Contributions E. Gourc wrote the paper and performed the calculations P.O. Mattei, R. Cote and M. Capaldo wrote the paper Data structure and information workflows: workflows - folder containing results for case 1 README.md - Instructions to regenerate figures reproduce.sh - script reproducing figures getSimPoints.m - Computation of the singular point of the SIM getFixedPoint.m - Get Fixed point from the analytical curve sizing.m - Compute the sizing curves (critical forcing amplitude VS nonlinear damping) plotFrc.m - Compute and plot the frequency response curve plotSim.m - Plot the SIM setParam.m - function that take the parameter of the system and put them inside structure getFixedPoint.m - Compute the fixed points of the slow flow equations Paper Description The paper deal with the passive control of resonant systems using nonlinear energy sink (NES).The objective is to highlight the benefits of adding nonlinear geometrical damping in addition tothe cubic stiffness nonlinearity. The behavior of the system is investigated theoretically by usingthe mixed harmonic balance multiple scales method. Based on the obtained slow flow equations,a design procedure that maximize the dynamic range of the NES is presented. Singularity theoryis used to express conditions for the birth of detached resonance cure independently of the forcingfrequency. It is shown that the presence of a detached resonance curve is not necessarily detrimentalto the performance of the NES. Moreover, the detached resonce curve can be completely suppressedby adding nonlinear damping. The results of the design procedure are the compared to numericalsimulations.