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While periodic responses of periodically forced dissipative nonlinearmechanical systems are commonly observed in experiments and numerics, theirexistence can rarely be concluded in rigorous mathematical terms. This lack ofa priori existence criteria for mechanical systems hinders definitiveconclusions about periodic orbits from approximate numerical methods, such asharmonic balance. In this work, we establish results guaranteeing the existenceof a periodic response without restricting the amplitude of the forcing or theresponse. Our results provide a priori justification for the use of numericalmethods for the detection of periodic responses. We illustrate on examples thateach condition of the existence criterion we discuss is essential.

When does a Periodic Response Exist in a Periodically Forced Multi-Degree-of-Freedom Mechanical System?