Principal Axes and Principal Moments of Inertia for a Rigid Body

                In this post, I will discuss the procedure for determining the principal axes and principal moments of inertia for a rigid body.  The principal axes of a body are a set of mutually perpendicular axes in which the resulting torques act independently of each other.  Furthermore, the principal moments of inertia are the respective inertial moments about these axes.

                Consider a cylinder of radius R and height h set such that the origin is at the center of the bottom base and the z axis is parallel with the height of the cylinder.  To begin, we will determine the center of mass of the cylinder.  This is done using the equation

where

Here, a represents the center of mass vector, r represents the unit vector pointing toward the center of mass, ρ is the mass density of the cylinder, V is the volume of the cylinder, and M is the total mass.  Additionally, as we are integrating over the volume of a cylinder, it is very helpful to convert to cylindrical coordinates.  For our cylinder, the center of mass is found to be

                The next step in our process is determining the inertial tensor for our rigid body.  As a result of a complex and lengthy derivation of which I will not be explaining in this post, we can construct this tensor using

In this equation J_ij represents our tensor with indices i and j, δ_ij is the kronecker delta operator, and x_k, x_i, and x_j represents our various axes with respect to the indices’ value.  Here, it is important to note that the kronecker delta operator has a value of 1 when i=j and a value of 0 when i≠j.  Upon computation of this tensor, we obtain

                Next, to determine our principal axes and principal moments of inertia, we must apply Steiner’s parallel-axis theorem to construct an inertial tensor relative to the center of mass of the cylinder.  This tensor can be solved for using the general equation

Here, I_ij represents the adjusted inertial tensor and a is our center of mass vector.  In application to our cylinder, we obtain

Finally, the principal axes of the body are the eigenvectors of tensor I_ij while the principal moments of inertia are the respective eigenvalues of I_ij.  For this body, the principal axes and principal moments of inertia are