## 3 Primary Resonance Analysis
### Periodic Motion
If response amplitude \(A\leq|u_{cr}|\), the system is not affected by the stiffness discontinuity; thus,  
\[H_{1}=(1-\alpha)A-\frac{3}{4}\beta A^{3}-\frac{5}{8}\gamma A^{5} \tag{21}\]  
while, if response amplitude \(A>|u_{cr}|\), assuming \(u_{cr}=A\cos\psi_{0}\), Integral (20) must be split over four intervals:  
* \(|u|\leq|u_{cr}|\ \forall\ \psi\in[-\psi_{0},\psi_{0}]\cup[\pi-\psi_{0},\pi+\psi_{0}]\),
* \(|u|>|u_{cr}|\ \forall\ \psi\in[\psi_{0},\pi-\psi_{0}]\cup[\pi+\psi_{0},2\pi-\psi_{0}]\). Hence,  
\[ \begin{array}{ll}H_{2}=(1-\alpha)A-&\frac{3}{8}\beta A^{3}-\frac{5}{8} \gamma A^{5}\\ &+\frac{1}{\pi}\big{[}\big{(}2(\alpha-1)A+\frac{3}{2}\beta A^{3}+\frac{5}{4} \gamma A^{5}\big{)}\psi_{0}\\ &+\Big{(}(\alpha-1)A+\beta A^{3}+\frac{15}{16}\gamma A^{5}\big{)}\sin 2\psi_{0} \\ &+\Big{(}\frac{1}{8}\beta A^{3}+\frac{3}{16}\gamma A^{5}\Big{)}\sin 4\psi_{0}+ \frac{1}{48}\gamma A^{5}\sin 6\psi_{0}\Big{]}\end{array}  \tag{22}\]  
In proximity of the primary resonance, stationary oscillations can be studied from the fixed points of (19) as  
\[ \begin{array}{ll}F_{0}\!\sin\phi+2\zeta\Omega A+\frac{4}{\pi}\mu=0\\ F_{0}\!\cos\phi+\sigma A+H_{i}=0\end{array}  \tag{23}\]  
Collecting trigonometric terms at the right-hand-side, squaring and adding side by side, the frequency-response curve is determined as follows:  
\[ \begin{array}{ll}\mathcal{F}(\Omega,A)=A^{2}\Omega^{4}+\big{(}4A^{2}\zeta^{ 2}+2A(H_{i}-A)\big{)}\Omega^{2}+\frac{16}{\pi}\mu\zeta A\Omega+\frac{16}{\pi^ {2}}\mu^{2}+(H_{i}-A)^{2}\\ -F_{0}^{2}=0\end{array}  \tag{24}\]  
which is a nonlinear algebraic equation. If \(\mu\
eq 0\), the solution must be obtained numerically while, in the case of negligible Coulomb friction, a closed-form solution can be obtained by solving a quadratic algebraic equation in \(\Omega^{2}\). The slowly varying phase \(\phi(\Omega)\) is retrieved from Equation (23) and it reads as follows:  
\[\tan(\phi)=\frac{2\zeta\Omega A+\frac{4}{\pi}\mu}{\left(\Omega^{2}-1\right)A+H _{i}} \tag{25}\]  
Using the definition of [PERSON] and [PERSON] [46], the backbone equation can be obtained from the second equation of (23) by assuming \(\cos\left(\phi\right)=0\) as  
\[A_{c}=\frac{H_{i}}{1-\Omega^{2}} \tag{26}\]  
Further considerations are possible from the evaluation of the critical forcing value to overcome the sticking condition, which can be obtained by the limit envelope under the hypothesis of \(A<|u_{cr}|\) where the isolator is not affected by the restoring force discontinuity. Setting \(\sin\left(\phi\right)=\pm 1\) from the first equation of (23), the limit envelope curve is  
\[A_{L}=\frac{F_{0}-\frac{4}{\pi}\mu}{2\zeta\Omega} \tag{27}\]Analyzing (27), it is possible to recall what is reported in Ref. [46]: in the absence of viscous damping, \(\zeta\to 0\), the limit envelope amplitude \(A_{L}\rightarrow\infty\); thus, dry friction alone cannot guarantee a bounded response in the case of resonant load with \(F_{0}>\frac{4}{\pi}\mu\). If the inequality is not satisfied, the excitation frequency is insufficient to induce vibrations of the isolator; thus, the system lies in the sticking region. The same condition could have been obtained equivalently by imposing \(A\to 0\) in the amplitude-frequency Equation (24) as shown in [28].