A two-stage multi-order probabilistic framework for instantaneous angular speed estimation with informative harmonic selection and bidirectional recursive smoothing

Accurate tacholess instantaneous angular speed (IAS) estimation is essential for condition monitoring of rotating machinery under non-stationary operating conditions. However, existing single-harmonic methods are vulnerable to local harmonic degradation, while conventional multi-harmonic fusion may incorporate weak or interference-dominated components and thereby reduce estimation reliability. In addition, conventional probability smoothing may suffer from boundary effects and dependence on predefined smoothing parameters. To address these issues, a Two-Stage Bidirectional Recursive Multi-Order Probabilistic Approach (Two-Stage BR-MOPA) is proposed. In the first stage, candidate harmonics derived from transmission kinematics are fused to obtain a coarse IAS trajectory. The corresponding theoretical harmonic ridges are then reconstructed and evaluated using a comprehensive criterion combining ridge energy and energy concentration, and informative harmonics are adaptively retained according to their cumulative contributions. In the second stage, the selected harmonics are used to reconstruct a refined joint IAS probability density function (PDF). The most concentrated IAS PDF is identified as a reliable anchor using a pq-mean-based sparsity measure, from which probability information is recursively propagated toward both temporal directions through Gaussian continuity constraints and Bayesian updating. This anchor-initialized recursive strategy alleviates boundary effects, eliminates the need for a predefined temporal neighborhood, and simplifies smoothing parameter selection. Informative harmonic selection further concentrates the joint PDF and reduces sensitivity to the acceleration parameter. The proposed method is validated using a simulated signal and two variable-speed gearbox datasets. Comparative, ablation, and parameter-sensitivity results demonstrate improved estimation accuracy, boundary stability, and parameter robustness under weak-harmonic, interference-contaminated, and rapidly varying-speed conditions.