## 3 Modal Identification
### Continuous Wavelet Transform Method for Modal Identification  
In order to obtain the experimental modal parameters of the beam, a modal analysis is carried out using a method based on the continuous wavelet transform (CWT) [25]. It consists in transforming a signal \(u(t)\) using functions, which are expressed as:  
\[\psi_{(b,a)}=\frac{1}{a}\psi\left(\frac{t-b}{a}\right) \tag{7}\]  
where \(a\) is a dilating scale parameter, \(b\) is a translation parameter (in time), and \(\psi\) is the mother wavelet. The mother wavelet must be a square-integrable and piecewise continuous function. For modal identification purposes using the free responses of a system with several degrees of freedom, the mother wavelet has to satisfy certain conditions, which are discussed in [25]. The Cauchy mother wavelet, defined by \(\psi(t)=(i/(t+i))^{n+1}\) with \(i\) the unit imaginary number, is then used in this study. The CWT can be expressed as:  
\[T_{\psi}[u](a,b)=\frac{1}{a}\int_{-\infty}^{+\infty}u(t)\overline{\psi}\left( \frac{t-b}{a}\right)dt \tag{8}\]  
where \(\overline{\psi}(. )\) is the complex conjugate of the function \(\psi(.)\). The local resolution of the CWT, in time and in frequency, depends on the dilatation parameter \(a\) and is determined, respectively, by the duration \(\Delta t_{\psi}\) and bandwidth \(\Delta\omega_{\psi}\) of the mother wavelet:  
\[\Delta t=a\Delta t_{\psi}\quad;\quad\Delta\omega=\frac{\Delta\omega_{\psi}}{a} \tag{9}\]  
The factor \(Q\) is introduced as the ratio of the center-frequency to the frequency bandwidth  
\[Q=\frac{\omega_{\psi}}{2\Delta\omega_{\psi}}=\frac{n+\frac{1}{2}}{\sqrt{2n+1}} \text{ for Cauchy mother wavelet} \tag{10}\]  
In order to avoid edge effects, \(Q\) needs to fulfill certain basic preconditions, which are:  
\[c_{f}\frac{\omega_{i}}{2d\omega_{i}}\leq Q\leq\frac{T_{f}\omega_{i}}{2c_{t}} \tag{11}\]  
where \(c_{t}\) and \(c_{f}\) are two chosen coefficients, here both equal to 5 (see [25]), \(T_{f}\) is the length of the signal, and \(d\omega_{i}=\min(\omega_{i+1}-\omega_{i},\omega_{i}-\omega_{i-1})\). For each mode, a local CWT is performed in a specific range of time and frequency by adjusting the length of the signal processed, the frequency bandwidth, and the factor \(Q\). Then, it is possible to extract the modal parameters from each measurement point to obtain the eigen frequencies, the modal damping ratios, and the mode shapes. In the case of free response at point \(j\) of the beam, of abscissa \(x_{j}\), Equation (2) leads to:  
\[ \begin{array}{ll}v(x_{j},t)=v_{j}(t)=\sum_{i=1}^{\infty}v_{ji}(t)&=\sum_{i= 1}^{\infty}\rho_{i}\exp(-\zeta_{i}\omega_{i}t)\cos(\hat{\omega}_{i}t-\varphi_{ i}(t))\cdot\phi_{i}(x_{j})\\ &=\sum_{i=1}^{\infty}A_{v_{ji}}(t)\cos(\alpha_{ji}(t))\end{array}  \tag{12}\]  
where \(\omega_{i}\) and \(\hat{\omega}_{i}\) are the undamped and damped vibration angular frequencies of the \(i\)th mode, \(\zeta_{i}\) is the damping ratio, \(\rho_{i}\) is a factor depending on initial conditions, and  
\[\left\{ \begin{array}{ll}A_{v_{ji}}(t)&=\ \ \left|\phi_{i}(x_{j})\right|\rho_{i} \exp(-\zeta_{i}\omega_{i}t)\\ \alpha_{ji}(t)&=\ \ \tilde{\omega}_{i}t-\varphi_{i}(t)+\frac{\pi}{2}(1-\text{ sign}(\phi_{i}(x_{j})))\end{array} \right. \tag{13}\]  
With the assumption of low damping (\(\xi_{i}<<1\)), each component \(v_{ji}(t)\) can be considered as asymptotic; thus, \(Z_{ji}(t)\approx A_{v_{ji}}(t)\exp(\alpha_{ji}(t))\) where \(Z_{ji}(t)\) is the analytical signal of \(v_{ji}(t)\). Then, the modal parameters can be identified using the CWT method. The instantaneous frequency is obtained using:  
\[\tilde{\omega}_{j}=\dot{a}_{ji}(t)=\frac{K}{a_{\tau_{ji}}(t)} \tag{14}\]  
with \(K\) a constant depending on the mother wavelet and \(a_{\tau_{ji}}(t)\) the time evolution of the ridge. For \(\dot{a}_{ji}(t)\), the dot represents the derivative with respect to time. The skeleton of the wavelet transform is the restriction of the CWT to each ridge and contains a maximum of information, very close to the signal components.