Moment of Inertia

Question

So I have had these questions confusing me for some time and tried googling but came up with long , hard-to-understand equations and notations and articles. So I would be glad if someone could clarify these things a bit more simpler.

I know (mass) moment of inertia = integrate(r^2) dm , where r is the perpendicular distance from the axis of rotation to the mass center of the elementary mass.

Product of inertia is the product of perpendicular distances to the same in two axis.

eg : Ixy, Ixz

What do we need to do to calculate principal moments of inertia. Does that mean finding Ixx, Iyy, Izz?

what do we need to do to find the direction of the principal axes? Does that mean finding the inertia matrix I,

( Ixx 0 0

I = 0 Iyy 0

0 0 Izz)

or?

Answer 1

For the case of a rigid body (such as a wooden top), the first step is to locate the center of mass (CM) of the object. This CM will be the origin of coordinates for the integrations below. There will be a mass density function ρ(x,y,z) such that ρ is zero outside of the object boundary. In many cases ρ will be a constant inside the object boundary ( for example the density of wood). The orientation of the x,y,z axes with respect to the structure of ρ is, in principle, arbitrary. However, if there is a symmetry axis (or axes), it is best to pick the z axis along the symmetry direction.

The next step is to compute the elements of the 3 x 3 moment of inertia tensor, I, as:

I_zz = ∫ ρ(x,y,x) (x² + y²) dx dy dz

I_xx = ∫ ρ(x,y,x) (z² + y²) dx dy dz

I_yy = ∫ ρ(x,y,x) (x² + z²) dx dy dz

I_xy = - ∫ ρ(x,y,x) (xy) dx dy dz = I_yx

I_xz = - ∫ ρ(x,y,x) (xz) dx dy dz = I_zx

Iyz _{= - ∫} ρ(x,y,x) (yz) dx dy dz = I_zy

These integrals are over the volume of the object. Evaluation of these integrals is very difficult if the object has no symmetry axes.

The next step is to note a general mathematical result. That is, for a symmetric tensor such as I, there will in general be a rotation matrix, Q, the brings this tensor into diagonal form. The equation for this is

Λ = Q I Q^T for some rotation matrix Q. Λ is diagonal matrix whose elements are the principle values of the moment of inertia tensor. The right hand side is the product of three 3x3 matrices. The ' T ' denotes transpose. Once Q is found, the elements of Λ are easy to work out. Finding Q is quite difficult in the general case. However, engineering codes, such a MATLAB, have built in routines for computing Q.

Hope this helps.