Computing tensor of Inertia in 2D

Question

I'm researching how to find the inertia for a 2D shape. The contour of this shape is meshed with several points, the x and y coordinate of each point is already known.

I know the expression of Ixx, Iyy and Ixy but the body has no mass. How do I proceed?

Answer 1

For whatever shape you have, you need to split it into triangles and handle each triangle separately. Then in the end combined the results using the following rules

Overall

% Combined total area of all triangles
total_area = SUM( area(i), i=1:n )
total_mass = SUM( mass(i), i=1:n )
% Combined centroid (center of mass) coordinates
combined_centroid_x = SUM( mass(i)*centroid_x(i), i=1:n)/total_mass
combined_centroid_y = SUM( mass(i)*centroid_y(i), i=1:n)/total_mass
% Each distance to triangle (squared)
centroid_distance_sq(i) = centroid_x(i)*centroid_x(i)+centroid_y(i)*centroid_y(i)
% Combined mass moment of inertia
combined_mmoi = SUM(mmoi(i)+mass(i)*centroid_distance_sq(i), i=1:n)

Now for each triangle.

Consider the three corner vertices with vector coordinates, points A, B and C

a=[ax,ay]
b=[bx,by]
c=[cx,cy]

and the following dot and cross product (scalar) combinations

a·a = ax*ax+ay*ay
b·b = bx*bx+by*by
c·c = cx*cx+cy*cy
a·b = ax*bx+ay*by
b·c = bx*cx+by*cy
c·a = cx*ax+cy*ay
a×b = ax*by-ay*bx
b×c = bx*cy-by*cx
c×a = cx*ay-cy*ax

The properties of the triangle are (with t(i) the thickness and rho the mass density)

area(i) = 1/2*ABS( a×b + b×c + c×a )
mass(i) = rho*t(i)*area(i)
centroid_x(i) = 1/3*(ax + bx + cx)
centroid_y(i) = 1/3*(ay + by + cy)
mmoi(i) = 1/6*mass(i)*( a·a + b·b + c·c + a·b + b·c + c·a )

By component the above are

area(i) = 1/2*ABS( ax*(by-cy)+ay*(cx-bx)+bx*cy-by*cx)
mmoi(i) = mass(i)/6*(ax^2+ax*(bx+cx)+bx^2+bx*cx+cx^2+ay^2+ay*(by+cy)+by^2+by*cy+cy^2)

Appendix

A little theory here. The area of each triangle is found using

Area = 1/2 * || (b-a) × (c-b) ||

where × is a vector cross product, and || .. || is vector norm (length function).

The triangle is parametrized by two variables t and s such that the double integral A = INT(INT(1,dx),dy) gives the total area

% position r(s,t) = [x,y]
[x,y] = [ax,ay] + t*[bx-ax, by-zy] + t*s*[cx-bx,cy-by]

% gradient directions along s and t
(dr/dt) = [bx-ax,by-ay] + s*[cx-bx,cy-by]
(dr/ds) = t*[cx-bx,cy-by]

% Integration area element
dA = || (dr/ds)×(dr/dt) || = (2*A*t)*ds*dt
%
%   where A = 1/2*||(b-a)×(c-b)||

% Check that the integral returns the area
Area = INT( INT( 2*A*t,s=0..1), t=0..1) = 2*A*(1/2) = A

% Mass moment of inertia components

         /  /  /  | y^2+z^2  -x*y    -x*z   |
I = 2*m*|  |  | t*|  -x*y   x^2+z^2  -y*z   | dz ds dt
        /  /  /   |  -x*z    -y*z   x^2+y^2 |

% where [x,y] are defined from the parametrization

Answer 2

I want to slightly correct excellent John Alexiou answer:

1. Triangle mmoi algorithm expects corners to be defined relative to triangle center of mass (centroid). So subtract centroid from corners before calculating mmoi
2. Shape mmoi algorithm expects centroids of each single triangle to be defined relative to shape center of mass (combined centroid). So subtract combined centroid from each triangle centroid before calculating mmoi.

So result code would look like this:

public static float CalculateMMOI(Triangle[] triangles, float thickness, float density)
{
    float[] mass = new float[triangles.Length];
    float[] mmoi = new float[triangles.Length];
    Vector2[] centroid = new Vector2[triangles.Length];
    
    float combinedMass = 0;
    float combinedMMOI = 0;
    Vector2 combinedCentroid = new Vector2(0, 0);
    
    for (var i = 0; i < triangles.Length; ++i) 
    {
        mass[i] = triangles[i].CalculateMass(thickness, density);
        mmoi[i] = triangles[i].CalculateMMOI(mass[i]);
        centroid[i] = triangles[i].CalculateCentroid();

        combinedMass += mass[i];
        combinedCentroid += mass[i] * centroid[i];
    }
    combinedCentroid /= combinedMass;


    for (var i = 0; i < triangles.Length; ++i) {
        combinedMMOI += mmoi[i] + mass[i] * (centroid[i] - combinedCentroid).LengthSquared();
    }

    return combinedMMOI;
}


public struct Triangle
{
    public Vector2 A, B, C;
    
    public float CalculateMass(float thickness, float density)
    {
        var area = CalculateArea();
        return area * thickness * density;
    }
    
    public float CalculateArea()
    {
        return 0.5f * Math.Abs(Vector2.Cross(A - B, A - C));
    }

    public float CalculateMMOI(float mass)
    {
        var centroid = CalculateCentroid()
        var a = A - centroid;
        var b = B - centroid;
        var c = C - centroid;
        
        var aa = Vector2.Dot(a, a);
        var bb = Vector2.Dot(b, b);
        var cc = Vector2.Dot(c, c);
        var ab = Vector2.Dot(a, b);
        var bc = Vector2.Dot(b, c);
        var ca = Vector2.Dot(c, a);
        return (aa + bb + cc + ab + bc + ca) * mass / 6f;
    }

    public Vector2 CalculateCentroid()
    {
        return (A + B + C) / 3f;
    }
}


public struct Vector2
{
    public float X, Y;
    
    public float LengthSquared()
    {
        return X * X + Y * Y;
    }
    
    public static float Dot(Vector2 a, Vector2 b)
    {
        return a.X * b.X + a.Y * b.Y;
    }

    public static float Cross(Vector2 a, Vector2 b)
    {
        return a.X * b.Y - a.Y * b.X;
    }
}