## 3 Applying Multiple Scales Lindstedt-Poincare
### Applying MSLP on the Damper with Cubic Nonlinearity  
In the following, the MSLP method is applied to the drill string model (Equation (10)) to find the analytical results to determine the steady-state amplitude. A small parameter \(\varepsilon\) is introduced, and Equation (10) is normalized regarding the inertia \(J\)  
\[\ddot{u}+\varepsilon^{2}d^{*}\dot{u}+\omega_{0}^{2}u+\varepsilon\alpha^{*}u^{ 3}=\varepsilon^{2}\Omega^{2}\varphi_{i;j}q^{*}cos(\Omega t) \tag{15}\]  
with \(d^{*}=\frac{d}{\varepsilon^{2}J}\), \(\alpha^{*}=\frac{\alpha}{\varepsilon J}\) and \(q^{*}=\frac{\dot{q}}{\varepsilon^{2}}\). Introducing MSLP coordinates leads to  
\[\omega^{2}\ddot{u}+\varepsilon^{2}\omega d^{*}\dot{u}+\omega_{0}^{2}u+ \varepsilon\alpha^{*}u^{3}=\varepsilon^{2}\Omega^{2}\varphi_{i;j}q^{*}cos \left(\frac{\Omega}{\omega}\tau\right). \tag{16}\]  
Using the detuning parameter \(\sigma\), \(\Omega\) can be written as \(\Omega=\omega\big{(}1+\varepsilon^{2}\sigma\big{)}\). Now, Equation (5) becomes  
\[\omega^{2}\ddot{u}+\varepsilon^{2}\omega d^{*}\dot{u}+\omega_{0}^{2}u+ \varepsilon\alpha^{*}u^{3}=\omega^{2}\Big{(}1+2\varepsilon^{2}\sigma+ \varepsilon^{4}\sigma^{2}\Big{)}\varepsilon^{2}\varphi_{i;j}q^{*}cos\left( \frac{\Omega}{\omega}\tau\right) \tag{17}\]  
and simplifies the agreement in analyzing only up to \(O\big{(}\varepsilon^{2}\big{)}\) to  
\[\omega^{2}\ddot{u}+\varepsilon^{2}\omega d^{*}\dot{u}+\omega_{0}^{2}u+ \varepsilon\alpha^{*}u^{3}=\omega^{2}\varepsilon^{2}\varphi_{i;j}q^{*}cos \left(\frac{\Omega}{\omega}\tau\right). \tag{18}\]  
A separation regarding the order \(\varepsilon\) leads to  
\[O(1):\ \omega^{2}D_{0}^{2}u_{1}+\omega^{2}u_{1}=0 \tag{19}\]  
\[O(\varepsilon):\omega^{2}D_{0}^{2}u_{2}+\omega^{2}u_{2}=-2\omega^{2}D_{0}D_{1 }u_{1}+\omega_{1}u_{1}-\alpha^{*}u_{1}^{3} \tag{20}\]\[ \begin{array}{ll}\mathcal{O}\big{(}\epsilon^{2}\big{)}:&\omega^{2}D_{0}^{2}u_{3}+ \omega^{2}u_{3}\\ &\quad=-2\omega^{2}D_{0}D_{1}u_{2}-\omega^{2}D_{1}^{2}u_{1}-2\omega^{2}D_{0}D_{2 }u_{1}+\omega_{2}u_{1}+\omega_{1}u_{2}-d^{*}\omega D_{0}u_{1}\\ &\quad-3\alpha^{*}u_{1}^{2}u_{2}+\omega^{2}\varphi_{i,j}q^{*}cos\Big{(}\frac{ \Omega}{\omega}\tau\Big{)}\end{array}  \tag{21}\]  
with \(D_{n}=\frac{\partial}{\partial\tau_{n}}\) representing the \(n^{\text{th}}\) time derivative. Equation (8) solves  
\[u_{1}=A_{c}exp(i\tau_{0})+cc \tag{22}\]  
with \(A_{c}\) being the complex amplitude with its complex conjugates \(\overline{A_{c}}\) and \(+cc\) denote the complex conjugate of the previous expression. The secular terms of the equation of \(O(\epsilon)\) are  
\[-2\omega^{2}iD_{1}A_{c}exp(i\tau_{0})+\omega_{1}A_{c}exp(i\tau_{0})-3\alpha^{* }A_{c}^{2}\overline{A_{c}}exp(i\tau_{0})=0 \tag{23}\]  
and the same for the complex conjugate expression. For secular terms in MSLP, \(D_{1}A_{c}\) is set to zero, giving  
\[\omega_{1}=3\alpha^{*}A_{c}\overline{A_{c}}. \tag{24}\]  
Now, Equation (20) results in the particular solution  
\[u_{2p}=\frac{\alpha^{*}}{8\omega^{2}}A_{c}^{3}exp(3i\tau_{0})+cc \tag{25}\]  
recalling again that the homogenous solution is already compensated by the result Equation (10) of Equation (8). The right-hand side of Equation (7) can be written as  
\[\varphi_{i,j}q^{*}cos\bigg{(}\frac{\Omega}{\omega}\tau\bigg{)}=\frac{\varphi_ {i,j}q^{*}}{2}exp(i(\sigma\tau_{2}+\tau_{0}))+cc \tag{26}\]  
using the detuning parameter to express \(\Omega\). By that, the secular terms of \(O\big{(}\epsilon^{2}\big{)}\) are  
\[ \begin{array}{ll}-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})\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{ \omega^{2}\varphi_{i,j}q^{*}}{2}\exp(i(\sigma\tau_{2}+\tau_{0}))=0.\end{array}