Mass moment of inertia (Ixx, Iyy, Izz)

The mass moments of inertia I_xx, I_yy, I_zz are the rotational inertias of the ship about its longitudinal, transverse, and vertical principal axes through the center of gravity, defined as I = the integral of the squared distance from the axis times dm, and related to the radius of gyration by I = m times k squared. They set the inertia terms in the roll, pitch, and yaw equations, so the roll natural period depends on I_xx plus roll added inertia over the roll restoring c_44 = displacement times g times GM. Typical gyradii are k_xx about 0.30 to 0.40 B, k_yy and k_zz about 0.25 L. Loading that redistributes weight changes these and shifts the natural periods.