How can I accurately calculate the moment of inertia for random 3D shapes?

[power11110]

Hello,

I am trying to find moment of inertia of various random 3D shapes and I have few general questions.

I take very thin 'plates' of the shape, calculate moment of inertia of that area, multiply it by thickness of one 'plate unit', also I add a square distance it is from the origin multiplied by a mass of that unit plate. Since I need to find out moment of inertia around z axis and the shapes are symmetrical by x and y-axis I get the general formula like this:

I = ƩdI = 2*(dI_area * density * t + distance^2 * dm);

where dI_area is a moment of inertia of unit plate
t is a thickness of unit plate
distance is a distance from origin
dm = Area*t*density; where Area is area of thin plate (or unit plate);

I would like someone to comment on my actions because I do not obtain right results using this method.

Thank you in advance

 

[Simon Bridge]

In general, for random shapes, the method won't work.
Presumably the shapes have a lot of rotational symmetry?

If you have a solid of rotation about the x-axis (say) and you need the moment of inertia about the z-axis, then you'd need to divide the solid into cylinders or rings - and work out the moment of inertia of each element (ring or cylinder) about the z axis ... then add them up.

i.e. what is the moment of inertia of a (hollow) cylinder length L, radius r, and thickness dr (density $\rho$) oriented along the x-axis and centered on the z-axis, about the z axis? What is if is not centered on the z axis?

i.e. what is the moment of inertia about the z axis of a disk radius r centered on the x-axis situated between x and x+dx?

It may be easier to find the moment of inertia about he x and y axis, and use the perpendicular axis theorem - but not usually.