Substituting a solid body with mass points

[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:

J_x J_xy J_xz
J_yx J_yy J_yz
J_zx J_zy J_z

I have to write an algorithm, which creates maximum 3 mass points x_i, y_i, z_i, m_i 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=1³m_i = M
- The location of the center of gravity must remain
x_M =∑_i=1³m_i x_i/M
y_M =∑_i=1³m_i y_i/M
z_M =∑_i=1³m_i z_i/M
- J_x must be unchanged:
J_xM = ∑_i=1³ (y_i² + z_i²) m_i
- The angle of the principal axes to x-axis should not change
J_xyM = ∑_i=1³ x_iy_im_i
J_xzM = ∑_i=1³ x_iz_im_i

Everything else is arbitrary.

What I did so far: with simple algebra I managed to find a solution for everything, except for J_xy and J_xz. 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.

 

[eljose79]

Dear fellow scientist,

Thank you for sharing your problem with us. I understand that you are trying to write an algorithm that creates three mass points in order to maintain certain properties of a solid body with known mass, center of gravity, and inertia matrix. It seems that you have made good progress so far, but are stuck on how to handle the Jxy and Jxz values.

One possible approach to consider is using the concept of principal axes. As you may know, the inertia matrix of a solid body can be diagonalized, meaning it can be transformed into a diagonal matrix with the principal moments of inertia on the diagonal. The principal axes are the axes along which these moments of inertia are maximum, minimum, and intermediate.

So, my suggestion would be to first diagonalize the given inertia matrix and find the principal axes and moments of inertia. Then, you can place the three mass points along these axes in a way that the sum of their masses is equal to the total mass of the body. This will ensure that the principal moments of inertia remain unchanged, including Jx, while satisfying the first two properties you mentioned.

Next, you can use the equations you have already derived for Jxy and Jxz, but instead of using the original coordinates, use the coordinates of the mass points along the principal axes. This should allow you to solve for the mass values of the points that will satisfy the required conditions for Jxy and Jxz.

I hope this suggestion helps you move forward with your algorithm. If you have any further questions or need clarification, please don't hesitate to reach out.

Best of luck with your work!