How to Calculate Mass Moment of Inertia of Irregular Shapes – Calculation Steps and Example of Mass Moment of Inertia

The mass moment of inertia is frequently used for mechanical design calculations of rotational bodies. Most of the time you will deal with regular geometries (like cylinders, spheres, etc.) for which mass moment of inertia values can be calculated using standard tables. For the irregular geometries, standard formulas are not available and you have to actually start from the basics. The example below will give you an idea of how to proceed for moment of inertia calculations of any irregular shapes.

Mass Moment of Inertia Example

Say, you need to calculate the mass moment of inertia about the side AE of the plate ABCDE . You will see this type of plate in a flat plate heat exchanger. The thickness of the plate is 5 and density is 1. You have to proceed as below:


Divide the whole area to some similar kinds of area. In our case we have divided the area (ABCDE) to one rectangle (ABDE) and a triangle (BCD).


Next, you have to use the mass moment of inertia equation:

I = ∫r2 dm………..eqn.1.1


dm – The mass of a very small slice, which is parallel to the axis about which the mass moment of inertia to be calculated

r – The distance of dm from the axis


But, since we have divided the whole area to two different areas, so, the eqn1.1 will be modified as:

I = 0 50∫x12 dm1 + 50 100 ∫x22 dm2 ………..eqn.1.2


dm1The mass of the small slice pqrs

x1The distance of the slice pqrs from the axis AE.

dm2The mass of the small slice uvwx

x2The distance of the slice uvwx from the axis AE.


Now, write “dm1 in terms of “dx1”. And it will become:

dm1 = volume * density

= 100 * 5 * dx1 * 1


Similarly write “dm2 in terms of “dx2 like below:

dm2 = {200 – (100/50)*x2} * 5 * dx2 *1


Now by putting the values of the dm1 and dm2 in eqn1.2 we can get the value of the mass moment of inertia of the plate about the axis AE as:

I = 500*[(x1)3 / 3] 0 50 + 500*[(x23 / 3) – (x24 / 4)] 50 100

= 78125000


In practical mechanical design calculations you may have to calculate mass moment of inertia for the irregular geometry. In that case you can either follow the basic calculation procedure as shown in this article or you can use CAD packages like ProE or AutoCAD. Or you can use both in order to ensure the accuracy of your calculation.

Related Readings

Calculate Area Moment of Inertia of Irregular Sections in Five Steps: This article will discuss the procedure to calculate area moment of inertia of different irregular cross sections.