## 4 The Phenomenon of Modal Coupling
### Top-Hysteresis Configuration (TC)
In Figure 4c, by means of a color map, the maximum displacements of the NC and HVA oscillators are compared for different forcing intensities and frequencies: blue dots refer to reductions of 40 to 85% with respect to the non-controlled case; green dots are related to slight variations and warm colors indicate the cases where the HVA is not beneficial and leads to an increase in the response, even though this is in a small range of low excitation intensities (yellow regions). In the previous case of internal resonance 1:1, a hysteretic nonlinear attachment used to reduce the vibrations of the primary structure, simply simulates the behavior of the classic TMD; therefore, the results described above are similar to what has already been observed within the framework of linear dynamics. Although the advantage of this device lies in its simplicity because hysteretic elements combine elastic and dissipative capacities without the need for a viscous damper [41], the effectiveness is, however, limited to definite excitation intensities. Furthermore, the nonlinearity resides in the constitutive law of the attachment. However, no typical phenomena of nonlinear dynamics are triggered; this novel contribution could be usefully investigated for the mitigation of vibrations, as illustrated below. Figure 4: TC1 oscillator close to a (1:1) internal resonance \(\omega_{2A}/\omega_{1A}\cong 1.3\)): **(a)** FRCs of oscillation amplitudes of mass \(m_{1}\) for increasing intensity (\(a_{g}=[0.005\)–\(0.125]g\)); **(b)** non controlled (black), HVA (red) and TMD (blue) responses at \(a_{g}=0.065g\); **(c)** color map comparing the maximum displacements of NC and HVA oscillators (cold colors reveal amplitude reductions and warm colors reveal amplitude increments). The occurrence of internal resonance conditions \(n\):1, with \(n>1\) integer [43; 44; 45; 46], is more interesting, and the relevant phenomena can be exploited to achieve a reduction in the forced response for certain values of excitation and frequency, as shown in the following. As the hysteretic restoring force is a highly nonlinear constitutive law, various internal resonance conditions are likely to occur. In the following, an oscillator that undergoes a 2:1 internal resonance was analyzed, Figure 5. Similar outcomes could be observed for \(n>2\) and also for other internal resonance conditions of the kind \(n\omega_{1}\cong m\omega_{2}\). On the same main structure of the TC1 system, an HVA was applied with mass \(m_{2}\), and the stiffness \(k_{2}\) was calibrated in order to obtain a frequency ratio close to 2:1; the mechanical parameters of this oscillator (TC2) are reported in Table A2 of the Appendix A. In Figure 5a, the FRCs of the primary mass displacement is reported with increasing excitation intensity. For low intensities, the resonance frequencies were close to those of linear system A (\(\omega_{1A}=5.35\) Hz, \(\omega_{2A}=11.5\) Hz, with an initial value \(r_{A}=\omega_{2A}/\omega_{1A}\cong 2.15\)); for high intensities, the resonance frequencies were close to those of linear system B (with a final ratio \(r_{B}=\omega_{2B}/\omega_{1B}\cong 1.5\)).