Robot manipulators have played an enormous role in the industry during the twenty-one century. Due to the advances in materials science, lightweight manipulators have emerged with low energy consumption and positive economic aspect regardless of their complex mechanical model and control techniques problems. This paper presents a dynamic model of a single link flexible robot manipulator with a payload at its free end based on the Euler-Bernoulli beam theory with a complete second-order deformation field that generates a complete second-order elastic rotation matrix. The beam experiences an axial stretching, horizontal and vertical deflections, and a torsional deformation ignoring the shear due to bending, warping due to torsion, and viscous air friction. The deformation and its derivatives are assumed to be small. The application of the extended Hamilton principle while taking into account the viscoelastic internal damping based on the Kelvin-Voigt model expressed by the Rayleigh dissipation function yields both the boundary conditions and the coupled partial differential equations of motion that can be decoupled when the manipulator rotates with a constant angular velocity. Equations of motion solutions are still under research, as it is required to study the behavior of flexible manipulators and develop novel ways and methods for controlling their complex movements.