We show that topological order and vibrational edge modes can exist in a classical mechanical system consisting of a two-dimensional honeycomb lattice of masses and springs. The band structure shows the existence of Dirac cones and unconventional edge states that are similar to the vibrational modes in graphene. Interestingly, as the system is placed on a constantly rotational coordinate system, the Coriolis force resulting from the non-inertial reference frame introduces time-reversal symmetry breaking and leads to topologically nontrivial band gaps. The nontrivial topological orders are further verified by the calculation of Chern numbers for corresponding bands.