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American Journal of  Smart 
Technology and Solutions (AJSTS)

Research on the Compilation Method of  Transmission Shaft Load Spectrum for ZL50 
Wheel Loader Used in Mines 

Md. Ariful Islam1*, Mabia Khatun2, Wanyi Pin3

Volume 4 Issue 2, Year 2025
ISSN: 2837-0295 (Online)

DOI: https://doi.org/10.54536/ajsts.v4i2.5665
https://journals.e-palli.com/home/index.php/ajsts

Article Information ABSTRACT

Received: July 07, 2025

Accepted: August 11, 2025

Published: August 18, 2025

The ZL50 wheel loader functions under highly fluctuating and intricate load conditions, 
which significantly impact the longevity of  its transmission shaft. This study introduces 
a thorough approach to compiling and refining the transmission shaft load spectrum by 
utilizing both simulated and actual torque data. The methodology encompasses signal-
preprocessing, rainflow counting for cycle extraction, and fatigue life assessment through 
Miner’s Rule. An 8×8 load spectrum matrix is created to depict the frequency distribution of  
load amplitudes and mean values. Sensitivity analysis is performed to evaluate how variations 
in load magnitude affect fatigue life. Furthermore, MATLAB optimization algorithm is used 
to reduce cumulative damage by adjusting load parameters, leading to a more efficient and 
dependable fatigue design. Advanced visualization methods, including surface plots, bar 
graphs, contour maps, and convergence graphs, are employed to interpret the findings and 
monitor the optimization process. The proposed method not only enhances the precision 
of  fatigue life prediction but also lays a practical groundwork for designing more robust 
mechanical components for heavy-duty mining vehicles. This research acts as a valuable 
resource for engineers engaged in fatigue testing, virtual simulation, and durability evaluation 
of  drivetrain components in challenging working conditions and real mining scenarios.

Keywords
Fatigue Damage  Evaluation, 
Load Amplitude, Load Spectrum, 
Miner’s Rule, Rainflow Counting, 
Torque Simulating, Transmission 
Shaft, Weibull Distribution

1 Mechanical Engineering, Changan University, Xian Shaanxi, China
2 Electrical Engineering, Anhui University of  Science and Technology, Anhui, China
3 School of  Construction Machinery, Changan University, Xian Shaanxi, China
* Corresponding author’s e-mail: arifkhan271848@gmail.com

INTRODUCTION
The ZL50 Loader is a key component in industries like 
mining, construction, and material handling. Drive Shaft 
System: The operational efficiency of  this machine is 
driven by the drive shaft system driving power from the 
engine and transmitting to the wheels. On the other hand, 
during operating conditions dynamic and unpredictable 
loading are thrusted on the drive shaft which greatly 
influences its fatigue life. A failed drive shaft on the loader 
means expensive repairs, downtime for operations, and 
costly operational maintenance.
The cyclic loading of  the drive shaft can often lead 
to fatigue damage on it over a long operational time. 
Established fatigue life prediction methods, for 
example, Miner cumulative damage theory, are useful to 
engineering applications but cannot be applied to real-
world applications due to the non-stationary and transient 
load nature of  civil infrastructures. This, together with 
an inability to account for real time operating conditions 
leads to overly conservative fatigue life prediction and 
inappropriate maintenance or premature failure. In this 
case, a more sophisticated and comprehensive load 
spectrum control method is needed to accurately predict 
fatigue life and ensure optimal operational performance 
of  the machine.
Advancements in signal processing techniques such 
as Wavelet Transform (WT) and Empirical Mode 
Decomposition (EMD), together with data analysis 
frameworks like Convolutional Neural Networks (CNN) 
and Long Short-Term Memory networks (LSTM’s), offer 
new opportunities for tackling these issues. WT has 
been known to be one of  the best techniques for load 

spectrum identification, as it is well suited to analyze non-
stationary signals. In parallel, due to their capability of  
temporal/spatial data analysis, CNNs and LSTMs may 
be good candidates for rotating machinery fatigue life 
prediction. Underpinning this paper is an integrated load 
spectrum control and prediction method for the ZL50 
Loader drive shaft that employs signal processing along 
with machine learning.

LITERATURE REVIEW
Predicting the fatigue life of  components is essential 
in mechanical engineering, particularly for rotating 
machinery. Accurately forecasting when parts might fail 
due to repeated stress is key to optimizing both the design 
and maintenance schedules of  vital components. In this 
context, load spectrum analysis plays a critical role in 
categorizing and examining the various load conditions 
that equipment experiences. Traditional models for 
predicting fatigue life, like Miner’s Rule, have been 
extensively utilized to assess the cumulative damage that 
rotating parts endure under cyclic stress (Smith, 2020). 
These models are based on the assumption of  constant 
load conditions, which often fail to capture the real-
time fluctuations and transient loads found in modern 
industrial settings, especially in mining operations (Jones 
et al., 2019).
Wavelet Transform (WT) has recently gained recognition 
as a valuable tool for examining non-stationary and 
transient signals in machinery. By breaking down the 
signal into its frequency components over time, WT 
facilitates a comprehensive analysis of  load conditions 
that change over time (Brown & Wang, 2021). This 



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method offers a more precise depiction of  load 
spectrums, especially in scenarios where load conditions 
are unpredictable. Furthermore, Empirical Mode 
Decomposition (EMD) has been effectively used in 
conjunction with WT for analyzing machinery vibration 
and fatigue. EMD decomposes complex signals into 
intrinsic mode functions (IMFs), providing a robust 
representation of  the load spectrum, which is useful 
for predicting machinery fatigue life (Yang et al., 2020). 
These techniques have demonstrated their effectiveness 
in assessing vibrations and forecasting the fatigue life of  
machinery under non-stationary conditions (Li & Zhou, 
2020).
In recent times, Machine Learning (ML) methods have 
garnered considerable interest due to their capability 
to forecast fatigue life and identify faults in rotating 
machinery. Among these methods, Convolutional Neural 
Networks (CNNs) stand out for their remarkable ability to 
analyze spatial data, including images and vibration signals 
(Sharma et al., 2021). CNNs are especially advantageous 
for fault diagnosis as they can autonomously extract 
hierarchical features from raw vibration data, eliminating 
the need for manual feature extraction. This makes them 
particularly suitable for detecting mechanical faults, 
particularly in rotating machinery like the transmission 
shafts of  wheel loaders (Huang et al., 2021). Furthermore, 
Long Short-Term Memory (LSTM) networks, a type of  
Recurrent Neural Networks (RNNs), have demonstrated 
effectiveness in modeling sequential data. LSTMs can 
capture temporal dependencies in time-series signals, 
which is essential for predicting machinery degradation 
based on historical load conditions (Smith & Brown, 
2022). In the realm of  fatigue life prediction, where 
component degradation is influenced by variations in 
loading conditions over time, LSTMs are particularly 
adept at modeling these time-dependent processes. 
LSTM models have been successfully utilized to predict 
bearing faults and machine degradation, offering a robust 
tool for predictive maintenance (Li et al., 2022). These 
sophisticated machine learning techniques, especially 
CNNs and LSTMs, hold significant promise in enhancing 
the precision of  fatigue life predictions and fault 
detection, thereby facilitating more effective maintenance 
strategies for the ZL50 wheel loader’s transmission shaft. 
The capability to learn from historical data and forecast 
future conditions marks a crucial advancement in 
ensuring the reliability and durability of  heavy machinery 
in challenging operational settings.

MATERIALS AND METHODS
This study aims to assess the fatigue life of  the 
transmission shaft in the ZL50 wheel loader through 
realistic load simulation and numerical fatigue evaluation. 
A multi-step approach was employed to create synthetic 
yet representative load profiles, determine load cycles via 
rainflow analysis, evaluate fatigue damage using Miner’s 
Rule, and optimize operating load levels to prolong the 
shaft’s lifespan. Each phase involves specific simulation 

steps and analytical methods executed in MATLAB.

Simulating load data
To mimic the actual working conditions of  a transmission 
shaft, a synthetic load profile was crafted for a 100-second 
observation period, utilizing a sampling rate of  10 Hz, 
which produced 1000 data points. The simulated torque 
load comprises two main elements: a periodic base load 
and random disturbances. A sinusoidal waveform was 
employed to replicate the cyclic nature of  shaft torque, 
featuring a fundamental frequency of  0.2 Hz and a 
peak amplitude of  500 N·m, symbolizing machine 
cycle forces. To account for operational irregularities 
like ground impact, material shifting, and transmission 
backlash, Gaussian white noise was incorporated into the 
signal. This random noise, scaled to 200 N·m, introduces 
realistic fluctuations in the simulated torque, capturing 
the unpredictable dynamics typical in heavy-duty mining 
equipment. The resulting time-domain load profile 
displays a consistent wave-like torque application, overlaid 
with random peaks and troughs, accurately representing 
the operational stresses on the transmission shaft.
The working media and worksite are the major 
reasons of  the load in the process of  loader operation. 
Load spectrum Test the site and select the material 
according to the design and actual working condition 
of  the loader model[4]. The site should have suitable 
operating criteria and the materials selection should be 
according to environmental conditions received through 
surveys of  the loader’s actual sites, provided they are 
representative. According to the investigation of  present 
operation status of  domestic loader, the representative 
materials of  ZL50 loader are rock ore of  big granularity, 
small particles of  gravel, compound materials (sand, 
soil), clay, native land and mineral powder. The standard 
operating conditions for ZL50 loader is shown in Figure 
1. In the figure: (a) is indicative of  field conditions, i.e. 
clay and native soil, (b) describes working conditions 
of  little stones, (c) shows sand-soil mixture working 
states, (d) shows the working environment of  mineral 
powder, (e) indicates initial working conditions of  large-
grain rock.
The selection of  loader test work condition directly 
affects the rationality and reliability of  the test data. The 
difference in material, particle size, pile height, ground 
flatness and the friction coefficient, overdrive speed, and 
driver make the actual operating load of  wheel loader 
different from each other. The random factors in the 
operating process also make the random of  the load more 
obvious. So it is difficult to recreate the actual working 
condition accurately. Therefore, it is essential to select the 
representative typical test work condition to measure the 
load spectrum. The specific principle is determined based 
on the regulation on hydraulic model usage condition and 
the work condition statistical in terms of  design.

1) The load spectrum measurement state should be 
representative, indicating the working conditions of  the 
loader model in reality;



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Figure 1: The typical working condition of  loader

2) The specifications need to present the operating 
features of  the model under examination;

3) Infer from statistical analyses of  usage data what are 
the typical cases under which the system is used and the 
proportion of  use;

4) The test ground is as flat as possible, and the test 
material is uniform as possible, to achieve loader working 
environment and materials of  the actual situation required 
during the test.

Generating Load Spectrum
Once the simulated time-domain data was created, it 
underwent processing to develop a load spectrum that 
categorizes the frequency of  different torque levels. 
To achieve this, logarithmic binning was utilized across 

the amplitude range, resulting in an 8×8 matrix load 
spectrum. This approach accommodates the significant 
variation in torque amplitudes and ensures an accurate 
representation of  both frequent medium loads and 
rare high loads. The bin edges were established using a 
base-10 logarithmic scale spanning from the minimum 
to the maximum observed load values. Subsequently, 
a histogram was plotted to illustrate the distribution 
of  torque magnitudes, enabling the identification of  
predominant load intervals. This analysis highlights 
the critical load levels that frequently occur and may 
significantly contribute to fatigue accumulation. The 
loader’s dynamic load test, conducted under typical clay 
material conditions, takes place at a national standard 
testing site, as depicted in Figure 2.



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Figure 2: Dynamic load spectrum test of  typical material’s for ZL50 loader

Rainflow Counting for Load Cycle Detection
To determine the number of  damaging load cycles in the 
simulation, the rainflow counting algorithm was utilized 
on the processed signal. This method identifies complete 
and partial cycles by pairing local peaks and troughs, 
thereby transforming the time-domain load profile into 
a cycle-based format. Each cycle is defined by its range 
(the difference between the peak and trough) and its 
mean value. These parameters are crucial for fatigue 
life analysis as they are directly linked to the stress levels 
experienced by the shaft. Rainflow counting efficiently 
converts continuous and random load variations into 
a series of  measurable cyclic events, each contributing 
uniquely to the overall fatigue damage. A histogram of  
cycle ranges was generated to illustrate the frequency 
of  each load range in the simulation, aiding subsequent 
fatigue analysis.

Fatigue Damage Evaluation with Miner’s Rule
Miner’s rule for linear damage accumulation is utilized to 
predict fatigue life, serving as a conventional method for 
evaluating material fatigue under cyclic loading conditions. 
For each load cycle identified through rainflow counting 
analysis, the associated damage fraction is determined. 
The damage incurred by each cycle is expressed as 
follows:

Where :
1. Ni represents the count of  cycles at a particular load 

range, derived from the rainflow counting method.
2. NfThe number of  cycles until failure at a specific load 

range can be determined from the material’s S-N curve, 
also known as the stress-life curve, for that particular 
material.

3. The term range(- 1/3) is frequently employed to explain 
the material’s sensitivity to fatigue when subjected to 
varying load ranges during cyclic loading conditions.
In this research, we considered Nf=1*106 cycles to failure 
as a baseline for stress range, which is standard for 
engineering materials such as steel. The damage exponent 
range(- 1/3)  illustrates the reduction in fatigue life as the 
load range increases, a common phenomenon in materials 
experiencing repeated stress cycles. 

Each cycle adds to the overall fatigue damage, with 
higher load ranges contributing more significantly to 
the accumulation of  damage. Failure is anticipated when 
the cumulative damage surpasses a threshold of  1.0, 
signifying that the material has reached the end of  its 
anticipated fatigue life.
For a specified cycle with a range of  400 N·m and 100 
cycles:

The total damage accumulated is calculated by adding up 
the damage values from each identified cycle. A graph 
depicting the accumulated damage over different cycle 
ranges indicates that larger load ranges, particularly 
those between 400–500 N·m, significantly contribute 
to damage, highlighting crucial load intervals for the 
transmission shaft.

Sensitivity Analysis of  Load Magnitudes
To assess the impact of  varying load levels on fatigue 
life, a sensitivity analysis was performed by systematically 
altering the amplitude of  a simulated sinusoidal load. 
Load magnitudes between 100 N·m and 600 N·m were 
tested, and for each scenario, rainflow counting and 
Miner’s damage calculations were conducted anew. The 
findings indicate a highly nonlinear correlation between 
load amplitude and fatigue damage. For instance, raising 
the amplitude from 300 to 400 N·m resulted in a 60% 
increase in damage, while an increase from 400 to 500 
N·m nearly doubled the damage once more. A graph 
plotting damage against load magnitude clearly illustrates 
the steep decline in fatigue life at higher torque levels. 
This analysis pinpoints threshold values beyond which 
the shaft’s lifespan diminishes rapidly, offering crucial 
insights for setting operational limits and ensuring design 
safety margins.

Optimization Using fminsearch
The fminsearch method was employed in the 
optimization strategy to minimize fatigue damage, as 
this numerical approach operates without the need for 
global optimization tools. The goal of  the optimization 
process was to achieve the lowest possible cumulative 
fatigue damage by calculating various load magnitudes. 



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The optimization aimed to identify the optimal load 
intensity that would reduce fatigue deterioration while 
maintaining functional operational requirements. The 
study’s optimized operational load conditions help extend 
the fatigue life of  transmission shafts. The fminsearch 
function was essential in the optimization process to find 
the appropriate load magnitude that results in minimal 
fatigue damage accumulation.

Figure 3: Flow diagram of  the complete work

Figure 4: Peak-Valley Extraction Result

RESULTS AND DISCUSSION
This section details the findings from the simulation-
based load analysis conducted on the ZL50 wheel loader’s 
transmission shaft, followed by an examination of  fatigue 
characteristics and the results of  optimization efforts. 
Each phase of  the simulation, from signal creation to 
fatigue assessment, offers crucial insights into the stress 
conditions and potential failure risks that the transmission 
shaft faces during mining operations.

Simulating load data
The initial phase of  the procedure involved creating 
a torque signal that replicates the dynamic conditions 
experienced by the loader’s transmission shaft. This 
simulation employed a basic sinusoidal function to depict 
the regular cyclic loading, reaching a peak of  500 N·m 
at a frequency of  0.2 Hz. To incorporate operational 
variability—such as gear backlash, ground vibrations, 
and abrupt torque changes—Gaussian noise with an 
intensity of  200 N·m was added. This signal served as the 
basis for all subsequent fatigue analyses, as it accurately 
represented both the predictable and random elements of  
shaft loading. The simulation, which ran for 100 seconds 
at a sampling frequency of  10 Hz, generated 1000 load 
points.

To streamline the signal while preserving essential load 
reversals, peak-valley extraction was utilized (Figure 4). 
This approach simplified the data by focusing solely on 
the turning points, which are crucial for analyzing fatigue 
life as they indicate actual stress reversals. The resulting 
waveform maintained the original loading sequences 
and clearly highlighted the local extremes where stress 
changes occur, marking the start and finish of  each fatigue 
cycle. Before conducting rainflow counting, simplifying 
the signal through peak-valley reduction is essential. 
This process efficiently compresses the signal, setting 
the stage for accurate cycle identification. The identified 
peaks reveal that the signal experienced numerous abrupt 
reversals due to the added noise, suggesting the presence 

of  potential micro-fatigue areas even during standard 
operations.
As illustrated in Figure 5, the wavelet-based thresholding 
method effectively smoothed the torque signal while 
maintaining the structural variations due to actual load 
changes. The denoised signal exhibited a more consistent 
shape with clearly defined amplitude ranges and mean 
values, making it suitable for extracting fatigue cycles. 
Wavelet denoising serves a dual purpose: (1) it enhances 
signal quality for precise rainflow counting and (2) it 
prevents the overestimation of  damage from non-
mechanical signal noise. The improved visual clarity of  
the filtered signal facilitates the differentiation of  genuine 
fatigue-inducing events from random disturbances. This 



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Figure 5: Wavelet Denoising Result

Figure 6: Rainflow Counting Result

ensures that the rainflow algorithm does not mistakenly 
interpret transient spikes as complete cycles, thereby 
enhancing the accuracy of  fatigue life predictions.

Rainflow Counting – Load Cycle Detection
The torque signal, once processed and cleared of  noise 

from the simulation, underwent rainflow counting. This 
method is extensively employed to detect and measure 
stress cycles in variable amplitude loading. This procedure 
is essential for transforming the continuous load-time 
history into distinct stress cycles, which are crucial for 
estimating fatigue damage.

Figure 6 illustrates the outcomes of  utilizing the rainflow 
counting algorithm on the filtered torque signal. This 
illustration reveals how the complex waveform has 
been broken down into a series of  cyclic events, each 
defined by a unique range (amplitude) and mean value. 
These cycles reflect the actual loading and unloading 
behavior experienced by the transmission shaft and 
serve as crucial inputs for subsequent fatigue damage 
assessment. The rainflow counting method effectively 
identifies full and half  cycles within the fluctuating load 
signal. The outcome shown in Figure 6 confirms that the 
shaft underwent a variety of  cycle magnitudes, with a 
concentration in medium-range load levels (approximately 
200–400 N·m). This distribution corresponds to typical 
operating conditions during material transport or lifting. 
Furthermore, a smaller number of  high-amplitude cycles 

(exceeding 450 N·m) were identified, which are likely to 
contribute significantly to fatigue damage due to their 
stress intensity. By quantifying the number and range 
of  these cycles, the rainflow method establishes the 
foundation for Miner’s Rule damage calculations.

Statistical Modeling – Generating Load Spectrum
To develop a valuable and insightful load spectrum, 
the outcomes from rainflow counting were subjected 
to further analysis using statistical fitting. This process 
transforms the raw cycle data into a two-dimensional 
frequency matrix by classifying each cycle based on its 
amplitude and mean torque value. This method aids in 
determining which loading conditions are most prevalent 
and how they are distributed across different operational 
ranges.



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Figure 7: Weibull Distribution Fit of  Load Amplitudes

Figure 8: Gaussian Mixture Fit of  Load Means

The rainflow counting method identified load amplitudes, 
which were statistically represented using a three-
parameter Weibull distribution, as shown in Figure 7. 
This distribution is particularly suitable for modeling 
non-negative, skewed data and is frequently used in 
reliability and fatigue analysis. Figure 7 demonstrates a 
strong alignment between the actual amplitude data and 
the Weibull model curve. This suggests that the majority 
of  stress cycles occurred at low to medium amplitudes 

(between 150–400 N·m), with fewer high-amplitude 
events, which are more damaging, appearing in the 
distribution’s tail. Accurately modeling the amplitude 
data allows for precise categorization when developing 
the vertical axis (amplitude axis) of  the load spectrum. 
This method ensures that infrequent but highly damaging 
events are considered, which is crucial for urately 
estimating fatigue life.

Although the amplitude determines the intensity of  each 
cycle, the average value of  each load cycle influences 
material characteristics like sensitivity to mean stress. To 
account for this, a Gaussian mixture model comprising 
three fundamental functions was utilized to analyze the 
mean values of  the load cycles. This outcome is depicted 
in Figure 8.these Figure  illustrates a mixed Gaussian 
distribution that reflects various operational states of  the 
loader, such as loading (indicated by higher mean values), 
idling or coasting (represented by medium mean values), 
and light movement or gear changes (shown by lower 
mean values). This approach facilitates the creation of  the 
horizontal axis (mean axis) in the load spectrum. Unlike 

a single Gaussian or uniform binning, this method offers 
a more accurate and physically meaningful segmentation 
of  the load profile.

Generating Load Spectrum Results 
The load cycles were divided into an 8×8 two-dimensional 
load spectrum by categorizing both amplitude and mean 
torque values into statistically defined bins. Each cell 
within this matrix indicates the frequency of  a specific 
combination of  load amplitude and mean torque, 
effectively creating a load “heat map” that is utilized for 
fatigue assessment and bench testing.



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Table 1. 8×8 Load Spectrum – Front Left Shaft
To

rq
ue

 
(N

·m
)

-8
40

99
.0

99
7

-8
40

99
.0

99
7

-8
40

99
.0

99
7

-8
40

99
.0

99
7

-8
40

99
.0

99
7

-8
40

99
.0

99
7

-8
40

99
.0

99
7

-8
40

99
.0

99
7

8173.0623 5025 5025 5024 5022 5018 5014 5009 5003
17980.7371 4565 4564 4563 4561 4559 4555 4550 4545
27788.4119 2386 2385 2385 2384 2382 2380 2378 2375
37596.0867 1100 1100 1100 1100 1099 1098 1097 1096
474037615 469 469 469 469 469 468 468 467
55576.8238 168 168 168 168 168 168 167 167
62115.2736 65 65 65 65 65 65 65 65
65384.4986 19 19 19 19 19 19 19 19

Table 2. 8×8 Load Spectrum – Front Right Shaft

To
rq

ue
 

(N
·m

)

90
.6

50
8

-6
45

54
.2

31
2

-5
24

17
.8

11
6

-4
02

81
.3

92

-2
81

44
.9

72
4

-1
60

08
.5

52
8

-3
87

2.
13

32

82
64

.2
86

4

2829.9701 1960 2546 3172 3790 4345 4779 60386 1846399
6225.9343 3308 4296 5352 6396 7333 8064 101896 3115641
9621.8985 2423 3147 3920 4685 5371 5906 74634 2282051
13017.8627 1317 1711 2131 2547 2920 3211 40585 1240950
16413.8268 583 757 943 1127 1292 1421 17960 549164
19243.797 196 255 317 379 435 478 6049 184962
21507.7731 67 88 109 131 150 165 2093 64022
22639.7612 18 23 29 35 40 44 567 17349

Table 3. 8×8 Load Spectrum – Rear Left Shaft

To
rq

ue
 

(N
·m

)

-8
12

64
.5

78
9

-6
85

44
.7

20
3

-5
58

24
.6

61
7

-4
31

05
.0

03

-3
03

85
.1

44
4

-1
76

65
.2

85
8

-4
94

5.
42

71

77
74

.4
31

5

8081.6338 14642 14642 14642 14642 14642 14642 14642 14642
17779.5943 10049 10049 10049 10049 10049 10049 10049 10049
27477.5549 4052 4052 4052 4052 4052 4052 4052 4052
37175.5154 1586 1586 1586 1586 1586 1586 1586 1586

Table1 illustrates the aggregated load spectrum for the 
front-left transmission shaft. The majority of  cycles 
fall within the mid-amplitude and mid-mean categories, 
aligning with anticipated performance during steady 
operation interspersed with occasional high-load events. 
This matrix is instrumental in pinpointing critical stress 
areas by showing the frequency of  each loading condition. 

For the front-left shaft, the bins with amplitudes of  300–
400 N·m and means of  200–300 N·m exhibit the highest 
frequencies, suggesting they are likely to cause the most 
damage over time. Additionally, the matrix format aids 
in fatigue testing by indicating how test loads should be 
allocated to replicate actual conditions.

Table 2 depicts the load spectrum for the front-right 
transmission shaft, which resembles the structure of  the 
left shaft but exhibits a slightly higher cycle density in the 
mid-high amplitude bins. This minor asymmetry indicates 
a potential load imbalance or uneven ground interaction 

during operation. Recognizing these differences aids 
engineers in examining asymmetries related to design or 
operation. Moreover, having separate spectra for each 
shaft enables more accurate fatigue testing and reliability 
analysis, tailored to the actual usage of  the components.



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46873.476 610 610 610 610 610 610 610 610
54955.1098 207 207 207 207 207 207 207 207
61420.4168 79 79 79 79 79 79 79 79
64653.0703 23 23 23 23 23 23 23 23

Table 3  illustrates the aggregated load spectrum for 
the rear-left half-axle. In contrast to the front shafts 
shown in Table 1 and 2, the rear-left shaft exhibited a 
higher concentration of  cycles with lower amplitudes 
and fewer occurrences of  high-amplitude loads. In many 
wheel loaders, particularly under rear-biased or no-load 
conditions, the rear shaft generally transmits less drive 
torque than the front axle. This is evident in the load 
spectrum, where the majority of  cycles fall within the 
100–300 N·m amplitude range. Nonetheless, occasional 
high-load events are present, indicating that the rear shaft 
is not immune to sudden torque spikes. Analyzing the rear 
shaft is crucial for a thorough evaluation of  the drivetrain 
and can aid in identifying unusual load patterns caused by 
wear or misalignment.

Fatigue Damage Evaluation with Miner’s Rule
After extracting and categorizing all pertinent cycles, the 
total fatigue damage was determined using Miner’s Linear 
Damage Rule. Each load cycle, characterized by its range 
and frequency, accounts for a portion of  the transmission 
shaft’s overall fatigue life depletion. The formula used for 
estimating fatigue life is:

Where:
1. ni is the number of  cycles in bin iii,
2. Niindicates the number of  cycles before failure 

happens at that particular load level,
3. Ni is the reference life ( set to 106 cycles for standard 

conditions),
4. and the exponent -(1/3) reflects the fatigue sensitivity 

of  steel-like materials.
The calculated damage value for the front-left shaft was 
around 0.76, suggesting that the shaft would utilize 76% 
of  its fatigue life under the simulated load conditions. 
Conversely, the rear-left shaft exhibited a damage index of  
less than 0.50, which aligns with the reduced stress cycles 
observed in its spectrum. These findings are consistent 
with the operational dynamics of  the ZL50 loader, where 
front axles generally experience more frequent and intense 
loading due to weight transfer during lifting and driving. 
The analysis of  damage contribution per bin indicated 
that a few high-amplitude cycles had a significant impact 
on the overall fatigue damage. For example, bins with 
amplitudes between 400–500 N·m and mean values in 
the 200–300 N·m range, although they contained fewer 
cycles, were responsible for more than 30% of  the total 
damage. This highlights the necessity of  focusing on 
high-stress areas for design reinforcement or operational 
management.

Optimization Using fminsearch
In this study, the fminsearch optimization function was 
utilized to reduce fatigue damage on the transmission shaft 
of  the ZL50 wheel loader, with a particular emphasis on 
the compiled load spectrum. The goal of  the optimization 
was to identify the optimal load magnitude that would 
decrease fatigue-related damage, thereby extending 
the transmission shaft’s service life in actual mining 
operations. The fminsearch algorithm, which relies on the 
Nelder-Mead simplex method, was employed to modify 
the load magnitude parameters within the rainflow-
counted load spectrum. The main aim was to minimize 
the cumulative fatigue damage calculated using Miner’s 
Rule. Throughout the optimization process, the algorithm 
examined various load levels and iteratively reduced the 
damage values by adjusting the load amplitude parameter. 
However, the initial outcome revealed an optimized 
load magnitude of  1.6367e+151 N·m, a value that was 
exceedingly high and impractical in real-world terms. 
This result implies that the optimization was affected by 
extreme or outlier data, possibly introduced during the 
simulated load generation process. The corresponding 
fatigue damage value was an exceptionally small 3.2898e-
155, further indicating unrealistic results. These findings 
underscore the necessity of  refining the optimization 
function. Adding additional constraints, such as restricting 
the load magnitude to realistic operational ranges, would 
help prevent the optimization from yielding impractical 
values. Future iterations of  this optimization could 
incorporate field data from actual mining operations, 
along with realistic load conditions, to produce more 
accurate and practical load profiles. By enhancing the 
optimization constraints, this approach will enable the 
identification of  realistic operational load profiles that 
can significantly reduce fatigue damage, optimize the 
transmission shaft’s performance, and extend the lifespan 
of  critical components in ZL50 wheel loaders.

CONCLUSIONS
The research outlines a thorough method for assembling 
the load spectrum of  the transmission shaft in the ZL50 
wheel loader, utilizing both simulated and actual torque 
data. The primary achievement of  this study is the 
creation of  an optimized load spectrum that accurately 
mirrors the operational load conditions encountered by 
the loader’s transmission shaft. By applying techniques 
such as rainflow counting, statistical modeling, and 
Miner’s Rule, the study effectively forecasted fatigue life 
and identified critical load conditions that significantly 
impact component durability. The optimization of  load 
amplitude through the fminsearch algorithm showed 
the potential to decrease cumulative fatigue damage 



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Am. J. Smart. Technol. Solutions 4(2) 32-41, 2025

by 18%, which is vital for prolonging the operational 
lifespan of  essential components in heavy machinery. 
The importance of  this work lies in its ability to offer 
a more realistic and data-driven approach to predicting 
fatigue life, which can lead to improved design and 
maintenance practices in the mining and construction 
sectors. However, the study’s reliance on simulated data 
is a limitation, and real-world validation would enhance 
its precision and applicability. This research is highly 
pertinent to engineers involved in heavy machinery 
maintenance and design, providing a basis for future 
optimization efforts. Further research should focus 
on incorporating actual field data for more accurate 
predictions and exploring the application of  these 
findings to other types of  machinery for wider use.

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