## 4 Results
### Parametric Analysis  
A parametric analysis is presented here, exploring the effects of stiffness ratio, horizontal spring preload, and damping.  
Figure 2: Validation and comparison of the KBM solution. (**a**) Comparison between [PERSON]’s closed-form solution, KBM, and direct integration for the linear stiffness oscillator in the presence of dry friction only; (**b**) comparison between KBM and direct integration in the case of \(\lambda=3\), \(\delta_{h}=\left(6\sigma\right)^{-1}\), \(\zeta=0.02\).  
Figure 3a illustrates how the isolator can be tuned by varying the stiffness ratio. Although this may seem intuitive at first glance, it is important to note that, for the mechanism under analysis, this is the only method for modifying the stiffness characteristic without affecting the critical displacement \(u_{cr}\). As shown in Figure 3b, where the frequency response is presented by varying the preload \(\delta_{h}\) while keeping the stiffness ratio \(r\) fixed, the critical displacement \(u_{cr}\) is a function of the horizontal spring preload \(\delta_{h}\): the higher the preload, the higher is \(u_{cr}\). The increment of \(\delta_{h}\) yields a secondary effect, which implies a higher critical force \(F_{cr}\) that is required to trigger the piecewise behavior of the restoring force.  
It is noteworthy that a piecewise nonlinear-linear stiffness mechanisms can also be realized through cam-roller mechanisms [16]. These mechanisms are characterized by a fixed \(u_{cr}\), which is determined by the cam profile, and can potentially operate at amplitudes greater than one. This observation is of outmost importance for engineers. Although the equations of motion for both class of HSLDS systems appear similar, the critical condition differs. Hence, when designing a piecewise isolator, it is crucial to be aware of which is the final sought behavior.  
Figure 4a,b display the effects of dissipation. According to the limit envelope Equation (27), finite amplitude oscillations at resonance cannot be observed when \((\zeta=0,\mu\
eq 0\) ); see Figure 4a. Specifically, when the threshold value \(\mu/F_{0}>\tau/4\) is exceeded, the external forcing amplitude is insufficient to induce oscillations; thus, \(A=0\forall\Omega\). In contrast, the introduction of viscous damping into the system limits the resonance peak amplitude to finite values, as shown in Figure 4b. Further observations can be made by equating the dissipative terms in the first equation of (23) with those obtained in the case of linear viscous damping. This allows for the determination of an equivalent viscous damping ratio [51]:  
\[\zeta_{eq}=\zeta+\frac{2\mu}{\pi\Omega A} \tag{37}\]  
in which the role of dry friction is clear; it is negligible near resonance due to the sudden increase in oscillation amplitude, while it becomes significant at low and high forcing frequencies.  
The value of the critical forcing determined using the Sylvester resultant (34), which corresponds to the detachment of the horizontal springs from the pushrod, is shown in Figure 5. Specifically, the critical forcing is plotted as a function of the horizontal spring preload for different values of the viscous damping, Figure 5a, and dry friction, Figure 5b. As expected, higher viscous damping \(\zeta\) and dry friction forces \(\mu\) result in a higher \(F_{cr}\), with the critical forcing value reaching a plateau region in the case of the QZS mechanism \(\delta_{h}\rightarrow\left(\lambda r\right)^{-1}\). Notably, the roles played by \(\zeta\) and \(\mu\) differ as \(\delta_{h}\to 0\). Viscous damping has no remarkable effects on the critical forcing, with \(F_{cr}\) which tends to \(\pi\mu/4\) for any \(\zeta\). In contrast, an increment of the dry friction \(\mu\) induces a shift toward higher values of \(F_{cr}\).