## 2 Materials and Methods
### Experimental Setup of Three-Degree-of-Freedom (3-DoF) Structure
\[\mathbf{M\ddot{x}}+\mathbf{Kx}=\mathbf{0} \tag{4}\]  
Assuming the solution of the form \(x(t)=\phi e^{i\omega_{n}t}\), where \(\phi\) is the mode-shape vector associated with the natural frequency \(\omega_{n}\), and replacing this in Equation (4) results in the following:  
\[\Big{(}\mathbf{K}-\mathbf{M}\omega_{n}^{2}\Big{)}\phi=\mathbf{0} \tag{5}\]  
For a non-trivial solution of Equation (5) and using the equation \(\mathbf{Av=}\lambda\mathbf{v}\) to find the eigenvalues and eigenvectors, the natural frequencies and mode shapes are obtained, where:  
* \(\mathbf{A}=\mathbf{M^{-1}K}\) corresponds to a symmetric matrix resulting from the combinations of masses and springs. * \(\mathbf{v}=\phi\), which is the mode shape of the vibration. * \(\lambda=\omega_{n}^{2}\) are the angular frequencies of the vibration. \[[\mathbf{v},\lambda]=eig(\mathbf{M^{-1}K}) \tag{6}\]  
\[\omega_{n}=\sqrt{\lambda} \tag{7}\]  
\[f_{n}=\frac{\omega_{n}}{2\pi} \tag{8}\]  
Figure 3 illustrates the vibration mode shapes for each \(f_{n}\) of the test rig structure, where the vectors \(\phi_{i}\) were normalized. The dataset was acquired using four mono-axial accelerometers PCB 333B30 installed on a three-degree-of-freedom (3-DoF) test rig structure. The sensors were positioned at the base (S1) and on the first, second, and third floors (S2, S3, and S4), as shown in Figure 4. The sensor locations were selected to capture the acceleration response associated with the displacement of each mass under different excitation scenarios. Although this  
Figure 3: Vibration mode shapes of 3-DoF structure. study does not attempt to determine the optimal sensor locations, it ensures consistent placement across floors. Each sensor was mounted directly on the mass of its respective floor to facilitate the analysis of vibration responses using power spectral density (PSD) and continuous wavelet transform (CWT) methods under different excitation conditions. The database was recorded using a SIRIUS(r) Mini 4 xACC data acquisition system (DAQ) from Dewesoft, Trbovljie-Slovenia featuring four channels and a sampling rate of 1000 Hz per channel. The system is equipped with a 24-bit delta-sigma dual-core analog-to-digital (A/D) converter and an anti-aliasing filter configured as a 5 th-order Bessel IIR filter with a cutoff frequency of 100 kHz. Each measurement was collected over a 2 min time window for each of the seven vibration scenarios: reference condition (ambient vibration with RB bar) without damage and six different excitations (ambient vibration, initial displacement, impact hammer, and three types of harmonic excitations with a shaker) with a HCB bar (\(k_{1}\)) featuring damage on one side, as is illustrated in Figures 2 and 5. These different excitation methods were selected based on tests performed in various articles to detect anomalies or damage in different types of structures. However, the primary focus of this article is the dynamical analysis of vibration signals under different excitations (free response, transient excitation, and forced vibration) of the test rig structure. Furthermore, a comparison of the spectral power and time-frequency energy behaviors from each sensor according to their location and the healthy condition of the structure using signal processing techniques is shown in Figure 6. These techniques (PSD and CWT) were evaluated using the previously calculated theoretical values of the natural frequencies and the measurements performed in the 3-DoF test rig. Figure 2:0 Excitation methods for the test rig structure. Figure 2:1 Specification of the half-cut bar (\(k_{1}\)). The length of the dataset is \(413\times 4\), which corresponds to the behavior of the structure for each excitation case, as mentioned below:  
1. AV Case (Ambient Vibration): Ambient vibration refers to the natural excitations present in the environment, which usually come from uncontrollable and unmeasurable sources [31].