## Appendix A Solution of the Nonlinear SDOF System  
For the purposes of the present analysis, it is supposed that the mass, initially at rest, that is, \(u(0)=0\) and \(\dot{u}(0)=0\), is subjected to a force \(f\) at time \(t=0\). For displacements \(u<u_{y}\), the connecting device behaves like an elastic spring and the elastic dynamic equilibrium equation is  
\[m\dot{u}+k_{e}u=f. \tag{1}\]  
Applying the initial conditions the solution of the differential equation is  
\[u=\frac{f}{k_{e}}\left[1-\cos\left(\omega_{e}t\right)\right], \tag{2}\]  
where \(\omega_{e}\) is the angular frequency of the system considering the elastic stiffness \(k_{e}\), that is, \(\omega_{e}^{2}=k_{e}/m\). The solution represents an oscillation centred in \(f/k_{e}\). Equation (2) holds if \(|u|\leq u_{y}\), that is, for \(f\leq 2F_{y}\). If \(f>2F_{y}\) the device would experience an initial elastic phase followed by a plastic deformation. The device yields at a time \(t^{\circ}\) obtained imposing \(u=u_{y}\) in Equation (2). The yielding point is marked with Y in Figure 3. Remembering that \(k_{e}u_{y}=F_{y}\), it results  
\[t^{\circ}=\frac{1}{\omega_{e}}\arccos\left(1-\frac{F_{y}}{f}\right). \tag{3}\]  
At \(t=t^{\circ}\) the speed of the mass is  
\[\dot{u}\left(t^{\circ}\right)=\frac{f}{k_{e}}\omega_{e}\sin\left(\omega_{e}t ^{\circ}\right). \tag{4}\]For \(t>t^{\circ}\) the system is in the plastic phase. The restoring force is  
\[F_{R}=F_{y}+k_{p}\left(u-u_{y}\right). \tag{10}\]  
The dynamic equilibrium equation of this phase is  
\[m\ddot{u}+k_{p}u=f-F_{y}+k_{p}u_{y}. \tag{11}\]  
This second order differential equation, that is, Equation (11), is solved considering the following Cauchy boundary conditions  
\[ \begin{cases}u\left(t^{\circ}\right)=u_{y}\\ \dot{u}\left(t^{\circ}\right)=\frac{f}{k_{e}}\omega_{e}\sin\left(\omega_{e}t ^{\circ}\right)\end{cases} . \tag{12}\]  
It results  
\[u=A\sin\left(\omega_{p}t\right)+B\cos\left(\omega_{p}t\right)+\frac{f-F_{y}}{ k_{p}}+u_{y} \tag{13}\]  
with  
\[A =\frac{F_{y}-f}{k_{p}}\sin\left(\omega_{p}t^{\circ}\right)+\frac{ \dot{u}\left(t^{\circ}\right)}{\omega_{p}}\cos\left(\omega_{p}t\right), \tag{14}\] \[B =\frac{F_{y}-f}{k_{p}}\cos\left(\omega_{p}t^{\circ}\right)-\frac{ \dot{u}\left(t^{\circ}\right)}{\omega_{p}}\sin\left(\omega_{p}t\right). \tag{15}\]  
The velocity of the mass is  
\[\dot{u}=A\omega_{p}\cos\left(\omega_{p}t\right)-B\omega_{p}\sin\left(\omega_{p }t\right). \tag{16}\]  
The system reaches its maximum elongation \(u_{+}\) (point U\({}_{+}\) in Figure 3) when \(\dot{u}=0\), that is, at time \(t_{st}\) computed as  
\[t_{st}=\frac{1}{\omega_{p}}\left[\arctan\left(\frac{A}{B}\right)+k\pi\right], \tag{17}\]  
where \(k\) is either \(0\) or \(1\) depending if \(A/B\) is positive or negative, respectively. The corresponding elongation \(u_{+}\) is obtained inserting \(t_{st}\) into Equation (13). The restoration force in the device evaluated through Equation (10) as  
\[F_{U_{+}}=k_{p}\left[A\sin\left(\omega_{p}t_{st}\right)+B\cos\left(\omega_{p} t_{st}\right)\right]+f-k_{p}u_{y}. \tag{18}\]  
The force in the device (\(F_{U_{+}}>f\)) calls back the mass \(m\), which starts moving with negative velocity. The process occurs along the unloading curve of Figure 3, which a linear elastic relationship (with tangent stiffness \(k_{e}\)) between position and restoring force. The residual displacement, that is, the displacement in the unloaded device is \(u_{r}=u_{+}-\frac{F_{U_{+}}-F_{y}}{k_{e}}+u_{y}\). The equation of motion describing this phase of the process is  
\[m\dot{u}+k_{e}\left(u-u_{r}\right)=f. \tag{19}\]  
The integral of the equation and its initial conditions are  
\[ \begin{cases}u=C\sin\left(\omega_{e}t\right)+D\cos\left(\omega_{e}t\right)+ \frac{f}{k_{e}}+u_{r}\\ u\left(t_{st}\right)=u_{+}\\ \dot{u}\left(t_{st}\right)=0\end{cases} . \tag{20}\]Equation (A15) represents an oscillatory solution between \(u_{+}\) and \(u_{-}=2u_{r}-u_{+}\) (points \(U_{+}\) and \(U_{-}\), respectively), that is, around the position \(u_{n}=f/k_{e}+u_{r}\) (point N in Figure 3).