## 3 Primary Resonance Analysis  
### Periodic Motion  
The periodic motion in the neighborhood of the primary resonance is analyzed using the Krylov-Bogoliubov-Mitropolsky (KBM) averaging method [36,43]. To this end, Equation (3) is rescaled with respect to a small parameter \(\epsilon\)[44,45] as  
\[u^{\prime\prime}+\Omega^{2}u=eG\left(u,u^{\prime},\Omega\tau\right) \tag{8}\]where  
\[G\big{(}u,u^{\prime},\Omega\tau\big{)}=F_{0}\text{cos}(\Omega\tau)-2\zeta u^{ \prime}-\mu\ \text{sign}(u)-F_{k}(u)+\sigma u \tag{9}\]  
and  
\[\overline{\zeta}=e\zeta,\ \overline{\mu}=e\mu,\ \overline{F}_{0}=eF_{0},\ \overline{F}_{k}=eF_{k},\ \Omega^{2}=1+e\sigma,\ \epsilon\ll 1 \tag{10}\]  
\(G(u,u^{\prime},\Omega\tau)\) is periodic with period \(T=2\pi/\Omega\), and it acts as a perturbation of the underlying linear oscillator. The first-order approximate solution of (8) is assumed to have slowly varying amplitude \(A=A(\tau)\) and phase \(\phi=\phi(\tau)\) as  
\[ \begin{array}{c}u=A\text{cos}\,\psi\\ u^{\prime}=-A\Omega\text{sin}\,\psi\end{array}  \tag{11}\]  
where \(\psi=\Omega\tau+\phi\). The time differentiation of the first equation of (11) leads to  
\[u^{\prime}=A^{\prime}\text{cos}\,\psi-A\Omega\text{sin}\,\psi-A\phi\prime\text {sin}\,\psi \tag{12}\]  
and comparing (12) to the second equation of (11), it requires the following condition to be satisfied:  
\[A^{\prime}\text{cos}\,\psi-A\phi\prime\text{sin}\,\psi=0 \tag{13}\]  
Differentiating the second equation of (11) gives  
\[u^{\prime\prime}+\Omega^{2}u=-A^{\prime}\Omega\text{sin}\,\psi-A\Omega\phi \prime\text{cos}\,\psi \tag{14}\]  
and substituting it into Equation (8) results in  
\[A^{\prime}\Omega\text{sin}\,\psi+A\Omega\psi\text{cos}\,\psi=-eG(A\text{cos} \,\psi,-A\Omega\text{sin}\,\psi,\psi-\phi) \tag{15}\]  
The set of Equations (13) and (15) is linear with respect to the variables \(A^{\prime}\) and \(A\phi\prime\):  
\[ \begin{pmatrix}\text{cos}\,\psi&-\text{sin}\,\psi\\ \text{sin}\,\psi&\text{cos}\,\psi\end{pmatrix}\begin{pmatrix}A^{\prime}\\ A\phi\prime\end{pmatrix} = \begin{pmatrix}0\\ -(\epsilon)G(A\text{cos}\,\psi,-A\Omega\text{sin}\,\psi,\psi-\phi)\end{pmatrix}  \tag{16}\]  
Solving the linear system (16), it is possible to obtain  
\[ \begin{array}{c}A^{\prime}=-\frac{\epsilon}{\Omega}G(A\text{cos}\,\psi,-A \Omega\text{sin}\,\psi,\psi-\phi)\text{sin}\,\psi\\ A\phi^{\prime}=-\frac{\epsilon}{\Omega}G(A\text{cos}\,\psi,-A\Omega\text{sin}\, \psi,\psi-\phi)\text{cos}\,\psi\end{array}  \tag{17}\]  
Averaging System (17) over period \(2\pi\), the slow variations of \(A\) and \(\phi\) are  
\[ \begin{array}{c}A^{\prime}=-\frac{\epsilon}{2\pi\Omega}\int_{0}^{2\pi}G(A \text{cos}\,\psi,-A\Omega\text{sin}\,\psi,\psi-\phi)\text{sin}\,\psi\,d\psi\\ A\phi^{\prime}=-\frac{\epsilon}{2\pi\Omega}\int_{0}^{2\pi}G(A\text{cos}\,\psi,-A \Omega\text{sin}\,\psi,\psi-\phi)\text{cos}\,\psi\,d\psi\end{array}  \tag{18}\]  
System (18) is affected by two nonlinearities: the Coulomb friction and the restoring force. Considering that \(u^{\prime}<0\ \forall\ \psi\in(0,\ \pi)\) and \(u^{\prime}>0\ \forall\ \psi\in(\pi,2\pi)\), the modulation Equation (18) is reduced to the standard form:  
\[\Gamma(\mathbf{q},\Omega)= \begin{cases}A^{\prime}=-\frac{\epsilon}{2\Omega} \Big{(}F_{0}\text{sin}\,\phi+2\zeta\Omega A+\frac{4}{\pi}\mu\Big{)}\\ A\phi^{\prime}=-\frac{\epsilon}{2\Omega}\Big{(}F_{0}\text{cos}\,\phi+\sigma A+H_ {i}\Big{)}\end{cases}  \tag{19}\]where \(\mathbf{q^{T}}=[A,\,\phi]\) and  
\[H_{i}=\frac{1}{2\pi}\int_{0}^{2\pi}f_{k}(A\cos\psi)\cos\psi\ d\psi \tag{20}\]  
with \(i\) = 1,2 depending on the value of \(A\).