## 5 Conclusions  
The focus of the present study is to develop an accurate yet simple model to investigate the dynamic behavior of HSLDS isolators. Experimental observations in [14; 39] have shown that, depending on the isolator's design, friction and discontinuous restoring forces significantly influence the vibration mitigation provided by the isolators. Therefore, these factors must be modelled to achieve an accurate representation of real systems. Under the assumption of harmonic load, the primary resonance analysis of HSLDS isolators with piecewise nonlinear-linear stiffness, viscous damping, and dry friction is conducted. Using the Krylov-Bogoliubov-Mitropolsky (KBM) averaging method, the theoretical formulation is derived, with particular attention to the parameters relevant for the design of the isolator. The study validates the KBM approach against closed-form solutions and direct integration of the governing equation, demonstrating its effectiveness in predicting the qualitative system's behavior. The conclusions are as follows:  
(1) To the best of the author's knowledge, this study presents the first investigation of the primary resonance analysis of HSLDS isolators with piecewise nonlinear-linear characteristic, viscous damping, and dry friction forces. The findings demonstrate that KBM method is well suited for qualitative analyses, as it provides algebraic equations that can be solved with ease, potentially aiding engineers in the design of systems for vibration isolation. (2) The parametric analysis shows that, for this type of HSLDS mechanism, increasing the preload of the horizontal springs \(\delta_{h}\) has a twofold effect: (i) it reduces the natural frequency of the system and (ii) it increases the critical displacement \(u_{cr}\) required to activate the piecewise restoring force. Additionally, a relationship for the equivalent viscous damping is provided, demonstrating the dependence between dry friction, forcing amplitude, and frequency, which explains the limited effect of dry friction near resonance conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{F_{0}}\)} & \multicolumn{2}{c}{**Jump-Up**} & \multicolumn{2}{c}{**Jump-Down**} \\ \cline{2-5}  & \(\mathbf{\Omega}\) & \(A\) & \(\mathbf{\Omega}\) & \(A\) \\ \hline \(F_{SN}\) & - & - & - & - \\
0.025 & 0.8023 & 0.1722 & 0.8318 & 0.2893 \\ \(F_{cr}\) & 0.8287 & 0.1842 & 0.9288 & \(u_{cr}\) \\
0.040 & 0.8672 & 0.2052 & 0.9938 & 0.5536 \\
0.050 & 0.8976 & 0.2222 & 1.0175 & 0.5876 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Saddle node bifurcation points. Figure 7: Steady-state dynamic response of the oscillator for different forcing amplitude vales \(F_{0}\). (a) Amplitude–frequency curves and stability region and, (b) phase. (3) The analysis of the Silvester resultant indicates that, regardless of the preload \(\delta_{h}\), higher dry friction forces are beneficial in increasing the critical forcing value \(F_{cr}\) required to activate the piecewise restoring force nonlinearity. In contrast, increasing viscous damping is effective only at higher \(\delta_{h}\) (i.e., for QZS systems), while its effect on \(F_{cr}\) is negligible as \(\delta_{h}\to 0\). (4) A detailed analysis, carried out on a specific test case, provides a comprehensive example on how to use the methodology object of the study. Leveraging the Sylvester resultant, critical forcing amplitudes, resulting in the onset of piecewise restoring force, are determined along with the condition to prevent jump-up/jump-down phenomena. Saddle-node bifurcations are determined through the stability analysis, and numerical values are reported in a table for comparison and validation of future studies. (5) The approach presented herein has two limitations: (i) the adopted friction model is suitable for the analysis of low-friction systems under the assumption of continuous motion; (ii) for friction-to-external force amplitude ratios close to the threshold \(\mu/F_{0}=\pi/4\), KBM performs poorly.