Figure 5 presents the theoretical basis with consistent steps for compiling the ground load spectrum of the whole tractor. Among the processes, the preprocessing section is essential for obtaining reliable acceleration load data. As shown, the process mainly includes three parts. First, there is a preprocessing section for dealing with the original ground load signals, including the welch power spectral density estimation of the six ground vibration loads, the CEEMDAN algorithm decomposition of the original signal, the wavelet threshold denoising algorithm, and the evaluation of the data denoising effect. Second, the study introduces a compilation method, including the time-domain extrapolation with the super-threshold model, GPD (Generalized Pareto Distribution) function, MEF (Mean Excess Function) threshold selection, and GPD parameter estimation. Third, the ground load spectrum is finalized for all ground testing and provides related software with multi-functions, including human–computer interaction, the visualization of real-time signal acquisition, data preprocessing display, and load spectrum compilation.
3.1. Data Preprocessing Method
The ground vibration signal received by the tractor during driving has the characteristics of nonlinearity and non-stationary. In the process of data acquisition and transmission, vibration signals will be subject to different types of noise interference, including external ground environment interference, internal sensor uncertainties, and other hardware system errors [
34], which will have a certain impact on the feature extraction and preprocessing analysis of acceleration signals. Therefore, it is necessary to denoise vibration signals and reduce abnormal points for ground load data preprocessing and load spectrum extrapolation.
(1) The CEEMDAN algorithm decomposition of the original signal
With regard to the acceleration signal processing problem, this paper adopts the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise algorithm (CEEMDAN algorithm). The CEEMDAN algorithm decomposes the vibration signal into a series of intrinsic mode components (IMF) through the EMD algorithm and adds the idea of adaptive noise and multiple superposition in the EEMD method to reduce mode aliasing [
35]. Then, the paper uses the wavelet threshold algorithm to denoise the high-frequency IMF components that carry more noise. The method aims to achieve a joint denoising effect and reduce the real information loss. The CEEMDAN method flowchart is shown in
Figure 6.
The CEEMDAN method is to add Gaussian white noise to the residual value every time the first order IMF component is calculated, calculating the mean value of the IMF component at this time, and iterating step by step. The process is as follows:
① Add Gaussian white noise w(i) of standard normal distribution to the collected original ground vibration signal x(i), perform J-times EMD decomposition of the newly generated signal, and repeat the overall average calculation of the generated modal component .
② Remove the residual of the first order component IMF1: .
③ Perform new EMD signal decomposition in r1(i) to obtain the first IMF1 component of the CEEMDAN decomposition. The second modal component and the residual r2(i) of the second order component are obtained by overall average calculation.
④ Repeat steps 2–3 until the residual signal obtained is a monotone function and the iteration cycle stops. Finally,
K-times intrinsic modal components
IMFk are obtained, and the original signal
x(
i) is decomposed into:
(2) Wavelet threshold denoising algorithm
The wavelet threshold denoising algorithm is used to reduce the order of IMF components of high-frequency noise through the Mallat tower algorithm, convert them into scale spaces of different scales, and obtain wavelet coefficients through threshold processing, filter and delete noise signals, and finally reconstruct signals. In this paper, the wavelet threshold is selected by the soft threshold function, where w is the wavelet coefficient and thr is the threshold.
Wavelet denoising soft threshold function expression is:
(3) Evaluation of data denoising effect
Reconstruct the processed signal components to obtain the complete signal after denoising. Through the comparative analysis of the waveform and spectrum before and after signal processing, the visual signal denoising effect is obtained. The preprocessing effect is measured by calculating the Signal Noise Ratio (SNR) and Root Mean Square Error (RMSE) of the vibration signal.
The
SNR calculation formula is:
The
RMSE calculation formula is:
where
is the original vibration signal and
is the processed signal. The smaller
RMSE and the larger
SNR indicate that the denoising effect of this method is better. Finally, the wavelet threshold denoising process is shown in
Figure 7.
3.2. Time-Domain Extrapolation Method of Ground Load
The load extrapolation method plays a vital role in the load spectrum compilation. The load extrapolation methods can be categorized as rainflow basin extrapolation and time-domain extrapolation. Skipping the rainflow counting steps, one can directly extrapolate a new ground load time series from the measured data while preserving the time history of the data to a great extent [
36]. Therefore, after the CEEMED and wavelet threshold denoising algorithms preprocess the original data, we develop a comprehensive process of spectrum compilation based on the time-domain extrapolation method. Due to the ground load intensity and complexity, we establish a POT model-based time domain extrapolation method and complete several tasks as follows: establish an extreme value sampling model of POT; obtain the GPD function and estimate its parameters; select the threshold by the MEF method and complete the fitting test of the GPD. To be noted, the paper conducts the fitting of the GPD function to find the distribution features of the measured ground vibration load. Since the selection of the load threshold can greatly affect the amount of data over the threshold, it is crucial to determine a reasonable threshold range for ensuring high accuracy and fast computing speed. After determining all the extrapolation steps, we will further embed it in our developed software for generating the ground load spectrum. The following describes the specific details for each task:
(1) Establishment of extreme value sampling model of POT.
In the POT model, the selected data are the samples within specific extreme values (between the upper and lower thresholds).
represents the ground vibration load data measured from the tractor’s front and rear axles. Each sample
is a time series and has
n random ground load data independent of each other while coming from the same distribution function
F(
x).
represents the threshold. The data greater than
(
) are the super-threshold data with the excess
. Moreover, the corresponding super-threshold distribution function
can be obtained as follows:
(2) Generalized Pareto Distribution
Since the ideal distribution function is unknown, the study analyzes the distribution of super-threshold data and the data amount to optimize and obtain a conditional distribution function. According to the theory, the super-threshold distribution is asymptotically distributed when the threshold is sufficiently large, which brings the GPD. Generally, the cumulative distribution function and the probability density function are used to describe the GPD, which can be expressed by Equations (7) and (8):
According to extreme value theory, if a random time series of loading sets has the characteristics of a fat tail, the excess of this dataset can be viewed as the GPD.
(3) Estimate parameters of GPD function
The research shows that it is more effective and consistent using the MLE method to find the solutions in the GPD function [
37]. The present paper adopts the MLE method for solving the GPD function and solves the scale parameter
and shape parameter
. The details are shown as follows:
① When
, the likelihood function is solved by taking the logarithm on both sides of the GPD function and gets:
By taking the partial derivatives of and and applying the Newton iteration method, Equation (9) is solved to obtain the likelihood estimates of and .
② When
, the logarithm form of the likelihood function for the GPD function is expressed as:
Taking the partial derivative of
on Equation (10) gets:
(4) The MEF threshold selection method
In the POT extreme value model, the threshold determines the quality of the fitting of the GPD function. If the selected threshold is too small, more super-threshold points will remain in the data sample, causing computation waste. If the selected threshold is too large, the obtained random data are deficient in fully reflecting the distribution features, causing poor accuracy. To find an appropriate threshold, we employ a graphical method with the use of the MEF method. According to the principle of the GPD, when the load data
obey the GPD function
, the distribution of the obtained data obeys the GPD function
, where
is the threshold. The average excess distribution function
of the obtained data is expressed by Equation (12).
When determining appropriate parameters
and
and obtaining the corresponding GPD function, the averaged excess distribution function
is linear to the selected threshold
. In the MEF diagram, the region closest to the fluctuation area ensures the linear relationship between the averaged excess and the threshold
. In this region, the relation expression between the average excess value and the threshold value is as follows:
Therefore, a threshold in the linear region of the MEF diagram is chosen as the optimal upper threshold for the GPD fitting.
(5) Goodness-of-fit test for GPD distribution
As discussed above, the extrapolated method finds the suggested scale and shape parameters , obtains the load data threshold, and then fits the extreme value data. However, the applicability and goodness of fitting the GPD is unclear. This task uses the Q-Q diagram method to assess the fitting effect and verify the proposed threshold.
After obtaining the distribution function, we obtain its uniformed distribution function
over the interval (0, 1).
represents the order statistic of the random time series and the uniformed distribution function can be expressed as:
Next, this study plots the Q-Q diagram for the extreme value data using the following equation:
This paper also draws a straight line of the selected distribution function on the Q-Q diagram to judge the quality of fitting the GPD. When the data points are close to the proposed line, the fitting agrees well with the original GPD.