## 2 Materials and Methods
### Experimental Setup of Three-Degree-of-Freedom (3-DoF) Structure
This decision was made considering that the objective of this study is not to perform an input-output analysis, but rather to evaluate the dynamic response of the structure in terms of its natural frequencies and energy distribution. Although the force was not controlled or precisely measured, the impact reliably produced broadband excitation, sufficient to excite the structure's natural frequencies and generate a free vibration response. Figure 11 represents the acceleration response of one of the recordings collected by the sensors installed in the structure under the impact of a hammer (impulse) with uncontrolled force input in the third mass \(M_{3}\). The highest peak of acceleration in all the sensors corresponds to the impact on the structure, where, according to mass, the amplitude of the accelerations changes, indicating their displacement. In addition, the lowest variation of acceleration is observed with the SI sensor, considering that its location is at the base, while the other sensors illustrate the damping response. Even though this case presents the free vibration of the structure, similar to the response of the previous case, this test shows a significant difference in the magnitude and duration of the vibration signals' oscillations compared to the ID case, where these parameters decrease more gradually. The last three cases (5, 6, and 7) involve the excitation of the structure using a shaker with controlled frequency and amplitude. Shakers have been widely used in SHM for damage detection and modal analysis. For instance, in [5; 11], a shaker is used for damage identification and location, where structures such as a grandstand simulator and a building prototype are excited with white noise signals or random vibrations. In contrast, [8] uses this system to determine the modal frequencies in beams of different materials working in a range of frequencies and computer vision techniques. The main difference between the experiments with transient excitations (ID case and IH  
Figure 2:6 Acceleration measurements with the Initial Displacement (ID) input. case) and harmonic excitations with a shaker lies in the way the structure was excited. While the transient excitations were applied manually under uncontrolled conditions, the shaker allows for precise control of both the excitation signal and the force level at a defined position on the structure. In these experiments, the shaker was driven by a signal generator producing a harmonic waveform of the form \(\sin(2\pi\times f_{d1}\times t)\), with a peak amplitude of 1 V. This signal was amplified using a power amplifier (output: 0.3 A and 1.1 V) which delivered an electric current proportional to the applied mechanical force. Each test was conducted with a duration of two minutes. 5. \(f_{d1}\) Case: Shaker Excitation With Frequency \(f=2.11\) Hz, \(f_{n1}<f_{d1}<f_{n2}\): In this case, the structure was excited with a shaker at a frequency approximately equal to \(1.1\times f_{n1}\) (2.11 Hz), i.e., about 10% above the first natural frequency \(f_{n1}\). Figure 12 shows the measured accelerations of the three masses \((M_{1},M_{2},M_{3})\) and the base of the structure under the harmonic excitation signal. The observed dynamic response patterns correspond closely to the theoretical vibration modes presented in Figure 2:7 For frequency excitation \(f_{d1}\), all three accelerometers (S2, S3, and S4) exhibit approximately sinusoidal accelerations, indicating that all masses vibrate in phase, in line with the first vibration mode. 6. \(f_{d2}\) Case: Shaker Excitation With Frequency \(f=6.41\) Hz, \(f_{n2}<f_{d2}<f_{n3}\): As in the previous case (Case 5), the structure was excited using the same shaker setup. Here, the excitation frequency was approximately equal to \(1.1\times f_{n2}\) (6.41 Hz), i.e., about 10% above the second natural frequency \(f_{n2}\). Figure 12 illustrates the first two seconds of the acceleration response for this case recorded by all sensors.