## 3 Primary Resonance Analysis
### Critical Forcing Amplitudes \(F_{s}\) and \(F_{cr}\)  
It is well known that nonlinear isolators governed by a Duffing-like equation may exhibit saddle-node bifurcations. If the forcing amplitude exceeds a critical value \(F_{SN}\), sudden oscillation jumps occur near resonance conditions. To determine the critical forcing value, the approach discussed by [PERSON] and [PERSON] [17], based on the Sylvester resultant of two polynomials [47], is adopted here. For excitation amplitudes below the detachment condition, i.e., \(A<\lvert u_{cr}\rvert\), the frequency response displays an inflection point \((\Omega_{s},A_{s})\), with Equation (24) providing three roots, each counted with its own multiplicity. Consideringthe derivatives of the frequency-response curve \(\mathcal{F}(\Omega,A)\) with respect the variable \(A\), the resultant \(\mathcal{R}\), expressed as a function of \(\Omega\), has one positive real root at the inflection point if  
\[\mathcal{R}\left(\frac{\partial\mathcal{F}}{\partial A},\frac{\partial^{2} \mathcal{F}}{\partial A^{2}}\right)\bigg{|}_{A=A_{s}}=\sum_{\begin{subarray}{ c}i=0,\\ i\ even\end{subarray}}^{36}b_{i}A_{s}^{i}=0 \tag{33}\]  
Polynomial (33) can be solved numerically, and the minimum real positive root provides \(A_{s}\). By substituting \(A_{s}\) into \(\frac{\partial^{2}\mathcal{F}}{\partial A^{2}}=0\), the critical frequency \(\Omega_{s}\) is obtained. Lastly, the forcing amplitude \(F_{SN}\) is provided by the evaluation of \(\mathcal{F}(\Omega_{s},A_{s})=0\).  
Another parameter useful to the design of the system is the critical forcing amplitude \(F_{cr}\) and frequency \(\Omega_{cr}\) leading to the detachment of the horizontal springs from the pushrod, which activates the restoring force piecewise nonlinearity. Those values can be determined by assuming \(A=u_{cr}\) within (24) and finding the stationary points of \(\frac{\partial\mathcal{F}}{\partial\Omega}=0\), which yields a depressed cubic in \(\Omega\), in which one root identifies the critical value \(\Omega_{cr}\). Substituting \(\Omega_{cr}\) into (24) leads to an equation of the variable \(F_{cr}\). This approach can be easily implemented, but it requires to work with a fixed value of the preload \(\delta_{h}\); therefore, it does not show the relationship \(F_{cr}=f(\delta_{h})\).  
A possible approach to determine \(F_{cr}=f(\delta_{h})\) is applying the Sylvester resultant as follows. Under the assumption of \(A=u_{cr}\), the polynomial resultant is constructed from the frequency response curve \(\mathcal{F}(\Omega,A)\) and its first derivative \(\frac{\partial\mathcal{F}}{\partial\Omega}\) as  
\[\mathcal{R}\Bigg{(}\left.\mathcal{F}\right|_{A=u_{cr}},\left.\frac{\partial \mathcal{F}}{\partial\Omega}\right|_{A=u_{cr}}\Bigg{)}\Bigg{|}_{\Omega=\Omega _{cr}}=\sum_{\begin{subarray}{c}i=0,\\ i\ even\end{subarray}}^{6}c_{i}(\delta_{h})F_{cr}^{i}=0 \tag{34}\]  
where \(c_{i}(\delta_{h})\) represents the \(i\)th polynomial in variable \(\delta_{h}\). Equation (34) is a nonlinear algebraic equation of variables \(F_{cr}\) and \(\delta_{h}\). The critical forcing \(F_{cr}\) can be determined by using numerical methods by spanning the preload in the range \(\delta_{h}\in\left[0,\left(\lambda r\right)^{-1}\right]\), which covers a wide range of cases, from HSLDS to the QZS mechanism.