Higher-order transmissibility and its linear approximation for in-service crack identification in train wheelset axles

1. Problem Statement

The structural integrity of train wheelset axles is critical for preventing derailments. Current industry practices rely on Non-Destructive Testing (NDT) (e.g., Ultrasound Testing) performed in workshops, which requires taking trains out of service and involves high costs and scheduled intervals. In-service Structural Health Monitoring (SHM) offers a continuous alternative but faces significant challenges:

• Data Scarcity: Real-world crack data is rare, making purely data-driven approaches difficult.
• Computational Cost: Physics-based identification requires solving inverse problems repeatedly. Modeling "breathing cracks" (cracks that open and close during rotation) introduces nonlinearities that require iterative, time-consuming numerical solutions (e.g., Newton-Raphson methods).
• Feature Sensitivity: Traditional linear damage indicators (like modal properties) are often insensitive to small, incipient cracks.

The core problem is developing a computationally efficient method to identify small, breathing cracks in rotating axles using in-service data without the prohibitive cost of solving nonlinear equations in real-time.

2. Methodology

The authors propose a novel framework combining a specific damage feature with a linearized approximation of the nonlinear system.

A. Damage Feature: Higher-Order Transmissibility (HOTr)

• Concept: Breathing cracks generate higher-order harmonics (e.g., 2nd, 3rd) in the vibration response due to the periodic variation in stiffness.
• Metric: The authors define Higher-Order Transmissibility (HOTr) as the frequency-dependent ratio of responses (e.g., strain or displacement) between two points at a specific harmonic order (specifically the second-order harmonic).
• Advantage: Unlike linear transmissibility, HOTr is highly sensitive to the location and size of incipient cracks and is independent of the magnitude of the external load (crucial for varying train speeds/loads).

B. Linear Approximation of Nonlinear Dynamics

To avoid solving the full nonlinear equation of motion iteratively, the authors derive a first-order perturbation-based approximation:

1. Primal Model: The system is modeled as a pristine (crack-free) axle with a constraint enforcing crack closure (using Lagrange multipliers).
2. Perturbation: The breathing crack is treated as a small perturbation ($\epsilon$) to the pristine stiffness. The nonlinear contact force is approximated by the constraint forces of a closed crack.
3. Decoupling: This transforms the nonlinear problem into a sequence of linear frequency-domain problems. The solution for the cracked state is derived from the pristine solution plus a correction term, eliminating the need for iterative nonlinear solvers.

C. Reduced-Order Modeling (ROM) & Substructuring

To further accelerate the process, two ROM strategies are employed:

• Rubin Method (RB): Reduces degrees of freedom (DoFs) while keeping crack interface and external force DoFs.
• Substructuring (SUB): Divides the axle into two parts:
  • Substructure A: The region where cracks may occur (kept as a full-order model).
  • Substructure B: The rest of the axle (reduced offline).
  • Benefit: The SUB model allows the reduced part of the axle to be pre-calculated, drastically reducing online construction time for different crack scenarios.

D. Crack Identification Algorithm

• Inverse Problem: Formulated as an optimization problem to minimize the error between measured and simulated HOTr.
• Solver: A Genetic Algorithm (GA) is used to search the parameter space (crack location, depth, and orientation) because the objective function is non-convex and non-smooth.
• Data Processing: Higher-order harmonics are extracted from noisy time-domain signals using a recursive correlation method.

3. Key Contributions

1. Proposal of HOTr: Introduction of Higher-Order Transmissibility as a robust, load-independent feature specifically tailored for detecting breathing cracks in rotating machinery.
2. Linear Approximation Scheme: Development of a first-order perturbation method that approximates the nonlinear crack response using linear system theory. This eliminates the need for iterative nonlinear solvers, reducing computational time by three orders of magnitude.
3. Efficient Substructuring Strategy: A novel combination of the perturbation method with substructuring (SUB) that allows for offline reduction of the non-cracked domain, making the method feasible for near real-time applications.
4. Validation: Comprehensive validation on both 2D beam and 3D hollow axle models, demonstrating accuracy in reproducing nonlinear dynamics and effectiveness in crack identification under noisy conditions.

4. Results

Computational Efficiency

• Speedup: The proposed linear approximation reduced solution times by factors of ~1,500x (2D) and ~3,200x (3D) compared to full nonlinear Multi-Harmonic Balance (MHB) solutions.
• Substructuring Advantage: In the 3D case, the SUB model was significantly faster than the RB model for online simulation because the large, non-cracked portion of the model was pre-reduced.

Accuracy

• Feature Reproduction: The linear approximation of HOTr showed excellent agreement with the full nonlinear solution.
  • 2D Case: Maximum RMSE of $3.8 \times 10^{-4}$.
  • 3D Case: Maximum RMSE of 0.22 (slightly higher due to model complexity, but sufficient for identification).
• Frequency Range: The method remained accurate across a wide frequency band, including sub-harmonics and resonances.

Identification Performance (Monte Carlo Analysis)

• Sensitivity: Second-order harmonics showed high sensitivity to crack location and depth, outperforming first-order harmonics.
• Noise Robustness:
  • Low Noise (<1%): The Genetic Algorithm successfully converged to the true crack parameters.
  • High Noise (5-10%): Identification accuracy decreased, particularly for very shallow cracks (e.g., 5% depth). However, even in high-noise scenarios, the identified parameters remained in close proximity to the true values.
• Trade-off: There is a clear trade-off between crack size, noise level, and localization precision. Cracks larger than 10% depth were identified with near-perfect accuracy even with moderate noise.

5. Significance

This study provides a viable pathway for real-time, in-service crack identification in train wheelsets. By decoupling the computational cost of nonlinear analysis from the identification process, the authors demonstrate that:

• Feasibility: Complex nonlinear damage detection can be performed within a near real-time context using standard computational resources.
• Safety: The ability to detect incipient cracks (1-5% depth) while the train is in operation could prevent catastrophic failures and optimize maintenance schedules, moving away from rigid, time-based inspections to condition-based monitoring.
• Generalizability: The framework is applicable to any rotating structure where breathing cracks are the dominant source of nonlinearity, provided the crack size is small enough to justify the perturbation assumption.

Limitations & Future Work:

• The method assumes the crack is small (perturbation theory); it may not be accurate for large, fully open cracks.
• Current validation is numerical; the authors are conducting experimental campaigns to verify the method on physical test rigs.
• Future work may explore meshless techniques (XFEM) for more flexible crack geometry representation.