## 5 Conclusions  
The aim of this paper is to investigate nonlinear modal interactions in multidegree-of-freedom systems with hysteresis in view of exploiting this phenomenon to achieve mitigation of the forced response. In particular, a simple 2 DOF structure that consisted of two masses and two elements, one elastic and one hysteretic, was investigated by considering two different configurations: top configuration, with the hysteretic element at the top (TC) between the two masses; and base configuration (BC), with the hysteretic element at the base. The nonlinearity of the hysteretic element is described by the Bouc-Wen model and can be classified as a strong nonlinearity due to the notable dependence of stiffness and damping on the oscillation amplitude. The main dynamic effects of hysteresis are first illustrated by the FRCs of a SDOF hysteretic system under harmonic excitation. The FRCs were always single-valued functions, which meant that the steady-state solutions were always stable for the full hysteresis model considered. Due to the reduction in stiffness with amplitude, the nonlinearity was of the softening type: the resonance frequency decreased from \(\omega_{A}\) for small amplitudes, to \(\omega_{B}\) for high amplitudes, where \(\omega_{A}\) and \(\omega_{B}\) were the frequencies of the linear oscillator with a stiffness that was, respectively, equal to the initial and post-elastic stiffness of the hysteretic element. The nonlinear behavior the 2 DOF system, as already observed for the SDOF, is again comprised between those of two linear systems: system A for small amplitudes, system B for large amplitudes. The modal properties of the system change with the oscillation amplitude. Accordingly, natural frequencies, and their ratio, changed and the conditions  
Figure 8: (**a**) color map of the maximum response of non-resonant system BC1n for different forcing frequencies and intensities; (**b**) non-resonant (BC1n) and resonant (BC1) FRCs responses at \(A=0.10\); (**c**) color map comparing the maximum displacements of BC1 and BC1n oscillators (cold colors reveal amplitude reductions, warm colors indicate amplitude increments, green indicates slight variations). Figure 9: BC2 oscillator close to (2:1) internal resonance (\(\omega_{2A}\)/\(\omega_{1A}\cong 2.09\)): (a) FRCs for increasing force intensity (\(A=\) 0.015–0.4), red \(A=\) 0.32; (**b**) color map highlighting the main branches. of internal resonance, where the phenomena of nonlinear interaction are exalted, were easily met. For the top configuration, first, the case where the hysteretic device had a frequency close to that of the main structure was considered, which recalled the case of the well-known tuned mass damper. The hysteretic element, which combined stiffness and dissipation characteristics, made it possible to avoid the viscous device, but the performance was similar to that of the viscoelastic damper, only in a certain range of force intensity, due to the dependence on the oscillation amplitude. However, the introduction of a hysteretic element could take advantage of the phenomena of nonlinear interactions, which occurred when the system was close to internal resonance conditions \(n\):1 with \(n>1\). The 2:1 condition was investigated for TC systems, and the 3:1 and 2:1 conditions were investigated for BC systems. A large variety of behaviors were observed; however, concerning the beneficial effects in vibration mitigation, similar outcomes were found for both configurations. At a certain force intensity, the FRCs around the resonance of the first mode exhibited a bifurcation with the arising of a novel mode. The existence of two resonant peaks around the first mode was the result of the involvement of the second mode, which strongly modified the modal trajectories and, mainly, the response amplitude.