## 3 Results
### Continuous Wavelet Transform (CWT)
#### 3.2.1 Scalogram Images
The main findings for each accelerometer are summarized below; the corresponding metric plots are presented in Figures 20-23:  
* S1 Sensor: In the AV case, the MaxEp is concentrated near 10 Hz, with numerous low-energy components clustered around 0.2. For the AVD case, the MaxEp at 10 Hz drops to approximately 0.4, and distinct frequency bands appear around 40 Hz, 125 Hz, and above 200 Hz. For frequencies above 1 Hz, the CV rises from 11% (AV case) to 33% (AVD case), indicating that the damage significantly increases the temporary energy fluctuations under ambient vibration excitation (Figure 20). * S2 Sensor: Both AV and AVD cases retain a dominant band near 5.75 Hz with changes in energy values over time, as indicated by the energy statistics (\(\mu\), \(\sigma\), and MaxEp). Figure 19: Scalogram images for the AV and AVD cases. However, secondary peaks at 1.77 Hz and 9.35 Hz keep MaxEp below 0.2. The CV for frequencies above 1 Hz rises from 12% (AV case) to 27% (AVD case), (Figure 21). * S3 Sensor: For both excitation cases, the MaxEp is concentrated near 9.35 Hz. All secondary bands exhibit MaxEp values below 0.3. For frequencies above 1 Hz, CV increases from 11% (AV case) to 27% (AVD case), confirming greater energy variability with damage in the structure (Figure 22). * S4 Sensor: Two dominant bands are observed at 5.75 Hz (shared with S2) and at 9.35 Hz (shared with S3). Secondary bands keep MaxEp below 0.4. The CV for frequencies above 1 Hz rises from 9% (AV case) to 38% (AVD case), indicating greater energy variability under conditions of damage (Figure 23). In general, the natural frequencies of the structure can be identified, but the damaged configuration exhibits systematically higher energy levels, particularly at the base of the structure (S1) and in the mass \(M_{3}\) (S4). These amplitude variations over time provide robust quantitative indicators for damage detection under ambient vibration excitations. Figure 21: Metric comparison \((\mu,\sigma,CV,MaxEp)\) for the S2 sensor in the AV and AVD cases. The transient excitation for the ID and IH cases was applied with the damage in the structure (spring \(k_{1}\) exhibiting a half-cut bar). Figure 24 presents the energy density variations of each frequency component over time with the scalogram images. Although the time-domain vibration responses (Figures 3 and 4) appear similar, these scalogram images reveal the frequency behavior of each sensor in greater detail. The figure illustrates that both cases reveal distinct energy-decay patterns, as follows:  
* S1 sensor: For both cases, the energy of the dominant frequency component decay around \(f_{ne}=5.72\) Hz indicates that the base of the structure is relatively insensitive to the transient excitations considering their location. * S2 sensor: In the ID case, the energy density near the experimental natural frequency \(f_{ne}=1.80\) Hz dissipates gradually compared to the faster energy density dissipation observed in the IH case at \(f_{ne}=5.72\) Hz. * S3 sensor: In the ID case, the energy density decay gradually approaches the experimental natural frequency \(f_{ne}=1.80\) Hz, in contrast to the faster energy decay observed in the IH case at \(f_{ne}=9.32\) Hz. * S4 sensor: A similar pattern is observed for sensor S3. The ID case produces a slow decay at \(f_{ne}=1.80\) Hz, whereas the IH case produces a fast decay at \(f_{ne}=9.32\) Hz. Additionally, a weak band near \(f_{ne}=5.72\) Hz is observed over time, but it has a significantly lower energy density than the two main frequencies. Figure 25 presents the harmonic excitation scalogram images for the three excitation cases, \(f_{d1},f_{d2}\), and \(f_{d3}\). As anticipated, each scalogram exhibits a dominant energy peak at the corresponding excitation frequency (\(2.11 Hz,6.41\) Hz, and \(9.68\) Hz) across most of the sensors. Notable exceptions are as follows:  
* S1 sensor: For the \(f_{d1}\) and \(f_{d3}\) cases, the third harmonic (i.e., 3 \(\times\) the excitation frequency) exhibits the highest energy peak, exceeding the fundamental.