## 2 Materials and Methods
### Experimental Setup of Three-Degree-of-Freedom (3-DoF) Structure  
The test rig is a structure with three degrees of freedom (3-DoF) that consists of a base and three vertically stacked floors. The assembly incorporates the following mechanical components:  
1. Bosch Rexroth aluminum profiles, \(45\times 45\times 220\) mm. 2. Bosch Rexroth angle brackets, \(45\times 45\) mm. 3. Bosch Rexroth T-head bolts. 4. Harmonic-steel bars. 5. Guidance system with a matched pair of Hiwin linear guides with profiled rails. 6. Base frame with fourth Bosch Rexroth aluminum profile joined to form a rectangle using angle brackets and a central profile on which the structure under study is anchored. The test rig comprises three vertically stacked floors, whose motion is governed by the applied excitation. The principal mechanical components of each floor are listed below and illustrated in Figure 1:  
* 8 T-head bolts
* 4 Angle brackets
* 1 Aluminum profile
* 2 Harmonic steel bars  
The experiments were conducted on the 3-DoF structure, presented in Figures 1 and 2, employing various excitation methods to analyze the dynamical response of the structure. In this context, the Reference Bar (RB) refers to the spring bar \(k_{1}\) under its  
Figure 1: Mechanical components of the test rig structure. healthy (undamaged) condition. The Half-Cut Bar (HCB) denotes the same spring element \(k_{1}\) modified to simulate structural damage, specifically with a cut of 1.5 mm in width and 0.75 mm in depth, corresponding to a 70% reduction in the cross-sectional height. This modification primarily reduces the local stiffness of the spring bar, with a comparatively minimal reduction in its mass. Figure 2 also indicates the specific bar that was replaced for each experimental scenario analyzed. The experimental scenarios illustrated in Figure 2 are summarized as follows:  
1. Ambient Vibration (AV)
2. Ambient Vibration with Structural Damage (AVD)
3. Initial Displacement of the mass \(M_{3}\) (ID)
4. Impact Hammer applied to the mass \(M_{3}\) (IH)
5. Harmonic excitation using a shaker at \(f=2.11\) Hz (\(f_{d1}\))
6. Harmonic excitation using a shaker at \(f=6.41\) Hz (\(f_{d2}\))
7. Harmonic excitation using a shaker at \(f=9.68\) Hz (\(f_{d3}\))  
The structural properties, such as mass \(\mathbf{M}\) and spring \(\mathbf{K}\), are represented using Equations (1) and (2), respectively. \[\mathbf{M}= \begin{bmatrix}M_{1}&0&0\\ 0&M_{2}&0\\ 0&0&M_{3}\end{bmatrix}  \tag{1}\]  
\[\mathbf{K}= \begin{bmatrix}k_{1}+k_{2}&-k_{2}&0\\ -k_{2}&k_{2}+k_{3}&-k_{3}\\ 0&-k_{3}&k_{3}\end{bmatrix}  \tag{2}\]  
Here, the \(\mathbf{M}\) values are \(M_{1}=0.960\) kg, \(M_{2}=0.960\) kg, \(M_{3}=0.885\) kg, while the values for \(\mathbf{K}\) are \(k_{1}=655\) N/m\({}^{2}\), \(k_{2}=655\) N/m\({}^{2}\), \(k_{3}=1177\) N/m\({}^{2}\). The \(k_{i}\) elements were calculated using Equation (3):  
\[k_{i}=2\bigg{(}12\frac{EI}{L_{i}^{3}}\bigg{)}\forall i=1,2,3 \tag{3}\]  
Each floor (\(i=1,2,3\)) of the test rig is supported by two identical harmonic-steel bars arranged in parallel denoted as 1a and 1b, 2a and 2b, 3a and 3b, respectively, as illustrated in Figure 2a. The lengths of these bars are \(L_{1}=0.282\) m, \(L_{2}=0.282\) m, and \(L_{3}=0.232\) m for the first, second, and third floors, respectively. All bars share the same rectangular cross-section with a moment of inertia \(I=2.917\times 10^{-12}\) m\({}^{4}\). The Young's modulus of the harmonic steel is \(E=210\) GPa. Figure 2: Structure of 3-DoF: (**a**) test rig structure, (**b**) mass-spring model. For the 3-DoF system, the free vibration equation is represented by (4), where \(\mathbf{x}=(x_{1},x_{2},x_{3})^{T}\). Natural frequencies \(f_{n}\) (\(f_{n1}=1.93\) Hz, \(f_{n2}=5.87\) Hz, \(f_{n3}=8.86\) Hz) were calculated using the eigenvalue analysis, and the vibration mode shapes \(\phi\) were found with the eigenvector from Equations (5)-(8), as mentioned in [30].