# Nonlinear Dynamics of an Internally Resonant Base-Isolated Beam under Turbulent Wind Flow
## 2 Model  
### Nonlinear Aeroelastic Model  
The aeroelastic behavior of a base-isolated tall building, with a square cross-section and subjected to a turbulent wind flow, is investigated. The system is similar to the one proposed in [37], but here the structural model, besides the aerodynamic one, is assumed to be nonlinear as well. The building is modeled by a planar visco-elastic Euler-Bernoulli beam (Figure 1), obeying the Kelvin-Voigt constitutive law. The beam is free at the top, and constrained at ground by a visco-elastic device, consisting of a rheological model made of a linear elastic spring of stiffness \(\kappa\) in-parallel with a nonlinear dashpot. This last is characterized by a linear viscosity coefficient \(\zeta_{1}\) and a nonlinear (van der Pol-like) viscosity coefficient \(\zeta_{3}\). The wind velocity \(U(t)\) is assumed to be uniform in space and equal to:  
\[U(t)=\bar{U}+\bar{u}\cos\Omega t \tag{1}\]in which \(\bar{U}\) is the (leading) steady-state wind velocity, and \(\hat{u}\left(\ll\bar{U}\right)\) the amplitude and \(\Omega\) the fundamental frequency of the turbulent component (the higher harmonics being neglected). The flow induces aerodynamic loads \(p_{a}(s,t)\)_nonlinearly_ depending on the structural velocity \(\hat{v}(s,t)\), described by the quasi-steady theory [44]. The relevant equations of motion and boundary conditions are derived in Appendix A, extending those obtained in [45] to the case of Kelvin-Voigt material and base isolation. Introduction of the following positions is done:  
\[v^{*} :=\frac{\
u}{l}, s^{*} :=\frac{s}{l}, t^{*} :=t\omega_{r}, \Omega^{*} :=\frac{\Omega}{\omega_{r}}, \omega_{r} :=\frac{1}{l^{2}}\sqrt{\frac{EI}{m}},\] \[\eta^{*} :=\eta\omega_{r}, c^{*}_{e} :=c_{e}\frac{\omega_{r}l^{4}}{EI}, b^{*}_{i} :=b_{i}\frac{\omega_{r}^{2}l^{5}}{EI}, \bar{U}^{*} :=\frac{\bar{U}}{\omega_{r}l}, \bar{a}^{*} :=\frac{\hat{u}}{\omega_{r}l}, \tag{2}\] \[\kappa^{*} :=\kappa\frac{l^{3}}{EI}, \zeta^{*}_{1} :=\zeta_{1}\frac{l^{3}\omega_{r}}{EI}, \zeta^{*}_{3} :=\zeta_{3}\frac{l^{5}\omega_{r}}{EI}\]  
where \(v(s,t)\) is the transverse displacement at the abscissa \(s\in[0,l]\) and time \(t\in[0,\infty)\), with \(l\) the length of the beam; \(EI\) is the flexural stiffness of the beam; \(m\) is the mass per unit length; \(\eta\) is the internal viscous damping coefficient; \(c_{e}\) is the external damping coefficient, accounting for dissipation of the beam in motionless air; \(b_{i}:=\frac{1}{2}\varrho_{a}DA_{i}\) (for \(i=1,3\)) where \(\varrho_{a}\) is the air mass density and \(A_{i}\) are dimensionless aerodynamic coefficients, depending on the cross-section shape, of which \(D\) is a characteristic length; \(\omega_{r}\) is a reference frequency; prime and dot denote differentiation with respect to the non-dimensional abscissa and time, respectively. Accounting for Equation (2) and omitting the asterisks, the nondimensional form of the partial differential equation is found to be:  
\[ \begin{split}&\bar{v}+(c_{e}+b_{1}\bar{U})\bar{v}+\left[v^{ \prime\prime}\bigg{(}1+\frac{v^{\prime 2}}{2}\bigg{)}+\eta\left[v^{\prime \prime}\bigg{(}1+\frac{v^{\prime 2}}{2}\bigg{)}\right]^{\star}\right]^{\prime\prime}+\left[(v^{ \prime\prime}+\eta v^{\prime\prime})^{\prime}\frac{v^{\prime 2}}{2}\right]^{ \prime}\\ &+\frac{\
u^{\prime}}{2}\left[\int_{0}^{s}\!\!v^{\prime 2}(\bar{s})d \bar{s}\right]^{\star\star}+\frac{v^{\prime\prime}}{2}\int_{1}^{s}\!\!\left[ \int_{0}^{\bar{s}}\!\!v^{\prime 2}(\bar{s})d\bar{s}\right]^{\star\star}d\bar{s}\\ &+\frac{b_{3}}{\bar{U}}\bar{v}^{3}+\left(b_{1}\bar{v}-\frac{b_{3} }{\bar{U}^{2}}\bar{v}^{3}\right)\hat{u}\cos(\Omega t)=0\end{split}  \tag{3}\]  
with boundary conditions:  
Figure 1: Base-isolated beam model under transverse uniform turbulent wind flow. \[