## 4 The Phenomenon of Modal Coupling
### Base-Hysteresis Configuration (BC)  
To explore the ability of a hysteretic isolator to reduce the transmission of excitation from the base to the superstructure by nonlinear modal coupling, the system with a hysteretic element at the base (Figure 3b) and with the BC1 characteristics (reported in Table A2 of the Appendix A), is considered in Figure 6. The small amplitude frequencies were \(\omega_{1A}=3.89\) Hz and \(\omega_{2A}=11.32\) Hz with a ratio \(\omega_{2A}/\omega_{1A}=2.90\), which are close to the internal resonance condition 3:1. Figure 6a focuses on the FRCs of the \(m_{2}\) displacement amplitude centered on the first resonance. Initially, for small excitation intensities, the single peak \(I\) is present close to the first natural frequency; however, with increasing intensity, at the critical value \(A=0.024\)  
Figure 6: BC1 oscillator close to a (3:1) internal resonance (\(\omega_{2A}/\omega_{1A}\cong 2.90\), \(\omega_{2B}/\omega_{1B}\cong 3.0\)): (**a**) FRCs around the first resonance for increasing excitation amplitude (\(A=0.007\)–\(0.14\)), optimal intensity \(A=0.10\) (red); (**b**) color map of the maximum response normalized to the excitation amplitude for different excitation frequencies and intensities (warm colors reveal peaks related to modes). a second peak \(I_{2}\) appears which grows until it reaches \(I_{1}\), at \(A=0.1\) (red _FRC curve_), and then overcomes it. As shown by the color map in Figure 6b, the fundamental branch \(\omega\)-A of the first resonance experiences a bifurcation and then shows a new branch in addition to the preexisting one, which is destined to disappear; the new branch \(I_{2}\) is the only surviving one for large excitation intensities. It is worth noticing that the nonlinear modal trajectories in the physical plane \(x_{1}-x_{2}\), related to the steady-state responses of \(I_{1}\) and \(I_{2}\), always exhibit opposite curvatures. This is shown in Figure 7: the novel mode \(I_{2}\) arises at \(A=0.024\), showing a \(\int\)-shaped modal trajectory, whereas the preexisting mode \(I_{1}\) exhibits a \(\int\)-shaped one; as soon as \(I_{2}\) prevails over \(I_{1}\) (\(A>0.1\)), its modal trajectory becomes \(\int\)-shaped, while the other turns \(\int\)-shaped and disappears. Finally, along both branches, the nonlinear modal shapes evolve, passing from the linear mode \(A\), for small excitation, to the linear mode \(B\), for high excitation, Figure 3d. The comparison of the dynamic behavior of two different systems, BC1, close to an internal resonance condition 3:1, and BC1n, far from an internal resonance condition (Table A2 in the Appendix A) with a frequency ratio \(\omega_{2A}/\omega_{1A}=3.40\), makes it possible to clearly illustrate the favorable effects of nonlinear modal interactions [1, 2, 29]. For BC1n, in this case, there is only a single branch related to the first resonance, Figure 8a, which, as expected, occurs at lower frequencies for increasing forcing intensities due to hysteretic softening. On the contrary, the maximum displacement of the two masses is much smaller for the system BC1 in conditions of internal resonance, with a reduction of about 70% for \(m_{2}\) with respect to the non-resonant case (BC1n), Figure 8b. As shown in Figure 8c, the closeness to internal resonance conditions leads to a relevant displacement reduction in a wide range of forcing frequency and amplitude in the blue regions. Internal resonance conditions \(n\):1 with \(n\) even can also occur. As an example, a BC2 system close to internal resonance 2:1 is considered in Figure 9.  
At small intensities, the amplification of the second mode \(II\) is much smaller than the first \(I_{1}\) due to the distribution of excitation. For increasing intensity, Figure 9a, the zone of the first resonance widens, and a novel peak \(I_{2}\) emerges and reaches the peak \(I_{1}\) near \(A=0.32\), leading to a flattening of the FRCs, with a notable reduction in the first resonance peaks in a large range of force intensities.