Motion of a Rigid Body, The Inertia Tensor & Euler's Equations (Part 1)

In classical mechanics courses rotations of a rigid body often occupy a good deal or course work.  The motion of a rigid body in space is determined by the two equations shown below:

1)  

2) 

Where: 

 and:  

  is the moment of inertia tensor.

Here, F is the total force on the body and N is the total torque about a suitable central point, 0.  The velocity about the center of mass is v, while:

  is the moment of inertia tensor and w  is the angular velocity about the point 0. For an unconstrained object moving in space, 0 is taken to be the center of mass.  Consider now two coordinate systems: One fixed to an object to allow calculation of I for the object, i.e.

The other (2nd system) primed for object for changes in orientation.

The difficulty that I changes as the body rotates can be avoided by referring eqn. (2) to a set of axes fixed in the body, which then yields:

or:   =

   

Since I is constant relative to the body axes. With proper choice of coordinate system we can write:

 =  

Þ  =

I₁₁ w₁  e1^ ₊   I₂₂ w₂  e2^  ₊  I₃₃ w₃  e3^ 

And: 

  =   

 e1^ w_yw_{z (}I_{33  -}  I_{22 )  +}  e2^ w_xw_{z (}I_{11  -}  I_{33 )  +} e3^ w_yw_x(I_{22  -}  I_11 )

Where:

_Nx =   w_yw_{z (}I_{33  -}  I_{22 )}   I₁₁ w_1'

_{Ny =   wywz (I11  -  I33 )   I22 w2'}

_{Nz =   wywz (I11  -  I33 )   I22 w2'}

for the rate of change of kinetic energy:

 

Þ 

  

Or:

Finally, for comparison, we examine the Euler's equations for the motion of a rigid body, assuming:

a) No torques,

b)  w₃   >>  w_2, w₁     and  w₃  = const.

Then we obtain 2 coupled differential equations:

I₁ w_{1'  +   (}I_{3  -}  I_{2 )} w₃w_{2  = 0}   

I₂ w_{2'   +   (}I_{1  -}  I_{3 )}w₁w_{3   = 0}   

_{-----------------}

Next:   I consider solutions for the specific differential equations for w_2, w₁