# Nonlinear Dynamics of an Internally Resonant Base-Isolated Beam under Turbulent Wind Flow
## 3 Perturbation Analysis  
The perturbation analysis is carried out here to evaluate the bifurcation equations and analyze in detail the post-critical solutions, into a small region of the parameter space. It is performed using the Multiple Scale Method in direct approach [46], i.e., directly applying the method to the PDE of motion (3) and (4), on the same line followed in [37], where a 1:2 resonance between the first mode of the beam and the turbulence excitation term is considered. However, here the procedure is extended to the concurrent 1:3 internal resonance condition between the first two modes of the beam, and taking into account the structural nonlinear contributions as well. Introducing a small and positive bookkeeping parameter \(\epsilon\), the following scaling of variable is chosen:  
\[v=\epsilon^{1/2}\vartheta \tag{8}\]  
consistently with systems with cubic nonlinear terms; for the parameters, the scaling is:  
\[c_{e}=c\hat{c}_{e},\ \ \ \eta=\epsilon\hat{\eta},\ \ \ b_{1}=\epsilon\hat{b}_{1},\ \ \ \zeta_{1}=e\hat{\zeta}_{1},\ \ \ \zeta_{3}=O(1),\ \ \ b_{3}=O(1) \tag{9}\]  
which brings linear damping and wind effect to the highest perturbation order here considered. Time scales are introduced, as \(t_{0}=t\), \(t_{1}=\epsilon t\), and time-derivatives consistently become:  
\[\frac{\partial}{\partial t}=\partial_{0}+\epsilon\partial_{1},\ \ \ \ \ \ \ \frac{\partial^{2}}{\partial t^{2}}=\partial_{0}^{2}+2\epsilon\partial_{0} \partial_{1} \tag{10}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{4}{c}{**Uncontrolled (U)**} & \multicolumn{2}{c}{**Controlled (C)**} \\ \hline  & **Mode 1** & **Mode 2** & **Mode 1** & **Mode 2** \\ \hline \(\omega\) & 3.52 & 22.03 & 2.17 & 6.51 \\ \(c_{1}\) & 0.37 & \(-\)0.51 & 0.52 & \(-\)0.15 \\ \(c_{2}\) & \(-\)0.50 & 0.50 & \(-\)0.17 & \(-\)0.70 \\ \(c_{3}\) & \(-\)0.37 & 0.51 & \(-\)0.52 & 0.15 \\ \(c_{4}\) & 0.50 & 0.50 & 0.68 & \(-\)0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequencies and coefficients of the natural modes 1 and 2 of both uncontrolled and controlled system; the modes are normalized according to \(\varphi(1)=1\). Internally resonant case for \(\kappa=6.5\). Figure 3: (**a**) First natural mode; (**b**) Second natural mode. Gray and black lines indicate the uncontrolled and controlled system, respectively. Internally resonant case for \(\kappa=6.5\). where \(\partial_{j}=\frac{\partial}{\partial t_{j}}\), \(j=0,1\). Moreover, series expansion of the dependent variable is applied:  
\[\vartheta=\vartheta_{0}+\epsilon\vartheta_{1} \tag{11}\]  
Resonance conditions read:  
\[ \begin{split}\Omega&=2\omega_{1}+\epsilon\sigma_{ \epsilon}\\ \omega_{2}&=3\omega_{1}+\epsilon\sigma_{i}\end{split}  \tag{12}\]  
where \(\sigma_{\epsilon},\sigma_{i}\) are the external and internal detuning parameters, respectively. Substituting Equations (8)-(11) in Equations (3) and (4), dividing by \(\epsilon^{1/2}\) and collecting terms multiplying \(\epsilon^{0}\) and \(\epsilon^{1}\), respectively, the following perturbation equations are obtained (omitting hat):  
\[ \begin{split}&\partial_{0}^{2}v_{0}+v_{0}^{\prime\prime\prime \prime}=0\\ & v_{0}^{\prime}=0\\ & v_{0}^{\prime\prime\prime}+\kappa v_{0}=0\\ &-v_{0}^{\prime\prime\prime}=0\\ & v_{0}^{\prime\prime}=0\end{split} \qquad\text{at}\;s=1 \tag{13}\]  
and  
\[