Viscoelastic materials (VEM) are often used in the ship floors to dampen the noise and vibration induced by machinery (engines, pumps, alternators) and live loads (human traffic and liquid movement in pipelines). Therefore, accurate modeling of such materials is important for the iterative design-analysis process. VEM exhibits both time and frequency-dependent properties. Traditionally, integer-order viscoelastic mechanical models are used to describe the rheological properties of the viscoelastic materials. However, such models find difficulties in predicting the rheological characteristics since their kernel functions are a combination of exponential functions. For instance, the integer-order Maxwell model is good at describing the stress relaxation behavior while is poor in capturing creep, and vice-versa for the integer-order Kelvin-Voigt model. However, the fractional-order mechanical model can overcome the above problem with a fewer number of model parameters. Therefore, in this paper, the fractional derivative-based Kelvin-Voigt mechanical model is employed to describe the time-dependent vibratory behavior of the VEM. To validate the effectiveness of the above model, a thin elastic plate bonded with a thin layer of VEM, subject to a concentrated impact load, is studied. Galerkin’s method and Triangular strip matrix approach are used to solve the partial fractional differential equations of motion. The semi-analytical approach of modal superposition is used to generate the response, whose first dynamic overshoot of displacement is crucial to contain the dynamic stress within the working stress level. This bypasses the computationally expensive Finite Element Method. Additionally, ANSYS is limited to the integer-order damping model only. This analysis gives insights into the efficacy of the VEM chosen for passive vibration control of structural components of ship hulls. A case study for free vibration is done with a commercially available VEM, which is used as surface flooring to cover steel plates of ship hulls, thereby acting as vibration dampers. Free vibration results show that the damping coefficient of the plate foundation system increases with increasing the order of the derivative. In addition, the amplitude of the transient response decreases with the order of the derivative. Thus, the classical integer-order mechanical model overestimates the damping of the viscoelastic materials, which leads to underestimating the displacement and associated stresses. The results are verified with literature and ANSYS.