# Nonlinear Dynamics of an Internally Resonant Base-Isolated Beam under Turbulent Wind Flow
## 3 Perturbation Analysis
\begin{split}&\partial_{0}^{2}v_{1}+v_{1}^{\prime\prime\prime \prime}=-2\partial_{0}\vartheta_{1}v_{0}-[v_{0}^{\prime}(v_{0}^{\prime}v_{0}^ {\prime\prime})^{\prime}-\eta\partial_{0}v_{0}^{\prime\prime\prime\prime}- \frac{v_{0}^{\prime}}{2}\partial_{0}^{2}\bigg{[}\int_{0}^{s}\!\!v_{0}^{2}( \xi)d\xi\bigg{]}\\ &-\frac{v_{0}^{\prime\prime}}{2}\int_{0}^{s}\!\!\partial_{0}^{2} \bigg{[}\int_{0}^{\xi}\!\!v_{0}^{2}(\xi)d\xi\bigg{]}d\xi-(c_{\epsilon}+b_{1}U )\partial_{0}v_{0}-\frac{b_{3}}{U}(\partial_{0}v_{0})^{3}\\ &-\bigg{(}b_{1}U\partial_{0}v_{0}-\frac{b_{3}}{U^{2}}(\partial_{ 0}v_{0})^{3}\bigg{)}\dot{a}\cos(\Omega t)\\ & v_{1}^{\prime}=0\\ & v_{1}^{\prime\prime\prime}+\kappa v_{1}=-v_{0}^{\prime}(v_{0}^{ \prime}v_{0}^{\prime\prime})^{\prime}-\eta\partial_{0}v_{0}^{\prime\prime \prime}-\zeta_{1}\partial_{0}v_{0}-\zeta_{3}\partial_{0}v_{0}v_{0}^{2}\bigg{\}} \quad\text{at}\;s=0\\ &-v_{1}^{\prime\prime\prime}=v_{0}^{\prime}(v_{0}^{\prime}v_{0}^ {\prime\prime})^{\prime}+\eta\partial_{0}v_{0}^{\prime\prime\prime}\\ & v_{1}^{\prime\prime}=-v_{0}^{\prime}\partial_{0}^{\prime\prime }-\eta\partial_{0}v_{0}^{\prime\prime}\end{split} \bigg{\}}\quad\text{at}\;s=1 \tag{14}\]  
Equation (13) provides the generating solution:  
\[v_{0}=A_{1}(t_{1})e^{i\omega_{1}t_{0}}\varphi_{1}(s)+A_{2}(t_{1})e^{i\omega_{ 2}t_{0}}\varphi_{2}(s)+\text{c.c.} \tag{15}\]  
where, due to the presence of damping, only internally resonant modes are retained, with \(A_{1}(t_{1}),A_{2}(t_{1})\) slow time varying complex amplitudes to be determined, c.c. standing for complex conjugate and \(i\) for the imaginary unit. Substitution of Equation (15) in Equation (14), with use of Equation (12), provides:  
\[ \begin{split}& v_{1}^{\prime\prime\prime\prime}-\partial_{0}^{2}v_{1}=q _{1}(s,t_{1})e^{i\omega_{1}t_{0}}+q_{2}(s,t_{1})e^{i\omega_{2}t_{0}}+\text{N. R.T}\\ & v_{1}^{\prime}=0\\ & v_{1}^{\prime\prime\prime}+\kappa v_{1}=Q_{1}(0,t_{1})e^{i \omega_{1}t_{0}}+Q_{2}(0,t_{1})e^{i\omega_{2}t_{0}}+\text{N.R.T}\bigg{\}}\quad \quad\text{at}\;s=0\\ &-v_{1}^{\prime\prime\prime}=Q_{1}(1,t_{1})e^{i\omega_{1}t_{0}}+Q _{2}(1,t_{1})e^{i\omega_{1}t_{0}}+\text{N.R.T}\\ & v_{1}^{\prime\prime}=C_{1}(1,t_{1})e^{i\omega_{1}t_{0}}+C_{2}(1, t_{1})e^{i\omega_{1}t_{0}}+\text{N.R.T}\end{split}  \tag{16}\]  
where \(q_{j},Q_{j},C_{j},j=1,2\), having the meaning of distributed loads, point-forces and couples, respectively, collect resonant terms and N.R.T. stands for non-resonant terms. For the sake of brevity, definition of \(q_{j},Q_{j},C_{j}\) is omitted here. Solvability conditions must be imposed in order to remove resonant terms at the right hand side of Equation (16). It assumes the form of the Virtual Power Equation, namely:  
\[ \begin{split}\int_{0}^{1}\!\!q_{1}(s,t_{1})\varphi_{1}(s)ds+Q_{1}(0,t_{1})\varphi_{1}(0)+Q_{1}(1,t_{1})\varphi_{1}(1)+C_{1}(1,t_{1})\varphi_{1}^{ \prime}(1)=0\\ \int_{0}^{1}\!\!q_{2}(s,t_{1})\varphi_{2}(s)ds+Q_{2}(0,t_{1}) \varphi_{2}(0)+Q_{2}(1,t_{1})\varphi_{2}(1)+C_{2}(1,t_{1})\varphi_{2}^{\prime}( 1)=0\end{split}  \tag{17}\]  
Coming back to the time \(t\) and reabsorbing \(\epsilon\), Equation (17) produces the following Amplitude Modulation Equations (AME):  
\[