Isaac Newton defines inertia as the tendency of objects in motion to stay in motion, and objects at rest to stay at rest, unless an outside force changes its speed or direction. However, if you’ve been in the realm of mechanical design for any length of time, you’ll likely have heard the term “moment of inertia”. But what does that even mean, and why is it so important? Read on to find out!
Moment of Inertia
In short, mass moment of inertia, or moment of inertia for short, is a measure of how resistant an object is to changes in rotational motion. The term was coined by Leonhard Euler and is used in his second law of motion. Its units are ML², and it is defined using the following formula for all point masses that make up the object, where m is the mass, and r is the distance from the axis of rotation:
It is also defined as the ratio of the angular momentum of an object to its angular velocity (kind of like a rotational mass):
Shown below is a table of the moment of inertia for different shapes and rotational axes:
This concept is used when designing rotating objects such as wheels and combat aircraft. In athletics, figure skaters reduce their moment of inertia by drawing their arms in, allowing them to spin faster. Moment of inertia is also why tightrope walkers carry long poles – it increases their moment of inertia, making them more stable.
Another property of moment of inertia is that is it additive; the overall moment of inertia of a composite system is the sum of the moments of inertia of its parts. As an example, let’s take an annular cylinder. An annular cylinder is simply one small cylinder subtracted from a larger cylinder. If we calculate the moments of inertia of each cylinder, we can subtract the moment of inertia of the small cylinder from the moment of inertia of the large cylinder to get the moment of inertia of the annular cylinder. In this case, we’ll assume a length and density of 1 so that cross-sectional area is equivalent to mass.
Since in this example, we used area in place of mass, that begs the question…can we define moment of inertia based on area? Leading to…
What About Moment of Area?
This is where the term “moment of inertia” gets confusing! The “area moment of inertia” is similar to the mass moment of inertia, but for a cross-sectional area instead of a whole object with mass. This is usually used to quantify an object’s resistance to bending. This is also known as the second moment of area. To show how this is derived, let’s take a rectangular cross section as shown below.
We’ll start with the moment of inertia of a thin rod about its center perpendicular to its length, which is ML²/12. We’ll replace the mass M with the area of the rectangle, which is bh, and then the length with the length of the rectangle, which is h, giving us a moment of area of bh³/12 (see later in this blog for the table of moment of areas).
To keep the concepts straight for the rest of this blog, I will be using moment of area to refer to this property throughout this blog. Its units are L⁴, and it can be defined using one of the following formulas (assume an xy planar cross section). Note how the formula is similar to that of moment of inertia; instead of mass, area is used.
To explain these formulas in more layman’s terms, moment of area is a measure of how a certain cross-sectional area’s points are distributed with regard to a certain axis. Most of the time, we’re defining this axis as the centroidal axis, which is the axis that goes through the geometric center. For beams composed of a single isotropic material, this would also be the neutral axis, which is the axis a cross section would bend about. Rather than doing calculus every time we do a bending calculation, we refer to a table for the moments of area of the most common shapes. Here’s one below:
The Parallel Axis Theorem
But what if a cross-section is more complex than these basic shapes? Or what if the axis isn’t the centroidal axis? No need to worry! You can use the parallel axis theorem! The parallel axis theorem states that the moment of area of a shape about a specified axis (I) is equal to the moment of the area about the shape’s centroidal axis (Ic) plus the product of the shape’s area (A) and the square of the distance between the two axes (d).
So we’ll quickly do an example. Let’s say we have a rectangular cross section but we want to calculate the moment of area about the bottom edge instead of the centroidal axis. We’ll use the rectangle from the table above, where b is the base and h is the height:
Now let’s say we wanted to add another rectangle on top of it to make an inverted T section. We can combine the parallel axis theorem and the additive property to calculate this new shape’s moment of area. First we will calculate the moment of area of the added (orange) section then add it to the original (blue) section:
The parallel axis theorem also works for mass moment of inertia. Use the moment of inertia formulae for a thin rod and try it for yourself!
Let’s Think Conceptually
(One shape is a lot more rigid than the other….)
So how can we optimize moment of area in our designs? We should aim for a higher moment of area (AKA more material away from the neutral axis) if we want more rigidity.
Shown below is a table of shapes in a 5×5 space (unitless). For any hollowed out or thin sections, a thickness of 1 was used. The moment of area in both the major and minor axes were calculated as well as the ratio between the moment of area and the cross-sectional area.
As expected, filling the entire space with material results in the overall highest moment of inertia. However, most of the time this is not practical in real life due to weight and cost concerns. I-beams and C-channels are very strong about their major axes but not as strong about their minor axes. Tubes are quite strong about both bending axes but will have more material overall than an equivalent I-beam or channel.
Not Complicated Enough Yet?
Let’s touch on two more concepts: the polar moment of inertia and the product of area. The polar moment of inertia, Iz, is used for torsion of a cross section and is a measure of how a cross sectional area’s points are distributed around a torsional center. It is calculated using the following equation:
If you have Ix and Iy already calculated, where x and y are perpendicular axes, Iz is the sum of those two quantities. This is known as the perpendicular axis theorem:
The product of area, Ixy, can be used as a measure of symmetry and rotational stability. It can also be used to rotate the reference axes in the following equations:
If you’re familiar with the stress transformation equations (and if you’re not, go here), these equations will look very similar to you.
Final Thoughts
Fully understanding moment of inertia and moment of area is a prerequisite to designing and analyzing structures – hopefully this blog helped! To quickly summarize the differences between the two concepts, refer to the table below:
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