# Combined inertia tensor of combined shapes

I have a solid cuboid of width $$w$$, height $$h$$, depth $$d$$, and mass $$m$$.

So inertia tensor for the cuboid is:

$$\begin{bmatrix} m(h*h+d*d)/12&0&0\\ 0&m(w*w+d*d)/12&0\\ 0&0&m(w*w+h*h)/12 \end{bmatrix}$$

Now I have a composite shape, which looks like this:

The red dot is actually inside that cuboid and that's the center of the point of origin.

How can I calculate the combined inertia tensor?

And how can I calculate combined inertia tensor if I set offset between these cuboids in $$z$$-axis?

I can calculate inertia tensor when the point is in the center of the gravity of that object, not from another random problem - that's the problem. I have done some googling this time

Can you help me out now?

The concept you need for problems of this type is the parallel-axis theorem, which relates the moment of inertia of an object about an axis to its moment of inertia about its center of mass.

• I tried to learn about the theorem from articles over the internet... Now I am stuck here, can you help me out ? Oct 27 '19 at 4:08
• thanks i think i now some math of parallel axis theorem.. took almost 3 months ... :'( Jan 3 '20 at 7:48

This figure can help you to solve your problem using parallel axis transformation . You can choose the x,y,z coordinate system at arbitrary point p, not necessarily center of mass, but all coordinate systems must be parallel and the same orientation.

• I tried to learn about the theorem from articles over the internet... Now I am stuck here, can you help me out ? Oct 27 '19 at 4:08

From wikipedia:

Let $$\mathbf {I_{0}}$$ be the inertia tensor of a body calculated at its centre of mass, and $$\mathbf {R}$$ be the displacement vector of the body. The inertia tensor of the translated body respect to its original center of mass is given by:

$${\displaystyle \mathbf {I} =\mathbf {I_{0}} +m[(\mathbf {R} \cdot \mathbf {R} )\mathbf {E_{3}} -\mathbf {R} \otimes \mathbf {R} ]}$$

where $$m$$ is the body's mass, $$\mathbf{E_3}$$ is the 3 × 3 identity matrix, and $$\otimes$$ is the outer product.

So for each sub-object you need to get the difference vector between its center of mass and the new combined center of mass. That is your displacement vector.

Once you have offset all the inertia tensors, you can simply add (+) them into a combined inertia tensor.