In this paper, we investigate the flexural vibrations of a cantilever beam of square cross section that is immersed in a quiescent viscous fluid. The cantilever beam is subject to base excitations with oscillation amplitudes comparable to the beam thickness. The structure is modeled using linear Euler-Bernoulli beam theory and the fluid-structure interaction is described via a nonlinear complex-valued hydrodynamic function which accounts for the added mass and damping contribution from the encompassing fluid. We formulate a hydrodynamic function that is appropriate for finite vibration amplitudes and a broad range of frequencies by conducting a 2D parametric computational fluid dynamics analysis. The proposed function is expressed in terms of the classical hydrodynamic function for unsteady Stokes flow plus a nonlinear correction that is a function of amplitude and frequency of vibration. Results from the 2D parametric analysis shows that moderately large amplitude oscillations promote nonlinear hydrodynamic damping. The proposed theoretical model is illustrated through the analysis of underwater vibrations and theoretical results are compared with experimental findings.