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PRINCIPAL AXES

A principal axis is an axis where the products of inertia vanish. Thus, the inertia matrix becomes purely diagonal. In the case of a three dimensional rigid body with the principal axes defined by x*,y* and z* and origin fixed to the mass center of the body, the products of inertia would be I_x^*y^* = I_y^*z^* = I_x^*z^* = 0. The resulting inertia matrix at the principal axes of bending would then be:

I = ⎡⎢⎢⎢⎢⎢⎣ I_x^*x^* 0 0 0 I_y^*y^* 0 0 0 I_z^*z^* ⎤⎥⎥⎥⎥⎥⎦

where I_x^*x^*,I_x^*x^*, I_x^*x^* are the principal moments of inertia.

USE OF PRINCIPAL AXES

Calculations done with respect to the principal axes of a body are greatly simplified compared to the inertial fixed frame of reference, in which the inertia matrix is fully populated rather than a diagonal matrix, as seen above. This diagonal matrix decouples the set of equations for angular momentum, reduces the number of terms considered, and makes calculation of the angular momentum much simpler.

CALCULATING THE PRINCIPAL AXES

To find the principal axes, starting with the inertia properties calculated with respect to the inertial frame, defined by x, y, and z, the the definition of angular momentum,

(1) H_c = (I^c_xxω_x + I^c_xyω_y + I^c_xzω_z)i + (I^c_xyω_x + I^c_yyω_y + I^c_yzω_z)j + (I^c_xzω_x + I^c_yzω_y + I^c_zzω_z)k

is used. The angular velocity ω can be written as a function of the directional cosines as ω = ωη_xi + ωη_yj + ωη_zk

(2) (I^c_xxω_x + I^c_xyω_y + I^c_xzω_z)i = Iωη_x, η_x = (ω_x)/(ω) (I^c_xyω_x + I^c_yyω_y + I^c_yzω_z)j = Iωη_y, η_y = (ω_y)/(ω) (I^c_xzω_x + I^c_yzω_y + I^c_zzω_z)k = Iωη_z, η_z = (ω_z)/(ω)

Simplifying these expressions by substituting for η:

(3) (I^c_xx − I)η_x + I^c_xyη_y + I^c_xzη_z = 0 I^c_xyη_x + (I^c_yy − I)η_y + I^c_yzη_z = 0 I^c_xzη_x + I^c_yzη_y + (I^c_zz − I)η_z = 0

This system is placed into matrix form, and the principal moments of inertia are calculated. Once the principal moment moments of inertia values are known, they can be replaced into Equation ↓ to find η = (η_x, η_y, η_z).

The principal axes are orthogonal to each other, meaning that the dot product of the unit vectors,e^*_x, e^*_y and e^*_z, equal zero unless it is a dot product with respect to itself. Thus we can use the definition of direction cosines to help understand the directions:

e^*_x⋅e^*_y = e^*_y⋅e^*_z = e^*_z⋅e^*_x = 0 and e^*_x⋅e^*_x = e^*_y⋅e^*_y = e^*_z⋅e^*_z = 1

So that:

(4) (I_x − I)η_z = 0 where I = I₁ or η_x = 0 Similarly, (I_y − I)η_y = 0 and (I_z − I)η_z = 0.

However, this assumes that the principal moments of inertia, I_x, I_y, and I_z are all different. If two of then are the same, they will coincide with the symmetry axis of the body.