{\omega_3}## is ##\dot{\omega_3}=-\Omega_2 \omega_2##.

Then substituting back into the first equation gives

$$\ddot{\omega_2}-\Omega_1 \Omega_2 \omega_2=0$$

Solving this differential equation then gives:

$$\omega_2=c_1e^{\sqrt{\Omega_1 \Omega_2} t}+c_2e^{-\sqrt{\Omega_1 \Omega_2} t}$$
 
Lambda96 said:
The equation ##\dot{\omega_2}+\Omega_1 \omega_3=0## derived in time is as follows, actually I would have to use the product rule, but since ##\dot{\omega_1}=0## I can leave this term out directly and get

$$\ddot{\omega_2}+\Omega_1 \dot{\omega_3}=0$$
ok

Lambda96 said:
The second equation then solved for ##\dot{\omega_3}## is ##\dot{\omega_3}=-\Omega_2 \omega_2##.
ok. It appears to me that you are still defining ##\Omega_2## as ##\frac{(I_2-I_1)\omega_1}{I_3}##. So, ##\Omega_2## is a negative number. That's alright, but then you need to keep in mind that ##\sqrt{\Omega_1 \Omega_2}## is imaginary.

Lambda96 said:
Then substituting back into the first equation gives

$$\ddot{\omega_2}-\Omega_1 \Omega_2 \omega_2=0$$

Solving this differential equation then gives:

$$\omega_2=c_1e^{\sqrt{\Omega_1 \Omega_2} t}+c_2e^{-\sqrt{\Omega_1 \Omega_2} t}$$
ok. The exponents on ##e## are imaginary. I think it would be nicer to rewrite this without imaginary quantities. You could use Euler's formula. Or, you could go back and define ##\Omega_2## such that it is positive and rederive the differential equation for ##\ddot \omega_2##. Then, you don't run into imaginary quantities.
 
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