## 4 Aeroelastic Stability Analysis
### Bifurcation Equation
Here, the internal damping \(\eta_{b}\) would trigger point-moments at the boundaries, equal to \(-\varrho^{2}\eta_{b}\partial_{0}v_{0}^{\prime\prime}\left(\pm 1/2\right)\); however, this last contribution is zero, because of the lower-order conditions \(\varrho^{2}v_{0}^{\prime\prime}\left(\pm 1/2\right)=0\). Equation (20) therefore read:  
\[\varrho^{2}v_{1}^{\prime\prime\prime\prime}-v_{1}^{\prime\prime}+\Lambda^{2} \int\limits_{-1}^{\frac{1}{2}}v_{1}ds+\partial_{0}^{2}v_{1}=q\left(s,t_{1} \right)e^{i\omega t_{0}}+\text{NRT}+\text{c.c.} \tag{22}\]  
\[ \begin{array}{ll}v_{1}\left(-1/2,t_{0},t_{1},\ldots\right)=0,&\varrho^{2}v_{1 }^{\prime\prime}\left(-1/2,t_{0},t_{1},\ldots\right)=0\\ v_{1}\left(1/2,t_{0},t_{1},\ldots\right)=0,&\varrho^{2}v_{1}^{\prime\prime} \left(1/2,t_{0},t_{1},\ldots\right)=0\end{array} \]  
where NRT stands for Non Resonant Terms, and  
\[ \begin{array}{ll}q\left(s,t_{1}\right):=&-i\omega\left[2\partial_{1}A\left(t _{1}\right)\phi(s)+\left(b_{1}u\phi(s)+c_{c}\phi(s)+\rho^{2}\eta_{b}\phi^{ \prime\prime\prime\prime}(s)+\Lambda^{2}\eta_{c}\int_{-\frac{1}{2}}^{\frac{1}{ 2}}\phi(s)\,ds\right.\\ &\left.-\eta_{c}\phi^{\prime\prime}(s)\right)A\left(t_{1}\right)+\frac{3b_{3} \omega^{2}\phi(s)^{3}}{u}A\left(t_{1}\right)^{2}\bar{A}\left(t_{1}\right)+ \frac{10b_{5}\omega^{4}\phi(s)^{5}}{u^{3}}A\left(t_{1}\right)^{3}\bar{A}\left(t _{1}\right)^{2}\right]\end{array}  \tag{23}\]  
Since the operator is singular (i.e., it admits a nontrivial homogeneous solution), the non-homogeneous problem cannot be solved for a generic know term. Therefore, a compatibility (orsolvability) condition, known as Fredholm alternative, has to be enforced on it. This condition requires the \(\omega\)-frequency forcing terms are orthogonal to the kernel of the self-adjoint operator, i.e.,  
\[\int\limits_{-\frac{1}{2}}^{\frac{1}{2}}\phi\left(s\right)q\left(s,t_{1}\right)=0 \tag{24}\]  
A detailed discussion on solvability conditions is provided in [51; 52], in the framework of perturbation methods. From Equation (24), an ordinary differential equation on the \(t_{1}-\)scale is obtained for the amplitude. By coming back to the true time \(t\) via backward rescaling, the equation assumes the well-known normal form for the Hopf bifurcation, i.e.,  
\[\dot{A}=\left(d_{0}+ud_{1}\right)A+\frac{d_{3}}{u}A^{2}\tilde{A}+\frac{d_{5}}{u ^{3}}A^{3}\tilde{A}^{2} \tag{25}\]  
where the real quantities \(d_{i}\) are defined as follows:  
\[ \begin{split} d_{0}&:=\frac{\int_{-\frac{1}{2}}^{ \frac{1}{2}}\left(-c_{e}\phi(s)^{2}-\eta_{b}\rho^{2}\phi(s)\phi^{\prime\prime \prime\prime}(s)-\eta_{c}\Lambda^{2}\phi(s)\int_{-\frac{1}{2}}^{\frac{1}{2}} \phi(s)\,ds+\eta_{c}\phi(s)\phi^{\prime\prime}(s)\right)\,ds}{2\int_{-\frac{ 1}{2}}^{\frac{1}{2}}\phi(s)^{2}\,ds}\\ d_{1}&:=-\frac{b_{1}}{2}\\ d_{3}&:=-\frac{3b_{3}\omega^{2}\int_{-\frac{1}{2}}^{ \frac{1}{2}}\phi(s)^{4}\,ds}{2\int_{-\frac{1}{2}}^{\frac{1}{2}}\phi(s)^{2}\,ds }\\ d_{5}&:=-\frac{5b_{5}\omega^{4}\int_{-\frac{1}{2}}^{ \frac{1}{2}}\phi(s)^{6}\,ds}{\int_{-\frac{1}{2}}^{\frac{1}{2}}\phi(s)^{2}\,ds }\end{split}  \tag{26}\]  
Finally, by letting \(A\left(t\right)=\frac{1}{2}a\left(t\right)e^{i\varphi\left(t\right)}\), and separating real and imaginary parts in Equation (25), the real form of the bifurcation equation is derived:  
\[ \begin{split}&\dot{a}=\left(d_{0}+ud_{1}\right)a+\frac{1}{4} \frac{d_{3}}{u}a^{3}+\frac{1}{16}\frac{d_{5}}{u^{3}}a^{5}\\ & a\phi=0\end{split}  \tag{27}\]  
Equation (27)-b supplies \(\varphi=\) const, since structural nonlinearities, that would be responsible for phase modulations, have been ignored. The steady solutions \(a=\) const of Equation (27)-a are periodic solutions for the original system; the \(T-\)periodic solutions \(a\left(t+T\right)=a\left(t\right)\) are quasi-periodic solutions for the original system. Due to the normalization adopted for the natural modes, the real amplitude \(a\) describes: (a) the maximum modulus of the displacement in the anti-symmetric modes, and, (b) the displacement at \(s=0\) in the symmetric modes.