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Euler's moment equations (Definition)

Euler's Moment Equations in terms of the principle axes is given by

$\displaystyle M_x = I_x \dot{\omega_x} + (I_z - I_y) \omega_y \omega_z $

$\displaystyle M_y = I_y \dot{\omega_y} + (I_x - I_z) \omega_x \omega_z $

$\displaystyle M_z = I_z \dot{\omega_z} + (I_y - I_x) \omega_x \omega_y $

In order to derive these equations, we start with the angular momentum of a rigid body

$\displaystyle \vec{H_B} = I \omega = \left [ \begin{array}{c c c} I_{xx} & -I_{... ...left [\begin{array}{c} \omega_x \ \omega_y \ \omega_z \end{array} \right ] $

Since the vector is in the body frame and we want the Moment in an inertial frame we need to use the transport theorem since our body is in a non-inertial reference frame to express the derivative of the angular momentum vector in this frame. So the Moment is given by

$\displaystyle \vec{M} = \dot{\vec{H_I}} = \dot{\vec{H_B}} + \vec{\omega} \times \vec{H_B} $

Since we are assuming the inertia tensor is expressed using the principal axes of the body the Products of Inertia are zero

$\displaystyle I_{yx} = I_{xy} = I_{xz} = I_{zx} = I_{zy} = I_{yz} = 0 $

and using the shorter notation

$\displaystyle I_{xx} = I_x $

$\displaystyle I_{yy} = I_y $

$\displaystyle I_{zz} = I_z $

Also since the moments of inertia are constant, when we take the derivative of the Inertia Tenser it is zero, so

$\displaystyle \dot{\vec{H_B}} = \left [ \begin{array}{c c c} I_x & 0 & 0 \ 0 ... ...left [\begin{array}{c} \omega_x \ \omega_y \ \omega_z \end{array} \right ] $

Carrying out the matrix multiplication

$\displaystyle \dot{\vec{H_B}} = \left [ \begin{array}{c} I_x \dot{\omega_x} \\ ... ...{array}{c} I_x \omega_x \ I_y \omega_y \ I_z \omega_z \end{array} \right ] $

after evaluating the cross product, we are left with adding the vectors

$\displaystyle \dot{\vec{H_B}} = \left [ \begin{array}{c} I_x \dot{\omega_x} \\ ... ..._x \omega_z (I_x - I_z) \ \omega_x \omega_y (I_y - I_x) \end{array} \right ] $

Once we add these vectors we are left with Euler's Moment Equations

$\displaystyle M_x = I_x \dot{\omega_x} + (I_z - I_y) \omega_y \omega_z $

$\displaystyle M_y = I_y \dot{\omega_y} + (I_x - I_z) \omega_x \omega_z $

$\displaystyle M_z = I_z \dot{\omega_z} + (I_y - I_x) \omega_x \omega_y $



"Euler's moment equations" is owned by bloftin.

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Cross-references: cross product, matrix multiplication, moments of inertia, inertia tensor, reference frame, theorem, vector, rigid body, angular momentum
There is 1 reference to this object.

This is version 4 of Euler's moment equations, born on 2005-03-10, modified 2006-03-28.
Object id is 36, canonical name is EulersMomentEquations.
Accessed 1749 times total.

Classification:
Physics Classification: 45.40.-f (Dynamics and kinematics of rigid bodies)
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