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Homework Statement
I am trying to understand euler's equations of motion.
First of all, relative to what point do we need to calculate the inertia tensor(J)?
Do we calculate it relativ to the world origin? or body origin?
The torque may be calculated as:
T = dL/dt = DL/Dt + wxL = D(Jw)/Dt + wx(Jw) =(1)= J*Dw/Dt +wx(Jw)
In the equality (1), why can we get J out of the derative?
This can be possible if we are dealing with a rigid body and the inertia tensor is calculated relative to the body origin.
But T and w are in world coordinates in this equation!
So:
T = J*Dw/Dt+wx(Jw) = J*dw/dt+wx(Jw)
Now we want to transform this equation from world coordinates to body coordinates, so we use the eigendecomposition of J, which is RMR(T).
However, what is R?
R is the rotation matric from body coordinates into world coordniates?
How can we see that this is true?
Does one of the R columns is the D? the axis the moment of inertia is measured relative to?
This implies that J is calculated relative to world origin and not body origin?
And what is M then? is M equal to J when calculated relative to body origin? or isn't it?
Lastly, M is a diagonal matrix, is a diagonal matrix comutative with other matrices?
Thank you.
Homework Equations
The Attempt at a Solution
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