A propeller shaft generally experiences three linear forces and three moments, the most important of which
are thrust, torque, and lateral forces (horizontal and vertical). Thus, we consider 4DOFs (degrees of freedom)
of propeller shaft vibrations. This paper is presented to obtain the vibration equations of the various coupled
vibrations of the propeller shaft at the stern of a ship (including coupled torsional-axial, torsional-lateral, axial-
lateral, torsional-axial, and lateral vibrations). We focused on the added hydrodynamic forces (added mass
and added damping forces) due to the location of the propeller behind the ship. In this regard, the 4DOFs of
the coupled vibration (torsional-longitudinal and lateral vibrations in the horizontal and vertical directions)
equations of shaft and propeller systems located behind a ship were extracted with and without added mass
and damping forces. Also, the effect of mass eccentricity was considered on vibrations occurring at the rear of
the ship.