Roll

Roll is the rotational motion of a ship about its longitudinal (x) axis. Its linear equation is (I_44 + a_44) phi-double-dot + b_44 phi-dot + c_44 phi = M_4(t), with restoring c_44 = displacement times g times GM, so the undamped natural roll period is T_phi = 2 pi times the square root of (I_44 + a_44) over c_44, often written about 2 pi k_xx over the square root of g times GM. Roll is lightly damped and nonlinear, with wave radiation, eddy, friction, lift, and bilge-keel components, so it can build to large amplitude near resonance in beam and oblique seas and through parametric excitation in head and following seas.