# I Substituting a solid body with mass points

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1. Oct 14, 2016

### Tawhiri

I ran into the following problem, and stuck for a couple of days now.

I have a solid body, rigid and and has uniform density. Its mass M, the location of the center of gravity x_M, y_M, z_M and its inertia matrix is known:

Jx Jxy Jxz
Jyx Jyy Jyz
Jzx Jzy Jz

I have to write an algorithm, which creates maximum 3 mass points xi, yi, zi, mi in a way, that the following properties of the solid body and the sum of the mass points are exactly the same:

- The mass must be identical ∑i=13mi = M
- The location of the center of gravity must remain
xM =∑i=13mi xi/M
yM =∑i=13mi yi/M
zM =∑i=13mi zi/M
- Jx must be unchanged:
JxM = ∑i=13 (yi2 + zi2) mi
- The angle of the principal axes to x axis should not change
JxyM = ∑i=13 xiyimi
JxzM = ∑i=13 xizimi

Everything else is arbitrary.

What I did so far: with simple algebra I managed to find a solution for everything, except for Jxy and Jxz. But I see no way to modify the variables to keep the good parameters and reach the missing ones at the same time.

Any suggestions are most welcome, many thanks in advance.

2. Oct 19, 2016