## 4 Results
### Numerical Example  
In this subsection, a detailed analysis of a specific mechanism is presented, providing an example on how the methods and approximate solutions derived previously can be used for the qualitative design and analysis of the isolator.  
The critical deflection \(u_{cr}\) and forcing amplitude \(F_{cr}\) are key parameters for evaluating the isolator's behavior. Assuming the parameters listed in Table 1 and using Equations (5) and (34), we obtain \(u_{cr}=0.3970\) and \(F_{cr}=0.0300\) as indicated by the red dots in Figure 6a,b.  
Figure 5: Critical forcing amplitude \(F_{cr}\) as a function of the horizontal spring preload \(\delta_{h}\). (**a**) Role of the viscous damping, with \(\lambda=3\), \(r=2.03\), \(\mu=0.012\); (**b**) role of the dry friction force \(\mu\), with \(\lambda=3\), \(r=2.03\), \(\zeta=0.02\).  
Substituting these values into (24) allows for the evaluation of the critical forcing frequency \(\Omega_{cr}=0.9288\). The onset of the piecewise restoring force activation is outlined by the black thick lines, beyond which the push-rod detaches from the horizontal spring for any forcing amplitude \(F\) within the hatched area. It is important to note that as \(u_{cr}\) increases with higher values of \(\delta_{h}\), the rate of change in \(F_{cr}\) decreases as \(\delta_{h}\) approaches \((3r)^{-1}\cong 0.16\). Beyond this threshold, the analysis is no longer of interest to the present study, as it would lead to the emergence of negative stiffness mechanism.  
Figure 6(a) presents the amplitude-frequency diagram for several forcing amplitudes. As shown in the validation analysis, the agreement between the approximate and numerical solution is acceptable. For \(F_{0}\leq~{}F_{cr}\), the system behaves as a nonlinear oscillator with quintic stiffness nonlinearity and mixed damping. For \(F_{0}>F_{cr}\), the piecewise nonlinearity of the restoring force is triggered, and the resonance peak shifts toward \(\Omega=1\). Using the procedure described in Section 3.2, the Sylvester criterion allows to determine the forcing amplitude \(F_{SN}=0.0217\) that prevents the system from experiencing saddle-node bifurcations. For \(F_{0}\leq~{}F_{SN}\), the isolator exhibits a predictable behavior without showing response multistability. When \(F_{0}=F_{SN}\), the limit inflection point occurs at \((\Omega_{SN},A_{SN})=(0.7762,0.1825)\). For \(F_{0}>F_{SN}\), the stability analysis is carried out to evaluate the occurrence of saddle-node bifurcations, and the jump-up/jump-down frequencies and amplitudes, evaluated using Equations (24) and (30), are listed in Table 2. In Figure 6(b), the phase diagram emphasizes the role of damping. At low forcing amplitude, the phase shift is primarily governed by dissipation while, as the forcing amplitude increases, the phase shift approaches that of an undamped Duffing oscillator.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\lambda\) & \(r\) & \(\delta_{h}\) & \(\mu\) & \(\xi\) \\ \hline
3 & 3747 & 1848 & \(\frac{1}{6r}\) & 1000 \\ \hline \end{tabular}
\end{table}
Table 1: Input parameters of the detailed analysis.  
Figure 6: Evaluation of the critical (**a**) displacement \(u_{cr}\) and (**b**) forcing amplitude \(F_{cr}\) with the set of parameters given in Table 1.