## 3 Test Results and Analysis
### Time-Frequency Characteristic Analysis  
The Hilbert-Huang transform (HHT) method was used to analyze the characteristics of the vibration signals in the joint time-frequency domain. The HHT method is a self-adaptive time-frequency  
Figure 10: Relationship between amplitude of harmonic and rolling compaction times. Figure 9: Variations of the fundamental and multiple harmonic amplitudes of each measuring point. analysis method suitable for nonlinear and unstable signal processing proposed by [PERSON] [44,45], which mainly includes two parts: EMD empirical mode decomposition and Hilbert spectrum analysis. The EMD method assumes that any signal is a composite signal composed of different Intrinsic Modal Functions (IMFs), and each IMF component must meet two conditions: (1) the number of extreme points and zero-crossing points were the same or at most one difference; (2) the mean of the upper and lower envelope of the signal was zero. In this way, any signal can be decomposed into multiple IMFs [44]. Then, for vibration signal F(T), which was decomposed into multiple IMFs, its Hilbert transformation is as follows:  
\[G(t)=\frac{1}{\pi}K\int_{-\infty}^{\infty}\frac{F(\delta)}{t-\delta}d\delta\]  
where K is a Cauchy principal value. Established analytic signal P(t):  
\[P(t)=F(t)+jG(t)=a(t)e^{i\Phi(t)}\]  
where a(t) is the amplitude function and \(\Phi\)(t) is the phase function. \[a(t)=\sqrt{F^{2}(t)+G^{2}(t)}\]  
\[\Phi(t)=arctan\frac{G(t)}{F(t)}\]  
The instantaneous frequency of the signal could be obtained by differentiating the phase [45]. Therefore, the Hilbert transform of the decomposed IMF component can obtain the distribution law of the signal on the time-frequency energy scale, which is the form of the Hilbert spectrum:  
\[H(\omega,t)=Re\sum_{i=1}^{n+1}a_{i}(t)\cdot e^{j\int\omega-i(t)dt}\]  
By integrating the time in the expression formula of the Hilbert spectrum, the distribution law of the signal on the frequency-amplitude scale can be obtained, which is the form of the corresponding marginal spectrum:  
\[h(\omega,t)=\int_{-\infty}^{\infty}H(\omega,t)dt\]  
Taking the acceleration of the first compaction vibrating wheel (Measuring Point 1) as an example, the original vibration signal is decomposed by EMD to obtain each IMF component, as shown in Figure 10. Fourier transform is used to obtain the spectrum curve of each IMF component, as shown in Figure 11. From Figures 10 and 11, the original vibration signal of the wheel is decomposed into 13 IMF components and one residual component, and each IMF component has a different amplitude and frequency component. Among each IMF component, the waveform of IMF6 is consistent with that of original signal, and its dominant frequency and amplitude are substantially consistent with that of the original signal; IMF5 mainly contains the same frequency as the fundamental wave, first a harmonic wave and second harmonic wave, but its dominant frequency and amplitude are the closest to the first harmonic. The three frequencies with the largest amplitude of IMF4 are basically consistent with the frequencies and amplitudes of second, third and fourth harmonic waves of the original vibration signal, and the remaining components are high-frequency or low-frequency interference signals. Therefore, the waveform of the wheel is composed of IMF6, IMF5 and IMF4. To some extent, EMD can identify the fundamental and harmonic components of vibration signals, and remove the mechanical and environmental noise interference. Based on the results of EMD, the energy spectrum is obtained by Hilbert transform, and the evolution characteristics of the vibration signal energy in the time-frequency domain are represented by a two-dimensional plane contour map, and Figure 12 is the Hilbert energy spectrum of each measuring point in the first pass and the eighth pass.