## 2 Response to Harmonic Excitation of a Hysteretic SDOF Oscillator  
Several models have been proposed in the literature to represent hysteretic behavior. For its simplicity and versatility, the differential model by Bouc-Wen is used here; by properly tuning the model parameters, it is possible to capture the real mechanical characteristics of many mechanical devices with hysteresis [29; 30; 31]. A single-degree-of-freedom (SDOF) oscillator with Bouc-Wen hysteresis and under harmonic excitation was first investigated, Figure 1a. The mass \(m_{1}\) was connected to the moving support by an element that provides a restoring force \(f\) due to an elastic and a hysteretic component:  
\[f=k_{1}x_{1}+z \tag{1}\]  
The elastic component was characterized by stiffness \(k_{1}\), whereas the hysteretic part \(z\) was provided by the Bouc-Wen law by solving the following nonlinear differential equation:  
\[\frac{dz}{dt}=\left\{k_{d}-\left[\gamma+\beta\,sgn\left(z\frac{dx_{1}}{dt} \right)\right]|z|^{n}\right\}\frac{dx_{1}}{dt} \tag{2}\]  
where \(k_{d},\ \gamma,\ \beta,\ n\) are the constitutive parameters of the Bouc-Wen law. The element stiffness depended on the oscillation amplitude, which varied from an initial value \(k_{A}=k_{1}+k_{d}\) to a final value \(k_{B}\) corresponding to the post-elastic stiffness, \(k_{B}=k_{1}\), Figure 1b. The yield-restoring force \(f_{y}\) and the yield displacement \(x_{y}=f_{y}/k_{A}\) are reported in Figure 1b. The ratio between the final and initial stiffness values is defined as the hardening coefficient and denoted by \(\delta\): \(\delta=k_{B}/k_{A}\). The parameter \(n\) in Equation (2) defines the degree of smoothness of the transition from the elastic to post-elastic branchesand is assumed here to be unitary, as in most mechanical applications. The parameters \(\beta\) and \(\gamma\) tune the extent of the restoring force loop; more precisely, their sum \((\beta+\gamma)\) affects the maximum value of the hysteretic force, while their ratio \(\gamma/\beta\) controls the shape of the cycles and, consequently, the dissipative capacity of the element, Figure 1c. In particular, for fixed \((\beta+\gamma)\), the maximum loop area, and therefore, the maximum energy dissipation is obtained when \(\gamma=\beta\)[7]; under this condition, the restoring force is defined as fully hysteretic. For \(\frac{\gamma}{\beta}>1\), there is a smaller cycle area and, therefore, decreasing dissipated energy; a reduced hysteresis is said to occur in this case. The equation of motion of a SDOF under a harmonic base excitation, which is characterized by amplitude \(a_{g}\) and frequency \(\Omega\), is:  
\[m_{1}\frac{d^{2}x_{1}}{dt^{2}}+k_{1}x_{1}+z=-m_{1}a_{g}\sin(\Omega t) \tag{3}\]  
where \(z\) is provided by Equation (2). It is useful to introduce some quantities related to the parameters of the Bouc-Wen hysteretic element with clear mechanical meanings, such as the hardening coefficient \(\delta\), the yield strength \(f_{y}\) and the yield displacement \(x_{y}\). These can be expressed as:  
\[\delta=\frac{k_{B}}{k_{A}},\ \ f_{y}=\frac{k_{A}}{k_{d}}\bigg{(}\frac{k_{d}}{ \beta+\gamma}\bigg{)}^{\frac{1}{\delta}},\ \ x_{y}=\frac{f_{y}}{k_{A}}, \tag{4}\]  
Additionally, the initial and final frequency, \(\omega_{A}\) and \(\omega_{B}\), which are associated with small and large oscillation amplitudes, the adimensional force intensity \(A\) and the driven frequency \(\eta\) are defined as:  
\[\omega_{A}=\sqrt{\frac{k_{A}}{m}},\ \ \omega_{B}=\sqrt{\frac{k_{B}}{m}},\ \ A= \frac{ma_{g}}{f_{y}},\ \ \eta=\frac{\Omega}{\omega_{A}} \tag{5}\]  
As demonstrated in [4], the stiffness and damping characteristics of the hysteretic absorber depend on the vibration amplitude \(\overline{x}\).