A linear model for the bending-bending-torsional-axial vibration of a spinning cantilever beam with a rigid body attached at its free end is derived using Hamilton's principle. The rotation axis is perpendicular to the beam (as for a helicopter blade, for example). The equations split into two uncoupled groups: coupled bending in the direction of the rotation axis with torsional motions and coupled bending in the plane of rotation with axial motions. Comparisons are made to existing models in the literature and some models are corrected. The practically important first case is examined in detail. The governing equations of motion are cast in a structured way using extended variables and extended operators. With this structure the equations represent a classical gyroscopic system and Galerkin discretization is readily applied where it is not for the original problem. The natural frequencies, vibration modes, stability, and bending-torsion coupling are investigated, including comparisons with past research.