The Load Cycle Amplitude Model: An Efficient Time-Domain Extrapolation Technique for Non-Stationary Loads in Agricultural Machinery

Abstract

In traditional time-domain extrapolation methods, the peak over threshold (POT) model is unable to accurately identify large load cycles in the load time history, resulting in distorted extrapolation results, particularly when addressing non-stationary loads. To address this problem, this paper proposes a time-domain extrapolation method based on the load cycle amplitude (LCA) model. The core of the method involves using load cycle amplitude features extracted from the measured loads as the basis for modelling, rather than extreme turning points based on threshold extraction. This approach prevents the load’s time-domain characteristics from compromising the accuracy of the extrapolation results. The case analysis results demonstrate that the extrapolation method based on the LCA model achieves more reliable results with both non-stationary and stationary loads. Furthermore, the streamlined modelling process results in reductions of 10.63% and 20.84% in the average computing time for the algorithm when addressing stress and vibration loads, respectively. The LCA model proposed in this paper further facilitates the integration of time-domain extrapolation methods into reliability analysis software.

1. Introduction

Today, to compete for a broader market share, companies must produce durable and reliable machines that can adapt to various conditions and operational requirements [1]. The unstable, randomly fluctuating, and time-varying loads that are characteristic of mechanical components are crucial factors that threaten machinery’s reliability [2]. Therefore, to accurately predict a product’s service life before market placement, it is first necessary to obtain the load characteristics of the product under actual operating conditions, i.e., the load spectrum.

The load spectrum is now widely used in aerospace, vehicles, construction machinery, and agricultural equipment [3,4,5]. The primary aspects of load spectrum compilation include measuring loads under typical operating conditions, preprocessing load data, and extrapolating loads [6]. In practice, obtaining the complete lifecycle load history of critical machine components via sensor installation is both costly and often infeasible [7]. The duration of the load data acquired through measurement is typically short relative to the object’s lifetime. Therefore, a critical technical challenge in compiling the load spectrum is extrapolating load variations over the lifecycle from short-term testing, particularly the extreme loads that significantly impact fatigue life.

Depending on the form of the load spectrum, load extrapolation methods are primarily classified into rainflow matrix extrapolation and time-domain extrapolation [8,9]. The method of rainflow matrix extrapolation, developed earlier, is now integrated into commercial software such as Ncode and TecWare. However, the rainflow matrix does not retain time–frequency information about the load, thereby limiting its application [10]. Johannesson proposed a theoretical framework for time-domain extrapolation methods [11]. As illustrated in Figure 1, the method defines loads exceeding a threshold in the time domain as extreme loads. The distribution of extreme loads is fitted and reconstructed through statistical methods to extrapolate these to the target length in the form of a load spectrum block. In comparison to the rainflow matrix extrapolation method, the time-domain method can obtain extrapolated loads with complete time–frequency information. Consequently, it offers greater potential for applications in reliability test validation and finite element simulations [12,13].

There are two fundamental technical issues with time-domain extrapolation methods based on dividing extreme loads by the POT model:

• Reasonable threshold selection methods;
• The method’s unsuitability for non-stationary loads with significant changes in mean value.

Research has been conducted to address these critical issues. The methods supporting threshold selection include the mean residual life (MRL) plot [14,15], Hill estimator [16,17], mean square error [18], correlation analysis [19,20,21], and empirical methods [10,22]. Among these, the MRL plot is the most widely used method.

When dealing with non-stationary loads, a common approach is to divide load segments with significant characteristic differences and extrapolate them separately [23]. On this basis, Markov chain Monte Carlo (MCMC) load simulation methods, capable of modeling the articulated variation of each load segment, have been proposed and validated for mechanical components with significant non-stationary load characteristics, including agricultural equipment [24] and construction machinery [25,26].

It should be noted that the aforementioned methods represent improvements to the traditional time-domain extrapolation method based on the POT model. This approach, which extracts extreme loads by setting a constant threshold, fails to accurately identify large load cycles within the load history, and errors increase significantly when non-stationary characteristics are pronounced. The core of the load cycle simulation method based on MCMC is to construct a Markov chain model that describes the switching characteristics between different operating conditions. On the basis of analyzing single-operating-condition loads with the POT extrapolation method, the Monte Carlo method is used to simulate the duration and switching sequence of each condition, thereby indirectly achieving the extrapolation simulation of all conditions. The method depends on a subjective division of operating conditions, and it still cannot avoid identification errors for extreme loads when a single condition exhibits non-stationary characteristics. Due to variations in the land area, crop characteristics, and operator actions, the switching of agricultural machinery operational states in the field often exhibits strong randomness. Therefore, relying on manual subjective judgment of the operating conditions makes it difficult to meet the requirements for compiling long-term load spectra for agricultural machinery.

In summary, this paper proposes a new time-domain extrapolation method for extending measured loads to a target length, taking into account large load cycles that are not captured by short-time measurements but may occur throughout the lifecycle. The innovation of the proposed method lies in using the load cycle amplitude characteristics extracted from the measured load as the modeling target, rather than extreme loads extracted based on time-domain signal thresholds. Moreover, the proposed method can fully preserve the mean information of the load cycle during the extrapolation of the load cycle amplitude. This method can perform extrapolation analysis on the complete time-domain signal containing multiple operating conditions, aiming to resolve the time-domain extrapolation challenges of non-stationary loads from a theoretical perspective, and further reduce computational costs.

This paper is organized as follows: First, the proposed framework for the LCA extrapolation approach is described in Section 2. Next, the selection of critical parameters in the extrapolation model is discussed in detail, drawing on extreme value theory. Finally, the method is applied to the measured load history of key components in agricultural equipment. The superiority of the method presented in this paper is demonstrated by comparing it with traditional time-domain extrapolation methods.

2. Materials and Methods

2.1. Principle of Load Cycle Amplitude Model Used for Extrapolation

Figure 2 presents a schematic of the time-domain extrapolation method proposed in this paper. It should be noted that the input data type comprises a sequence of turning points obtained after peak and valley extraction, rather than the original sampled signal. This approach effectively filters out small load cycles that minimally impact reliability before extrapolation. The proposed method, based on extreme value theory, achieves load extrapolation by modeling the characteristic distribution of extreme loads. In contrast to traditional time-domain extrapolation methods, this approach treats load cycles with larger amplitudes as extreme loads, rather than turning points above a given threshold. In this case, the mean and amplitude information for each load cycle is extracted by the rainflow counting method [27].

As shown in Figure 2, the turning points with large load values do not correspond to the generation of large load cycles, which is the primary source of error in traditional time-domain extrapolation methods. To address this issue, we first discretize the sequence of time-domain turning points into distinct load cycles. Subsequently, load cycle amplitudes exceeding a predetermined threshold are treated as extreme loads and modeled using an appropriate distribution. The original load cycle amplitudes are replaced by a series of randomly generated extreme loads. Finally, by combining the mean information of load cycles recorded by the rainflow counting and the extrapolated amplitude, we generate the extrapolated time-domain load signal.

2.1.1. Modeling of Extreme Loads

The purpose of load extrapolation is to extrapolate the extreme loads that may occur during the product’s lifecycle based on limited test data. Therefore, the focus needs to be on the extreme loads that significantly impact reliability rather than on the total load sample when modelling loads. The generalized extreme value distribution (GEVD) and the generalized Pareto distribution (GPD) are two widely used extreme value distribution functions in statistical theory [28].

Assuming that the extreme-amplitude sample X in the load cycle is an independent random variable obeying the GEVD, it can be modelled by the following probability distribution function:

$$F(X)=\exp \left(-{\left(1-\xi \left({\displaystyle \frac{X-\mu }{\sigma }}\right)\right)}^{{\displaystyle \frac{1}{\xi }}}\right)$$

(1)

where µ is the position parameter, and ξ and σ are the shape and scale parameters, respectively. When ξ < 0, the GEVD is a type II extreme value distribution, i.e., a Frechet distribution. When ξ > 0, the GEVD is a type III extreme value distribution, i.e., a Weibull distribution. When ξ = 0, the GEVD takes the following form:

$$F(X)=\exp \left(-\exp \left(-\left({\displaystyle \frac{X-\mu }{\sigma }}\right)\right)\right)$$

(2)

The probability distribution function for modelling extreme-amplitude samples using the GPD is of the following form:

$$F(X,{\mu }_0)=\{\begin{array}{c}1-{\left(1+\xi \left({\displaystyle \frac{X-{\mu }_0}{\sigma }}\right)\right)}^{-{\displaystyle \frac{1}{\xi }}}, \xi \ne 0\\ 1-\exp \left(-\left({\displaystyle \frac{X-{\mu }_0}{\sigma }}\right)\right), \xi =0\end{array}$$

(3)

where μ₀ is the threshold for classifying the extreme-amplitude samples, ξ is the shape parameter, and σ is the scale parameter. Similar to the GEVD, depending on the value, ξ < 0 or ξ > 0 means that the extreme-amplitude samples have a thin-tailed or thick-tailed distribution, respectively, and for ξ = 0, the GPD is equivalent to an exponential distribution.

2.1.2. Algorithm

The time-domain extrapolation method based on the LCA model can extrapolate the original load sample to the target length in the form of a spectrum block via the following detailed steps.

• Measured load signals are acquired from target components. It should be noted that the time-domain load samples need to be representative of the load characteristics of the current operating conditions. The specific sample length selection method can be found in the literature [29].
• The turning point sequences in the time-domain signals are extracted. Small load cycles that contribute less to fatigue damage can be filtered out by a rainflow filter.
• The relationship between load cycle characteristics and time-domain loads is established using the rainfall counting method. The period characteristics (full or half cycles), start position, end position, mean value, and amplitude information are recorded for each load cycle.
• The amplitude of each load cycle is analyzed individually as a statistic, and a threshold is set to extract the extreme load cycles with larger amplitudes.
• A suitable distribution function is selected to fit the extreme-amplitude samples according to extreme value theory.
• Monte Carlo methods are used to randomly generate extrapolated amplitudes consistent with the number of extreme-amplitude samples and satisfy the homogeneous distribution requirement. It should be noted that each half-load cycle will have a common peak point or valley point with another half-load cycle. Therefore, the extrapolation results of the common inflection points should be compared; this paper takes the larger amplitude as the extrapolated result for the amplitude of the current half-load cycle.
• The extrapolated eigenvalues of each load cycle are restored to the load time history according to the mapping relationship established in step (3). The specific algorithm is as follows (Algorithm 1):

Algorithm 1: Reconstructing the time-domain signal of turning points.

Instruction:   {a1,…, ai,…, an}: The amplitude of the ith load cycle;   {m1,…, mi,…, mn}: The mean value of the ith load cycle;   pi: The peak value for the ith load cycle;   vi: The valley value for the ith load cycle;   for the ith load cycle, i = 1:n     if the ith load cycle is a full cycle       $p_i=m_i+{\displaystyle \frac{a_i}{2}}$;       $v_i=m_i-{\displaystyle \frac{a_i}{2}}$;     else the ith load cycle is a half cycle       Assume that the jth load cycle has a common peak or valley with the ith cycle;       $p_i=\max \left(m_i+{\displaystyle \frac{a_i}{2}},m_j+{\displaystyle \frac{a_j}{2}}\right)$;       $v_i=\min \left(m_i-{\displaystyle \frac{a_i}{2}},m_j-{\displaystyle \frac{a_j}{2}}\right)$;     end if   end for

8

Steps (1–7) are repeated until the target extrapolation length is satisfied.

2.2. Source of Data

Two representative load data were selected to validate the method proposed in this paper, including a stress signal where the load fluctuates sharply with time (non-stationary load) and a vibration signal where the average is relatively constant (stationary load), as shown in Figure 3.

Compared to general power machinery, agricultural equipment operates in harsher environments, which subjects its key components to more complex loads. In order to verify the applicability of the method proposed in this paper, the most widely used agricultural equipment, i.e., tractors, was selected as the research subject. Figure 4 illustrates the two forms of the tractor during ploughing operations. Depending on the plough’s position relative to the soil, these forms are defined as the ploughing stage and the field adjustment stage. These stages alternate to form a complete operation cycle in the target field.

As illustrated in Figure 4, even within the same operating condition, the characteristic differences between the load segments change dramatically with shifts in the tractor’s operational patterns. During the ploughing stage, the tractor experiences a high traction load. The ploughing depth can be kept relatively consistent once the plough has been adjusted. As a result, the load magnitude of the ploughing stage is high compared to that of the field adjustment stage, but there is less variation between the individual operating sections. In the field adjustment stage, the implements are in the lifting position and the traction resistance is low. Load variation primarily results from road excitation. The load magnitude is smaller than that in the ploughing stage but is influenced by road leveling and the vehicle speed, resulting in more dramatic fluctuations and significant differences among operating sections. In subsequent chapters, both the non-stationary load, represented by the stress signal, and the stationary load, represented by the vibration signal, are used to verify the method’s feasibility.

2.3. Construction of the Extrapolation Model

Figure 5 shows the process of constructing the extrapolation model. The goal is to obtain thresholds and corresponding distribution fitting parameters that accurately describe the characteristics of the extreme load distribution. The effectiveness of this threshold selection method, which combines MRL plot analysis and a goodness-of-fit test, has been confirmed by many researchers [19,20,21].

First, the MRL plot analysis method is used for the preliminary selection of thresholds, subjectively identifying the potential range for the optimal threshold. On this basis, the GPD and GEVD are used to perform fitting analyses on the discretized thresholds within the optimal threshold range. The quantitative selection of thresholds is achieved based on the results of the goodness-of-fit test. The existing literature has confirmed that extreme loads extracted using the POT model in traditional time-domain extrapolation methods tend to follow the GPD [30]. The time-domain extrapolation method proposed in this paper extracts the components with higher cycle amplitudes in the time-domain signal as extreme loads, which is significantly different from the POT model in traditional methods. To thoroughly compare the potential fitting approaches, both extreme value distributions widely used in engineering were applied to analyze the extreme loads.

MRL plots were analyzed for each of the two load signals presented in Section 3. For a sample of extreme load cycle amplitudes (X₁, …, X_i, …, X_n) that exceed a given threshold μ, the mean of the exceedances can be calculated by the following equation:

$$e_n(\mu )={\displaystyle \frac{1}{N_{\mu }}}{\displaystyle \sum _{i=1}^{N_{\mu }}\left(X_i-\mu \right)}, X_i>\mu$$

(4)

where X_i − μ is the exceedance, and N_μ is the number of load cycles whose amplitude exceeds the threshold value μ. The MRL plot is a curve with the threshold μ as the horizontal coordinate and the mean of exceedances as the vertical coordinate. The fluctuation degree of the curve can reflect the stability of the parameters in the distribution fit for the extreme load cycle corresponding to the current threshold.

As the threshold value increases, the number of extreme load cycle samples gradually decreases. As illustrated in Figure 6, to prevent distortion of parameter estimates in the fitted distribution due to a low sample number, the parts of the MRL plot with sharp curve fluctuations should be prefiltered. The relatively smooth threshold ranges [0, 25] (non-stationary load) and [0, 8] (stationary load) were extracted.

To evaluate the fit of the GEVD and the GPD to samples with extreme load cycles exceeding the amplitude threshold, we further discretized the threshold interval. In this study, the discretization gradient of the threshold was set to 0.02 times the unit physical quantity. The empirical cumulative distribution functions of the extreme samples corresponding to each threshold were calculated, and the cumulative distribution functions were fitted using the GEVD and GPD, respectively. A correlation analysis was conducted on the cumulative distribution functions of the extreme-value samples and the fitted results, where the coefficient of determination was used as a goodness-of-fit evaluation criterion. As illustrated in Figure 7, the GPD fit the extreme samples significantly more accurately than the GEVD for both stationary and non-stationary loads. Therefore, the GPD was chosen to model the extreme loads in this paper.

Once a suitable distribution function has been identified, the next step is to determine the optimal threshold value for each type of load individually. To avoid errors introduced by subjective factors in the graphical method, a computational method was used to select the optimal threshold by comparing the fitting accuracy of different thresholds corresponding to extreme loads. For the detailed screening algorithm, please refer to the literature [23]. The final results of the parameter fitting are shown in

Table 1. The results of the GPD parameter fitting.

Load Type	Amplitude Threshold μ	Shape Parameter ξ	Scale Parameter σ
Stress (non-stationary load)	14.42 (MPa)	0.0108	7.5688
Vibration (stationary load)	4.62 (g)	−0.0389	0.8236

.

It is important to clarify that the GPD is not the designated distribution fitting function for the method proposed in this article. Extreme load samples from different agricultural machinery operating conditions will inevitably have characteristic differences. Therefore, in practical applications, a detailed analysis based on load characteristics is required. The appropriate probability distribution function should be selected with the goal of achieving fitting accuracy.

We thus completed the modeling of extreme load cycles with amplitudes above the threshold. The amplitude information for extrapolated load cycles can be obtained using a Monte Carlo sampling method based on the fitted cumulative distribution function. Extrapolated load cycle characteristics can be recorded as a rainflow matrix or restored to a time-domain load signal.

3. Results and Discussion

The two load signals discussed in Section 2.2 were extrapolated separately using the method proposed in this paper. As an example of a complex non-stationary load, Figure 8 illustrates the stress load history before and after equal-length extrapolation. It is evident that the time-domain extrapolation method based on the LCA model can accurately identify the turning points that cause large load cycle amplitudes and extrapolate the turning points according to the amplitude distribution. The primary focus of the extrapolation model is the amplitude of the extreme load cycles. This ensures that the accuracy of the extrapolation results is not compromised by the load’s time-domain characteristics. Consequently, the method is more effectively applied to non-stationary loads.

To demonstrate the superiority of the method, the traditional time-domain extrapolation method based on the POT model [10] and the method proposed in this paper were applied to extrapolate stress and vibration loads, respectively. The time-domain extrapolation results, of equal length, are presented as mean–amplitude rainflow matrices, as illustrated in Figure 9.

In terms of rainflow matrix consistency between the extrapolation results and the original load, the time-domain extrapolation method based on the LCA model outperforms the traditional method for both non-stationary and stationary loads. A key reason for this superiority is that the LCA model is constructed directly from the mean–amplitude information of the load cycles. This approach can avoid errors in the extrapolation results caused by the non-correspondence between the turning points extracted by the POT model and the actual extreme load cycles. Additionally, the POT model fails to accurately identify characteristics of extreme loads during phases with significant non-stationarity, such as the ploughing and field adjustment stages of stress loads. This further exacerbates the distortion of the mean–amplitude information of the load cycle in the extrapolation results.

In practice, developers often focus on the impact of the load cycle amplitude on the product’s reliability. In this process, statistical analysis of the load time-domain signal, recording the amplitude of each load cycle and the corresponding frequency, is a widely used technique for compiling the load spectrum. For quantitative assessments of consistency among rainflow statistical results, dimension reduction is performed on the rainflow counting results of equal-length extrapolation, categorizing them by their mean and amplitude. Figure 10 shows the distribution curves of the mean–frequency and amplitude–frequency for stress load cycles corresponding to different methods. Based on this, the root-mean-square error (RMSE) was used to quantitatively assess the differences in distribution curves between the extrapolated results and original loads. The formula for the RMSE is as follows:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left(y_i-x_i\right)}^2}}{n}}}$$

(5)

In the equation, y_i and x_i are the statistical characteristic distribution sequences for the extrapolated results and the original loads, respectively, and n is the number of mean and amplitude groups set in the rainflow counting process.

Compared to the original stress loads, the RMSEs of the mean–frequency distribution curves for the extrapolated results using traditional and improved methods are 42.59 and 0.49, respectively, while for the amplitude–frequency distribution curves, they are 103.71 and 2.41, respectively. Similarly, when processing vibration loads, the RMSEs of the mean–frequency distribution curves for the extrapolated results from the traditional and improved methods are 90.85 and 0.52, respectively, and for the amplitude–frequency distribution curves, they are 34.98 and 11.49, respectively. From the root-mean-square error calculations, it is evident that the extrapolation results based on the LCA model have a higher consistency with the original signal in the mean–amplitude statistical distribution of the load cycles.

Further, 10³ equal-length extrapolations were carried out for each of the two load types to verify the method’s reliability. The first 10 extrapolations of each method were selected to plot the amplitude cycle cumulative frequency curves for visual comparison, as shown in Figure 11.

It is evident that after several extrapolations, both methods exhibit large amplitude cycles that are not present in the measured loads. For stress loads with significant non-stationary characteristics, the method based on the LCA model proves more efficient than the traditional method based on the POT model. The method proposed in this paper achieves extrapolation results that are more consistent with the amplitude distribution characteristics of the measured loads. However, the amplitude distribution in the extrapolation results of the traditional method is severely distorted, resulting in an aggressive load spectrum. For smoother vibration loads, both methods can produce extrapolation results that more closely match the amplitude distribution of the measured load cycles.

Furthermore, eigenvalues such as the maximum amplitude, mean value, standard deviation, damage consistency of load cycles, and algorithm running time were selected for analysis. A quantitative comparison of the 10³ extrapolation results obtained by various methods against the measured load is presented in

Table 2. Comparison of extrapolation results with original load eigenvalues.

Data	Extrapolation Method	Eigenvalue [5th Percentile, 95th Percentile]
		Deviation of Load Cycles Maximum Amplitude [%]		Deviation of Load Cycles Mean [%]		Deviation of Load Cycles Standard Deviation [%]
Stress	POT	[9.6563, 118.4116]		[3.2717, 3.5620]		[8.3915, 13.7504]
	LCA	[−20.3089, 19.9517]		[−0.1603, 0.1669]		[−1.9154, 1.9026]
Vibration	POT	[−6.8915, 5.6357]		[−0.0081, 0.0068]		[−0.2955, −0.1697]
	LCA	[−8.3770, 9.4668]		[−0.0118, 0.0121]		[−0.1043, 0.1057]
Data	Extrapolation Method	Eigenvalue [5th Percentile, 95th Percentile]
		Pseudo-Damage Deviation [%]			Algorithm Runtime [s]
Stress	POT	[64.0209, 298.7662]			[0.7882, 0.8483]
	LCA	[−16.7792, 17.5977]			[0.7039, 0.7426]
Vibration	POT	[−1.1506, −0.7140]			[1.9110, 2.0259]
	LCA	[−0.3564, 0.3628]			[1.5109, 1.5984]

. The deviation value is expressed as a percentage of the original load’s characteristic value. A deviation value of zero indicates that the extrapolation result is precisely consistent with the original load eigenvalues.

The damage consistency deviation was calculated based on the pseudo-damage theory [31]. Rainflow counting was performed on the target signal to extract the mean–amplitude information of load cycles. Based on the modified Miner’s rule, the damage of all load cycles was accumulated.

$$D={\displaystyle \sum _{i=1}^z\frac{1}{{\delta }_i}}={\displaystyle \frac{1}{C}}{\displaystyle \sum _{i=1}^z{S}_i^k}$$

(6)

In the formula, 1/δ_i represents the damage value caused by the ith load cycle as measured, z is the total number of load cycles, k is the inverse slope coefficient of the material’s S-N curve, S_i is the amplitude of the ith load cycle, and C is a constant variable.

Since there is no strict correspondence between pseudo-damage and material structural parameters, the pseudo-damage d can be simplified to the following equation.

$$d={\displaystyle \sum _{i=1}^z{S}_i^k}$$

(7)

It is evident that the pseudo-damage d is only related to the load cycle amplitude and the inverse slope coefficient k, which reflects the fatigue characteristics of the material. The pseudo-damage for both the extrapolated and original loads was calculated. The formula for calculating the deviation in damage consistency Q is as follows, where d₀ is the pseudo-damage corresponding to the original load.

$$Q={\displaystyle \frac{d-d_0}{d_0}}\times 100\%$$

(8)

It is evident that for stress loads, the maximum amplitude and standard deviation of the extrapolation results obtained using the traditional POT method significantly exceed the corresponding eigenvalues of the original load. At this point, the extrapolation results become unreliable. Indiscriminate use of the POT method results in an excessively aggressive load spectrum, potentially causing damage up to three times greater than that of the original load. Compared to POT extrapolation, the LCA method is better suited to non-stationary loads. The deviation of each eigenvalue approximates a normal distribution with a mean of zero and is more stable for each eigenvalue range.

For vibration loads, although the mean value remains relatively constant, the standard deviation varies considerably over time. The randomly reconstructed turning points in the traditional POT method cannot accurately match the large load cycles in the original load, resulting in the standard deviation of the load cycles in the extrapolation results consistently being lower than the corresponding eigenvalues of the original load. Although the results may indicate large load cycles not observed in actual measurements, the POT method falls short of achieving the extrapolation objective concerning the damage consistency caused by extrapolated loads. The pseudo-damage from the extrapolated results consistently appears more conservative than that of the original load.

In summary, compared with the traditional POT method, the LCA method proposed in this paper can obtain more reliable extrapolation results. It is valuable that the LCA method is applicable to time-domain loads with different characteristics. In addition, the traditional POT extrapolation method needs to extract and model the extreme loads in the time-domain signal based on the upper and lower thresholds. In contrast, the LCA-based extrapolation method only requires modelling of the extreme amplitudes extracted from the time-domain signal, making the modelling process more streamlined. Compared to traditional methods, the LCA extrapolation method reduced the average computation time for equal-length extrapolation of stress and vibration loads by 10.63% and 20.84%, respectively, further reducing computational costs.

This paper focused on the principles of a load cycle amplitude model applied to time-domain extrapolation, and it validated the method’s effectiveness using the stress and vibration loads of a tractor front axle as an example. It should be noted that the validation dataset used in this study is not sufficient to fully demonstrate the generalizability of this method to other types of agricultural equipment loads. Further research on this method will focus on obtaining more comprehensive load information for typical agricultural machinery. Based on the characteristics of long-term and short-term load spectra and actual application scenarios, such as simulation analysis and bench testing, application studies on the LCA-based time-domain extrapolation method will be conducted.

4. Conclusions

In traditional time-domain extrapolation methods, the POT model is unable to accurately identify large load cycles within the load history, resulting in distorted extrapolation results, especially when dealing with non-stationary loads. To address this issue, this paper proposes a time-domain extrapolation method based on the LCA model. The method was validated with two typical load signals obtained during field trials involving agricultural equipment. The main conclusions this study are as follows:

• Compared to the POT model used in the traditional time-domain extrapolation methods, the LCA model proposed in this paper more accurately identifies the large load cycles within the load history. It is valuable that the LCA model is applicable to time-domain loads with different characteristics.
• In constructing the load extrapolation model for tractors, it is recommended to use the GPD to fit extreme-amplitude samples exceeding the threshold. This recommendation is based on the GPD’s ability to provide a more stable and accurate fit than the GEV distribution.
• The case analysis results demonstrate that the extrapolation method based on the LCA model consistently achieved more reliable results with both non-stationary and stationary loads. Additionally, the streamlined modeling process led to a significant reduction in computing time, decreasing it by 10.63% and 20.84% for stress and vibration loads, respectively.

This paper introduces a new approach to time-domain load extrapolation. The time-domain characteristics of the load do not affect the accuracy of the extrapolation results. This method effectively addresses the limitations of traditional time-domain extrapolation methods and further reduces computational costs. This facilitates the integration of time-domain extrapolation methods into reliability analysis software.