# Nonlinear Dynamics of an Internally Resonant Base-Isolated Beam under Turbulent Wind Flow
## Appendix A Nonlinear Equations of Motion  
The nonlinear equations of motion of the cantilever, realized with viscous-elastic material and equipped with base isolation device, are evaluated here as an extension of what was done in [45] to the case under analysis. Being \(u(s,t),v(s,t)\) the longitudinal and transversal displacements of the axis-line points, respectively, the kinematic problem of the cantilever reads:  
\[\varepsilon =u^{\prime}+\frac{v^{\prime 2}}{2} \tag{11}\] \[k =v^{\prime\prime}-(u^{\prime}v)-v^{\prime 2}v^{\prime\prime} \tag{12}\]  
where \(\varepsilon\) is the axial strain and \(k\) the bending curvature, Taylor-expanded up to the quadratic and cubic terms, respectively. Essential boundary conditions are written at cross-section \(A\), namely:  
\[u_{A} =0 \tag{13}\] \[v^{\prime}_{A} =0 \tag{14}\]  
As typical for the case of longitudinally movable end (here at cross-section \(B\)), the internal constraint \(\varepsilon=0\) is applied and it is implemented introducing the Lagrange multiplier \(\lambda(s,t)\), which represents the reactive axial force along the beam. Moreover, a response function relevant to linear Kelvin-Voigt material is assumed, namely:  
\[M=Elk+\eta k \tag{15}\]  
being \(EI\) the bending stiffness and \(\eta\) the internal viscous factor. The extended Hamiltonian Principle, consequently, reads:  
\[ \begin{split}\delta H&=\int_{t_{1}}^{t_{2}}\Bigl{[} \int_{0}^{l}\bigl{[}m(\dot{u}\delta\dot{u}+\bar{v}\delta\dot{v})-(Elk+\eta\dot {k})\delta k-(c_{e}\dot{v}-p_{a})\delta v+\lambda\delta\varepsilon+\varepsilon \delta\lambda\bigr{]}ds\\ &+(\kappa v_{A}+\zeta_{1}\dot{v}_{A}+\zeta_{3}\dot{v}_{A}v_{A}^{ 2})\delta v_{A}\Bigr{]}dt=0,\qquad\forall\delta u,\delta v,\delta\lambda\end{split}  \tag{16}\]  
where \(m\) is the mass per unit length, \(c_{e}\) the external damping coefficient, \(p_{a}\) the aerodynamic force, \(\kappa\) the base device stiffness, \(\zeta_{1},\zeta_{3}\) the linear and cubic damping coefficients of the device, respectively, \(\delta\) is the variation operator and \(t_{1},t_{2}\) two generic time instants, at which \(\delta u,\delta v,\delta\lambda\) all vanish. After substitution of Equations (11) and (12) in (16) and application of the part integration formula, the equations of motion are obtained. Among them, the one relevant to \(\delta u\) reads:  
\[m\dot{u}+[\lambda-Et\dot{v}^{\prime}v^{\prime\prime\prime}+\eta v^{\prime}v^{ \prime\prime\prime}]^{\prime}=0 \tag{17}\]  
which can be integrated to evaluate the expression of the Lagrange multiplier:  
\[\lambda=\int_{1}^{s}[(EIv^{\prime}v^{\prime\prime\prime}+\eta v^{\prime}v^{ \prime\prime\prime})^{\prime}-mi]d\xi \tag{18}\]  
where the boundary condition \(\lambda(1,t)=0\) is used. The equation relevant to \(\delta\lambda\) reproduces the constraint \(\varepsilon=0\). Then, using Equation (11) for \(\varepsilon\) and after integration, the expression for \(u\) is obtained:  
\[u(s,t)=-\frac{1}{2}\int_{0}^{s}\!\!v^{\prime}(\zeta,t)^{2}d\xi \tag{19}\]where the boundary condition (A3) is used. Substitution of Equations (A8) and (A9) in the equation relevant to \(\delta v\), provides the following condensed equation of motion, where only the variable \(v\) appears and terms up to the cubic order are retained:  
\[ \begin{split} m\ddot{v}&+c_{e}\dot{v}+\frac{m}{2}v^{ \prime}\left[\int_{0}^{s}v^{\prime}(\xi,t)^{2}d\xi\right]^{\bullet\bullet}+ \frac{m}{2}v^{\prime\prime}\int_{1}^{s}\left[\int_{0}^{\xi}v^{\prime}(\xi,t)^ {2}d\xi\right]^{\bullet\bullet}d\xi\\ &+(EI+\eta\partial_{t})\left[v^{\prime\prime}\left(1+\frac{v^{ \prime 2}}{2}\right)\right]^{\prime\prime}+\left[\frac{v^{\prime 2}}{2}(EI+\eta \partial_{t})v^{\prime\prime\prime}\right]^{\prime}-p_{a}=0\end{split} \] (A10)  
with boundary conditions:  
\[