Keywords

1 Introduction

The testing of vehicle ride comfort still relies on subjective evaluations by test drivers. There are a lot of advantages to using the same method in vehicle tests as the general customers to evaluate cars. However, obtaining accurate physical information such as frequency and amplitude of vibration from subjective evaluation is usually challenging. The development of a quantitative analysis method for ride comfort vibration is not only useful for improving ride comfort but also development of new suspension components and their control logic. Although the magnitude squared coherence (the coherence) is commonly used to examine the correlation between the input from road surfaces and the output which is vehicle body vibration, the spectrum and phase obtained from the frequency analysis used in the method are averaged values within the window. That makes it difficult to capture the characteristics of transient vibrations. Moreover, the calculation of the coherence further deteriorates the temporal resolution by averaging over a constant window for the spectrum and phase. On the other hand, the Hilbert-Huang transform (HHT) which is suitable for non-stationary vibration was proposed [1]. HHT consists of the empirical mode decomposition (EMD) and the Hilbert transform (HT), and the instantaneous amplitude and the instantaneous phase of the target signal are calculated by HT. In addition, the instantaneous frequency is obtained from the instantaneous phase. Since the signal that can be transformed by HT has to have only a single frequency component, the target signal is first decomposed into the intrinsic mode functions (IMFs) by EMD. EMD acts as a data-adaptive high-pass filter and has high temporal resolution. However, the drawback called mode mixing when EMD is applied to signals with intermittent signals is indicated by the author. In this study, we describe a method for mitigating the drawback when applying it to vehicle vibration and a correlation analysis with high temporal resolution using HHT.

2 Applying EMD to a Vehicle Vibration

2.1 Verification Method

The goal is to identify the characteristics of the instantaneous vibration when the vehicle behavior that the passengers perceive as comfortable or uncomfortable and to identify the components contributing to the vibration. For that goal, a road surface with a single convex input occurring on the ISO class D is used to conduct a full vehicle simulation using VI-CarRealTime. Figure 1a shows the road input and the simulation condition. The vertical displacement at the point of contacting the front tire and the road is shown in Fig. 1b. The vehicle parameters used in the simulation are the preset “Sedan car” model in VI-CarRealTime.

Fig. 1.
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Condition and road profile for simulation

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2.2 Masking EMD

The procedure for processing EMD is first to take the difference between the average of the upper and lower envelope curves for the target signal. This process is repeated iteratively on the residual until achieving a threshold, at which point the residual is extracted as an IMF. Then, the difference between the extracted IMF and the target signal is taken, and the same process is repeated to sequentially decompose the signal into IMF of progressively lower frequencies. On the other hand, EMD has been pointed out to have the problem of mode mixing where it decomposes signals with intermittent components into IMFs with non-monocomponents as mentioned earlier. As the vibration of a vehicle varies significantly depending on the driving location and conditions, the characteristics and frequency of inputs from the road are not constant. This becomes a challenge when EMD is applied to vehicle vibrations. In this study, the masking EMD [2] which is modified for intermittency signals is used to mitigate the drawback. The masking EMD adds a sinusoidal signal with single-frequency component across the entire range of the target signal as the masking signal. By executing EMD on the target signal, the components close to the frequency of the masking signal are extracted as an intrinsic mode along with the imposed signal. By removing the added masking signal from the intrinsic mode, the IMF of the target signal is obtained alone. According to the boundary map for masking EMD described by Fosso et al. [3], a smaller amplitude of a mask signal than the target signal is considered better. However, if the amplitude is too small compared to the target signal, the imposed signal may not work as a mask. Thus, we propose adapting the amplitude of the mask signal to an intermittency target signal using the sliding filter. This also acts to mitigate the drawback of EMD. Figure 2 shows an example of the target signal and a masking signal that adapts to the local amplitude of the target signal. In addition, by Fosso et al., the conditions of frequency between the mask signal and the two components that are to be extracted and separated are proposed with f1/fm > 0.7 and f2/fm < 0.6, where f1, f2, and fm represent the frequency of the signal to be extracted, the frequency of one to be separated and the masking frequency respectively. This study determines the masking frequencies are set to 1, 2, 4, 8, 16, 32, and 64 Hz by taking into account the logarithmic relationship of human sensory perception of vibration while meeting the aforementioned conditions. Figure 3a and Fig. 3b show the instantaneous frequency of IMFs obtained by HHT with the conventional EMD and with the masking EMD using the configuration mentioned above respectively. Instantaneous frequency from HHT is shown with lines in red. Furthermore, the background shows the scalogram using CWT for the same vibration data. The instantaneous frequency around 10 Hz in Fig. 3b which represents the maximum amplitude of the convex input is finely decomposed compared to Fig. 3a. In addition, the instantaneous frequency lines in Fig. 3b are less crossed compared to Fig. 3a, and the proposed method effectively decomposes the signal.

Fig. 2.
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An example of mask signal

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Fig. 3.
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Apply EMD to vehicle vibration

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3 Correlation Analysis of Input and Output in Vehicle Vibration by HHT

The coherence is a common method for correlation analysis. It is calculated from the Fourier transform (FT), and the wavelet transform (WT) is represented by Eq. (1), where Cxy represents the cross spectrum of the input and the output signals, Pxx and Pyy represent the power spectrum of the input and output signal respectively, Ax and Ay represent spectra of input and output, θx and θy represents the phase of input and output, and the E represents an expected value.

$$ coh^2 = \frac{{|E(C_{xy} )|^2 }}{{E(P_{xx} )E(P_{yy} )}} $$
(1)
$$ = \frac{|E(A_x )|^2 |E(A_y )|^2 }{{E(P_{xx} )E(P_{yy} )}} \cdot |E(e^{i(\theta_x - \theta_y )} )|^2 $$
(2)

The first part of Eq. 2 is calculated based on the ratio of input and output amplitudes, while the second part is determined by the phase difference between the input and output. In the case of transient responses, although the amplitude ratio is not constant, the phase difference remains constant. Thus, the correlation analysis using HHT utilizes the second part of Eq. 2. Specifically, it is determined that input and output are correlated when the instantaneous frequency of input signals corresponds to the one of output for a certain duration. This method is possible because the instantaneous frequency can be obtained by HT, whereas the method with FT or WT deteriorates the accuracy of coherence. Figure 4a shows the correlation analysis of vertical acceleration on the front wheel and vertical acceleration of the center of gravity on the vehicle body obtained from the aforementioned simulation using HHT. The lines in orange represent the spots having the correlation of the input which is the acceleration of the wheel and output which is the acceleration of the vehicle body, and the coherence using CWT is shown in the background. The lines are within a region of high coherence using CWT in the time axis direction, thereby correlation analysis using HHT exhibits high temporal resolution compared to coherence using CWT. Figure 4b illustrates the correlation analysis when the front bump rubber is replaced with the shorter one. It indicates that the correlation is detected at the moment when the bump rubber reaction force peaks around 30 - 40 Hz. Figure 4c illustrates the correlation when the greater damping force on the front damper is applied. It also indicates that the correlations affected by damping force around 10 - 20 Hz, 60 Hz are detected. Furthermore, features of Fig. 4b and Fig. 4c can be observed in Fig. 4a. This observation in Fig. 4a corresponds to the similarity of the bump rubber reaction force in Fig. 4a and Fig. 4b, and the similarity of the generated damping force in Fig. 4a and Fig. 4c. Consequently, HHT allows to find the characteristics of the vibration at the moment of forced vibration and the contribution of components. Figure 5 shows the characteristics of bump rubbers and dampers used in Fig. 4.

Fig. 4.
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Correlation analysis using HHT and CWT

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Fig. 5.
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Specifications of vehicle components used for the correlation analysis

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4 Conclusion

This study presents the method for detecting the correlation between road inputs and vehicle behavior using HHT with the adaptive amplitude masking EMD. The adaptive amplitude masking EMD was demonstrated to alleviate mode mixing for intermittent road inputs and their corresponding vehicle vibrations. Furthermore, the correlation detection method using HHT has superior temporal resolution compared to the coherence with CWT, allowing for more localized correlation detection. This advantage enables the identification of components affecting vehicle vibration.