## 2 Multiple Degrees of Freedom Mechanical Systems  
The vibrating mechanical system shown in Figure 1 is a schematic representation of one possible discrete form of a generic wind turbine blade considered as a mechanical structure with a finite number of degrees of freedom. It is well known that its corresponding dynamic response is determined in terms of the parameters \(m_{i}\), \(b_{\varphi,i}\), \(k_{p,i}\), where the last two parameters can be expressed as a combination of a linear part and a non-linear part (viscous damping plus dry friction, Coulomb friction, etc., as well as linear stiffness plus a non-linear polynomial stiffness, duffing stiffness for example). It is important to remark that the connecting or coupling discrete elements \(b_{\varphi,i}\) and \(k_{p,i}\) do not follow a determined pattern or an assembling rule. The configuration shown in Figure 1 is a schematic diagram of the discrete form, considering a finite number of degrees of freedom, let it be \(n_{dof}\). In addition to the before mentioned, let us assume that the non-linear effects can be concentrated into a function \(\Phi(x,\dot{x})\), which can take a structure such that it contains or concentrates the sum of the non-linearities mentioned before. Thus, we can express the system dynamics behavior, in the conditions of free vibrations (no exogenous forces acting on the system), of this non-linear system by the following differential equation:  
\[M\ddot{x}+B\dot{x}+Kx=-\Phi(x,\dot{x}) \tag{1}\]  
where, the vector \(x\in R^{n_{def}}\) denotes the physical displacements of the masses as a function of time \(t\), and the function \(\Phi(x,\dot{x})\in R^{n_{def}}\) is a non-linear restoring force, commonly depending on the displacements and velocities of the considered \(n_{def}\) degrees of freedom. In Equation (1), the dynamic response of the linear part is determined by the matrices \(M\in R^{n_{def}\times n_{def}}\), \(B\in R^{n_{def}\times n_{def}}\), and \(K\in R^{n_{def}\times n_{def}}\), which are the mass damping and stiffness matrices, respectively, given by:  
\[M=\left[ \begin{array}{cccc}m_{11}&m_{12}&\cdots&m_{1n_{def}}\\ m_{21}&m_{22}&\cdots&m_{2n_{def}}\\ \vdots&&\ddots&\\ m_{n_{def}/1}&m_{n_{def}/2}&\cdots&m_{n_{def}/n_{def}}\end{array} \right],B= \left[ \begin{array}{cccc}b_{11}&b_{12}&\cdots&b_{1n_{def}}\\ b_{21}&b_{22}&\cdots&b_{2n_{def}}\\ \vdots&&\ddots&\\ b_{n_{def}/1}&b_{n_{def}/2}&\cdots&b_{n_{def}/n_{def}}\end{array} \right]\]  
\[K=\left[ \begin{array}{cccc}k_{11}&k_{12}&\cdots&k_{1n_{def}}\\ k_{21}&k_{22}&\cdots&k_{2n_{def}}\\ \vdots&&\ddots&\\ k_{n_{def}/1}&k_{n_{def}/2}&\cdots&k_{n_{def}/n_{def}}\end{array} \right] \tag{2}\]  
As mentioned before, it is common for the non-linear restoring force to assume a structure such that it contains the sum of non-linearities like dry friction, non-linear damping, and some polynomial stiffness as follows [18]:  
\[\Phi(x,\dot{x})=K_{ip}x^{r}+K_{(i-1)p}x^{r-1}+\cdots+K_{2p}x^{2}+F_{d}\text{ sign}(x)+C_{d}\text{sign}(\dot{x}) \tag{3}\]  
Figure 1: Generic linear/nonlinear vibrating mechanical system. where \(r\) is the order or the polynomic stiffness, and \(K_{ip}\in R^{n_{def}\times n_{def}}\) are the polynomial stiffness matrices, with \(i=2,3,...,r\) given by:  
\[K_{ip}=\left[ \begin{array}{cccc}k_{ip11}&k_{ip12}&\cdots&k_{ip1n_{def}}\\ k_{ip21}&k_{ip22}&\cdots&k_{ip2n_{def}}\\ \vdots&&\ddots&\\ k_{ip_{n_{def}1}}&k_{ip_{n_{def}2}}&\cdots&k_{ip_{n_{def}n_{def}}}\end{array}  \right],\ \ \ i=2,3,...,r \tag{4}\]  
The matrices \(F_{d}\in R^{n_{def}\times n_{def}}\) and \(C_{d}\in R^{n_{def}\times n_{def}}\) are the Coulomb dry friction and nonlinear damping matrices, respectively, with the following structure. \[F_{d}=\left[ \begin{array}{cccc}f_{d11}&f_{d12}&\cdots&f_{d1n_{def}}\\ f_{d21}&f_{d22}&\cdots&f_{d2n_{def}}\\ \vdots&&\ddots&\\ f_{dn_{def}1}&f_{dn_{def}2}&\cdots&f_{dn_{def}n_{def}}\end{array} \right],C_{d} =\left[