Learning topic
Product of Inertia and Axis Rotation
Learn product of inertia Ixy and transform Ix, Iy, and Ixy under axis rotation, use the invariant Ix + Iy, and determine principal centroidal axes.
The product of inertia $I_{xy}$ characterizes the combined distribution of area relative to two mutually perpendicular axes. Unlike $I_x$ and $I_y$, it may be positive, negative, or zero.
The second moments of area about the x and y axes are:
$$I_x=\int_A y^2\,dA,\qquad I_y=\int_A x^2\,dA.$$
The product moment of area is:
$$I_{xy}=\int_A xy\,dA.$$
The polar second moment of area about the intersection of orthogonal axes is:
$$J_p=\int_A(x^2+y^2)dA=I_x+I_y.$$
Second and polar moments of area have dimensions of length to the fourth power, for example mm⁴.
Symmetry
If one centroidal axis is an axis of symmetry, the product of inertia with respect to that axis and the perpendicular centroidal axis is zero. However, $I_{xy}=0$ by itself does not necessarily imply geometric symmetry.
Axis rotation
When orthogonal axes are rotated through an angle θ, the second moments and product of inertia transform as:
$$I_{x'}=\frac{I_x+I_y}{2}+\frac{I_x-I_y}{2}\cos2\theta-I_{xy}\sin2\theta,$$
$$I_{y'}=\frac{I_x+I_y}{2}-\frac{I_x-I_y}{2}\cos2\theta+I_{xy}\sin2\theta,$$
$$I_{x'y'}=\frac{I_x-I_y}{2}\sin2\theta+I_{xy}\cos2\theta.$$
Principal centroidal axes satisfy $I_{x'y'}=0$. Their orientation follows from:
$$\tan2\theta_p=-\frac{2I_{xy}}{I_x-I_y},$$
with the quadrant and the adopted positive direction of angle measured consistently.
Under rotation of axes, the sum $I_x+I_y$ remains unchanged. This invariant is useful for checking calculations.
Principal axes
Orientations for which $I_{xy}=0$ and $I_x$ and $I_y$ take extreme values are called principal axes of inertia. If they pass through the centroid, they are principal centroidal axes.
Engineering significance
For unsymmetrical sections, principal axes are required in unsymmetrical bending and other problems where the loading direction does not coincide with convenient geometric axes.
About this topic
This topic explains the product of inertia Ixy and how second moments of area depend on coordinate-axis orientation. It presents the transformation equations for Ix, Iy, and Ixy, the invariant Ix + Iy, and the condition used to determine principal centroidal axes.