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In Pressurized Water Reactors (PWR) assemblies are exposed to challenging thermal, mechanical, and irradiation loads during operation. Global core and local fuel assembly flow fields coupled with seismic excitation result in fuel assembly and fuel rod vibrations. The fact that vibrations may become excessive in certain conditions has consequences on operational safety margins in fuel assemblies designs. In order to understand how the fuel assembly responds dynamically to an external excitation, it is important to identify the main characteristics of the structures. Among them, the fuel assembly system damping is a fundamental parameter that is usually identified by a number of experiments involving fluid-structure interaction. Recent studies have shown that the damping ratio increases with the excitation force when the structure is entering large-amplitude vibrations, in which case the geometric non-linearities have to be taken into account. The present paper presents an advanced identification procedure developed to identify the system characteristics from experimental non-linear response curves obtained from forced vibration tests, accounting for fluid-structure interaction, at different excitation levels. Furthermore, the numerical tool developed in this analysis is capable of working with systems presenting one-to-one internal resonance, i.e. systems with symmetry such as circular tubes and circular cylindrical shells. The method relies on a harmonic decomposition of the displacement to cope with the data usually available by vibration measurements.