free vibrations or forced vibrations, by substituting the proper function $g$ in equations (\[eq:SlowMoy\]).

In the system subject of this paper, the general term $g({\mathbf{z}},\dot{{\mathbf{z}}},t)$ includes, along with linear terms, nonlinear dissipative terms defined by the differential Bouc-Wen model. As these terms are difficult to average in close form, we have used a numeric approach. However, in order to proceed with the analytical calculations, we introduce a similar notation to equation (\[eq:solapprox\]) for the averaged nonlinear restoring force $r$ of the Bouc-Wen model. The resulting amplitude $a_r$ and phase $\varphi_r$ are derived numerically by substituting $v(t)=a_v\cos(\tau+\varphi_v)$ in relation (\[eq:BoucWen\]) and performing an averaging on the fast time scale (as in equation (\[eq:SlowMoy\])): $$\label{eq:TermNLHarm}
    a_r e^{i\varphi_r} = \dfrac{1}{2\pi} \int_0^{2\pi} r(\dot v,t) e^{-i\psi} d\psi$$ This step introduce no additional approximation or simplification to the averaging procedure of equation (\[eq:SlowMoy\]). Details on the variations of the averaged hysteretic restoring force with the non-linear displacement $v$ are provided in section \[ssect:NLM\_Num\] where the equivalent stiffness and damping are studied.

Non-linear modes {#sect:NonlinearModes}
================

In section, we perform a modal analysis of the system (\[eq:mvtR\]) unforced and undamped. Modal studies of non-linear systems are numerous in the literature [@Rosenberg-Ndof; @Shaw-Pierre; @Vakakis-NNM; @Wanda; @Wanda-NLM] and the 