## 3 Algebraic and Online Modal Parameters Identification  
Consider, as mentioned in the last section, that the mechanical system shown in Figure 1 is a general graphic representation of a wind turbine blade, whose dynamic response can be determined by Equation (1). The classic experimental modal analysis paradigm for mechanical engineering is to consider that the influence of the non-linear part can be neglected, that is, \(\Phi(x,\dot{x})\approx 0\) or, in the best case, it is such that: \(\Phi(x,\dot{x})\equiv 0\). When possible, considering the dynamic behavior of the system as linear results in an uncoupled \(2n_{def}\) grade, which can be expressed in terms of the principal or generalized coordinates [3]\(q_{i}\) as follows:  
\[\ddot{q_{i}}+2\xi_{i}\omega_{ni}q_{i}+{\omega_{ni}}^{2}q_{i}=0\] \[x(t)=\Psi q(t) \tag{6}\]  
where \(\omega_{ni}\) and \(\xi_{i}\) are the natural frequencies and damping ratios associated with the i-th vibration mode, respectively, and \(\Psi\) is the modal matrix with components \(\psi_{ij}\) that contains the column space of the eigenvectors of the system (1) given by  
\[\Psi=\left[ \begin{array}{cccc}\psi_{11}&\psi_{12}&\cdots&\psi_{1n_{def}}\\ \psi_{21}&\psi_{22}&\cdots&\psi_{2n_{def}}\\ \vdots&&\ddots&\\ \psi_{n_{def}1}&\psi_{n_{def}2}&\cdots&\psi_{n_{def}n_{def}}\end{array} \right] \tag{7}\]  
In notation of the [PERSON]'s operational calculus [20; 21], the modal analysis representation or modal model (6) is:  
\[(s^{2}+2\xi_{i}\omega_{ni}s+{\omega_{ni}}^{2})q_{i}=0 \tag{8}\]The combination of Equations (6) and (8) leads to  
\[x_{i}(s)=\sum_{j=1}^{n_{def}}\frac{\psi_{ij}(\alpha_{0,i}+\alpha_{1,i}s)}{(s^{2}+2 \xi_{i}\omega_{ni}s+{\omega_{n}}^{2})} \tag{9}\]  
where the constants \(\alpha_{0,i}\) and \(\alpha_{1,i}\) depend on the initial conditions. Therefore, the physical displacements \(x_{i}\) are given by  
\[p_{c}(s)x_{i}(s)=\rho_{0i}+\cdots+\rho_{1i}s+\rho_{2n_{def}-1i}s^{2n_{def}-1} \tag{10}\]  
with:  
\[p_{c}(s)=s^{2n_{def}}+a_{2n_{def}-1}s^{2n_{def}-1}+...+a_{3}s^{3}+a_{2}s^{2}+a _{1}s+a_{0} \tag{11}\]  
where \(p_{c}(s)\) is the characteristic polynomial of the vibrating mechanical system, \(\rho_{ij}\) are constants that depend on the initial conditions and the modal matrix components \(\psi_{ij}\). The components of the characteristic polynomial (11) provide the damping factors and damped natural frequencies, and hence, the natural frequencies and damping ratios of the flexible structure. Moreover, since the characteristic polynomial of the system is unique, it does not make any difference in which output \(x_{i}\) is selected. Note that the mathematical model (1) can describe any \(N\) DOF vibrating mechanical system, with \(N\) being an arbitrarily large integer. In previous works, the direct parameter identification of the system parameters such as mass, stiffness, and damping ratios have been realized using algebraic parameter identification methods as described in [20; 25], where some comparisons with ARX and ARMAX methods are discussed, consequent works as [26; 27] report the modal parameters identification schemes, evaluated in time domain and online. In this work, we propose a reliable extension of those algebraic parameter identification methods [20; 26; 27] for general \(N\) DOF modal models. Herein, \(n_{dof}<N\) is the number of degrees of freedom to be considered, and the parameters \(a_{i}\) belong to the characteristic polynomial of the system. The components of this polynomial contain the modal parameters of the system \(\omega_{i}\) and \(\xi_{i}\). As reported in [20; 25; 25; 27], the general expression for the proposed time domain, algebraic identification scheme for modal parameters in the case of multiple degrees of freedom, is given by  
\[\Theta=G^{-1}V=\frac{1}{\Delta} \begin{bmatrix}\Delta_{1}\\ \Delta_{2}\\ \vdots\\ \Delta_{n}\end{bmatrix}