## 2 Response to Harmonic Excitation of a Hysteretic SDOF Oscillator
In particular, the equivalent stiffness \(k_{e}\) as a function of \(\overline{x}\) can be expressed in the following form:  
\[k_{e}(\overline{x})=k_{1}+\frac{k_{d}}{\overline{x}(\beta+\gamma)}\Big{(}1-e ^{-\overline{x}(\beta+\gamma)}\Big{)} \tag{6}\]  
According to Equation (6), it decreases with the excitation intensity varying from \(k_{A}\) to \(k_{B}\). To characterize the damping properties, the equivalent damping coefficient \(\zeta_{e}\) was introduced and defined as the value of damping of a viscoelastic oscillator that dissipates the same energy \(E_{BW}\) of the hysteretic device in an oscillation with the same frequency and amplitude \(\overline{x}\):  
\[\zeta_{e}(\overline{x})=\frac{E_{BW}}{2\pi k_{e}\overline{x}^{2}} \tag{7}\]  
Figure 1: (**a**) SDOF Bouc–Wen oscillator; (**b**) first loading branch; (**c**) restoring force loops for \(\gamma/\beta=1\), \(10\), \(50\). The frequency response curves (FRCs) of the hysteretic SDOF (Table A1 in the Appendix A) with \(\gamma/\beta=1,\ 10\) are reported in Figure 2a,b for increasing values of the excitation intensity. In Figure 2a the case of full hysteresis (BW1) is illustrated. The nonlinearity of hysteresis is of the softening type, and the frequency-amplitude curve drawn by the resonance peaks is bent on the left. The nonlinear frequency of the oscillator is close to \(\omega_{A}\) for low amplitudes and decreases, moving towards the post-elastic frequency \(\omega_{B}\) for increasing response amplitudes. The curves are made by stable periodic solutions; the branch nearly vertical which connects the non-resonant and resonant branches is marginally stable [33]. Figure 2b shows a response with reduced hysteresis loops, \(\gamma/\beta=10\) (BW2). For the same excitation intensity, greater response amplitudes are obtained at resonance with respect to the previous case BW1, as evident in Figure 2d. This is expected behavior related to a lower level of equivalent damping, Figure 2c. A more specific difference is the coexistence of the resonant and non-resonant branches in a small range of frequency, leading to multivalued curves with the classical jump phenomenon. This range of coexisting solutions widens with \(\gamma/\beta\), while disappearing with increasing force intensity, as evidenced by different constitutive laws [33]; this is because the dissipation capacity between full and reduced hysteresis is not more appreciable at high amplitudes. The trend of the equivalent damping coefficient \(\zeta_{e}\) versus the response amplitude (Equation (7)) is reported in Figure 2c for fully and reduced hysteretic systems; \(\zeta_{e}\) increases with amplitude, reaches a maximum, and then decreases, which means that in the intermediate range of amplitudes, the dissipation capacity is greater. This is the reason why, although the equivalent stiffness decreases with amplitude, the relation between force and resonance response amplitude is steeper with respect to the ranges of small and large values, Figure 2d. Figure 2: SDOF oscillator: frequency–response curves for varying excitation intensity: (**a**) \(\gamma/\beta=1\), \(A=0.01\)–\(0.63\); (**b**) \(\gamma/\beta=10\), \(A=0.01\)–\(0.57\) (\(\omega_{A}=7.83\) Hz, \(\omega_{B}=4.31\) Hz); (**c**) equivalent damping coefficient \(\zeta_{e}\) vs. response amplitude; (**d**) excitation intensity vs. response amplitude.