Decoupling Steady-State and Transient Switching Effects: A Mode-Decomposed Fatigue Analysis of Planetary Gears in Power-Split Hybrid Buses

Abstract

To address the prominent fatigue failure risk of planetary gears in power-split hybrid buses and the lack of quantitative damage analysis across various operating modes in existing studies, this paper focuses on the front planetary gear set of a power-split hybrid bus. Based on a full-vehicle co-simulation model, loads under full operating conditions are decomposed into 11 operating modes, mode-switching loads are analyzed and extracted, and mode-decomposed and mode-switching fatigue loading spectra are compiled. Fatigue simulation is then conducted using Miner’s linear damage accumulation rule. Results show that the sun gear directly coupled to motor is the system’s most fatigue-susceptible component, exhibiting significant asymmetric unilateral tooth flank damage. The hybrid electric vehicle (HEV) mode contributes approximately 88% of total damage to the sun gear’s right flank, dominating system fatigue damage. Transient mode-switching conditions account for approximately 60% of total damage to the sun gear’s left flank, serving as the core damage source. Compared with the traditional full-condition merging method, the proposed mode-decomposed method improves the conservatism of life prediction. This work provides methodological support for refined strength design and targeted optimization of power-split hybrid transmission systems.

1. Introduction

Driven by global carbon neutrality targets and stringent emission regulations, the low-carbon transformation of commercial vehicles has become a core trend in the transportation industry [1,2]. Electric technology faces multiple challenges in range and cost due to limitations imposed by high-load and frequent start–stop scenarios in commercial vehicle applications, making hybrid electric technology the mainstream solution for low-carbon transition. Power-split hybrid systems are widely adopted, among which the clutchless configuration demonstrates significant advantages and has achieved large-scale implementation [3]. In this configuration, the engine is rigidly connected to the transmission system without clutch-based buffering and vibration absorption, causing transient torque and speed fluctuations during start–stop and mode switching to act directly on planetary gears, triggering severe load variations [4]. Planetary gear sets are subjected to combined alternating steady-state and transient impact loads over extended periods, making tooth surface contact fatigue the predominant failure mode of such systems, which seriously threatens system reliability and operational safety [5]. Therefore, research on fatigue life prediction and damage evolution patterns of planetary gears constitutes a critical theoretical foundation for refined strength design and full-lifecycle service safety assurance of hybrid transmission systems [6]. Existing studies have established a fatigue calculation framework centered on S-N curves and linear damage criteria, providing mature theoretical support for strength design of conventional steady-state transmission systems [7].

However, this framework exhibits limitations when applied to clutchless power-split hybrid systems. On one hand, existing studies have established a framework for the synchronous coupling between real-time data and high-fidelity physical models [8], which can capture the transient mode switch effects of the planetary gear train in hybrid power systems [9,10]. In addition, dynamic stress analysis techniques and fatigue life assessment methods based on multi-body co-simulation have been well established [11,12]. However, most existing studies focus on the gear load characteristics and fatigue damage distribution laws under steady-state driving conditions [13,14]. For hybrid power systems for commercial vehicles, especially those with a clutchless power-split configuration, relevant research has mostly concentrated on the optimization of energy management strategies, the improvement of control algorithms for mode switch smoothness [15,16,17,18,19,20], and the redesign of safety-critical components using multi-level topology optimization methods [21]. Very few studies have systematically quantified the contribution of different power transmission modes under full working conditions to the fatigue damage of the planetary gear train in this configuration. Furthermore, the formation mechanism and evolution law of differential gear damage under various working conditions have not been revealed in existing research. Moreover, the urban bus operation scenario for commercial vehicles is characterized by frequent start–stop cycles, high mode-switching frequency, and severe load fluctuations. Under different operating modes, the meshing working surfaces, load transmission paths, and force characteristics of planetary gears exhibit essential differences. Fatigue analysis without mode decomposition cannot accurately capture the damage evolution process of gears, nor can it identify the core control operating conditions that determine gear service life [22,23,24].

On the other hand, as the core input for gear fatigue damage analysis and life prediction, the compilation accuracy of fatigue loading spectra directly determines the reliability and engineering applicability of fatigue analysis results [6,25]. Existing studies predominantly adopt traditional integrated load spectra, in which gear loads across the full-time domain are uniformly aggregated, classified, and statistically compiled according to different road condition types and power source output characteristics [14,26,27]. This method has been widely applied in strength design and reliability verification of general mechanical transmission systems, owing to its core advantages of simplified calculation procedures, high computational efficiency, and compatibility with load characteristics of conventional steady-state transmission systems [28,29,30,31,32]. However, for the service characteristics of clutchless power-split hybrid systems featuring multi-mode-switching and significant transient load impacts, this method exhibits inherent and unavoidable defects. For instance, full-condition wide-range load classification merges low-proportion, high-amplitude transient impact loads during mode switching and engine start–stop processes with a large number of steady-state low-amplitude loads into the same load level, severely diluting the nonlinear damage contribution of transient loads. Meanwhile, existing studies lack analysis of the impact of mode switching process loads on planetary gear fatigue life in hybrid systems, and also lack research on mode-specific damage patterns and quantitative analysis for each operating mode of hybrid systems. Integrated statistical compilation cannot reflect the essential differences in gear meshing, loads, and rotational speeds under different operating modes, resulting in distorted load cycle counting and an inability to distinguish the differential damage influences of each mode, ultimately leading to underestimated damage calculation and over-optimistic life prediction, which fails to provide accurate support for refined transmission system design.

To address the abovementioned research gaps, this paper takes the front planetary gear train of a clutchless power-split hybrid electric bus as the research object, and proposes a fatigue damage analysis method for planetary gear trains based on operating condition decomposition. Compared with previous studies, this paper separately extracts the loads of 11 operating modes and the transient loads during mode switch processes of the power-split hybrid power system, calculates the fatigue damage induced by each individual operating mode and mode switch process respectively, and quantifies the damage contribution ratio of loads from each operating mode and mode switch process. On this basis, this paper reveals the evolution law of gear fatigue damage under multiple working conditions, identifies the core damage-dominant working conditions, and provides a targeted optimization direction for the reliability optimization of the hybrid power system.

Following the technical roadmap of “model construction–simulation calculation–pattern discovery,” this study first analyzes the system configuration and power transmission characteristics of 11 operating modes, establishes an AVL CRUISE R2019.2 + MATLAB/Simulink R2018b full-vehicle co-simulation model, and completes accuracy verification. Subsequently, loads for each operating mode and mode-switching conditions are extracted, and full-condition, mode-decomposed, and mode-switching fatigue loading spectra are compiled based on the histogram counting method. Thereafter, a rigid-flexible coupling model of planetary gear set is constructed, and fatigue simulation is conducted based on Miner’s criterion to quantify the damage contribution of each operating condition. Finally, results from the two methods are compared to reveal mode-specific damage patterns and propose engineering optimization directions. The paper is organized as follows: Section 1 presents the introduction; Section 2 describes the system configuration, simulation model establishment, and validation; Section 3 completes load characteristic analysis and fatigue loading spectrum compilation; Section 4 conducts mode-decomposed fatigue simulation and damage pattern analysis; Section 5 discusses methodological advantages, application value, and limitations; and Section 6 summarizes the conclusions. The overall technical roadmap is shown in Figure 1.

2. Power-Split Hybrid Powertrain System and Co-Simulation Model

2.1. Configuration and Working Principle of Power-Split Hybrid System

2.1.1. Vehicle Core Parameters and Hybrid System Configuration

The research object of this paper is a power-split hybrid bus. The schematic diagram of the hybrid system structure is shown in Figure 2, and the main vehicle parameters are listed in

Table 1. Vehicle model parameters.

Item	Parameter Value
Mass of vehicle	12,582 kg
Frontal area	7.5 m2
Air resistance coefficient	0.52
Rotational mass coefficient	1.1
Rolling resistance coefficient	0.0075
Wheel radius/wheel inertia	0.464 m/0.51 kg·m2
Characteristic parameters of PG1 and PG2	2.11
Rotational inertia of PG1 and PG2	0.0015 kg·m2
Transmission ratio of main reducer	5.31
Peak torque/maximum speed of engine	851 N·m/2300 r/min
Moment of inertia of engine	1.534 kg·m2
Peak torque/maximum speed of MG1	400 N·m/5000 r/min
Peak torque/maximum speed of MG2	700 N·m/7500 r/min
Moment of inertia of MG1/MG2	0.0226 kg·m2/0.0601 kg·m2

[20]. The system primarily consists of one engine, two motor-generators, and two planetary gear sets, with the clutch eliminated and the engine directly connected to the transmission system. The front planetary gear set (PG1) connects the engine and front motor (MG1), serving the function of power split; the rear planetary gear set (PG2) connects with the rear motor (MG2), functioning as a reducer; MG1 is used to start the engine, charge the battery, and supplement system power; MG2 serves as the main power source of the system for vehicle propulsion; and the engine provides additional power demand.

2.1.2. Co-Simulation Foundation Model Architecture

Figure 3 shows the full-vehicle model established in AVL CRUISE, and Figure 4 presents the control strategy model constructed in MATLAB/Simulink. Communication between the two models is realized via the “cruise_m.dll” module: the full-vehicle model transmits operating conditions and real-time vehicle state information to the control strategy model, which in turn feeds control signals back to the full-vehicle model.

The full-vehicle model was developed using AVL CRUISE, while the control strategy model was established in MATLAB/Simulink. Co-simulation was performed under synthesized driving cycles combining the China Heavy-Duty Commercial Vehicle Test Cycle for Coach (CHTC-C) and the China Heavy-Duty Commercial Vehicle Test Cycle for Bus (CHTC-B) [33]. Figure 5 presents representative vehicle speed data extracted from the synthesized driving cycle.

The mechanical connections among hybrid powertrain components were established in AVL CRUISE. Data communication with MATLAB/Simulink was achieved through the CRUISE Interface, enabling vehicle operating data to be transmitted to the control strategy model in MATLAB/Simulink. After processing, the control strategy outputs control signals back to the AVL CRUISE full-vehicle model for vehicle control execution. The control strategy encompasses three working states: stationary, drive and brake. The switching logic among these three working states is illustrated in Figure 6.

2.2. Architecture of Hybrid Powertrain Control Strategy

2.2.1. Analysis of Operating Mode Characteristics

The three working states—stationary, drive, and brake—are further subdivided into 11 working modes. The stationary state corresponds to only one working mode, namely M1 (Stationary). When the vehicle is in the stationary mode, the output power of each power source is zero in the initial stage. Additionally, the working modes corresponding to the drive and brake states, as well as their mode-switching logic, are illustrated in Figure 7 and Figure 8, respectively.

After the vehicle transitions from the stationary state to the drive state, the power sources begin to output power, which is specifically subdivided into five working modes. When the system operates in M2 (One Motor), MG2 serves as the power source for output. The power output from MG2 is transmitted to the sun gear of PG2, and after speed reduction through PG2, the power is output from the planet carrier of PG2 to the drive axle, thereby propelling the vehicle.

When the system operates in M3 (Dual Motor), the power output from MG2 is insufficient. MG1 provides a power supplement through PG1. At this time, Brake 1 is engaged. The power from MG1 is input into the sun gear of PG1, and then output through the ring gear of PG1 to the planet carrier of PG2, jointly driving the vehicle with MG2.

After the vehicle transitions from the driving mode to the braking mode, the braking state is further subdivided into five working modes. M7 (Choose) serves as the braking-type choose mode. In this state, Brake 2 is engaged to prepare for entering the braking mode. At this point, three scenarios are distinguished:

When the vehicle speed is ≤100 km/h and the engine speed is relatively low, the system switches to M10 (Comp Brake). This mode represents the composite braking state (motor braking + mechanical braking), during which the vehicle decelerates while recovering braking energy. If the battery is found to be fully charged during the braking process, the system switches to M11 (Mech Brake); otherwise, it switches back to M10.

When the vehicle speed is ≤100 km/h and the engine speed is relatively high, the system switches to M8 (Engine Down). Under M8 mode, if the engine shuts down within 0.5 s, the system directly switches to M10. Otherwise, the system first switches to M9 (Engine Off), and then switches to M10.

When the vehicle speed is high (>100 km/h) and significant braking force is required, the system directly transits to M11.

2.2.2. Theoretical Verification of Powertrain Load Output

This study verifies the accuracy of the model by comparing the differences between theoretical calculations and simulation test results. The verification process calculates the rotational speed and torque for steady-state conditions M2, M3, and M6 and transient conditions M5 and M8, respectively.

The theoretical calculation formulas for the front sun gear rotational speed under both steady-state and transient conditions are identical [34], as follows:

$$n_{s,calcu}=(1+p)\times n_c-p\times n_r$$

(1)

where n_s,calcu is the calculated rotational speed of the sun gear, p = 2.11 is the characteristic parameter of the front planetary gear set, n_c is the rotational speed of the planet carrier, and n_r is the rotational speed of the ring gear.

The theoretical calculation formula for the front sun gear torque under steady-state conditions is as follows [28]:

$$T_{s,calcu}=-{\displaystyle \frac{T_c}{1+p}}$$

(2)

where T_s,calcu is the calculated torque of the sun gear, and T_c is the torque of the planet carrier.

The theoretical calculation formula for the front sun gear torque under transient conditions is as follows [34]:

$$T_{s,calcu}=T_{s,sim}-T_{s,dyna}$$

(3)

where T_s,sim is the simulated torque of the sun gear at the subsequent time step, $T_{s,dyna}=J_s\times {\alpha }_s$ is the dynamic torque of the sun gear, J_s is the moment of inertia of the sun gear, and α_s is the angular acceleration of the front sun gear.

Table 2. Rotational speed error.

Mode	M2	M3	M5	M6	M8
Average error (%)	0.02	0.05	0.05	0.02	0.04
Maximum error (%)	0.11	0.08	0.09	0.06	0.08

and

Table 3. Torque error.

Mode	M2	M3	M5	M6	M8
Average error (%)	0.07	0.07	7.17	0.02	5.37
Maximum error (%)	0.13	0.18	9.36	0.08	7.79

compare the rotational speed errors and torque errors of sun gears in models M2, M3, M5, M6, and M8. According to the computational results, the rotational speed errors of sun gears in M3, M5, M6, and M8 are all below 0.1%, and the torque errors in M2, M3, and M6 are also less than 0.1%, satisfying the precision requirements. However, the maximum torque errors under transient operating conditions in M5 and M8 reach 9.36% and 7.79%, respectively. To verify whether these torque errors meet the precision criteria, an error propagation analysis is conducted subsequently.

Preliminary fatigue simulation tests were conducted on the planetary gear using simulated loads and theoretical loads. The contact fatigue damage results of the sun gear under both loading conditions are presented in

Table 4. Fatigue damage discrepancy between simulated and theoretical loads.

Theoretical Load Damage (-)	Simulated Load Damage (-)	Error (%)
8.38 × 10−5	7.99 × 10−5	4.6%

. The fatigue damage calculated from the simulated loads exhibits an error of 4.6% relative to the theoretical loads, which is below the 5% threshold. This confirms that the torque error meets the precision requirements for fatigue life analysis and can be used in the subsequent simulation calculation of tooth surface contact fatigue damage.

3. Fatigue Damage Analysis of Planetary Gears

3.1. Fatigue Damage Analysis Under Full Operating Conditions

3.1.1. Load Spectrum Compilation

The accuracy of gear fatigue life prediction highly depends on the precision of the fatigue loading spectrum. Based on the hybrid electric vehicle model established through CRUISE/Simulink co-simulation, this study will extract the actual operating load data of the planetary gear set and subsequently compile the fatigue loading spectrum. Compared with traditional assumptions or single-operating-condition data, the load data obtained by this method are closer to the actual vehicle operating conditions, and simultaneously cover both steady-state and transient conditions, particularly capable of capturing the gear load characteristics under multiple operating modes.

During gear rotation, each tooth undergoes a periodic “engagement–meshing–disengagement” process, and the load exhibits significant pulsating characteristics (i.e., one loading cycle corresponds to one revolution), rather than continuous variation. To accurately capture the pulsating characteristics of gear loads, this study employs the histogram counting method for planetary gear load analysis [35]. This method divides the torque load into n amplitude intervals, and calculates the number of cycles in each interval by combining rotational speed and action time, thereby constructing a fatigue loading spectrum that conforms to gear transmission characteristics. The principle and the calculation formula for the number of cycles are shown in Figure 9 and Equation (4), respectively.

$$N_i={\displaystyle {\sum }_{i=1}^m{\displaystyle \underset{\Delta t_i}{\int }n(t)dt}}$$

(4)

where N_i is the number of revolutions corresponding to the nth-level torque load, n (t) is the rotational speed of the gear, and m is the number of time intervals.

The full-operating-condition full-load data obtained from the CRUISE/Simulink co-simulation were statistically processed using the histogram counting method. To determine the optimal number of load division levels, a sensitivity analysis of the level count was conducted. Preliminary fatigue damage simulation results, as shown in Figure 10, indicate that the fatigue damage on both the left and right tooth surfaces fully converges after 60 levels. Therefore, the load is ultimately divided into 60 levels.

As indicated by the statistical results of the sun gear fatigue life loading spectrum in Figure 11, this loading spectrum fully covers the full operating range of the planetary gear set under all operating conditions, with a torque range from −254 N·m to 350 N·m and a rotational speed range from −1258 r/min to 1259 r/min. It encompasses all types of operating conditions including driving, braking, and steady-state cruising under forward and reverse meshing.

3.1.2. Simulation Calculation and Damage Analysis of Tooth Surface Fatigue

The planetary gear parameters designed and verified according to the hybrid system parameters and ISO 6336 are shown in

Table 5. Planetary gear parameters.

Parameters	Planet	Sun	Ring
Number of teeth	23	41	87
Modulus (mm)	2.3	2.3	2.3
Helix angle (°)	20	20	20
Pressure angle (°)	20	20	20
Tooth width (mm)	35	35	35

[36], and the constructed rigid–flexible coupling dynamics model of the planetary gear set is shown in Figure 12.

As the planetary gears are subject to high service loads under actual operating conditions, 20CrMnTi was selected as the gear material, with its performance parameters presented in

Table 6. Performance parameters of 20CrMnTi.

Elastic Modulus	Yield Strength	Tensile Strength	Permissible Contact Stress	Permissible Bending Stress
2.12 × 105 MPa	993 MPa	1164 MPa	1300 MPa	300 MPa

[37].

Bearings are added to provide constraints for the gear shafts. All bearings adopted are tapered roller bearings that have passed the verification, and their specific designations are listed in

Table 7. Bearing model.

Arrangement Position	Planet Carrier	Sun Gear	Ring Gear
Brand	Koyo	Koyo	SKF
Designation	32911JR	30218JR	30218

.

The S-N curve of 20CrMnTi material [38] is shown in Figure 13. This study calculates fatigue damage based on the aforementioned S-N curve, with the specific procedure as follows:

The fatigue loading spectrum developed in this paper is input into the planetary gear dynamic simulation model. Maximum contact stress values at critical contact zones under each load level are extracted via finite element analysis. Leveraging the stress-life mapping relationship of the S-N curve, the material’s ultimate cyclic life corresponding to each stress level is determined. Combined with the actual cyclic count of that load level, the fatigue damage per single load level is computed. Finally, applying Miner’s linear fatigue damage accumulation criterion, the fatigue damage across all load levels is aggregated to derive the total fatigue damage per single complete loading cycle. This value is then extrapolated to obtain the total contact fatigue life of the gear structure.

The SAE 85W-90 API GL-5 gear lubricant is adopted in the simulation to match the heavy-duty operating conditions of the planetary gear train. As a core factor affecting gear dynamic load and fatigue life, lubricant temperature directly changes the kinematic viscosity of the oil, and further affects the oil film thickness and friction coefficient of the tooth surface under elastohydrodynamic lubrication. Specifically, rising oil temperature will reduce viscosity, thin the oil film, aggravate friction loss and transmission error, intensify the dynamic excitation of the gear system, and eventually lead to an increase in the dynamic load factor K_v and a decline in gear contact and bending fatigue life. The reference oil temperature for conventional steady-state conditions is set to 70 °C in this study, which corresponds to the typical operating oil temperature of the gearbox for the target hybrid electric bus in urban scenarios [39].

The calculation of fatigue damage is based on the linear fatigue damage accumulation theory. The Miner’s rule is adopted in this paper to calculate the fatigue damage, with its principle shown in Equation (5).

$$D={\displaystyle \sum _{i=1}^n\frac{n_i}{N_i}}={\displaystyle \sum _{i=1}^n{d}_i{n}_i}$$

(5)

where D is the cumulative value of fatigue damage, n_i is the number of load cycles of the material at the i-th stress level, N_i is the number of cycles to fatigue failure of the material at the i-th stress level, and d_i is the single-cycle fatigue damage at the i-th stress level.

This study applies ISO 6336 [37] and Miner’s rule to independently calculate cumulative fatigue damage on the left and right gear flanks under bidirectional loading. Each working flank is treated as an independent unit, with decoupled meshing conditions, stress states, and load histories between flanks—no cross-flank damage interaction or stress superposition occurs.

Based on the fatigue life loading spectrum, the power loads of rotational speed and torque are applied to the sun gear and planet carrier of the planetary gear model. The software is run to generate the analysis results of the gear load spectrum. The simulation results show that the bending fatigue damage of the planetary gears is zero, and the contact fatigue damage is presented in the table below.

As shown in

Table 8. Fatigue damage results of the PG1.

Gear	Contact Damage (-)
	Left Tooth Flank	Right Tooth Flank
Ring gear	0	0
Sun gear	4.68 × 10−7	7.01 × 10−5
Planetary gear	2.51 × 10−7	3.12 × 10−5

and

Table 9. Maximum contact stress of the PG1.

Gear	Contact Stress (MPa)
	Left Tooth Flank	Right Tooth Flank
Ring gear	706.78	676.46
Sun gear	834.03	877.30
Planetary gear	834.03	877.30

, the right tooth flank of the PG1 sun gear exhibits the maximum contact fatigue damage with a value of 7.04 × 10⁻⁵, making it the critical component that determines the fatigue life of the planetary gear train. This is followed by the planetary gears, with a right tooth flank damage value of 3.14 × 10⁻⁵. The ring gear has the minimum contact fatigue damage, with a damage value of 0 for both the left and right tooth flanks.

Combined with the architecture analysis of the power-split hybrid powertrain system, the PG1 sun gear is directly connected to MG1 and bears the high-amplitude, high-frequency loads output by MG1, with a maximum contact stress of 951 MPa, thus resulting in the most significant fatigue damage. Additionally, during gear meshing, positive torque corresponds to the sun gear’s right flank engagement under load, while negative torque corresponds to the left flank engagement. Based on full-condition load history statistics, the ratio of cumulative load duration for negative torque (left flank damage) to positive torque (right flank damage) is 1:102. Positive torque accounts for over 99% of the total load duration. According to Miner’s linear fatigue damage accumulation rule, fatigue damage is linearly proportional to the number of load cycles. Consequently, the cumulative fatigue damage on the right flank vastly exceeds that on the left flank.

Figure 14 and Figure 15 show the maximum contact stress and maximum bending stress of the PG1 sun gear under various load levels, while Figure 16, Figure 17 and Figure 18 show the maximum contact stress nephograms of the sun gear, planetary gear, and ring gear respectively. It can be seen from the figures that the load on the tooth surface of the planetary gear is obviously uneven, resulting in severe stress concentration.

Based on the statistical results of the maximum tooth flank contact stress and bending stress of the PG1 sun gear under full operating conditions, significant differences are observed in the loaded meshing flank of the sun gear under different operating modes: For load level 1–25, only the left tooth flank bears the meshing load, while the stress on the right tooth flank is 0. The maximum contact stress of the left tooth flank reaches 893 MPa, with a corresponding maximum bending stress of 179 MPa. Load reversal occurs under load level 26–60, where only the right tooth flank participates in meshing. The contact stress and bending stress increase gradually with the rise in operating load, and reach the full-condition peak under load level 60. The maximum contact stress of the right tooth flank is 951 MPa, with a corresponding bending stress of 215 MPa, which is significantly higher than the peak level of the left tooth flank.

Combined with the contact stress contour plots in Figure 16, Figure 17 and Figure 18, the tooth flank contact stress exhibits a significant non-uniform distribution along the face width and tooth depth directions. Affected by the gear helix angle, an obvious stress concentration occurs at the tooth flank end, where the local stress peak is significantly higher than the average level of the entire tooth flank.

In summary, the analysis of the combined full-condition loading spectrum identifies the PG1 sun gear as the critical weak component of the planetary gear train, and reveals the distribution characteristics of asymmetric contact fatigue damage on its tooth flanks. However, this method cannot separate the damage contribution from each operating mode and mode-switching process, making it difficult to locate the core damage-causing operating conditions and clarify the damage formation mechanism. To break through this limitation, this paper splits the full-condition loads and carries out an analysis of the load characteristics and fatigue damage laws of each operating mode.

3.2. Mode-Decomposed Load Analysis of Planetary Gears

3.2.1. Modes Load Extraction and Compilation of Mode-Decomposed Fatigue Loading Spectrum

The loads corresponding to working modes M1 to M11 were extracted separately, and the histogram method was adopted to perform statistical analysis on the loads of each mode. Finally, 11 groups of fatigue loading spectra were obtained, each of which contains 10 load levels. The first-level load with the maximum torque among the 11 sets of fatigue life loading spectra, as well as its corresponding rotational speed, is presented in Figure 19.

Statistical results indicate that the maximum torque of the PG1 sun gear across the 11 modes ranges from −253 N·m to 350.24 N·m. The maximum positive torque occurs in M8, while the maximum negative torque is observed in M5. The corresponding rotational speed spans from −2583 r/min to 1167 r/min, with its positive and negative fluctuations fully covering the forward and reverse operation scenarios of the mechanism.

Torque and rotational speed under each mode exhibit a strong coupling characteristic. Specifically, high-torque modes are concentrated in the medium and low rotational speed ranges, whereas high rotational speed modes mostly correspond to low torque output. The obtained loading spectra accurately reflect the load fluctuation law of the PG1 sun gear in actual operation, and provide a reliable load input basis for subsequent fatigue life analysis.

3.2.2. Mode-Switching Load Extraction and Fatigue Loading Spectrum Compilation

Based on the characteristics of each mode switch, the dynamic loads during the switching process shall be extracted separately. The specific extraction method is defined as follows: the load is extracted from the torque variation interval between the steady-state load prior to the mode switch and the steady-state load after the mode switch. For example, when the system switches from mode M2 to M6 in Figure 20a, the mode switch is initiated at 0.38 s, where the torque exhibits dynamic changes until it re-attains the steady-state value at 1.81 s; accordingly, the load within the interval of 0.38 s to 1.81 s is extracted as the mode switch load. For the switching process of the system from M6 to M2 in Figure 20b, the mode switch is initiated at 0.11 s; the torque changes immediately and returns to the steady state at 0.77 s. Thus, the time window for extracting the mode switch load in this process is determined as 0.11 s to 0.77 s.

After all mode switch loads are completely extracted, the mode switch fatigue loading spectrum is compiled in accordance with the histogram method, and the results are shown in Figure 21. The statistical results show that the loads during the mode-switching process cover a wide range, with a torque range of −254 to 350 N·m and a rotational speed range of −2934 to 549 r/min, comprehensively covering the full-range transient characteristics of load reversal, amplitude mutation, and speed jump during the mode-switching process. Compared with the steady-state operating condition fatigue load spectrum, the mode-switching process contains a large number of high-amplitude, high-change-rate shock loads. Even though some load cycle frequencies are relatively low, they still significantly accelerate the cumulative process of tooth surface contact fatigue.

In summary, this paper has completed the load separation for M1–M11 operating modes and the mode-switching process, and has compiled the corresponding mode-decomposed fatigue loading spectra and mode-switching fatigue loading spectrum. The differentiated load characteristics between steady-state and transient operating conditions have been clarified, providing accurate load inputs for subsequent refined fatigue damage analysis. Based on the aforementioned fatigue loading spectra, fatigue damage simulation calculations will be conducted in the following sections to quantify the damage contribution of each operating condition and systematically reveal the mode-specific fatigue damage patterns of planetary gear set in this hybrid system.

4. Fatigue Damage Results Analysis for Operating Modes and Mode Switching

4.1. Damage Contribution and Damage Mechanism Analysis

4.1.1. Damage Characteristics and Cause Analysis

Different from the traditional full-condition method which treats the full-process loads as an integral whole, the proposed mode-decomposed method decomposes the total fatigue damage into the sum of damage induced by 11 operating modes, and can further decouple the total damage into the superposition of damage caused by steady-state operating conditions and transient operating conditions:

$$D_{total}={\displaystyle \sum _{i=1}^N{D}_i}={\displaystyle \sum _{i=1}^N{D}_{Steady(i)}}+{\displaystyle \sum _{i=1}^N{D}_{Transient(i)}}(i=1\cdots 11)$$

(6)

where D_total is the total damage, D_i is the damage of the i-th mode, D_steady(i) is the steady-state operating condition damage of the i-th mode, and D_Transient(i) is the transient operating condition damage of the i-th mode.

The fatigue loading spectra of operating modes and mode switching were imported into the planetary gear power load for simulation analysis, and the fatigue damage results are shown in

Table 10. Fatigue damage of PG1 sun gear in each mode.

Mode	Contact Damage (-)		Max Contact Stress (MPa)
	Left Tooth	Right Tooth	Left Tooth	Right Tooth
M1	1.89 × 10−9	1.65 × 10−10	851.63	847.51
M2	0	0	0	582.42
M3	2.69 × 10−9	1.54 × 10−6	819.8	900.44
M4	0	0	0	773.27
M5	4.05 × 10−7	0	890.81	0
M6	2.09 × 10−8	7.07 × 10−5	839.77	894.72
M7	0	8.87 × 10−9	0	862.58
M8	0	7.65 × 10−6	0	985.36
M9	0	1.42 × 10−7	0	911.46
M10	0	0	0	586.76
M11	0	0	0	141.04
Switch	2.58 × 10−7	1.45 × 10−5	897.65	985.19

. The results in Table 10 show that the contact fatigue damage of the sun gear left and right tooth flanks exhibits significant single-side-dominated and asymmetric distribution characteristics: fatigue damage occurs on only one side of the tooth flank in most operating modes, and the overall cumulative damage of the right tooth flank is higher than that of the left tooth flank. The working tooth flank of the sun gear is jointly determined by the direction of load torque and the direction of rotational speed. When the system operates under driving conditions (such as M6), the right tooth flank of the sun gear is the working flank; when the system operates under reverse braking/dragging conditions (such as M5), the left tooth flank of the sun gear is the working flank.

From the fatigue damage results, it can be seen that the damage characteristics of different operating modes exhibit significant differences.

(1) Under each operating mode

The M6 (HEV) driving condition is the core contributing condition for sun gear fatigue damage, with its right tooth flank contact fatigue damage reaching 7.07 × 10⁻⁵, the highest value across all operating conditions, corresponding to a maximum right tooth flank contact stress of 894.72 MPa. In this mode, engine power is transmitted through PG1 to PG2, working together with MG2 to drive the vehicle. Furthermore, this condition accounts for the highest proportion during the entire vehicle driving process, with high load cycle frequency. The high-frequency medium-to-high amplitude meshing loads are the dominant factor for tooth surface contact fatigue cumulative damage, directly determining the overall fatigue life of the planetary gear set.

M5 (Engine Up), M8 (Engine Down), and M9 (Engine Off) are typical transient impact damage conditions. All three modes require MG1 to provide instantaneous high torque to drag the engine through PG1. Among them, M5 is a reverse torque transient condition, with left tooth flank damage reaching 4.05 × 10⁻⁷, the highest value across all operating conditions for the left tooth flank, corresponding to a maximum left tooth flank contact stress of 890.81 MPa. The reverse meshing load impact during the engine speed increasing process is the core cause of damage in this condition. The right tooth flank damage under M8 and M9 conditions reaches 7.65 × 10⁻⁶ and 1.42 × 10⁻⁶ respectively, and the dynamic load fluctuations during the engine speed increase/decrease process further aggravate the tooth surface fatigue accumulation.

The left and right tooth flank damage under M2 (One Motor), M4 (MG1 Down), and M11 (Mech Brake) conditions are all zero, corresponding to extremely low tooth surface contact stress levels. According to the power-split configuration of the hybrid structure, when the system operates in M2 mode, MG2 serves as the sole power source for the entire vehicle, with MG2’s power transmitted through PG2 to the drive axle without passing through PG1; therefore, the PG1 sun gear experiences no damage in M2 mode. MG1 is connected to the sun gear of PG1. When the system operates in M4, MG1 will reduce its speed and give PG1 a large reverse torque. The maximum stress in M4 is 773.27 MPa; however, the M4 duration is too short at only 0.16 s. Although the stress is relatively high, the cycle count is too low, resulting in no damage to the sun gear. When the vehicle undergoes mechanical braking (M11), PG1 does not participate in the entire process, thus causing no damage to the sun gear. For M1, M3, M7, and M10, fatigue damage exists on only one side of the tooth flank at extremely low levels. Among them, the right tooth flank damage under M3 (Dual Motor) is 1.54 × 10⁻⁶, while the damage under other modes is below the 10⁻⁸ magnitude. These modes either have low load amplitude or low cycle frequency, and their contribution to the overall fatigue life can be neglected.

(2) Mode-switching process

During this process, the right tooth flank damage reaches 1.45 × 10⁻⁵, corresponding to a maximum right tooth flank contact stress of 985.19 MPa. This indicates that during mode switching, the high-amplitude stress impact caused by instantaneous load variation, even with relatively low cycle frequency, can still cause significant fatigue damage.

4.1.2. Quantitative Analysis of Damage Contribution

(1) Quantitative analysis of total damage contribution

To quantify the influence of each operating mode on planetary gear fatigue damage, this paper conducts a statistical analysis of the sun gear tooth flank contact fatigue damage contribution for modes M1 through M11, with results shown in Figure 22.

As shown in the figure, the damage contribution of both the left and right tooth flanks exhibits a highly concentrated operating condition distribution characteristic, with damage proportion directly related to operating condition load characteristics and meshing force mechanisms. The left tooth flank serves as the meshing working surface only under system reverse torque conditions. Among these, M5 (Engine Up) shows a damage contribution of up to 94.2%, making it the absolute dominant source of left tooth flank fatigue damage. The reverse high-amplitude impact load caused by engine speed increase is the core inducing factor for left tooth flank fatigue accumulation. This result indicates that in control strategy design, priority should be given to optimizing the torque smooth transition algorithm during mode switching. Reducing the torque change rate during the switching process can significantly extend planetary gear life. Next is M6 (HEV) with a contribution of 4.9%, while the combined contribution of all other modes is only 0.9%, having minimal impact on total damage.

The right tooth flank is the main meshing working flank of the system under driving operating conditions. Among them, the damage contribution of the M6 (HEV) steady-state driving operating condition reaches 88%, which is the core source of the total fatigue damage of the entire planetary gear train. This mode has the highest time proportion and the most cycle counts during the whole vehicle driving process, and the medium–high amplitude meshing loads with high cycle counts produce a strong fatigue accumulation effect. This is followed by the M8 (Engine Down) transient operating condition, with a contribution of 9.6%; M3 (Dual Motor EV) accounts for 1.9%, and the total contribution of the remaining modes is only 0.5%. The three modes M6, M5 and M8 together contribute more than 99% of the tooth flank fatigue damage of the sun gear and are the core critical operating conditions that determine the fatigue life of the planetary gear set.

(2) Quantitative analysis of damage contribution per unit time

To further investigate the influence of operating modes on the fatigue damage of planetary gears, a quantitative analysis of the fatigue damage contribution per unit time for each operating mode is conducted, and the results are shown in Figure 23.

As shown in the figures, the damage contribution per unit time presents a concentration characteristic of absolute dominance by transient operating conditions, which is significantly different from the steady-state dominant law of total damage. For the left tooth flank, the damage contribution per unit time of the M5 (Engine Up) transient operating condition is as high as 99.98%, while the total of the remaining operating conditions is only 0.02%. This is because the reverse high-amplitude impact load caused by engine speed-up under the M5 operating condition leads to an instantaneous jump of the tooth flank contact stress in an extremely short time, and the damage rate per unit time is much higher than that of other steady-state low-load operating conditions.

For the right tooth flank, the damage contribution per unit time of the M8 (Engine Down) transient operating condition reaches 94.9%, which is the core source of instantaneous damage on the right tooth flank. The M6 (HEV) steady-state operating condition accounts for 3.8%, and the total of the remaining operating conditions is only 1.3%. The core cause is that the dynamic load fluctuation during engine speed-down under the M8 condition brings high-cycle and high-amplitude stress impacts, and the fatigue accumulation effect per unit time is much higher than that of steady-state driving operating conditions. The transient operating conditions of engine speed-up and speed-down are the core sources of damage per unit time, and their instantaneous impact loads are the key inducing factor for the acceleration of tooth flank contact fatigue damage.

(3) Quantitative analysis of damage contribution under mode-switching conditions

To investigate the difference in the influence of transient and steady-state operating conditions on fatigue damage, a quantitative analysis of the damage contribution of transient operating conditions (mode-switching operating conditions) and steady-state operating conditions is carried out herein, and the results are presented in Figure 24.

As shown in the figures, the damage contributions of the two types of operating conditions to the left and right tooth flanks exhibit significant asymmetric characteristics. For the left tooth flank, the damage contribution of transient operating conditions reaches 60%, which is significantly higher than the 40% contribution of steady-state operating conditions. The core cause is that the left tooth flank only serves as the meshing working flank when the system is under reverse torque operating conditions. The frequent abrupt reverse load changes and instantaneous impacts during mode switching will cause a significant jump in the tooth flank contact stress in a short time, making them the dominant source of fatigue damage on the left tooth flank.

For the right tooth flank, the damage contribution of steady-state operating conditions reaches 82%, while transient operating conditions only account for 18%. This is because the right tooth flank is the main meshing working flank of the system under forward driving conditions. The steady-state driving operating condition, which accounts for the highest time proportion during the whole vehicle driving process, has the meshing characteristics of high cycle counts and long duration. The long-term accumulated fatigue damage is much higher than that of short-term transient impacts, and it is the core source of the total damage on the right tooth flank. The results show that the transient mode-switching operating conditions are the core inducing factor for the damage of the left tooth flank, while the steady-state operating conditions determine the overall damage level of the right tooth flank.

4.2. Comparison of Damage Prediction Between Full-Condition Merging Method and Mode-Decomposed Method

4.2.1. Comparison of Total Damage and Mileage Life Prediction Results Between Two Methods

The two fatigue loading spectra analyzed in this section are explicitly defined as follows:

(1) Full-Condition Merging Method: All planetary gear loads obtained from CHTC-C/CHTC-B synthesized condition simulation are merged into a single entity, without distinguishing steady-state operating modes, transient conditions, or mode-switching processes. A single full-condition fatigue loading spectrum is compiled for fatigue damage calculation.

(2) Mode-Decomposed Method: The full-condition loads are decomposed into M1 through M11 independent operating modes, and fatigue loading spectra are compiled separately according to the load characteristics of different modes. Meanwhile, based on the unilateral meshing force mechanism of planetary gear sets, independent load cycles of the left and right tooth flanks are statistically analyzed separately. The histogram counting method adapted to gear meshing characteristics is employed to accurately count cycle numbers. After calculating the damage for each operating condition separately, the total damage is obtained through linear superposition based on the Miner criterion.

The driving mileage of the CHTC-C/CHTC-B composite operating conditions in this paper is 25 km, and the calculation formula for the fatigue life of the planetary gear is as follows:

$$L_m={\displaystyle \frac{m}{d}}$$

(7)

where L_m is the mileage life of the planetary gear, m is the simulated driving mileage of a single CHTC-C/CHTC-B composite operating condition, and d is the simulated fatigue damage of a single CHTC-C/CHTC-B composite operating condition.

The total fatigue damage of the PG1 under CHTC-C/CHTC-B synthesized conditions was analyzed using both the full-condition merging fatigue loading spectrum and the mode-decomposed fatigue loading spectrum, respectively. The total damage results and mileage life obtained from the two methods are shown in

Table 11. Fatigue damage and mileage life obtained by the two methods.

Analysis Method	Full Operating Condition	Mode-Decomposed
Fatigue Damage (-)	7.04 × 10−5	8.00 × 10−5
Mileage Life (105 km)	3.55	3.13

. As can be seen from the statistical results in the table, the total fatigue damage for a single CHTC-C/CHTC-B synthesized condition cycle calculated using the conventional full-condition merging fatigue loading spectrum method is 7.04 × 10⁻⁵, corresponding to a planetary gear mileage life of 3.55 × 10⁵ km. The total fatigue damage for a single condition cycle calculated using the mode-decomposed fatigue loading spectrum method proposed in this paper is 8.00 × 10⁻⁵, corresponding to a mileage life of 3.13 × 10⁵ km.

The comparative analysis reveals that the fatigue damage calculated via the full-condition method is about 12% lower than that of the mode-decomposed method, yielding a relatively optimistic prediction. To validate the parameter sensitivity of this conclusion, this study further investigates the effects of planetary gear material properties and load sampling step size.

The permissible contact stress of the material is a critical parameter influencing gear contact fatigue life. To assess how material property uncertainty affects the discrepancy between the two methods, the permissible contact stress of the gear material was perturbed by ±10% and ±20% from the baseline value (1300 MPa). Fatigue damage values from both methods were then compared, with results presented in

Table 12. Fatigue damage comparison between two methods at four permissible contact stress.

Permissible Contact Stress (MPa)	1040	1170	1430	1560
Full-condition (A)	3.34 × 10−3	6.89 × 10−4	3.37 × 10−6	3.19 × 10−7
Mode-decomposed (B)	3.68 × 10−3	7.54 × 10−4	3.85 × 10−6	3.56 × 10−7
(B − A)/B	9.2%	8.6%	12.5%	10.4%

.

Across all material variants, the mode-decomposed method consistently yields higher fatigue damage values than the traditional full-condition method. The percentage difference in damage values remains stable between 8.6% and 12.5%. This demonstrates that while variations in material properties simultaneously affect the absolute damage values of both methods, the mode-decomposed method persistently predicts higher damage than the full-condition method. Thus, the conclusion exhibits robustness to changes in material properties.

Additionally, the discretization precision of the load time history—quantified by the load sampling step size—is another critical factor influencing the accuracy of fatigue damage accumulation calculations. To evaluate its impact on the conclusion, this study conducts a comparative analysis by refining the sampling step size from the baseline of 0.01 s to 0.005 s and 0.001 s while holding all other conditions constant. Results are summarized in

Table 13. Fatigue damage comparison between two methods at varying load sampling step sizes.

Load Sampling Step Size (s)	0.01	0.005	0.001
Full-condition (A)	7.04 × 10−5	7.04 × 10−5	7.02 × 10−5
Mode-decomposed (B)	8.00 × 10−5	8.00 × 10−5	8.01 × 10−5
(B − A)/B	12%	12%	12.4%

.

When the sampling step size varies from 0.01 s to 0.001 s, the fatigue damage values computed by both methods exhibit negligible change. The relative difference rate—defined as (B − A)/A, where A denotes the full-condition method’s damage and B denotes the mode-decomposed method’s damage—fluctuates within a narrow band of 12.0% to 12.4%, with a maximum variation amplitude of merely 0.4%. This demonstrates that within the selected step size range, the results are insensitive to temporal discretization, and the damage discrepancy between the two methods exhibits robust stability.

4.2.2. Causes of Damage Underestimation by the Traditional Method and Accuracy Advantages of the Proposed Method

The core difference between the two methods lies in the boundaries of load statistics and grading. The traditional full-operating-condition method takes the complete load of the full operating condition as a whole, uniformly divides 10 load intervals based on the full-operating-condition torque extreme value, generates a single overall fatigue loading spectrum, and obtains the total damage by superposition after calculating the single-level damage. The mode-decomposed method proposed in this paper first extracts the independent loads of 11 operating modes M1–M11, takes the single-mode load as a unit, divides 10 load intervals respectively based on the single-mode torque extreme value, generates 11 independent fatigue loading spectra, and obtains the total damage by linear superposition after calculating the damage of each mode separately. The underestimation of fatigue damage by the traditional method essentially originates from the excessively wide statistical boundary of the overall unified grading, resulting in the distortion of load characteristics and insufficient matching between simulation calculation and actual service characteristics. The core causes are summarized in three aspects:

(1) The overall grading under full operating conditions eliminates the differences in operating characteristics of loads within the same torque interval under different modes, leading to distortion in the statistics of load cycle counts. Gear fatigue damage is strongly correlated with torque amplitude, rotational speed, and load duration. The traditional method merges loads in the same torque interval from different modes, calculates cycle counts with the average rotational speed and total duration, loses the real operating characteristics of loads, causes inconsistency between cycle statistics and actual meshing conditions, and ultimately underestimates the total damage.

The wide-interval grading under full operating conditions dilutes the damage contribution of low-proportion, high-amplitude transient loads, which is the core reason for damage underestimation. This method divides wide intervals based on the full-operating-condition torque extreme value. The transient high-amplitude loads generated during mode switching and engine speed-up/down are merged into adjacent intervals and mixed with low-amplitude steady-state loads, which cannot be counted independently and accurately. The nonlinear damage effect of high-amplitude loads is severely diluted, resulting in a significant underestimation of the damage caused by transient loads.

The overall full operating condition fatigue loading spectrum has insufficient matching with the boundary conditions of gear fatigue simulation, which amplifies the calculation deviation of single-level damage. The accuracy of gear fatigue simulation is highly dependent on the stability of load boundaries. The loads at the same level in the traditional method include loads from multiple modes and multiple meshing states, so only averaged boundaries can be adopted, which cannot match the real meshing characteristics. The single-level calculation deviation is further amplified after linear superposition, ultimately leading to the underestimation of total damage.

Aiming at the inherent defects of the traditional method, the core accuracy advantages of the mode-decomposed fatigue loading spectrum method proposed in this paper are reflected in three aspects. First, it has finer granularity of load statistics. Independent grading and statistics are performed with a single operating mode as the unit, avoiding the merging and averaging of loads from different modes. The load parameters of each level are completely matched with the actual operating state of the corresponding mode, ensuring statistical accuracy from the source. Second, it accurately captures the nonlinear damage contribution of transient high-amplitude loads. The independent grading for a single mode enables transient high-amplitude loads to form independent load levels without being merged and diluted by wide intervals, solving the core limitation of transient damage underestimation in the traditional method. Third, the fatigue loading spectrum is highly matched with the simulation boundaries. The single-mode fatigue loading spectrum corresponds to a unique operating mode, with a stable meshing state and force boundary, which significantly improves the simulation calculation accuracy. The final obtained total damage and life prediction results are more conservative and offer higher engineering safety, which can provide reliable support for the design of transmission systems and the formulation of maintenance cycles.

5. Discussion

This study focuses on the fatigue damage problem of the front planetary gear set in clutchless power-split hybrid electric buses, proposes a mode-decomposed fatigue loading spectrum method based on operating mode separating, and systematically reveals the mode-decomposed damage law of planetary gears in this configuration. The results show that the rigid connection between the engine and the transmission system in the clutchless configuration causes the planetary gears to simultaneously withstand the high-cycle-count load accumulation from steady-state driving, and the transient bidirectional load impacts from mode switching and engine start–stop, forming a unique damage distribution characteristic. The PG1 sun gear directly connected to MG1 is the weakest component of the system in terms of fatigue life. Its right tooth flank, as the main meshing surface for forward driving, has a total damage much higher than that of the left tooth flank, presenting a significant asymmetric unilateral damage characteristic.

Compared with the traditional full-condition merging fatigue loading spectrum method, the method proposed in this paper solved the core limitations of the traditional method, namely the dilution of transient impact damage and the distortion of load characteristics caused by wide-interval load grading. Quantitative results show that the mileage life prediction result of planetary gear set by the traditional method is about 12% over-optimistic compared to the proposed method, which verifies that the traditional method significantly underestimates the transient load damage of the clutchless hybrid powertrain system, and proves the accuracy advantage of the proposed method in load statistics and damage calculation. Meanwhile, this study quantifies the differentiated damage contribution of steady-state and transient operating conditions. The steady-state M6 (HEV) operating mode contributes about 88% of the total damage to the right tooth flank, which is the determining factor of the overall system life. The transient operating moded M5 and M8 and the mode-switching process are the absolute dominant factors of damage per unit time. Among them, the mode-switching operating condition contributes about 60% of the total damage to the left tooth flank, which is the core inducing factor for tooth flank microcrack initiation and early failure.

At the gear structural design level, the four identified core damage working conditions can be used as limit inputs for planetary gear strength design in the future. Stress concentration on the sun gear tooth flank can be alleviated through tooth tip modification and helix angle optimization. At the control strategy optimization level, the torque smooth transition algorithm for M5 and M8 transient conditions can be optimized to reduce transient impact during mode switching. Meanwhile, the power source load distribution for M6 can be optimized to reduce fatigue accumulation from high-frequency loads.

Based on the research results of this study, further deepening and improvement can be carried out from the following dimensions in the future. First, at the load input level, full-condition load data collected from actual vehicle road tests can be supplemented in the future to better align with the real service characteristics of urban transit buses and optimize load input accuracy. Second, at the model accuracy level, multi-factor coupling effects such as operating temperature, time-varying lubrication and tooth flank wear can be incorporated to construct a multi-field coupled refined fatigue model, improving the comprehensive characterization of the gear fatigue damage evolution mechanism. Third, at the result verification level, this study has revealed the core patterns of multi-mode fatigue damage through simulation analysis. In the future, systematic experimental verification and refinement can be conducted through bench fatigue tests and actual vehicle operational data tracking to validate and improve the research conclusions.

6. Conclusions

In this paper, the front planetary gear set of a clutchless power-split hybrid electric bus is taken as the research object. Aiming at the service characteristics of this configuration, namely no clutch buffering and significant transient abrupt load changes, a fatigue damage analysis method for planetary gears based on operating mode splitting is proposed. Through complete vehicle co-simulation, compilation of the mode-decomposed fatigue loading spectrum, and rigid–flexible coupled dynamic simulation, the fatigue damage law of planetary gears under full operating conditions is systematically revealed. The main conclusions are drawn as follows:

In the front planetary gear train of the clutchless power-split hybrid powertrain system, the PG1 sun gear is the weakest component in terms of fatigue life, and the contact fatigue damage of its right tooth flank is the maximum value in the whole system. The left and right tooth flanks of the sun gear present a significant asymmetric unilateral damage characteristic. The total damage of the right tooth flank under driving operating conditions is much higher than that of the left tooth flank under reverse dragging operating conditions, and the stress concentration effect at the tooth end further amplifies this damage difference.

The fatigue damage of the planetary gear set presents a highly concentrated distribution characteristic across operating modes. The steady-state M6 (HEV) contributes 80% of the total damage to the right tooth flank, which is the dominant source of the total fatigue damage of the system. The transient operating conditions of M5 (Engine Up) and M8 (Engine Down) are the absolute dominant factors of the damage per unit time for the left and right tooth flanks, respectively. The transient mode-switching operating condition contributes 60% of the total damage to the left tooth flank. These four types of operating conditions jointly determine the overall fatigue life of the planetary gear train.

The mode-decomposed fatigue loading spectrum method proposed in this paper breaks through the inherent defects of the traditional method. It effectively avoids the core accuracy limitations of the traditional method. Specifically, it resolves the dilution of transient load damage and the distortion of load characteristics. The prediction result of the planetary gears’ mileage life by the traditional method is about 12% higher than that by the proposed method. The calculation results of the proposed method are more conservative and more consistent with the actual service state of the gears, which can provide reliable theoretical and data support for the design optimization and full lifecycle fatigue life prediction of transmission components in the same type of hybrid powertrain systems.