To address the diverse needs of modern-day structural applications, mechanical metamaterials with architected microstructure found immense attention. Anti-curvature lattice metamaterial is one such class of microstructural design which provides on-demand mechanical properties. The major focus of the existing studies on the different classes of metamaterials including anti-curvature lattices has been tailoring linear and nonlinear static properties through microstructure architecturing. In engineering applications, the structures made of metamaterials frequently encounter dynamic loading environments. Lack in estimation and tailoring of nonlinear dynamic properties of metamaterials leaves avenue for the present study. In this work, nonlinear dynamics of anti-curvature lattice cell walls under far-field dynamic stresses are analyzed by modeling it as doubly curved beam with fixed-rotationally restrained boundaries subjected to harmonic loading. A semi-analytical framework through variational energy principle is developed for analyzing large amplitude vibration of the doubly curved beam. As the large amplitude curved beam vibration problem simulates the nonlinear dynamics of anti-curvature metamaterial cell walls, the developed curved beam framework leads toward the development of a multi-scale dynamics framework estimating nonlinear dynamic properties of anti-curvature lattices. The parametric investigations on the influences of the direction of applied dynamic load along with the degree and nature of curvature of doubly curved beam provide foundation for nonlinear dynamic property tailoring of anti-curvature lattices.