A novel two-dimensional Hilbert transform method for structural vibration measurement and modal analysis

The phase motion estimation (PME) method gains attention in structural vibration measurement due to its intrinsic properties, including non-surface preparation and full-field measurement capacity. This offers great potential in capturing the full image of structural modes, particularly for structures with large and complex geometries. However, phase estimation using the PME method requires convolving complex Gabor wavelets with a well-defined wavelength/frequency of the moving object in the video/frames from the prior knowledge, and this is still challenging in practical campaign settings. The consequence of setting an inappropriate wavelength for Gabor wavelets may lead to low-fidelity measurements. In contrast, the Hilbert transform is able to obtain the instantaneous phase of the signal without this condition, making it a preferable alternative to the Gabor transform based PME methods. However, the Hilbert transform based on low observational dimension (e.g., one-dimensional Hilbert transform) for phase motion estimation may result in measurement outcomes that suffer from a low Signal-to-Noise Ratio (SNR) and unsatisfactory accuracy. To tackle this issue, a new phase motion estimation method, so-called two-dimensional Hilbert transform motion estimation, termed 2D-HPME, is proposed in this paper. Compared to the traditional PME and HPME methods, the proposed method employs a two-dimensional Hilbert transform on the video sequences to estimate instantaneous phase information per frame. This allows for deriving vibrations in the time domain via instantaneous phase difference. Furthermore, the feasibility of using Empirical Mode Decomposition (EMD) in modal parameters estimation (e.g., resonance frequencies and mode shapes) is also discussed. The performance of the proposed method in vibration measurements and analysis is evaluated and validated through laboratory tests on a steel cantilever beam, comparing it with the conventional PME methods. Acceleration signals from the accelerometer is used as a reference. The results indicate that the proposed 2D-HPME method achieve comparable results compared to that of the accelerometer and outperforms the conventional PME methods.