Load Analysis and Test Bench Load Spectrum Generation for Electric Drive Systems Based on Virtual Proving Ground Technology

Abstract

The reliability of the EDS (Electric Drive System) in electric vehicles is crucial to overall vehicle performance. This study addresses the challenge of acquiring high-fidelity internal load data in the early development phase due to the absence of prototypes, overcoming the limitations of traditional road tests, which are costly, time-consuming, and unable to measure gear meshing forces. A method based on a VPG (Virtual Proving Ground) is proposed to acquire internal loads of a dual-motor EDS, analyze the impact of typical virtual fatigue durability road conditions on critical components, and generate load spectra for test bench experiments. Through point cloud data-based road modeling and rigid-flexible coupled simulation, dynamic loads are accurately extracted, with pseudo-damage contributions from eight intensified road conditions quantified using pseudo-damage calculations, and equivalent sinusoidal load spectra generated using the rainflow counting method and linear cumulative damage theory. Compared to the limitations of existing VPG methods that rely on simplified models, this study enhances the accuracy of internal load extraction, providing technical support for EDS durability testing. Building on existing research, it focuses on high-fidelity acquisition of EDS loads and load spectrum generation, improving applicability and addressing deficiencies in simulation accuracy. This study represents a novel application of VPG technology in electric drive system development, resolving the issue of insufficient early-stage load spectra. It provides data support for durability optimization and bench testing, with future validation planned using real vehicle data.

1. Introduction

With the rapid development of the electric vehicle industry, the reliability of the EDS has become a critical factor affecting vehicle performance and market competitiveness. Compared to traditional internal combustion engine vehicles, the EDS in electric vehicles exhibits distinct dynamic characteristics: instantaneous high torque output and frequent fluctuations subject transmission components to severe alternating loads; the demand for high-speed operation results in a highly compact system structure, leading to size constraints and pronounced stress concentration issues in components such as gears and transmission shafts [1]; and lightweight design further reduces material thickness, diminishing strength redundancy. These characteristics impose higher requirements on fatigue durability and reliability validation of the EDS during the development phase.

Although traditional vehicle testing on the road can provide real load data, it is costly, time-consuming, and affected by environmental and human factors, making it difficult to ensure test consistency and meet the efficiency demands of modern platform-based development, particularly in the early stages when prototype vehicles are not available [2,3]. Moreover, conventional road spectrum collection can only capture external connection channel loads, such as those at suspension mounts, lugs, or shaft systems, but fails to directly measure internal loads critical to EDS durability analysis, such as gear meshing forces [4]. VPG technology, through digitized road modeling and multibody dynamics simulation, offers an efficient alternative to generate high-fidelity load spectra [5,6,7]. Several studies have advanced the application of VPG technology. Tasci et al. (2011) [2] developed a 3D digital road scanning system at General Motors, using laser-based measurement to create high-precision CRG models compatible with the FTire model (flexible tire model) and ADAMS simulations, enabling accurate load prediction for vehicle systems and highlighting the superiority of 3D road profiles over 2D profiles in capturing realistic tire–road interactions. Haga et al. (2006) [4] proposed a time-domain evaluation method using a tire suspension test rig, validating the accuracy of the FTire model in predicting spindle forces at various cleat heights, demonstrating Nissan’s application of VPG for durability analysis. Zhou et al. (2021) [8] described a digital road construction method for VPG, employing CRG models derived from point cloud data for vehicle dynamics simulation, validating its application in Chongqing Changan Automobile. Wu et al. (2017) [9] applied the VPG simulation to evaluate knuckle impact fatigue, integrating 3D pebble road models and FTire model, confirming the practical importance of the technology for structural optimization. Minakawa et al. (2004) [10] used simplified velocity-driven models in VPG simulations, but neglecting motor or transmission dynamics limited the accuracy of internal load extraction, such as gear meshing forces. Riepl et al. (2004) [11] documented Fiat’s use of VPG with FTire and RGR (Regular Grid Road) models to enhance load spectrum generation, although challenges remained to capture internal gear dynamics. These studies indicate that leading companies like General Motors, Nissan, and Fiat have widely adopted VPG technology, but existing methods often rely on simplified velocity-driven models, making it challenging to accurately extract internal EDS loads, thus limiting the reliability of the test bench design.

To address these challenges, this study focuses on the dual-motor EDS of electric vehicles, as its critical role in vehicle performance and complex dynamic load characteristics make it an ideal subject for studying internal load extraction and durability validation. The dual-motor EDS, due to its high torque output, frequent load fluctuations, and multi-power-source interactions, generates unique internal load patterns, posing higher demands on fatigue durability analysis. The secondary reducer, differential, transmission shaft, input shaft, and output half-shaft are selected as critical components, as they serve as primary load-transmitting elements, directly bearing motor torque and road-induced alternating stresses. The integration of steering, suspension, and body subsystems ensures that the rigid-flexible coupled model accurately simulates vehicle dynamics, meeting the high-fidelity simulation requirements of VPG technology. This study proposes a high-precision method based on a VPG for load acquisition and equivalent load spectrum generation. The method employs finite element modeling and rigid-flexible coupled multi-body dynamics simulation, integrating CRG road models generated from point cloud data and flexible tire models. It uses HyperMesh to perform finite element modeling of critical components, generating MNF files; subsequently, an EDS subsystem is constructed in ADAMS, integrated with steering, suspension, and body subsystems to establish a rigid-flexible coupled vehicle dynamics model, enhanced by FTire model for high-fidelity simulation. Road models are processed from point cloud data through statistical outlier filtering, voxel grid downsampling, RANSAC plane fitting, and PCA (Principal Component Analysis) coordinate alignment, utilizing multi-resolution grid division and inverse distance weighted interpolation to create CRG files. For eight intensified rough road conditions, nCode 11.1 software is used to calculate pseudo-damage, quantifying pseudo-damage contributions; the rainflow counting method and Miner’s linear cumulative damage theory are then applied to transform complex load spectra into equivalent sinusoidal load spectra suitable for test bench testing [12,13]. This study overcomes the limitations of traditional models in internal load extraction, laying a data foundation for durability optimization design and test bench experiments.

2. Virtual Model Construction

2.1. Road Surface Model

Currently, commonly used road surface models are divided into RGR models and CRG models. RGR models feature uniform node spacing, making them suitable for simulating simple terrains. However, they face challenges in accurately describing complex roads with curves, slopes, and textures, resulting in cumbersome modeling processes. In contrast, CRG models allow flexible adjustment of lateral node spacing, enabling precise representation of complex road surfaces with high computational efficiency. Additionally, CRG is a universal format that facilitates the processing of various road data types. Therefore, this study adopts road surfaces converted to CRG format for VPG simulation analysis.

A vehicle-mounted laser scanning system was employed to collect high-density point cloud data from the durability test roads of the proving ground, achieving a density exceeding 50 points per square meter. The raw LAS format data includes multidimensional information, such as spatial coordinates, reflection intensity, and classification labels [14]. Preprocessing involves a multi-stage filtering and denoising mechanism to achieve data refinement. A statistical outlier filtering algorithm is applied, which calculates the distance distribution within the point cloud neighborhood and removes discrete noise points by setting a threshold of three standard deviations. Voxel grid downsampling is utilized to reduce the data volume while preserving road surface features. Depending on the complexity of the road features and computational resource constraints, a compression ratio of 40% is applied for flat roads, while a 60% ratio is used for roads with complex textures, ensuring precision loss remains within 5% to effectively reduce subsequent computational loads. For road areas in the virtual proving ground with complex conditions, such as water accumulation or debris, the RANSAC plane fitting algorithm is iteratively applied to separate ground point clouds. By setting a maximum distance tolerance to meet specified precision requirements, the road surface is accurately identified, and effective point sets are extracted. To eliminate deviations between the original coordinate system and the vehicle dynamics simulation coordinate system, a coordinate alignment transformation matrix is constructed using PCA. This is achieved by calculating the eigenvectors of the point cloud covariance matrix, where the direction of the largest eigenvalue is designated as the longitudinal axis, aligned with the primary road texture direction; the second-largest eigenvalue direction is set as the lateral axis, and the smallest eigenvalue direction as the normal axis. This operation aligns the primary road texture direction with the coordinate axes, providing a unified geometric reference for subsequent grid division [15].

The OpenCRG toolchain, developed based on MATLAB R2016a and adhering to the ISO 15037-3 [16] standard, converts preprocessed point cloud data into parameterized road surface models. This process employs multi-resolution grid division, with spatial resolution determined based on research objectives. For standard resolution, the planar computational domain is determined by calculating the extrema of point cloud coordinates, and an equal-interval grid division method is applied. The number of grid nodes is calculated using the formula:

$$N_u={\displaystyle \frac{u_{\max }-u_{\min }}{\Delta u}}-1,N_v={\displaystyle \frac{v_{\max }-v_{\min }}{\Delta v}}-1,$$

(1)

where $\Delta u$ and $\Delta v$ represent the grid step sizes in the longitudinal and lateral directions, respectively.

For each grid node, an inverse distance weighted interpolation algorithm is used to calculate the normal height, balancing interpolation smoothness and feature preservation by adjusting the weight exponent. The resulting CRG model contains a standardized data structure: at the geometric level, it includes axis ranges, grid step sizes, and an elevation matrix. To support the requirements of multi-body dynamics simulation in ADAMS, the parameterized road model defines periodic characteristics through the mods field at the physical level: Repeat Period specifies the lateral repeat period, defining the length of repeating texture units along the road; Repeat Times indicates the number of longitudinal repetitions, defining the spatial expansion factor of the model during simulation. At the quality control level, the crg_check function provided by the OpenCRG toolbox is used to perform integrity checks on the road model, including grid continuity, elevation outliers, and boundary consistency, ensuring model errors are controlled within acceptable limits. As shown in Figure 1, a schematic comparison of the fish-scale pit CRG reinforced road surface file and point cloud data for a typical virtual test field fatigue durability road. As shown in Figure 2, a schematic of other processed typical fatigue durability CRG road surface files.

2.2. Flexible Tire Model

The FTire model is highly favored in VPG applications due to its capability to output high-frequency response data. VPG requires the simulation of various complex road conditions, such as short-wavelength surfaces like cobblestone roads and densely pitted roads, where the high-frequency dynamic response characteristics of tires are critical. For instance, when a vehicle travels on rough cobblestone or densely bumped roads, the tire must rapidly respond to high-frequency road excitations. The FTire model accurately captures such high-frequency dynamic responses, playing a pivotal role in precisely simulating vehicle handling stability, ride comfort, and measuring road loads under complex conditions.

In contrast, magic formula tire models, such as the PAC model, have their own advantages but exhibit limitations in simulating high-frequency excitations. Magic formula tire models fit experimental data using combinations of trigonometric functions, primarily to describe mechanical properties such as longitudinal force, lateral force, and aligning torque during normal vehicle operation. However, for the high-frequency, short-wavelength excitations common in VPG, these models struggle to accurately reflect the dynamic mechanical behavior of tires.

The FTire model is not only suitable for vehicle simulations under short-wavelength road irregularities but also serves as a strongly nonlinear dynamic model based on physical mechanisms, enabling the study of vehicle handling characteristics under complex excitations. Based on flexible ring theory, it simulates nonlinear stiffness and damping characteristics by discretizing the tire structure. Its rigorous physical modeling framework allows integration into multi-body dynamics and finite element analysis environments. Even in the absence of experimental data, the FTire model ensures reasonable and consistent simulation results. The model covers an extensive range of frequencies and wavelengths, comprehensively accounting for various excitation sources and nonlinear transmission mechanisms. The relative error between model parameter identification results and experimental values does not exceed 12% [3], meeting accuracy requirements and ensuring the reliability of tire force calculations in subsequent experiments.

2.3. Electric Drive Vehicle Model

Based on vehicle parameters, an EDS model, including a two-stage reducer, differential, transmission shaft, input shaft, and output half-shaft, is constructed in ADAMS using the electric drive system template. The typical structure is shown in Figure 3. This EDS model is interconnected with other critical vehicle subsystems, including steering, braking, front and rear suspensions, and the vehicle body, to establish a complete vehicle model.

Considering the significant influence of components such as the front and rear stabilizer bars in the vehicle model, as well as the reducer gears, housing, transmission shaft, input shaft, and output half-shaft in the EDS, on the loads at the connection points between the transmission system and the vehicle body frame, finite element mesh models of these critical components are first constructed in HyperMesh, as illustrated in Figure 4 for the housing finite element model. Material properties, node distribution, and element types are precisely defined, and appropriate boundary conditions and control cards are set to simulate actual assembly relationships. Subsequently, free modal analysis is performed using the OptiStruct solver to extract modal parameters and generate MNF required by ADAMS. This enables the construction of a rigid-flexible coupled vehicle model incorporating a flexible transmission system [17,18], as shown in Figure 5.

3. Materials and Methods

3.1. Road Surface Model

The vehicle virtual road simulation selects representative high-frequency, low-frequency, and intensified impact rough road conditions. Eight types of CRG models are generated from point cloud data, with their types, simulation speeds, and durations listed in

Table 1. Road conditions for VPG simulation.

Road Type	Road Length (m)	Simulation Speed (km/h)	Simulation Duration (s)
Uniform wavy road	110	40	9.9
Large cobblestone road	252	30	30.2
Small cobblestone road	120	40	10.8
Belgian road	100	30	12.0
Washboard road	80	40	7.2
Twisted road	95	20	17.1
Fish-scale pit	220	40	19.8
Repaired asphalt road	140	40	12.6

. The simulation schematic is shown in Figure 6.

From the perspective of dynamic transmission characteristics, the dense pulse excitations of high-frequency washboard roads, the quasi-static large-amplitude force deviations of low-frequency twisted roads, and the narrow pulse impacts of intensified rough roads are distinctly characterized by the time-domain features of wheel center vertical forces. The wheel center loads for the three different road surface types are shown in Figure 7.

To validate the accuracy of the rigid-flexible coupled model in VPG simulations, a comparative analysis with the rigid body model was conducted, focusing on the time-domain characteristics of the left front wheel vertical force and suspension point longitudinal force under the washboard road condition (40 km/h, 6 s). The rigid body model assumes all components are rigid, neglecting material damping and deformation effects (axial stretching of the input shaft and contact strain of gears). The comparison results are presented in Figure 8 and

Table 2. Comparison of dynamic responses between rigid-flexible coupled model and rigid body model under washboard road condition.

Parameter	Rigid-Flexible Coupled Model	Rigid Body Model
Peak front wheel vertical force (N)	10,416.29	12,568.80
Average front wheel vertical force (N)	6344.27	7059.27
Standard deviation of wheel vertical force (N)	2116.78	2456.76
Peak suspension point longitudinal force (N)	6641.29	6791.81
Average suspension point longitudinal force (N)	4975.01	4992.91
Standard deviation of suspension longitudinal force (N)	1077.37	1142.10

.

Figure 8 illustrates the time-domain response curves of the left front wheel vertical force and suspension point longitudinal force for both the rigid-flexible coupled model and the rigid body model under the washboard road condition. The rigid body model exhibits greater amplitude and variation range in wheel vertical force, attributed to its neglect of damping and vibrational modes of flexible components. The rigid-flexible coupled model, utilizing MNF files, captures these dynamic effects, significantly reducing wheel vertical force fluctuations and providing a more realistic mechanical response. The suspension point longitudinal force curves show smaller differences between the models, though the rigid body model displays slightly higher fluctuations due to its inability to simulate flexible interactions between the suspension and transmission system.

Table 2 indicates that under high-frequency washboard road excitation, the rigid body model’s peak wheel vertical force (12,568.80 N) is approximately 20.7% higher than that of the rigid-flexible coupled model (10,416.29 N), and its standard deviation (2456.76 N) is 16.1% higher, reflecting a larger force variation range. The peak suspension point longitudinal force (6791.81 N for the rigid body model vs. 6641.29 N for the rigid-flexible model) and standard deviation (1142.10 N vs. 1077.37 N) show smaller differences (2.3% for peak and 6.0% for standard deviation), suggesting that suspension point forces are less influenced by flexibility effects, though the rigid body model still exhibits slightly higher fluctuations. These findings align with the conclusions of Liao et al. (2023) [19], which reported that dynamic forces in rigid-flexible coupled models are 10–20% lower than those in rigid models, as rigid models tend to amplify responses under high-frequency excitation. This comparative analysis demonstrates that the rigid-flexible coupled model more accurately captures the dynamic behavior of the electric drive system under high-frequency washboard road excitation, avoiding the overestimation of wheel vertical force fluctuations inherent in the rigid body model. This provides a more reliable foundation for optimizing design and conducting fatigue analysis.

3.2. Critical Components Analysis

To ensure the modeling accuracy of key EDS components in Virtual VPG simulations, the technical characteristics of the input shaft, reduction helical gear, intermediate gear shaft, differential bevel gear, and output half-shaft (including material, physicochemical properties, and geometric features) have been precisely defined. Geometric parameters are derived from actual measurements, and material selection is informed by Lipman and Maier (2021) [20], balancing high strength, fatigue resistance, and cost-effectiveness. Alloy steels 42CrMo4 and 40Cr offer excellent toughness and fatigue resistance, being suitable for withstanding axial forces, torque, and cyclic loads; carburized steel 18CrNiMo7-6, with surface hardening (60 HRC), significantly enhances wear resistance and contact fatigue performance, making it ideal for gear components; 20MnCr5 alloy steel combines strength and machinability, making it appropriate for the intermediate gear shaft.

Table 3. Technical characteristics of key EDS components.

Component	Material	Geometric Features
Input shaft	Alloy steel (42CrMo4)	Maximum radius: 30 mm, gear connection radius: 20.645 mm, length: 377.1 mm
Reduction helical gear	Carburized steel (18CrNiMo7-6)	Module: 2.2, helix angle: 23.5°, number of teeth: 25/89
Intermediate gear shaft	Alloy steel (20MnCr5)	Large end radius: 65 mm, small end radius: 25.915 mm, length: 250 mm, helix angle: 23.5°
Differential bevel gear	Carburized steel (18CrNiMo7-6)	Pitch angle: 45°, tooth face width: 25.4 mm, number of teeth: 18, module: 3
Output half-shaft	Alloy steel (40Cr)	Radius: 17.5 mm, spline length: 50 mm

summarizes the technical characteristics of these key components, ensuring simulation results align with real-world conditions.

The simulation defines material properties during the generation of the flexible body MNF file, and the strain distribution cloud maps of the components are obtained from the results after VPG simulation, and are shown in Figure 9.

To thoroughly investigate the damage mechanisms of various loads on critical components, the study initially considers the damage contribution of each load independently. The excitations from low-frequency, high-frequency, and intensified rough roads exhibit distinct load mechanisms on the aforementioned critical components during simulation. Specifically:

Low-frequency road excitations cause axial misalignment, generating axial tensile or compressive forces, which are the primary factors for damage on twisted roads. High-frequency road excitations transmit vertical impacts through the suspension, inducing vertical force resonance. Radial forces, mainly caused by gear meshing misalignment or bearing installation errors, have a minor impact. The material, 42CrMo4 alloy steel, may contain inclusions or microcracks that, under axial and vertical forces, could initiate cracks at the shaft journal. Surface scratches reduce the fatigue limit. Therefore, fatigue analysis of the input shaft should focus on the damage effects of axial and vertical forces.

As the core of power transmission, the helical gear bears the combined effects of torque, tangential force, axial force, and radial force. Torque, influenced by road conditions and driving behavior, fluctuates significantly during rapid acceleration or rough road impacts, easily causing fatigue cracks at the gear root. Tangential force, correlated with torque, exacerbates gear surface wear and contact stress. Axial force, due to the helix angle design, alternates frequently under steering or bumpy conditions, leading to bearing misalignment and reduced meshing accuracy. Radial force, positively correlated with tangential force, is intensified by high-frequency excitations or manufacturing errors, affecting bearing life. The material, 18CrNiMo7-6 carburized steel, may have residual stresses or microcracks in the carburized layer, readily initiating cracks at the gear root, while uneven hardening may lead to pitting. Therefore, a comprehensive evaluation of the damage effects of torque, tangential force, axial force, and radial force is necessary.

This component receives input torque and transmits it to the next stage. Intensified rough road conditions cause torque fluctuations, while the helical gear’s helix angle generates axial force, and radial force is significantly affected by meshing characteristics. The material, 20MnCr5 alloy steel, may have surface defects or grain boundary weakening, which could initiate cracks under torque and axial force. Vertical force, attenuated through multiple stages (tire–suspension–housing), has a relatively small amplitude (approximately 5% to 10% of the wheel center vertical force) and can be simplified in analysis. Therefore, fatigue analysis should focus on torque, axial force, and radial force.

Responsible for torque distribution, the tangential force varies significantly with road resistance. Sudden torque spikes under rough road impacts [21] (reaching 3–5 times the rated value) are the primary causes of gear surface pitting and spalling. Radial force, correlated with tangential force, exacerbates meshing errors and bearing fatigue under high-frequency vibrations. Axial force, due to the bevel gear design, alternates frequently on low-frequency twisted roads, affecting meshing stability. The material, identical to the reducer helical gear, may have microcracks or inclusions in the carburized layer, potentially leading to pitting or gear root cracks. Therefore, the damage effects of tangential force, axial force, and radial force should be prioritized in analysis.

As the terminal of power transmission, torque surges instantaneously under rough road or hill-climbing conditions, easily causing fatigue fractures at the spline root or shaft journal. Vertical force, transmitted through the wheel to the outer end of the half-shaft, generates alternating bending stress, leading to surface cracks. Axial force intensifies universal joint wear during vehicle body torsion, with a damage contribution secondary to torque and vertical force. The material, 40Cr alloy steel, may contain inclusions or surface defects that could initiate cracks under torque and bending stress, with uneven heat treatment exacerbating fatigue failure. Therefore, fatigue analysis should focus on the damage effects of torque, vertical force, and axial force.

3.3. Road Surface Pseudo-Damage Contribution

Based on the simulation load data from eight types of CRG models and the force characteristics analysis of critical components, this study employs nCode software to perform pseudo-damage calculations and achieves systematic quantification of the pseudo-damage contribution of different roads to each component through normalized road excitation processing. Due to variations in road length resulting from the conversion of raw point cloud data and differences in simulation speeds set for each condition, the time histories of the load spectra obtained from simulations lack a unified reference for travel distance. To ensure the comparability of load data across different road conditions, the original load spectra require normalized truncation processing. Given that a fixed calculation step size and sampling frequency were uniformly set during the simulation process, this study standardizes the load spectra by defining an equivalent travel distance for the vehicle model under each condition, using a proportional time truncation method based on the target road length and calculating the corresponding truncation duration according to the simulation speed of each condition.

$$t_{\mathrm{cut}}={\displaystyle \frac{L_{\mathrm{target}}}{v}},$$

(2)

where $L_{\mathrm{target}}$ is the target road length (m) and v is the simulation speed (m/s). Reasonable truncation is performed based on the effective simulation duration and speed. Due to the unstable vehicle dynamic response in the initial stage of simulation and boundary condition interference in the final stage, it is necessary to exclude the transitional segment data with indistinct characteristics at both ends, thereby obtaining a standardized load time series under equal-distance travel conditions, as shown in

Table 4. Standardized load time series.

Road Type	Simulation Speed (km/h)	Standard Time Series (s)
Uniform wavy road	40	6
Large cobblestone road	30	8
Small cobblestone road	40	6
Belgian road	30	8
Washboard road	40	6
Twisted road	20	12
Fish-scale pit	40	6
Repaired asphalt road	40	6

.

To calculate the pseudo-damage effects that require focused attention for the components, namely the pseudo-damage contribution of the target channels under eight different road conditions, based on the linear cumulative principle, the pseudo-damage contribution of the j-th road condition is defined as:

$${\eta }_j={\displaystyle \frac{D_j}{D_{\mathrm{total}}}}\mathrm{and}\sum _{j=1}^8{\eta }_j=1,$$

(3)

where ${\eta }_j$ represents the normalized proportion of the road condition’s contribution to the overall pseudo-damage.

In nCode, pseudo-damage is calculated, and the relationship between material fatigue life N and stress amplitude is parameterized using the S–N curve in logarithmic form:

$$log\left(N\right)=\mathrm{SNIntercept}-\mathrm{SNSlope}·log\left(\sigma \right),$$

(4)

where SNSlope is the slope of the S–N curve, reflecting the material’s sensitivity to stress variations, and SNIntercept is the intercept of the S–N curve, a material constant.

In the EDS of electric vehicles, critical components such as the input shaft, reducer helical gear, intermediate transmission gear shaft, differential bevel gear, and output half-shaft are typically made of steel or alloy steel materials. These materials, with high strength, good toughness, and wear resistance, ensure efficient and stable power transmission. However, under complex and variable road conditions, the dynamic loads borne by each component vary significantly, and the stress response characteristics of different channels contribute differently to fatigue damage. To accurately quantify the damage extent of each critical component under different road excitations and identify key factors affecting system durability, this study employs the nCode platform, combined with Miner’s linear cumulative damage theory, to perform pseudo-damage calculations. It systematically analyzes the damage contribution of different channels for each component across eight CRG conditions, providing data support for the optimization design and reliability enhancement of the EDS.

Based on the comprehensive analysis of

Table 5. Input shaft pseudo-damage contribution.

Road Type	Torque	Axial Force	Radial Force
Uniform wavy road	0.065	0.042	0.068
Large cobblestone road	0.220	0.160	0.230
Small cobblestone road	0.040	0.029	0.037
Belgian road	0.120	0.100	0.132
Washboard road	0.180	0.125	0.198
Twisted road	0.095	0.360	0.100
Fish-scale pit	0.200	0.155	0.205
Repaired asphalt road	0.080	0.069	0.130

,

Table 6. Reducer helical gear pseudo-damage contribution.

Road Type	Torque	Tangential Force	Radial Force	Axial Force
Uniform wavy road	0.058	0.060	0.062	0.048
Large cobblestone road	0.210	0.220	0.225	0.190
Small cobblestone road	0.037	0.035	0.033	0.030
Belgian road	0.108	0.110	0.115	0.105
Washboard road	0.162	0.173	0.190	0.140
Twisted road	0.085	0.082	0.090	0.340
Fish-scale pit	0.190	0.200	0.205	0.180
Repaired asphalt road	0.080	0.080	0.080	0.087

,

Table 7. Intermediate transmission gear shaft pseudo-damage contribution.

Road Type	Torque	Axial Force	Radial Force
Uniform wavy road	0.060	0.046	0.056
Large cobblestone road	0.215	0.195	0.220
Small cobblestone road	0.041	0.029	0.038
Belgian road	0.115	0.118	0.103
Washboard road	0.165	0.150	0.172
Twisted road	0.090	0.400	0.073
Fish-scale pit	0.195	0.185	0.195
Repaired asphalt road	0.080	0.077	0.083

,

Table 8. Differential bevel gear pseudo-damage contribution.

Road Type	Tangential Force	Axial Force	Vertical Force
Uniform wavy road	0.058	0.052	0.055
Large cobblestone road	0.210	0.190	0.220
Small cobblestone road	0.037	0.028	0.030
Belgian road	0.108	0.095	0.103
Washboard road	0.162	0.128	0.180
Twisted road	0.085	0.310	0.080
Fish-scale pit	0.190	0.180	0.200
Repaired asphalt road	0.080	0.067	0.082

and

Table 9. Output half-shaft pseudo-damage contribution.

Road Type	Torque	Axial Force	Vertical Force
Uniform wavy road	0.060	0.047	0.052
Large cobblestone road	0.220	0.190	0.210
Small cobblestone road	0.037	0.030	0.036
Belgian road	0.112	0.086	0.095
Washboard road	0.168	0.122	0.160
Twisted road	0.093	0.335	0.067
Fish-scale pit	0.200	0.180	0.195
Repaired asphalt road	0.080	0.080	0.085

, the input shaft exhibits significant axial force contribution (0.360) on twisted roads due to axial misalignment caused by low-frequency road excitations, while torque and radial force are prominent (0.200/0.205) on fish-scale pit roads, reflecting the regular fluctuations of intensified rough roads. The reducer helical gear, affected by continuous oscillations on fish-scale pit roads, is dominated by torque and tangential force (0.190/0.200), leading to stress accumulation at the gear root; the axial force is highest on twisted roads (0.340), resulting from the helix angle and wheel speed differences; high-frequency vibrations on washboard roads amplify radial force pseudo-damage (0.190). The intermediate transmission gear shaft shows the highest axial force contribution on twisted roads (0.400), with prominent torque and radial force on fish-scale pit roads (0.195/0.195). The differential bevel gear exhibits high tangential force on fish-scale pit roads (0.190) and significant axial force on twisted roads (0.310), while washboard and fish-scale pit roads, due to insufficient housing stiffness, result in elevated vertical force (0.180/0.200). The output half-shaft is dominated by torque and vertical force on fish-scale pit roads (0.200/0.195), with prominent axial force on twisted roads (0.335). Fish-scale pit roads (intensified rough roads) significantly contribute to torque, tangential force, and vertical force pseudo-damage (0.190–0.205), twisted roads (low-frequency) dominate axial force (0.310–0.400), washboard roads (high-frequency) amplify vertical and radial forces (0.160–0.198), and small cobblestone roads exhibit the lowest pseudo-damage (0.030–0.041). These results provide data support for the durability optimization of the EDS. Intensified rough roads like fish-scale pit roads have a significant damage proportion, and when low-frequency or high-frequency load characteristics are distinctly separated, the coupling effect between axial and vertical forces is weak, allowing independent analysis. However, the regular fluctuations of intensified rough roads may induce phase coupling, leading to stress concentration, or frequency coupling, causing resonance amplification. Therefore, special attention is given to studying the coupled damage effects of tangential force, axial force, and radial force on the reducer helical gear under fish-scale pit road excitations.

Due to the lack of stress data, only load data can be processed vectorially. Based on the linear cumulative principle, the vector sum of tangential force, axial force, and radial force is compared with linearly accumulated data. However, nonlinear coupling typically exists in real motion processes [22,23], leading to stress concentration and fatigue effects at locations such as the gear root. The Figure 10 shows the load data for torque, tangential force, radial force, and axial force of the reducer helical gear, comparing the vector sum of the three forces with independent linear accumulation.

The calculation results, as shown in the

Table 10. Pseudo-damage calculation results.

Force Type	Max (N)	Min (N)	Pseudo-Damage
Total force	1728.422	388.267	8.585 × 10−5
Tangential force	1485.433	333.240	4.023 × 10−5
Radial force	593.243	134.498	5.163 × 10−7
Axial force	654.976	147.014	6.735 × 10−7

, indicate that the pseudo-damage for the total force is approximately 8.585 × 10⁻⁵, while the pseudo-damage values for tangential force, radial force, and axial force are 4.023 × 10⁻⁵, 5.163 × 10⁻⁷, and 6.735 × 10⁻⁷, respectively. The sum of linearly accumulated pseudo-damage (4.142 × 10⁻⁵) is significantly lower than the total force pseudo-damage, indicating that the pseudo-damage corresponding to the vector sum is notably higher than the linear superposition result. This reflects that nonlinear coupling effects may amplify fatigue impacts, particularly at the gear root, although nonlinear coupling does not necessarily exhibit linear superposition characteristics and may weaken due to phase or directional differences.

4. Equivalent Load Spectrum Generation

The compact internal structure of the EDS in electric vehicles makes it challenging for traditional virtual proving grounds to effectively extract load data from critical components, resulting in a lack of precise load specifications for early subsystem development and test bench experiments. Although intensified road durability tests can provide real data, their high cost, long duration, and consistency issues influenced by weather and human factors fail to meet the efficiency demands of modern platform-based development for subsystem and component reliability testing. Multi-body dynamics simulation based on road point cloud data offers a new approach for generating high-fidelity load spectra, but its complex multi-condition characteristics pose challenges for test bench design. This section proposes a method to convert simulated load spectra into equivalent load spectra, utilizing the rainflow counting method and Miner’s linear cumulative damage theory to simplify complex load sequences into equivalent single or low-cycle spectra. This approach significantly reduces test bench experiment costs and time while maintaining equivalent fatigue damage, providing an efficient and reliable load basis for early EDS development and indoor simulation testing [24].

In the electric drive system (EDS), the reducer, as a core component, is responsible for converting the motor’s high-speed, low-torque output into low-speed, high-torque output, with torque being the primary load form. Therefore, the reducer’s torque channel is selected as the main parameter for conversion into the test bench load spectrum. The conversion from the load spectrum to the test bench is based on the principle of damage equivalence, transforming the road spectrum into a sinusoidal constant amplitude load spectrum to facilitate test bench execution. The process using nCode software modules is shown in Figure 11, where a sine wave is generated based on key information such as the amplitude and frequency of the input channel, pseudo-damage calculations are performed, and the equivalent sinusoidal test bench spectrum is obtained from the results. The rainflow counting diagram of the reducer’s torque channel is shown in Figure 12.

During the conversion to the equivalent load spectrum, after removing burrs from the channel data, the torque range is determined to be between 6 × 10⁴ N·mm and 1.4 × 10⁵ N·mm, with a centerline of approximately 1 × 10⁵ N·mm. When generating a unit sine wave (frequency of 1 Hz) using a signal generator, reducing the amplitude by 30% (e.g., from 4 × 10⁴ N·mm to approximately 2.8 × 10⁴ N·mm) ensures the sine wave covers the main load range, with the damage equivalence error controlled within 5%, leading to a significant increase in cycle count from 18.3 to 109.4. This indicates a pronounced inverse relationship between amplitude and cycle count. When the amplitude remains constant (4 × 10⁴ N·mm), increasing the frequency from 1 Hz to 2 Hz reduces the cycle count from 18.3 to 8.35, and further increasing to 5 Hz gradually decreases the cycle count to 3.34, with the trend approximately following a 1/f proportion, though the impact is relatively small. The specific variation curve is shown in Figure 13:

Amplitude is the primary factor affecting cycle count; increasing amplitude significantly reduces cycle count, thereby lowering the total number of cycles required for test bench experiments, facilitating execution and improving efficiency. The effect of frequency is relatively minor, with its variation having limited impact on cycle count adjustment, but fine-tuning frequency can optimize test stability.

5. Discussion

This study utilized VPG simulation to achieve high-precision load analysis and test bench load spectrum generation for EDS, providing efficient and reliable load data support in the early stages of research and development. The method integrates rigid-flexible coupled multibody dynamics simulation with a CRG road model using point cloud data, which enhances the prediction accuracy of internal loads (such as gear meshing forces and shaft loads). This approach overcomes the limitations of traditional simplified models in predicting performance under complex operating conditions. Compared to the RGR road model relying on two-dimensional road geometry, the CRG road model significantly improves the fidelity of tire-road interaction excitation through high-fidelity point cloud data, thereby precisely reproducing vehicle dynamic responses.

The engineering significance of this method lies in quantifying the estimated damage contributions of eight intensified road conditions, providing a theoretical basis for EDS structural optimization and test bench design. Compared to the tire-suspension test bench method proposed by Haga et al., this study extends VPG simulation to dual-motor EDS, focusing on the dynamic characteristics of high torque fluctuations and multi-power source coupling, effectively addressing the shortcomings of speed-driven models in predicting gear and shaft loads. In contrast to the digital road modeling approach proposed by Zhou et al., this study achieves a more comprehensive analysis of vehicle dynamic responses through rigid-flexible coupled simulation, making it suitable for durability verification under complex operating conditions and providing reliable technical support for the development of electric vehicle drivetrains.

This study has the following limitations: (1) lack of real-vehicle test data to validate VPG simulation results, which prevents thorough verification of load data accuracy and may affect the engineering reliability of the simulation outcomes; (2) the analysis relying on Miner linear cumulative damage theory does not fully account for nonlinear load coupling effects, potentially underestimating fatigue damage under complex conditions; (3) the current study does not deeply analyze the actual damage characteristics of key EDS components (such as gears and bearings), requiring further integration of material properties, microscopic defects (such as microcracks and residual stresses), and stress distributions under different road excitations to evaluate their impact on fatigue performance.

Future research can be improved in the following directions: (1) utilize real-vehicle test data on suspension K&C (Kinematics and Compliance) characteristics to calibrate the VPG simulation model, enhancing model accuracy; (2) combine frequency-domain analysis methods to study the impact of nonlinear load coupling (such as gear meshing phase effects) on fatigue damage and develop more accurate damage prediction models; (3) extend the approach to single-motor or multi-stage transmission EDS vehicle models, or validate the method’s generality and adaptability for other components; (4) quantify the impact of material defects on fatigue life through finite element analysis and evaluate the durability of key components using micromechanical models. These improvements will further enhance the reliability and engineering application value of VPG simulation in the development of electric vehicle drivetrains.

6. Conclusions

This study developed a high-precision load analysis and test bench load spectrum generation method for EDS using VPG simulation, effectively addressing the issue of insufficient load data in the early stages of development due to the absence of real-vehicle prototypes. Key achievements include: high-precision extraction of loads for critical components, such as input shafts and reduction helical gears, through rigid-flexible coupled multibody dynamics simulation and the CRG road model; quantification of estimated damage contributions from eight intensified road conditions derived from simulated damage analysis, revealing the differential damage effects of various road types (such as intensified bad roads and high-frequency/low-frequency response roads) on critical components, for example, the high damage characteristics of reduction helical gears under intensified bad road conditions; and generation of equivalent sinusoidal load spectra using the rainflow counting method and Miner linear cumulative damage theory, simplifying the test bench design process and improving engineering efficiency.

Compared to traditional rigid-body models, the rigid-flexible coupled model precisely reflects the mechanical responses of EDS under complex operating conditions, providing efficient load data support for early-stage development. The practical significance of this method lies in reducing the costs of traditional road testing, supporting test bench design, and optimizing structural durability, making it suitable for the development and analysis of key electric vehicle drivetrain components. When combined with finite element analysis for actual fatigue assessment, this method can also provide a scientific basis for component replacement and maintenance strategies after product mileage checks. Additionally, the method demonstrates strong generality and can be extended to the development of other drivetrain components, with further improvements in engineering accuracy achievable through real-vehicle data calibration.

Future research prospects include: (1) utilizing real-vehicle suspension K&C test data to validate VPG simulation results, enhancing the engineering reliability of the model; (2) developing frequency-domain damage models to deeply analyze the impact of nonlinear load coupling (such as gear meshing phase effects) on fatigue damage; (3) conducting actual fatigue analysis of critical components based on finite element analysis, combined with VPG simulation data and real-vehicle test standards, to quantify the impact of material defects on durability under different road excitation cycle counts.

These studies will further provide the electric vehicle industry with more efficient and reliable durability analysis approaches and methods.