## 4 The Phenomenon of Modal Coupling
### Top-Hysteresis Configuration (TC)  
The top-hysteresis configuration models a primary structure, represented by the mass \(m_{1}\), connected to the base, and an attachment, represented by the mass \(m_{2}\), connected to the main structure with a hysteretic element. The hysteretic device, when properly designed, can mitigate the vibrations of the main structure using a mass \(m_{2}\) which is much smaller than \(m_{1}\) (light attachment). In the linear field, the importance of the [PERSON] viscoelastic tuned mass damper (TMD) [34; 35] is well recognized. By adding a linear attachment with proper parameters, the peak exhibited by the FRC of the primary system response could be divided into two much lower peaks of equal value. In recent years, much work has been devoted to nonlinear attachments after the pioneering work of [36]; however, few of these have dealt with hysteretic devices [3; 4; 6; 37; 38; 39; 40; 41; 42] or related to friction damping [43]. In the following, a hysteretic vibration absorber (HVA) was adopted, and therefore, the calibration of the parameters should be more accurate. In fact, as seen in the previous section, the 2 DOF resonance frequencies depend on the oscillation amplitude; thus, at the design stage, it is necessary to take into account the oscillation amplitudes reached by the primary system to obtain the suitable frequency ratio \(\omega_{2}/\omega_{1}\) that is close to optimal. In order to simulate the behavior of the TMD, it is necessary to tune the characteristics of the added mass \(m_{2}\) and of the constitutive law so that the frequency of the primary system is similar to that of the hysteretic attachment, approaching the 1:1 internal resonance condition. This requires greater attention, as shown by the analysis of the following cases. Figure 4 depicts the response to the harmonically excited 2 DOF system TC1; its mechanical characteristics are provided in Table A2 of the Appendix A. The system presents a ratio of the initial frequencies \(\omega_{2A}/\omega_{1A}\cong 1.3\), which is greater than the optimal value of 1.25 of the TMD. As the amplitude increases, the frequency ratio approaches optimum, leading to a reduction in the response for a definite range of forcing amplitudes. The frequency response curves of the primary structure displacement are reported in Figure 4a with increasing excitation. For the optimal operation of the HVA, in analogy with the behavior of the TMD, the external excitation should generate two equal peaks in the FRC of \(m_{1}\); this occurs for \(a_{g}=0.065g\), which corresponds with the red curve in Figure 4a,b. For lower intensities, the equivalent stiffness \(k_{e}\) of the device is lower than the optimal stiffness value of TDM, and as a result, the first peak exceeds the second. On the other hand, for larger amplitudes, the equivalent stiffness of the device exceeds the optimal stiffness of the TMD, and consequently, the second peak predominates. When the amplitude of the external excitation is close to \(a_{g}=0.065g\), the ratio \(\frac{\omega_{2}}{\omega_{1}}\) of the system with a hysteretic attachment is close to the value of the optimal visco-elastic tuned mass damper. At this level of excitation, the HVA is as effective as the TMD; in fact, in Figure 4b, there is an 80% reduction in the oscillation amplitudes compared to those that would occur on the non-controlled primary structure (NC). As shown in Figure 4c, the effectiveness of the HVA is maintained even for higher values of the external intensity, provided that the driven frequency is close to the first resonance.