Moments of Inertia about inclined axis
In structural and mechanical design, it is sometimes necessary to calculate the moment of inertia with respect to a set of inclined u, v, axes when the values of q , Ix, Iy, Ixy are known.
To do this we will use transformation equations which relates the x, y, and x’, y’ coordinates.
From Figure, these equations are:
Moments of Inertia about inclined axis,, continue
we wish to determine moments and product of inertia with respect to new axes x’ and y’.
|Jo = Ix’ + Iy’ = Ix + Iy|
Principal Axes and Principal Moments of Inertia
We will now determine the orientation of these axes about which Ix’ , Iy’ are maximum and minimum. This particular axes are called principal axes By differentiating the first of Eqs. 10-9 with respect to q and setting the result to zero. Thus;
Therefore, at θ = θp ;
By substituting for θ in Ix’ , Iy’ and Ix’y’ equations and simplifying, we obtain;