# Nonlinear Dynamics of an Internally Resonant Base-Isolated Beam under Turbulent Wind Flow
## 3 Perturbation Analysis
\begin{split}& a_{1}=\Big{(}d_{10}+\bar{d}d_{10}\Big{)}a_{1}+\bar{d}d _{12}\cos\psi_{1}a_{1}+\frac{1}{4}a\frac{d_{1}}{\bar{D}}\cos\psi_{1}a_{1}^{3}+ \frac{1}{4}\Big{(}\frac{d_{2}}{\bar{D}}+\bar{d}_{2}\Big{)}a_{1}^{3}+\frac{1}{4 }a\frac{d_{4}}{\bar{D}}\cos\psi_{1}a_{1}^{3}\\ &\quad+\bar{u}d_{11}\cos(\psi_{1}+\psi_{2})a_{2}+\frac{1}{4}\bar{ u}\frac{d_{3}}{\bar{D}}\cos(\psi_{1}+\psi_{2})a_{1}^{2}a_{2}+\frac{1}{4}\Big{(} \Big{(}\frac{d_{3}}{\bar{D}}+\bar{d}_{3}\Big{)}\cos\psi_{2}-\bar{d}_{3}\sin \psi_{2}\Big{)}a_{1}^{2}a_{2}\\ &\quad+\frac{1}{4}\bar{u}\cos(\psi_{1}+\psi_{2})\frac{d_{6}}{a_{1 }^{2}a_{2}}+\frac{1}{4}\Big{(}\frac{d_{7}}{\bar{D}}+\bar{d}_{7}\Big{)}a_{1}^{2 }+\frac{1}{4}a\frac{d_{9}}{\bar{D}}\cos\psi_{1}a_{1}a_{2}^{2}+\frac{1}{4}\bar{ u}\frac{d_{8}}{\bar{D}}\cos(\psi_{1}+\psi_{2})a_{2}^{2}\\ &\quad a_{2}=\bar{u}d_{21}\cos(\psi_{1}+\psi_{2})a_{1}+\frac{1}{4 }\Big{(}\Big{(}\frac{d_{3}}{\bar{D}}+\bar{d}_{13}\Big{)}\cos\psi_{2}+\frac{1}{ 4}\sin\psi_{2}\Big{)}a_{1}^{3}+\frac{1}{4}\bar{u}\frac{d_{15}}{\bar{D}}\cos( \psi_{1}+\psi_{2})a_{1}^{3}\\ &\quad+\Big{(}d_{22}+\bar{d}d_{22}\Big{)}a_{2}+\frac{1}{4}\bar{ u}\frac{d_{14}}{\bar{D}}\cos\psi_{1}a_{1}^{2}a_{2}+\frac{1}{4}\Big{(}\frac{d_{ 16}}{\bar{D}}+\bar{d}_{16}\Big{)}a_{2}^{2}a_{1}+\frac{1}{4}\bar{u}\frac{d_{18} }{\bar{D}}\cos\psi_{1}a_{1}^{2}a_{2}\\ &\quad+\frac{1}{4}\bar{u}\frac{d_{17}}{\bar{D}}\cos(\psi_{1}+\psi _{2})a_{1}a_{2}^{2}+\frac{1}{4}\bar{u}\frac{d_{19}}{\bar{D}}\cos(\psi_{1}+\psi _{2})a_{1}a_{2}^{2}+\frac{1}{4}\Big{(}\frac{d_{20}}{\bar{D}}+\bar{d}_{20} \Big{)}a_{2}^{3}\\ &\quad a_{1}\psi_{1}=-e_{c}a_{1}-2 ad_{12}\sin\psi_{1}a_{1}+\frac{1 }{2}\frac{d_{1}}{\bar{D}}\sin\psi_{1}a_{1}^{3}-\frac{1}{2}\frac{d_{4}}{\bar{D }}\sin\psi_{1}a_{1}^{3}+2 ad_{11}\sin(\psi_{1}+\psi_{2})a_{2}\\ &\quad+\frac{1}{2}\Big{(}\Big{(}\frac{d_{5}}{\bar{D}}+\bar{d}_{ 5}\sin\psi_{2}+\bar{d}_{5}\cos\psi_{2}\Big{)}a_{1}^{2}a_{2}+\frac{1}{2}\bar{ u}\frac{d_{3}}{\bar{D}}\sin(\psi_{1}+\psi_{2})a_{1}^{2}a_{2}\\ &\quad-\frac{1}{2}\bar{u}\frac{d_{6}}{\bar{D}}\sin(\psi_{1}+\psi _{2})a_{1}^{2}a_{2}-\frac{1}{2}\bar{u}\frac{d_{9}}{\bar{D}}\sin\psi_{1}a_{1}^{ 2}+\frac{1}{2}\frac{d_{8}}{\bar{D}}\sin(\psi_{1}+\psi_{2})a_{2}^{3}+\frac{1}{2 }\bar{d}_{2}a_{1}^{3}+\frac{1}{2}\bar{d}_{2}a_{1}^{3}+\frac{1}{2}\bar{d}_{2}a _{1}^{2}a_{2}^{2}\\ &\quad a_{1}a_{2}\psi_{2}=-\bar{u}d_{21}\sin(\psi_{1}+\psi_{2})a_ {1}^{2}-\frac{1}{4}\Big{(}\Big{(}\frac{d_{13}}{\bar{D}}+d_{13}\Big{)}\sin\psi_{ 2}-\bar{d}_{13}\sin\psi_{2}\Big{)}a_{1}^{4}-\frac{1}{4}\bar{u}\frac{d_{15}}{ \bar{D}}\sin(\psi_{1}+\psi_{2})a_{1}^{4}\\ &\quad+\sigma_{i}a_{1}a_{2}+3\bar{u}d_{12}\sin\psi_{1}a_{1}a_{2}- \frac{3}{4}\bar{u}\frac{d_{1}}{\bar{D}}\sin\psi_{1}a_{1}^{3}a_{2}+\frac{3}{4} \bar{u}\frac{d_{4}}{\bar{D}}\sin\psi_{1}a_{1}^{3}a_{2}+\frac{1}{4}\bar{u}\frac{ d_{14}}{\bar{D}}\sin\psi_{1}a_{1}^{3}a_{2}\\ &\quad-\frac{1}{4}\bar{u}\frac{d_{18}}{\bar{D}}\sin\psi_{1}a_{1}^{ 3}a_{2}-3d_{11}\sin(\psi_{1}+\psi_{2})a_{2}^{2}-\frac{3}{4}\Big{(}\Big{(} \frac{d_{5}}{\bar{D}}+\bar{d}_{5}\Big{)}\sin\psi_{2}+\bar{d}_{5}\cos\psi_{2} \Big{)}a_{1}^{2}a_{2}^{2}\\ &\quad-\frac{3}{4}\bar{u}\frac{d_{3}}{\bar{D}}\sin(\psi_{1}+\psi _{2})a_{1}^{2}a_{2}^{2}+\frac{3}{4}\bar{u}\frac{d_{6}}{\bar{D}}\sin(\psi_{1}+ \psi_{2})a_{1}^{2}a_{2}^{2}+\frac{1}{4}\bar{u}\frac{d_{7}}{\bar{D}}\sin(\psi_{1}+ \psi_{2})a_{1}^{2}a_{2}^{2}\\ &\quad-\frac{1}{4}\bar{u}\frac{d_{19}}{\bar{D}}\sin(\psi_{1}+\psi _{2})a_{1}^{2}a_{2}^{2}+\frac{3}{4}\bar{u}\frac{d_{9}}{\bar{D}}\sin\psi_{1}a_{1} ^{3}-\frac{3}{4}\bar{u}\frac{d_{8}}{\bar{D}}\sin(\psi_{1}+\psi_{2})a_{2}^{4}\\ &\quad-\frac{1}{4}(3\bar{d}_{2}-\bar{d}_{16})a_{1}^{3}a_{2}+\frac{1}{4 }\bar{u}a_{1}^{3}-\frac{1}{4}(3\bar{d}_{7}-\bar{d}_{20})a_{1}a_{2}^{2}\\ \end{split} \]  
Equilibrium solutions are sought imposing \(a_{1}=a_{2}=\psi_{1}=\psi_{2}=0\) in Equation (19), and their stability is evaluated through the sign of the real part of the eigenvalues of the Jacobian matrix of the same system.