*2.3. VMD–HHT Model*

Since VMD could weaken the inherent end effect and mode aliasing of EMD, a new approach of VMD–HHT time–frequency analysis is suggested in this paper by replacing EMD with VMD. First, the acceleration data were decomposed using VMD. Secondly, the power spectrum density (PSD) was used to extract each mode frequency. As a result, the frequency range of dynamic responses could be defined, and the noise modes were eliminated. Then, the time–frequency–energy spectrum could be gained by employing HHT to extracted modes. Lastly, assisted by the spectrum to adjust and determine feature modes, characteristic information of dynamic responses with high accuracy could be obtained in all directions. Figure 1 shows the process of the VMD–HHT-based characteristic extraction model.

To verify the VMD–HHT-based characteristic extraction model for dynamic responses, a set of data was simulated to compare the performance between traditional HHT and VMD–HHT in extraction. The analog data consisted of five sinusoidal signals with different amplitudes and frequencies (the frequencies were 1 Hz, 5 Hz, 10 Hz, 15 Hz, and 35 Hz, respectively) at a sampling frequency of 100 Hz, and the effect of random noise was also added, as shown in Figure 2.

As shown in Figure 3, HHT cannot extract characteristic information of accelerometer signals. In contrast, the VMD–HHT model can accurately separate frequencies of analog signals into each band (separation frequency is consistent with setting frequency) and qualitatively analyze frequency distribution of energy based on color. Moreover, the analog data were made up of time-invariant stationary signals with constant statistics along the time axis. According to previous analyses and research, even for accessible composite signals, HHT cannot separate characteristic information effectively. Its algorithm EMD would cause mode aliasing, while the VMD–HHT would not. Therefore, this paper proposes an HHT method using VMD instead of EMD, from now on referred to as VMD– HHT time–frequency analysis model.

**Figure 1.** VMD–HHT-based characteristic extraction model.

**Figure 2.** Analog time series of acceleration.

**Figure 3.** Comparison between HHT and VMD–HHT: (**a**) characteristic information extraction of analog signals using HHT; (**b**) characteristic information extraction of analog signals using VMD–HHT.
