This paper proposes novel unit cells that can be arranged periodically constructing disparate auxetic metamaterials with negative Poisson’s ratio for engineering applications. The two-dimensional auxetic unit cells are designed utilizing the mathematical model of the modified Solid Isotropic Material with Penalization (SIMP) topological optimization method with the objective of maximizing the magnitude of the negative Poisson’s ratio. The deformation mechanisms of the auxetic structures are controlled effectively by straight and oblique struts attached at rotation joints since the optimum designs are a combination of chiral and re-entrant unit cells. Three structures with optimum auxetic designs are chosen to verify the modified SIMP method using numerical simulation. Additionally, the energy absorption and vibration mitigation of auxetic metamaterials have been extensively studied. The results showed a significant impact of negative Poisson’s ratio on the considered dynamic properties of the auxetic structures.