The study of interactions through direct contact between blade-tips and outer casings in modern turbomachines may be very time-consuming when the classical finite element method is used. The construction of reduced-order models using component mode synthesis methods generally allows for dramatic increase in computational efficiency and may be used in order to improve the knowledge over these interaction phenomena. Among the available approaches, both a fixed-interface method and a free-interface method are considered here in an original manner to reduce the size of a realistic two-dimensional model. The equations of motion are solved using an explicit time integration scheme with the Lagrange multiplier method where friction is accounted for. This method offers energy momentum conserving which is a critical point to ensure the convergence of the algorithm. Moreover, it is shown that even in a non-linear framework the reduced-order models converge to the finite element solution as the number of modes included in the models increases. Considering the fixed-interface method of Craig-Bampton and the free-interface method of Craig-Chang-Martinez, it is shown that a method with fast displacement convergence may be less efficient in terms of motion convergence.