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I am interested in finding a formula for the inertia matrix of a rigid body about its center of mass. This particular rigid body is composed of other rigid bodies with known inertia matrices about their center of masses. An example is more cubes and some spheres welded together in some shape. Supposing the mass and center of mass of each of the component rigid body is known, then I find the following result:

Theorem:

Let $B$ be a rigid body made out of $N$ distinct rigid bodies $\{B_1, ..., B_N\}$. Let $\overrightarrow{OC_i}$ denote the center of mass in world frame $\{O,\vec{i},\vec{j}, \vec{k}\}$ of the rigid body $B_i$ and $m_i$ denote its mass. Let also $I_i$ denote the inertia matrix of rigid body $B_i$ about its center of mass $C_i$. Then the center of mass of the rigid body $B$ is $$ \overrightarrow{OC_B} = \frac{m_1 \overrightarrow{OC_1} + ... + m_N\overrightarrow{OC_N}}{m_1 + ... + m_N}$$ and its inertia matrix $I_B$ about its center of mass $C_B$ is $$ I_B = I_1 + ... + I_N + I_{points}$$ where $I_{points} = -\sum_{i=1}^N m_i \left[\overrightarrow{OC_i} - \overrightarrow{OC_B}\right]^2$ with $[v]$ denoting the skew-symmetric matrix constructed from the vector $v$.

Proof:

The formula for the center of mass is well known. For the inertia matrix suppose each rigid body $B_i$ is composed of $N_i$ particles of mass $m_{ij}$ at position $\overrightarrow{Or_{ij}}$. Then by definition $$ I_B = -\sum_{i=1}^N \sum_{j=1}^{N_i} m_{ij} \left[ \overrightarrow{Or_{ij}} - \overrightarrow{OC_B}\right]^2 = -\sum_{i=1}^N \sum_{j=1}^{N_i} m_{ij} \left[ \overrightarrow{Or_{ij}} - \overrightarrow{OC_i} + \overrightarrow{OC_i} - \overrightarrow{OC_B}\right]^2$$ hence

$$ I_B = -\sum_{i=1}^N \left( \sum_{j=1}^{N_i} m_{ij} \left[ \overrightarrow{Or_{ij}} - \overrightarrow{OC_i}\right]^2 +\sum_{j=1}^{N_i} m_{ij} \left[ \overrightarrow{Or_{ij}} - \overrightarrow{OC_i}\right]\left[ \overrightarrow{OC_i} - \overrightarrow{OC_B}\right] + \sum_{j=1}^{N_i} m_{ij} \left[ \overrightarrow{OC_i} - \overrightarrow{OC_B}\right] \left[ \overrightarrow{Or_{ij}} - \overrightarrow{OC_i}\right] + \sum_{j=1}^{N_i} m_{ij} \left[ \overrightarrow{OC_i} - \overrightarrow{OC_B}\right]^2 \right) = \sum_{i=1}^N I_i + I_{points}$$

Is this correct? Is this property known and used? Can someone give me a reference for this ... I want to use this in a paper ...

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It is called 'parallel axis theorem'. Here is the wikipedia entry for it: https://en.wikipedia.org/wiki/Parallel_axis_theorem

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