Abstract: Vibrations of a floating ice cover on top of a two-dimensional ideal fluid of arbitrary depth are studied when the effects of nonlinearity, inertia, and damping are all considered. We reduce the fully nonlinear problem to a cubic-truncation system involving variables on the free surface by expanding the relevant pseudo-differential operators and retaining nonlinear terms up to the third order. To validate the accuracy of the reduced model, we focus on the free wavepacket solitary wave solutions. In the absence of damping, the normal form analysis is performed to derive the cubic nonlinear Schrödinger equation, which predicts the existence of free wavepacket solitary waves in the primitive equations and the accuracy of the cubic-truncation model. The main advantage of the cubic-truncation approximation over the quadratic-truncation model is that the resultant NLS equation has correct coefficient of the nonlinear term, which allows a better approximation of dynamic responses of the ice cover near the phase speed minimum. Solitary waves are then numerically computed, and it is shown that the cubic-truncation approximation agrees well with the full Euler equations for bifurcation curves and wave profiles, indicating that the reduced model is more accurate than the quadratic truncation model. The nonlinear dynamic response of a floating ice sheet to a fully localized constant-moving load is investigated based on the cubic-truncation model. The time-dependent solutions are compared with the data from the field measurements, and good agreement is achieved between the numerical results and experimental records.