- #1
Tawhiri
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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.
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.