to take place and leading a fast energy dissipation. Example 2 displays a case where a high dissipation rate inhibits the energy transfer whereas example 1 shows an optimal design, in term of dissipation, which ensures that both the energy transfer and the energy dissipation are achieved.

Transient free response {#sect:Transient}
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In this section, we provide examples of transient free responses which highlight the energy pumping phenomenon and underline the efficiency of the previous modal analysis predictions. We focus on the first example nonlinearity ($\gamma=10^{-3}$) which appears to be more efficient according to the non-linear modes predictions. The exact system of equations (\[eq:mvtR\]) (with $f(t)=0$) was used with the Bouc-Wen restoring force defined by equation (\[eq:BoucWen\]); we impose the initial following initial conditions: $$\dot x(0) = \sqrt{\dfrac{2 \mathcal{H}_0}{M}}, \ x(0)=0, \ \dot v (0) = 0 \ \mbox{ and } v (0) = 0$$ where $\mathcal{H}_0$ is the initial energy of the system. These initial conditions simulate an impact on the main mass. The results of two simulations with different initial energy input are depicted in figures \[fig:TransH15-x\] to \[fig:TransH60-v\]. Beside from the displacements $x$ and $v$ history, the history of the instantaneous frequency is also plotted for both 