I need to input inertia data into a bio-mechanical simulation program. I can find inertia data for parts of the human body, around the normal axes.
I know the parallel axis theorem, so I can compute the inertia for the body parts in different locations.
But if the body part is angled to the normal axes, can you then compute the inertia, using its known inertia around the x, y and z axes?
Let's say that a human arm and hand has the following inertia values (around its center of gravity) $$I_x = I_y = 0.3\ kg \cdot m^2$$ $$I_z = 0.02\ kg\cdot m^2$$
This would be the values if the arm is hanging down. But let's say I angle the arm outwards ("abduct" it) by 30°, can I then use $I_x,\ I_y$ and $I_z$ to find the new values? (As the angle increases, the $I_y$ value would gradually fall and reach the old $I_z$ value at 90°. Is there maybe a trigonometric relation for this??) OK, I am not a physicist, I only do bio-mechanics occasionally :)
Edit 19/7:
Just to clarify:
What I wish to find out is how angling of the body parts affects their inertia. To use a diver that is somersaulting forward as an example : If the diver keeps the arms straight down the body, the values given above would be correct. But the diver can reduce his/her inertia by angling the arms outwards. (You sometimes see divers do that in the pike position.) I wish to compute the inertia for different angles, all for rotation forward. I wish to start with just the body part alone. (After that you can apply the parallel axes theorem.)
I wonder if there could exist a simple trigonometric relationship since one can divide the mass into point mass elements? I can't prove this is the correct answer but I plotted $0.3\cos(\alpha) + 0.02\sin(\alpha)$. It would show how the moment of inertia of the arm, for rotation forward (like in somersaulting), would decrease, as the arm is abducted (by $\alpha^{\circ}$). (In this example the arm alone is "somersaulting".) Intuitively it looks "OK".
(The other alternative is that this can't be computed. You must make a new measurement. But in the case of the arm (looking a little like a rod), I think at least a reasonable estimate could be computed from the x, y and z values.
But what leads me to believe that it is computable, is that you can input $I_x,\ I_y$ and $I_z$ to computer programs and they can compute rotations along oblique axes.)