Non-smooth modal analysis is an extension of modal analysis to non-smooth systems, prone to unilateral contact conditions for instance. The problem of a one-dimensional bar subject to unilateral contact on its boundary has been previously investigated numerically and the corresponding spectrum of vibration could be partially explored. In the present work, the non-smooth modal analysis of the above system is reformulated as a set of functional equations through the use of both d'Alembert solution to the wave equation and the method of steps for Neutral Delay Differential Equations. The system features a strong internal resonance condition and it is found that irrational and rational periods of vibration should be carefully distinguished. For irrational periods, the displacement field of the non-smooth modes of vibration consist in piecewise-linear functions of space and time and such a motion is unique for a prescribed energy. However, for rational periods, new periodic solutions are found analytically. They belong to families of iso-periodic solutions with piecewise-smooth displacement field in space and time.