Physics-constrained identification of equivalent interfacial friction coefficients and load-transfer behavior in a wedge-loading planetary traction drive

Technical Summary: Physics-constrained identification of equivalent interfacial friction coefficients and load-transfer behavior in a wedge-loading planetary traction drive

Problem Statement
Wedge-loading planetary traction drives (WPTDs) transmit torque through high-pressure rolling contacts within elastohydrodynamic lubrication (EHL) regimes. While these systems offer high power density and low vibration, their internal state variables—specifically the equivalent interfacial friction coefficients of multiple traction interfaces and the real-time contact participation states of adaptive loading branches—are difficult to measure directly. Existing mechanical models are primarily forward-looking, treating structural configurations and friction properties as known inputs to predict system responses. Consequently, there is a lack of physically consistent inversion methods capable of identifying these internal, state-dependent parameters from measurable system-level variables (speed and torque). Pure data-driven approaches often fail to produce physically consistent results when data is sparse or noisy, while high-fidelity finite-element (FE) models are computationally prohibitive for rapid parameter inversion across dense operating grids.

Methodology
The study proposes a physics-constrained inversion framework that integrates quasi-static mechanical modeling, finite-element analysis, and Physics-Informed Neural Networks (PINNs).

1. Quasi-Static Mechanical Modeling: A quasi-static model was established to describe the coupling between normal loads, tangential traction forces, contact participation, and input–output torque equilibrium. The model defines five core traction contact interfaces ($C_1$–$C_5$) and introduces contact participation coefficients ($\lambda$) to represent the state (contact or separation) of the two adaptive loading branches.
2. Finite-Element (FE) Verification and Sample Generation: An FE model was developed to verify contact-load distributions and extract configuration compensation information for the loading mechanism (specifically the variation in the center distance between loading rollers). This process corrected deviations caused by rigid-body assumptions. Using the verified model, 720 physics-enhanced theoretical samples were generated, covering input speeds from 500 to 6000 rpm and output torques from 5 to 350 N·m.
3. PINN Architecture and Training: A PINN was constructed with input variables (input/output speeds and torques) and direct outputs (five equivalent interfacial friction coefficients and two contact participation states). The network was trained using a hybrid dataset comprising the theoretical samples and an experimental anchoring set (88 samples) that provided operating conditions but no ground-truth labels for internal parameters.
4. Physics-Constrained Loss Function: The training objective minimized a composite loss function consisting of:
   • Friction-coefficient supervision: Based on theoretical reference values from the quasi-static model.
   • Mechanical equilibrium residuals: Eleven component-level residuals (force and moment equilibrium) ensuring the inverted parameters satisfy physical laws.
   • Contact-state regularization: A soft prior guiding the contact participation states to remain near unity (bilateral contact) at low loads and allowing separation at high loads.
   • Crucially, experimental transmission efficiency was not used as a supervised label during training.

Key Contributions

• Inversion Framework: Development of a non-invasive method to identify equivalent interfacial friction coefficients and infer contact participation states in a complex WPTD without requiring direct measurement of internal tribological states.
• Physics-Enhanced Data Generation: Creation of a dataset augmented by FE verification to compensate for configuration variations in the loading mechanism, providing physically consistent training targets.
• Post-Processed Efficiency Validation: A strategy where transmission efficiency is calculated a posteriori from the inverted internal states and compared against experimental data solely for validation, ensuring the model learns physical mechanisms rather than merely fitting efficiency curves.
• Relative Friction–Load Coupling Index: Introduction of a metric ($I_{c,i} = \mu_i \sigma_i$) to rank the contribution of different traction interfaces to the system's overall friction-load coupling, facilitating interface-level optimization.

Results

• Model Accuracy: The quasi-static model showed strong agreement with FE results, with normal load errors below 2% for traction interfaces and a maximum relative error of 7.99% for the compensated loading-roller center distance.
• Efficiency Prediction: In posterior validation against experimental data at 500, 3000, and 6000 rpm, the proposed PINN significantly outperformed a baseline Artificial Neural Network (ANN) that lacked physical constraints. The PINN reduced the Root Mean Square Error (RMSE) of post-processed transmission efficiency by 52.86%, 64.27%, and 63.06% respectively, while maintaining $R^2$ values above 0.980.
• Physical Consistency: The inverted friction coefficients and post-processed Hertzian contact stresses followed theoretical trends within the in-sample region and maintained continuity in out-of-sample regions.
• Interface Analysis: The analysis identified that the sun roller–planet roller interface ($C_1$) and the right loading branch interfaces ($C_3$, $C_5$) are the dominant contributors to friction-load coupling, accounting for approximately 47%, 29%, and 21% of the total coupling intensity, respectively. The left loading branch interfaces ($C_2$, $C_4$) contributed less than 1.2%.
• High-Speed Extrapolation: The model was extrapolated to 6500–10000 rpm. The results indicated that the dominant contributing interfaces remained stable ($C_1$, $C_3$, $C_5$) and that relative friction-load coupling intensity decreased with increasing speed under constant load, though these high-speed results are presented as model-based trends rather than experimental confirmations.

Significance and Claims
The paper claims that the proposed framework successfully addresses the inverse problem of identifying internal tribological states in WPTDs under sparse experimental conditions. By embedding mechanical equilibrium constraints directly into the learning process, the method ensures that the identified parameters are physically consistent, thereby improving the reliability of post-processed system-level efficiency predictions. The study emphasizes that the results regarding interface-level parameters serve as physical consistency checks and interpretative tools for load transfer behavior, rather than direct experimental measurements of local friction or stress. The authors position the high-speed extrapolation (6500–10000 rpm) as preliminary guidance for future experimental planning and structural optimization, explicitly noting that definitive conclusions regarding high-speed efficiency and friction power loss require further high-speed bench testing and coupled thermo-lubrication-mechanical analysis.