## 1 Introduction
Their proposed SMF method segments the vibration signal using a sliding window, expressing it as a sum of complex exponentials, which are then regrouped into energy grids to form single-mode functions. This technique effectively overcomes the mode mixing problem, preserving true modal content even for closely spaced frequencies, and it avoids the low-frequency spectral leakage inherent in CWT. However, the authors did not investigate scenarios dominated by large-amplitude broadband noise or quantify the energy of the dominant component. The authors in [28] investigated damage detection in reinforced concrete girders using a Local Time-Frequency Transform (LTFT) with the least squares method to determine time-varying Fourier coefficients. They processed both synthetic and laboratory vibration signals employing Renyi entropy to quantify the spectrum, enabling a comparison of structural states before and after crack-induced damage. Their method provided high resolution in spectrum representation and effective noise suppression compared to the Short-Time Fourier Transform (STFT) and S Transform (ST). However, although this method was able to qualitatively identify the existence of damage, it lacked the quantification of its degree or the precise location of the damage. This limitation is due to the absence of analysis considering varied sensor placements and detailed energy analysis of frequency components. Beyond vibration signals, other diagnostic techniques include optical methods and strain gauge analysis. For instance, machine vision methods, such as those evaluating cross-correlation coefficients, have demonstrated high accuracy (errors below 1%) in measuring dynamic displacement of large-scale steel bridge models under shaking with white noise and seismic excitations [29]. However, these applications often necessitate specific controlled conditions that influence measurement parameters due to variations in image capture. While the measure of spectral energy concentration is a suitable feature for damage detection, its analysis often remains qualitative, making rigorous comparison between different techniques in time-frequency spectra challenging. Furthermore, the evaluation of different excitation methods in structural analysis is often limited, with most studies focusing on a single excitation type for damage assessment. Nevertheless, frequency domain analysis in several articles examines the displacement of structural natural frequencies based on ambient vibration excitation or hammer excitations. Similarly, time-frequency domain analysis offers valuable insights into energy distribution for damage detection, although its effectiveness is influenced by the trade-off between time and frequency resolution, which is determined by the chosen window function and mother wavelet. This study examines the vibration response of a three-degree-of-freedom (3-DoF) structure subjected to seven different excitation scenarios. These include ambient free vibration, two types of transient excitations, and forced harmonic excitations at three frequencies aligned with the natural frequencies of the test rig. Mono-axial accelerometers are mounted at the base and on each floor of the structure to capture the dynamic response. The analysis integrates both spectral power, determined by the power spectral density (PSD), and time-frequency energy, estimated using the continuous wavelet transform (CWT), to characterize the dynamic behavior of the structure under varying conditions. This article is structured based on three primary analyses:  
* The influence of structural damage is assessed by comparing the healthy and damaged states of the test rig under ambient vibration excitation. Specifically, variations in the spectral power and time-frequency energy associated with the experimentally identified natural frequencies are analyzed according to sensor location. * The effect of damage on the dynamic characteristics of the structure is explored using two types of transient excitations.