## 2 Materials and Methods
### Continuous Wavelet Transform (CWT)  
The wavelet transform, in general, decomposes time signal variation into wavelets (short-duration waveforms whose frequency content varies over time). This property provides the location of the time and frequency, particularly for analyzing transient signals. In this study, the CWT is used to illustrate the changes in the energy of the signal's frequency components over time. Equation (11) computes the coefficients in two dimensions (time, scale) of the CWT of a signal \(x(t)\). \[CWT(\tau,s)=\frac{1}{\sqrt{s}}\int_{-\infty}^{\infty}x(t).\Psi^{*}\left(\frac{ t-\tau}{s}\right)dt=\left\langle x(t),\Psi_{\tau,s}(t)\right\rangle \tag{11}\]  
Here, \(x(t)\) is the acceleration signal, which is preprocessed via min-max normalization. The parameter \(\Psi^{*}\) is the complex conjugate of the mother wavelet function used (Morlet wavelet). The translation parameter \((\tau)\) controls time shifting to examine different points in time. The scaling coefficient \((s)\) is the wavelet width to extract frequency information [24; 39]. The Morlet wavelet is used because of its suitability for examining nonstationary signals, such as the vibration signals considered in this study. This wavelet is an analytical wavelet with a spectrum that concentrates energy on positive frequencies (\(\omega>0\)). This wavelet combines a symmetric Gaussian envelope function with a modulating sinusoidal function, as presented in Figure 13 and defined by Equation (12). The parameter \(\beta\) is the number of cycles considering wavelet shape, balance time, and frequency resolution [40]. \[\Psi(t)=e^{-\frac{\beta^{2}t^{2}}{2}}\cdot cos(\pi t) \tag{12}\]  
The real and complex Morlet wavelets are defined by Equations (13) and (14) and Figure 13:  
\[\Psi(t)=\bigg{(}\frac{\beta}{\sqrt{2\pi}}e^{-\frac{\beta^{2}t^{2}}{2}}\bigg{)} \cdot cos(\omega_{c}t) \tag{13}\]  
\[\Psi(t)=\frac{1}{\sqrt{\pi f_{b}}}e^{\omega_{c}t}e^{-t^{2}/f_{b}} \tag{14}\]  
where:  
* \(\omega_{c}=2\pi f_{c}\) is the central wavelet angular frequency and determines the number of oscillations of the complex sinusoid inside the symmetric Gaussian envelope function. For engineering applications, \(f_{c}=6\) is widely used. * \(f_{b}\) controls the wavelet shape (bandwidth). The Morlet wavelet family is the translated and scaled representation of the mother wavelet used to capture signal characteristics across multiple time-frequency resolutions in Equation (15). \[\Psi_{\tau,s}(t)=e^{-\frac{\beta^{2}(t-\tau)^{2}}{s^{2}}}cos\bigg{(}\frac{\pi (t-\tau)}{s}\bigg{)} \tag{15}\]  
This study employs the scalogram image, a 2D contour plot of the energy density of vibration signals in the time-frequency domain, to characterize the dynamic response of the test rig structure [40]. For each sensor \(i\), the scalogram is defined by the squared modulus of CWT. Equation (16) computes the scalogram for a sensor \(i\) and \(k\) sample of the \(m\) observations collected during the experiment. Figure 14 shows the acceleration of S2 for the ID case in the time-frequency domain (scalogram image) and its original time-domain waveform (acceleration signal). \[\text{SC}_{i}^{(k)}(\tau,s)=\left|\text{CWT}x_{i}(\tau,s)_{i}\right|^{2},\ \ \ \ i=1,2,3,4\text{ and }k=1,2,\ldots,m \tag{16}\]  
Figure 13: Complex Morlet wavelet. To evaluate the influence of different excitation types on vibrational energy distribution, scalogram images were generated for each excitation case and each sensor. The excitation cases are grouped according to their excitation method or dynamic behavior:  
\[\{AV,AVD\},\quad\{IH,ID\},\quad\{f_{d1},f_{d2},f_{d3}\}\]  
For each excitation scenario, \(m\) observations were collected per sensor. Equation (17) computes the median energy resulting in a single scalogram image. \[\widetilde{SC}_{i}(\tau,s)=\underset{k=1,\ \ldots,\ m}{\text{median}}\left(SC_{i}^{ (k)}(\tau,s)\right) \tag{17}\]  
Each median scalogram \((\widetilde{SC})\) comprises \(N_{t}\) time samples per scale, corresponding to a 2 min recording acquired at sampling rate \(fs\).