## 3 Algebraic and Online Modal Parameters Identification
#### Non-Linearity Analysis  
In order to enhance the reliability of the modal parameters identification scheme proposed here, it is important to verify the presence of the significant non-linearities in the  
Figure 2: Algebraic identification approach.  
inherent system dynamic response that could compromise the performance of the algebraic identifier (21). Even though the algebraic identification technique has been numerical and experimentally verified for non-linear mechanical systems, see [20; 29], the main aim of this work is to analyze the linear part of the dynamic response for modal parameters identification purposes. To achieve this particular objective, there are numerical methods for determining the influence of non-linearities present in the system dynamics, and these methods are well reported in [17] and references therein. Despite the numerous advantages of the linearity assumption on mechanical systems, there exist cases where the linear methods are ineffective or inoperative; therefore, we need to analyze and quantify the presence of non-linearities on the particular mechanical system numerically.  
The Hilbert transformation [18; 29; 30] is a popular and well-founded mathematical tool widely used to analyze the dynamic behavior of mechanical systems due to its particular properties when it is applied to experimental frequency response functions. Since the Hilbert transform maps the functions under consideration into the same domain, when applying the Hilbert transform to a FRF, let it be \(\mathcal{R}(j\omega)\), then, the imaginary and the real parts of the FRF are related as follows:  
\[Re(\mathcal{R}(j\omega))=-\frac{1}{\pi}\eta\int_{-\infty}^{\infty}\frac{Im( \mathcal{R}(j\omega))}{\omega-\omega_{c}}dj\omega \tag{25}\]  
\[Im(\mathcal{R}(j\omega))=\frac{1}{\pi}\eta\int_{-\infty}^{\infty}\frac{Re( \mathcal{R}(j\omega))}{\omega-\omega_{c}}dj\omega \tag{26}\]  
where \(j=\sqrt{-1}\) and \(\eta\) denotes the Cauchy principal value of the integral, which is necessary to consider due to the singularity of (25) and (26) present at \(\omega=\omega_{c}\). The equations or relations (26) and (25) are known as the Hilbert transform pairs [18; 30]. It is well known that, for non-linear systems, the pairs do not comply and the Hilbert transform will return a distorted version of \(\mathcal{R}(j\omega)\). As a result, we have a non-linearity indicator by analyzing the level of distortion on the original FRF \(\mathcal{R}(j\omega)\) by using cross-correlation as follows:  
\[\sigma_{\mathcal{H}}=\left\|Q_{\mathcal{HR}}(0)\right\|^{2} \tag{27}\]  
where \(\left\|Q_{\mathcal{HR}}(0)\right\|\) is the normalized cross-correlation coefficient and \(Q_{\mathcal{HR}}\) is given by  
\[Q_{\mathcal{HR}}(\Delta j\omega)=\int_{-\infty}^{\infty}\mathcal{H}(j\omega )\mathcal{R}(j\omega+\Delta j\omega)dj\omega \tag{28}\]  
where \(\mathcal{R}(j\omega)\) is the original FRF of the system and \(\mathcal{H}\) is the Hilbert transformation of \(\mathcal{R}\). This coefficient is a numerical indicator of the non-linear behavior of the system at a specific amplitude of the input force. We use this index to study the presence of non-linearities in the system under study, such that when \(\sigma_{\mathcal{H}}\) diverges from 1, the system shows a behavior that is mainly non-linear; that is, for a linear system, the expected value of \(\sigma_{\mathcal{H}}\) is precisely 1. It is important to consider that the Hilbert transform is a numerical method; then, we need to consider some linearity criterion. Therefore, as reported in [17; 29; 30], we can consider the set of values \(0.9\leq\sigma_{\mathcal{H}}\leq 1\) for a linearity assumption of the dynamic behavior of the system.