In this study, an improved mathematical representation of a drill-string assembly is developed to provide an alternative assessment on vibration irregularities proliferating downhole due to bit-rock interference. Lateral vibrations receive particular attention due to their high frequency content which alter the dynamic response of the drill-string, instigate casing damage, and impede optimal penetration rates. The response of the drill-string is captured by synthesizing compatible stationary bit excitations, via an auto-regressive digital filter, and implementing Monte Carlo simulation, while the power spectral density function is approximated to elucidate the dynamic characteristics during drilling. Formulating adequate physical parameters for the equation of motion implies incorporating a finite element technique, where the flexibility of the drill-string and elastic characteristics of the well-bore are accounted for. In conjunction with the stochastic nature of the excitation, the mathematical representation accounts for rig structural parameters, drilling fluid circulating within annulus/casing extremities, and a nonlinearity exhibited through a contact force generated between the well-bore and drill-string segment. To address the nature of the nonlinearity, the method of statistical linearization is incorporated to establish an equivalent linear system.