## 4 Aeroelastic Stability Analysis
### Bifurcation Equation  
According to the MSM, a small dimensionless parameter \(\epsilon\) is introduced by a suitable ordering of variables and parameters. First, the displacement is rescaled as \(v\left(s,t\right)\rightarrow\epsilon v\left(s,t\right)\), so that \(\epsilon\) is of the order of the amplitude of motion. Then, damping and aerodynamic coefficients are rescaled as: \(\eta_{b}\to e\eta_{b}\), \(\eta_{c}\to e\eta_{c}\), \(\epsilon_{c}\to e\epsilon_{c}\), \(b_{1}\rightarrow\epsilon b_{1}\), \(b_{3}\to e^{-1}b_{3}\), \(b_{5}\to e^{-3}b_{5}\), i.e., nonlinear coefficients are taken much larger than the linear ones (as suggested by the numerical application illustrated ahead), in order linear and nonlinear aerodynamic forces appear at the same level in the perturbation scheme. Moreover, independent time-scales \(t_{0}:=t\), \(t_{1}:=et\),..., are introduced, so that\(v\left(s,t_{0}\left(t\right),t_{1}\left(t\right),\ldots;\epsilon\right)\). Hence, by exploiting the chain rule, the following formal expansions for the time differential operators hold:  
\[\partial_{t}=\partial_{0}+\epsilon\partial_{1}+... \tag{17}\] \[\partial_{t}^{2}=\left(\partial_{0}+\epsilon\partial_{1}+...\right) ^{2}\]  
in which \(\partial_{j}:=\partial/\partial t_{j}\), \(j=0,1,\ldots\). Finally, the independent variable is expanded in series of \(\epsilon\) as:  
\[v\left(s,t;\epsilon\right)=v_{0}\left(s,t_{0},t_{1},\ldots\right)+\epsilon v_{ 1}\left(s,t_{0},t_{1},\ldots\right)+... \tag{18}\]  
By introducing rescaling and expansions in the equations of motion, and collecting the coefficients of the different powers of \(\epsilon\), the following perturbation equations are derived. Order \(\epsilon^{0}\):  
\[ \begin{array}{ll}\varrho^{2}v_{0}^{\prime\prime\prime\prime}-v_{0}^{\prime \prime}+\Lambda^{2}\int\limits_{-1/2}^{1/2}v_{0}ds+\partial_{0}^{2}v_{0}=0\\ v_{0}\left(-1/2,t_{0},t_{1},\ldots\right)=0,&\varrho^{2}v_{0}^{\prime\prime} \left(-1/2,t_{0},t_{1},\ldots\right)=0\\ v_{0}\left(1/2,t_{0},t_{1},\ldots\right)=0,&\varrho^{2}v_{0}^{\prime\prime} \left(1/2,t_{0},t_{1},\ldots\right)=0\end{array}  \tag{19}\]  
Order \(\epsilon\):  
\[ \begin{array}{ll}\varrho^{2}v_{1}^{\prime\prime\prime\prime}-v_{1}^{\prime \prime}+\Lambda^{2}\int\limits_{-1/2}^{1/2}v_{1}ds+\partial_{0}^{2}v_{1}=&- \varrho^{2}\eta_{b}\partial_{0}v_{0}^{\prime\prime\prime\prime}+\eta_{c} \partial_{0}v_{0}^{\prime\prime}-\Lambda^{2}\eta_{c}\partial_{0}\int\limits_{ -1/2}^{1/2}v_{0}ds-2\partial_{0}\partial_{1}v_{0}+\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\left(c _{c}+b_{1}u\right)\partial_{0}v_{0}-b_{3}\frac{1}{u}\left(\partial_{0}v_{0} \right)^{3}-b_{5}\frac{1}{u^{3}}\left(\partial_{0}v_{0}\right)^{5}\\ v_{1}\left(-1/2,t_{0},t_{1},\ldots\right)=0,&\varrho^{2}v_{1}^{\prime\prime} \left(-1/2,t_{0},t_{1},\ldots\right)=-\varrho^{2}\eta_{b}\partial_{0}v_{0}^{ \prime\prime}\left(-1/2,t_{0},t_{1},\ldots\right)\\ v_{1}\left(1/2,t_{0},t_{1},\ldots\right)=0,&\varrho^{2}v_{1}^{\prime\prime} \left(1/2,t_{0},t_{1},\ldots\right)=-\varrho^{2}\eta_{b}\partial_{0}v_{0}^{ \prime\prime}\left(1/2,t_{0},t_{1},\ldots\right)\end{array}  \tag{20}\]  
The generating problem, Equation (19), admits the monomodal solution:  
\[v_{0}\left(s,t\right)=A\left(t_{1}\right)\phi\left(s\right)e^{i\omega t_{0}}+ \text{c.c.} \tag{21}\]  
in which \(A\left(t_{1}\right)\) is an unknown complex modulating function, c.c. stands for complex conjugate terms, and \(\left(\omega,\phi\left(s\right)\right)\) is a real eigenpair of the undamped and unloaded model, determined in Section 3. When the \(\epsilon\)-order problem, Equation (20), is considered, damping and aerodynamics terms appear on the right hand side of the equation, to be evaluated via Equation (21).