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.

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:

## Step-1

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**).

## Step-2

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

**I = ∫**r^{2}** dm………..eqn.1.1**

Where,

**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

## Step-3

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

**I =**_{ 0}^{50}∫x_{1}^{2} dm_{1} + _{50}^{100} ∫x_{2}^{2} dm_{2}** ………..eqn.1.2**

** **

Where,

**dm**_{1}** – **The mass of the small slice **pqrs**

**x**_{1}** – **The distance of the slice **pqrs **from the axis **AE.**

**dm**_{2}** – **The mass of the small slice **uvwx**

**x**_{2}** – **The distance of the slice **uvwx **from the axis **AE.**

## Step-4

Now, write “**dm**_{1}**"** in terms of **“dx**_{1}**". **And it will become:

**dm**_{1}** = volume * density **

** = 100 * 5 * dx**_{1}** * 1**

## Step-5

Similarly write “**dm**_{2}**"** in terms of **“dx**_{2}**"** like below:

**dm**_{2}** = {200 – (100/50)*x**_{2}**} * 5 * dx**_{2}** *1**

## Step-6

Now by putting the values of the **dm**_{1}** **and **dm**_{2} in **eqn1.2** we can get the value of the mass moment of inertia of the plate about the axis **AE** as:

**I = 500*[(x**_{1}**)**^{3}** / 3] **_{0}** **^{50}** + 500*[(x**_{2}^{3}** / 3) – (x**_{2}^{4}** / 4)] **_{50}^{100 }** **

= **78125000**

## Conclusion

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.