The reliable analysis of the non-stationary response of mechanical and structural systems under transient excitations, and in particular the tracking of the time-varying evolution of the dominant modal frequencies during damaging events, remains a long-standing challenge for vibration-based monitoring. A persistent practical limitation is that the time-frequency map on which any subsequent automatic or learning-based identification stage relies is built on a frequency grid that depends on implementation choices. This means that records acquired on different sensors - from conventional accelerometers to MEMS and vision-based measurement chains - or under different events cannot be compared on a common substrate. The present article addresses the limitation at the representational level by introducing the Stockwell-Ditommaso-Ponzo representation (SDP), obtained by sampling the Stockwell transform on an ordered Farey grid of primitive rational frequencies. Each coefficient is assigned a rational label, an ordered Voronoi frequency cell and an exact logarithmic quadrature weight induced by the Stockwell measure. Subsequent to the declaration of the sampling frequency, the analysed band and the maximum conductor, the support, the cells and the weights are uniquely determined prior to the processing of any record. A time-varying modal frequency, such as that observed when the apparent lateral stiffness of an instrumented building degrades under strong-motion excitation and partially recovers afterwards, is therefore encoded as a deterministic path on a geometry shared across sensors, repeated events and simulations of the same structure. A distinctive property of this construction is that the storage grid is invariant to the length of the analysed record: because the support depends only on the sampling frequency, the analysed band and the maximum conductor, and not on the number of samples, two recordings of different duration, once brought to the same sampling frequency, are projected onto an identical frequency lattice, whereas the grids of the short-time Fourier, wavelet and conventional Stockwell transforms, in their standard FFT-based forms, shift from one record to the next unless an identical support is explicitly prescribed. The representation is validated on the real strong-motion recording acquired on the top floor of a four-storey reinforced-concrete building in Bonefro (Italy) during the 2002 Molise earthquake: the SDP cell-path recovers the published apparent-frequency sequence (pre-event 2.5 Hz, softened minimum about 1.3 Hz, post-event 1.9 Hz) on the declared rational geometry, in agreement with the published estimates and with three reference time-frequency techniques (the short-time Fourier, wavelet and conventional Stockwell transforms) applied to the same record, and it recovers the orthogonal-direction sequence as well. It is exercised on a noisy swept-sine, a controlled multi-component signal, alternative rational grids and the simulated response of a five-storey reinforced-concrete infilled frame, and is compared quantitatively with the short-time Fourier, wavelet and conventional Stockwell transforms before the real-data case study. The practical value of this invariance is quantified directly: when the same physical window is read from records of different duration, the coefficients stored on the rational support agree to machine precision and keep identical positions, whereas on the record-dependent FFT grid of the conventional Stockwell transform a coefficient-wise comparison is in error by up to 40%, with frequency mismatches of several hundred mHz, and the maps are not even of equal size; the computational cost is measured against the short-time Fourier and conventional Stockwell transforms under a common input, frequency band, hardware and execution protocol, and is found to be comparable per frequency node to, and lower overall than, the conventional transform on the same band. The proposed representation is intended to function as a reproducible, transform-level front end on which calibrated automatic identification and damage-assessment procedures, including artificial-intelligence-based pipelines, can be built in a controlled manner.
Time-frequency representation of nonlinear structural response through rationally sampled Stockwell analysis
Rocco Ditommaso
;Felice Carlo Ponzo
2026-01-01
Abstract
The reliable analysis of the non-stationary response of mechanical and structural systems under transient excitations, and in particular the tracking of the time-varying evolution of the dominant modal frequencies during damaging events, remains a long-standing challenge for vibration-based monitoring. A persistent practical limitation is that the time-frequency map on which any subsequent automatic or learning-based identification stage relies is built on a frequency grid that depends on implementation choices. This means that records acquired on different sensors - from conventional accelerometers to MEMS and vision-based measurement chains - or under different events cannot be compared on a common substrate. The present article addresses the limitation at the representational level by introducing the Stockwell-Ditommaso-Ponzo representation (SDP), obtained by sampling the Stockwell transform on an ordered Farey grid of primitive rational frequencies. Each coefficient is assigned a rational label, an ordered Voronoi frequency cell and an exact logarithmic quadrature weight induced by the Stockwell measure. Subsequent to the declaration of the sampling frequency, the analysed band and the maximum conductor, the support, the cells and the weights are uniquely determined prior to the processing of any record. A time-varying modal frequency, such as that observed when the apparent lateral stiffness of an instrumented building degrades under strong-motion excitation and partially recovers afterwards, is therefore encoded as a deterministic path on a geometry shared across sensors, repeated events and simulations of the same structure. A distinctive property of this construction is that the storage grid is invariant to the length of the analysed record: because the support depends only on the sampling frequency, the analysed band and the maximum conductor, and not on the number of samples, two recordings of different duration, once brought to the same sampling frequency, are projected onto an identical frequency lattice, whereas the grids of the short-time Fourier, wavelet and conventional Stockwell transforms, in their standard FFT-based forms, shift from one record to the next unless an identical support is explicitly prescribed. The representation is validated on the real strong-motion recording acquired on the top floor of a four-storey reinforced-concrete building in Bonefro (Italy) during the 2002 Molise earthquake: the SDP cell-path recovers the published apparent-frequency sequence (pre-event 2.5 Hz, softened minimum about 1.3 Hz, post-event 1.9 Hz) on the declared rational geometry, in agreement with the published estimates and with three reference time-frequency techniques (the short-time Fourier, wavelet and conventional Stockwell transforms) applied to the same record, and it recovers the orthogonal-direction sequence as well. It is exercised on a noisy swept-sine, a controlled multi-component signal, alternative rational grids and the simulated response of a five-storey reinforced-concrete infilled frame, and is compared quantitatively with the short-time Fourier, wavelet and conventional Stockwell transforms before the real-data case study. The practical value of this invariance is quantified directly: when the same physical window is read from records of different duration, the coefficients stored on the rational support agree to machine precision and keep identical positions, whereas on the record-dependent FFT grid of the conventional Stockwell transform a coefficient-wise comparison is in error by up to 40%, with frequency mismatches of several hundred mHz, and the maps are not even of equal size; the computational cost is measured against the short-time Fourier and conventional Stockwell transforms under a common input, frequency band, hardware and execution protocol, and is found to be comparable per frequency node to, and lower overall than, the conventional transform on the same band. The proposed representation is intended to function as a reproducible, transform-level front end on which calibrated automatic identification and damage-assessment procedures, including artificial-intelligence-based pipelines, can be built in a controlled manner.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
