## 3 Applying Multiple Scales Lindstedt-Poincare
### Including Friction in MSLP  
Including friction damping in the analysis creates the needs for approaching the sign function that changes the sign regarding the relative angular velocity, which characterizes the friction damping, by, e.g., polynomial expressions. Many ways of this are shown in the literature--for example, by [PERSON] in [26]. One way is to use a Fourier series to approximate the friction term \(sgn(\dot{u})\). Regarding the analysis, only multiples of the angular natural frequency are considered so that the expression can be simplified using only \(exp(in\omega_{0}\tau_{0})\). The term  
\[2 iD_{x}A_{c}=\frac{\omega_{0}}{2\pi}\int\limits_{0}^{2\pi/\omega_{0}}fexp(-i \omega_{0}\tau_{0})d\tau_{0} \tag{40}\]  
with \(n=1\) is of interest for the secular terms. Introducing a new variable \(\phi=\omega_{0}\tau_{0}\) shortens Equation (40) to  
\[\frac{\omega_{0}}{2\pi}\int\limits_{0}^{2\pi/\omega_{0}}fexp(-i\omega_{0}\tau _{0})d\tau_{0}=\frac{1}{2\pi}\int\limits_{0}^{2\pi}fexp(-i\phi)\frac{1}{ \omega_{0}}d\phi. \tag{41}\]This expression can now be placed in the term of the corresponding order of \(\varepsilon\). This way of approximating the sign function also shows that considering analyzing higher orders of \(\varepsilon\) makes the analysis much more complicated, as then higher terms of the Fourier series are needed. The MSLP analysis of the combination of cubic stiffness and the friction contact (Equation (1)) lead to  
\[\ddot{u}+\varepsilon^{2}d^{*}\dot{u}+\omega_{0}^{2}u+\varepsilon\alpha^{*}u^{3 }+\varepsilon^{2}\mu^{*}sgn(\dot{u})=\varepsilon^{2}\Omega^{2}\varphi_{i,j}q^ {*}cos(\Omega t) \tag{42}\]  
with \(\mu^{*}=\frac{\mu}{\varepsilon^{2}\dot{J}}\). By using \(\Omega=\omega\big{(}1+\varepsilon^{2}\sigma\big{)}\) and MSLP coordinates,  
\[\omega^{2}\ddot{u}+\varepsilon^{2}\omega d^{*}\dot{u}+\omega_{0}^{2}u+ \varepsilon\alpha^{*}u^{3}+\varepsilon^{2}\mu^{*}sgn\big{(}\omega\dot{u}\big{)} =\omega^{2}\varepsilon^{2}\varphi_{i,j}q^{*}cos\bigg{(}\frac{\Omega}{\omega} \tau\bigg{)} \tag{43}\]  
result in the same for \(O(1)\) and \(O(\varepsilon)\) as the MSLP analysis for the damper with cubic stiffness (Equations (8) and (20)) and the friction damping terms affect solutions of \(O\big{(}\varepsilon^{2}\big{)}\) and higher. Solving \(O\big{(}\varepsilon^{2}\big{)}\) yields the secular terms  
\[ \begin{split}-2i\omega^{2}D_{2}A_{c}exp(i\tau_{0})+\omega_{2}A_{ c}exp(i\tau_{0})-d^{*}i\omega A_{c}exp(i\tau_{0})-\frac{3\alpha^{*2}}{8\omega^{2}} A_{c}^{3}\overline{A_{c}}^{2}exp(i\tau_{0})\\ -\frac{1}{2\pi}\int\limits_{0}^{2\pi}fexp(-i\phi)\frac{1}{\omega_ {0}}d\phi exp(i\tau_{0})+\frac{\varphi_{i,j}q^{*}\omega^{2}}{2}exp(i(\sigma \tau_{2}+\tau_{0}))=0\end{split}  \tag{44}\]  
where \(\omega_{2}\) is set to zero; otherwise, it would be complex, giving an expression for \(D_{2}A_{c}\)  
\[2 iD_{2}A_{c}=-\frac{id^{*}A_{c}}{\omega}-\frac{3\alpha^{*2}A_{c}^{3}\overline {A_{c}}^{2}}{8\omega^{4}}-\frac{1}{2\pi\omega^{2}\omega_{0}}\int\limits_{0}^{2 \pi}fexp(-i\phi)d\phi+\frac{\varphi_{i,j}q^{*}}{2}exp(i\sigma\tau_{2}). \tag{45}\]  
The term resulting from the Fourier series must now be further analyzed. Therefore, it needs to be considered that  
\[f\big{(}\dot{u}\big{)}=\left\{ \begin{aligned} \mu^{*}\dot{u}>0\\ -\mu^{*}\dot{u}<0\end{aligned} \right. \tag{46}\]  
by which  
\[\int\limits_{0}^{2\pi}fexp(-i\phi)d\phi \tag{47}\]  
can be separated into its real and imaginary parts, leading towards  
\[Im:\int\limits_{0}^{2\pi}-sin(\phi)f\ d\phi=\int\limits_{0}^{\pi}sin(\phi)\mu^ {*}\ d\phi-\int\limits_{\pi}^{2\pi}sin(\phi)\mu^{*}\ d\phi=4\mu^{*} \tag{48}\]  
\[Re:\int\limits_{0}^{2\pi}cos(\phi)f\ d\phi=\int\limits_{0}^{\pi}cos(\phi)(- \mu^{*})\ d\phi+\int\limits_{\pi}^{2\pi}cos(\phi)\mu^{*}\ d\phi=0 \tag{49}\]  
so that the whole expression can be written as \(i4\mu^{*}\). This yields, similar to obtaining Equation (30) in the earlier MSLP, the combined derivative of the complex amplitude by  
\[