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Is there an easy way to calculate the moment of inertia of an arbitrary 2-dimensional polygon, with respect to an axis perpendicular to the plane of the polygon (passing through the center of mass)?

What I intended to do:

  1. Derive a formula for the moment of inertia of a triangle, with respect to an axis through a vertex of the triangle.
  2. Apply this formula on every triangle center of mass - vertex - next vertex in the polygon and add the results.

What I achieved:

  • For a triangle with vertices $(0,0)$, $(a,0)$ and $(b,c)$, the moment of inertia with respect to the axis $x=0\wedge y=0$ is:
    $$\int_0^{b}\int_0^{\frac{c}{b}\cdot x}x^2+y^2\text{ }dy\text{ }\text{ }\text{ }dx - \int_a^b\int_0^{\frac{c}{b-a}\cdot(x-a)}x^2+y^2\text{ }dy\text{ }\text{ }\text{ }dx$$ $$=\int_0^{b}\dfrac{\left(c^3+3b^2c\right)x^3}{3b^3}dx - \int_a^b\dfrac{c\left(\left(c^2+3b^2-6ab+3a^2\right)x^3+\left(-3ac^2-3ab^2+6a^2b-3a^3\right)x^2+3a^2c^2x-a^3c^2\right)}{3\left(b^3-3ab^2+3a^2b-a^3\right)}dx$$ $$=\dfrac{b\left(c^3+3b^2c\right)}{12} - \dfrac{\left(b-a\right)c\left(c^2+3b^2+2ab+a^2\right)}{12}$$ $$=\frac{c^3a+ca^3+bca^2+b^2ca}{12}$$ $$=\frac{ac}{12}\cdot\left(a^2+b^2+c^2+ab\right)$$
  • Now we can calculate the moment of inertia of a triangle with vertices $A(x_1,y_1)$, $B(x_2,y_2)$ and $C(x_3,y_3)$ with respect to the axis through A as follows:
    The Euclidean transformation that takes $A$ to $(0,0)$ and $B$ to $(\sqrt{x_2^2+y_2^2},0)$, takes $C$ to $(x_3',y_3')$ (which can be calculated quite easily).
    As a Euclidean transformation preserves the moment of inertia, we can apply the preceding formula.
  • Now we can apply this to every triangle center of mass - vertex - next vertex in the polygon and add the results to get the total moment of inertia.


I haven't tested this yet (I'll do that later), but could someone please check my calculations and my method?
Furthermore, if there is an easier way to calculate this, please let me know ;)

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  • $\begingroup$ What if your object can't be broken into triangles like this? What if the center of mass isn't located in the polygon. Are you only considering regular polygons? $\endgroup$ Jul 26, 2019 at 13:14
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    $\begingroup$ Please look also at this (Polygone) en.wikipedia.org/wiki/Second_moment_of_area $\endgroup$
    – Eli
    Jul 26, 2019 at 14:54
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    $\begingroup$ @Aaron Stevens I'm only considering convex polygons. $\endgroup$ Jul 26, 2019 at 15:30
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    $\begingroup$ Just as a side note, I think arbitrary polygon would be a better descriptor than random polygon. An arbitrary polygon is a polygon whose details have not been specified; a random polygon is one whose details are blindly picked out of a hat. In other words, randomness refers to a lack of predictability, which is not what you mean here. $\endgroup$
    – J. Murray
    Jul 27, 2019 at 0:37
  • $\begingroup$ @J.Murray you're right. I didn't know there was a difference between both words. Fixed it now. $\endgroup$ Jul 27, 2019 at 11:09

3 Answers 3

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Turning Eli's comment into an answer to my own post:

Apparently, there exists something called the second moment of area (see also: https://en.wikipedia.org/wiki/Second_moment_of_area).
This is the mathematical definition of the physical moment of inertia.

The second moment of area for an arbitrary shape $R$ with respect to an arbitrary axis $a$ is defined as: $$J_{a} = \iint\limits_{R}\rho^2 dA$$

When the axis is the $x$-axis, the second moment of area can be calculated as follows: $$J_x=\iint\limits_R y^2 dx dy$$ And when it's the $y$-axis: $$J_y=\iint\limits_R x^2 dy dx$$

For a polygon $A_0A_1A_2\cdots A_{n-1}$ with $A_i=(x_i,y_i)$ and $A_n=A_0$, those are equal to: $$J_x = \frac{1}{12}\sum_{i=0}^{n-1}\left(x_iy_{i+1}-x_{i+1}y_i\right)\left(y_i^2+y_iy_{i+1}+y_{i+1}^2\right)$$ $$J_y = \frac{1}{12}\sum_{i=0}^{n-1}\left(x_iy_{i+1}-x_{i+1}y_i\right)\left(x_i^2+x_ix_{i+1}+x_{i+1}^2\right)$$

Now, denote by $z$ the line perpendicular to the $xy$-plane that passes through the origin. Then: $$J_z = \iint\limits_R\rho^2 dA = \iint\limits_Rx^2+y^2\text{ }dA = \iint\limits_Ry^2\text{ }dxdy + \iint\limits_Rx^2\text{ }dydx = J_x+J_y$$ (this is the so-called perpendicular axis theorem)
Thanks to this, we can calculate $J_z$!


I strongly believe that I was going to end up with this as well, due to the $12$ in the denominator and the fact that only subsequent $x$- and $y$-values are multiplied.

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We can also derive the formula for the second moment of area w.r.t to the axis through the origin directly.

Let $A_0(=A_n), A_1\ldots, A_{n-1}$ be the vertices of your polygon $P$ where $A_i = (x_i, y_i)$.

For $\vec{r} = (x,y)$ notice that $\operatorname{div}(r^2\vec{r}) = 4r^2$ so using the two-dimensional divergence theorem we get $$J_z = \int_{P} r^2\,dA = \frac14\int_P \operatorname{div}(r^2\vec{r})\,dA = \frac14\int_{\partial P}r^2\,\vec{r}\cdot \hat{n}\,d\gamma$$

where we integrate over the boundary $\gamma = \partial P$ of the polygon with $\hat{n}$ being the outward pointing normal.

Notice that for $i=0,\ldots, n-1$ you can parameterize the line segment $[A_i, A_{i+1}]$ with $\gamma_i : [-1,1] \to \mathbb{R}^2$ given as $$\gamma_i(t) = \frac{A_{k+1}+A_k + t(A_{k-1}-A_k)}2, \quad t\in[-1,1]$$

and the outward pointing normal is $\hat{n} = \frac{(A_{k+1}-A_k)\times \hat{z}}{\|A_{k+1}-A_k\|}$ where $\hat{z} = (0,0,1)$. We have $\gamma_i'(t) = \frac12(A_{k-1}-A_k) $ so $\hat{n}\,d\gamma_i = \frac12 (A_{k+1}-A_k)\times \hat{z}\,dt$. Therefore

\begin{align} J_z &= \frac14\int_{\partial P}r^2\,\vec{r}\cdot \hat{n}\,d\gamma \\ &= \frac14\sum_{i=0}^{n-1}\int_{\gamma_i}r^2\,\vec{r}\cdot \hat{n}\,d\gamma_i \\ &= \frac1{64}\sum_{i=0}^{n-1}\int_{-1}^1 \big\|A_{k+1}+A_k + t(A_{k-1}-A_k)\big\|^2\,\big(A_{k+1}+A_k + t(A_{k-1}-A_k)\big)\cdot (A_{k+1}-A_k)\times \hat{z}\,dt \end{align}

Using the triple product rule we get \begin{align}\big(A_{k+1}+A_k + t(A_{k-1}-A_k)\big)\cdot \left((A_{k+1}-A_k)\times \hat{z}\right) &= \left(\big(A_{k+1}+A_k + t(A_{k-1}-A_k)\big)\times (A_{k+1}-A_k)\right)\cdot \hat{z}\\ &= 2(A_{k+1}\times A_k)\cdot \hat{z}\\ &= 2(x_{k}y_{k+1}-x_{k+1}y_k) \end{align} Hence by integrating it follows \begin{align} J_z &= \frac1{32}\sum_{k=0}^{n-1}(x_{k}y_{k+1}-x_{k+1}y_k)\int_{-1}^1 \big\|A_{k+1}+A_k + t(A_{k-1}-A_k)\big\|^2\,dt\\ &= \frac1{12}\sum_{k=0}^{n-1}(x_{k}y_{k+1}-x_{k+1}y_k)\left(x_{k+1}^2+x_{k+1}x_k+x_k^2 + y_{k+1}^2+y_{k+1}y_k+y_k^2\right) \end{align} which recovers your formula.

Inspired by this more general calculation.

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Go through all the edges of the polygon and form a triangle with the origin

fig1

The area of the triangle is

$$ {\rm area}(OAB) = \frac{ x_A y_B - x_B y_A}{2} $$

The center of mass of the triangle is

$$ {\rm \vec{cm}}(OAB) = \pmatrix{ \frac{x_A+x_B}{3} & \frac{y_A+y_B}{3} } $$

The MMOI of the triangle about the origin is

$$ {\rm mmoi}(OAB) = \frac{ (x_A y_B - x_B y_A) \left( (x_A-x_B)^2 + (y_A-y_B)^2 + x_A x_B + y_A y_B \right)}{12} $$

When you go through all the edges, some of the areas are going to be positive and some negative values, and when you add them all up the total area of the polygon is calculated.

$$ {\rm area} = \sum_i {\rm area}_i $$

where ${\rm area}_i$ is the area formed from the i-th edge as show above.

The center of mass of the polygon is calculated with

$$ {\rm \vec{cm}} = \frac{1}{\rm area} \sum_i {\rm area}_i \; {\rm \vec{cm}}_i $$

Similarly when you add up all the mass moment of inertia values the total mmoi of the polygon about the origin is calculated.

$$ {\rm mmoi}_{\rm origin} = \sum_i {\rm mmoi}_i $$

To get the mmoi about the center of mass apply the parallel axis theorem

$$ {\rm mmoi}_{\rm cm} = {\rm mmoi}_{\rm origin} - {\rm area} \| {\rm \vec{cm}} \|^2 $$

where $\|\,\|$ denotes the eucledian norm $\sqrt{x^2+y^2}$ of a vector.

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