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Intermediate Axis Theorem

One of the interesting physics phenomena is intermediate axis theorem, if the rigid object has increasing moment of inertia for each orthogonal axis and you spin it axis of intermediate moment of inertia it oscillates.

Here is the video:

The analysis of system using rigid body equations:

Screen Shot 2017-10-24 at 12.51.55 AM

In that case M_1, M_2 and M_3 are zero. $latex I_1 < I_2 0, w_1\approx 0 $ and w_3\approx 0 , in order to solve higher order differential equation MATLAB can be used. Here is func.m file:

function dydt = func(t,y)
I1=1;
I2=2;
I3=3;
dydt = zeros(3,1);
dydt(1) = (I2-I3)*y(2)*y(3)/I1;
dydt(2) = (I3-I1)*y(1)*y(3)/I2;
dydt(3) = (I1-I2)*y(1)*y(2)/I3;

from command line or other .m file run the following lines.

clear all;
[t, y]=ode45(@func,[0:0.1:100],[.01 4 .01]);
plot(t,y(:,1),t,y(:,2),t,y(:, 3))

Numerical solution of differential equations shows w_2 is oscillating between -4 and 4. While changing rotation, very short amount of time w_1 and w_3 are non zero.
data1 -> w_1
data2 -> w_2
data3 -> w_3

intermediate_axis

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