*4.2. Moving Cart Configuration*

Time-frequency planes obtained from the moving cart experiment are shown in Figure 6. Results also show the estimated true temporal variation of the beam's fundamental frequency (in dashed blue), obtained by assuming linearity of the system and interpolating between the measured frequencies from the FRF (Figure 2c) at the 50 mm and 200 mm positions. The STFT and WT were conducted with a sliding window of length 4096, corresponding to 164 ms, and the overlap size is half

of the window length. Data were down-sampled from 25 kHz to 1.25 kHz to conduct the WVD in order to reduce the computational burden by maintaining a low-size window of 512 data points and improve the frequency resolution to bin sizes of 2.4 Hz, instead of 12.2 Hz under 25 kHz. The SST and MSST were processed with a non-overlapping sliding window size of 1024 data points at a reduced sampling rate of 1.25 kHz, corresponding to a duration of 819 ms. The WT was conducted using Morse wavelets, the SST used a Gaussian window, and the MSST was iterated five times, as for the fixed cart configuration. Table 2 lists results for performance metrics *J*<sup>1</sup> and *J*3, along with a summary of the processing window lengths used in the study.

**Figure 6.** Time-frequency planes from the fixed cart configuration obtained using the (**a**) STFT; (**b**) WT; (**c**) WVD; (**d**) SST; and (**e**) MSST method with extracted frequencies (red solid lines) and estimated true frequencies (dashed blue lines).

A visual comparison of the fundamental frequency extracted by the TFR methods (solid red line) with the estimated true frequency (dashed blue line) in Figure 6 shows that the SFTF provided the

most precise estimation of the beam's fundamental frequency that can be linked to the cart's location, followed by the WT. The WT showed more chattering in the results, but with a better adaptation to the varying frequency. This is confirmed by performance metric *J*<sup>1</sup> (Table 2), which also shows that the SST underperformed with respect to the other TFR methods. The computation time per iteration (*J*3) was significantly faster for the STFT and WT methods, smaller than the window hopping time (82 ms). For the WVD, the down-sampling strategy enabled a computation time of 262 ms per window, instead of approximately 10 s using a window size of 2048 data points. However, despite such improvement in the frequency resolution and computational time, the WVD failed at identifying the beam frequency during the movement of the cart, as observable in Figure 6. The MSST's computation time is significantly longer than for the other methods, attributable to the longer window lengths that were necessary in implementing the methods.


**Table 2.** Moving cart configuration time-frequency analysis comparison.

Results from the moving cart experiment showed that both the STFT and WT were adequate methods through their fast computational speed and acceptable precision on the frequency estimation. The three other methods did not provide adequate performance in terms of computation time. Moreover, the WVD did not succeed at extracting the fundamental frequency with acceptable precision. Overall, compared with results obtained under the fixed cart configuration experiment, it can be concluded that both the STFT and WT methods have good promise for real-time application to high-rate state estimation due to their fast computation time and level of precision. It is worth remarking that the WT's precision relative to the STFT is approximately the same, unlike results seen under the fixed cart configuration where the WT's estimation error was close to three times that of the STFT. This can be attributed to the faster convergence speed of the WT, whereas the WT was capable of adapting more quickly to a change in the system's frequency under the moving cart configuration. Thus, it appears that the WT is more applicable to the high-rate problem, given its faster convergence speed, but this may come at the cost of lower precision on the estimation depending on circumstances.
