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Bridge Modal Identification using Acceleration Measurements within Moving Vehicles

Soheil Sadeghi Eshkevari, Thomas J. Matarazzo, Shamim N. Pakzad

arXiv:2001.01797v1eess.SPstat.AP

TL;DR

Moving-vehicle acceleration contains bridge vibrations mixed with vehicle suspension and road-roughness effects, complicating comprehensive bridge system identification. The paper proposes two decontamination approaches followed by STRIDEX; both identify the first four modes with reported frequency errors below 1.2% and 1.3%, respectively.

  • Problem

    Mobile-sensor studies have often identified only partial modal properties, while vehicle-carried bridge measurements are contaminated by suspension and road-profile effects.

    Source: §1.1 · p. 2

  • Method

    Two pipelines deconvolve vehicle dynamics using either its frequency response function or EEMD, separate roughness with SOBI, and analyze the resulting mobile data with STRIDEX.

    Source: §1.5 · p. 5, §4.3 · p. 14, §8 · p. 21

  • Results

    Both methods estimated the bridge’s first four modes; frequency errors were below 1.2% for FRF and 1.3% for EEMD, and all estimated mode-shape MAC values exceeded 0.94.

    Source: §8 · p. 22

  • Takeaways & Limitations

    The study presents a methodology for complete bridge modal identification from moving-vehicle acceleration, including frequencies, damping ratios, and mode shapes.

    Source: §8 · pp. 21–22

    The methods work best when vehicle and bridge frequency bands do not overlap; common vehicles may overlap with bridge frequencies.

    Source: §6 · p. 17

Abstract

from arXiv · show

Vehicles crossing bridge structures respond dynamically to the bridge's vibrations. An acceleration signal collected within a moving vehicle contains a trace of the bridge's structural response, but also includes other sources such as the vehicle suspension system and surface roughness-induced vibrations. This paper introduces two methods for the bridge system identification using data collected by a network of moving vehicles. The contributions of the vehicle suspension system are removed by deconvolving the vehicle response in frequency domain. The first approach utilizes the vehicle transfer function, and the second uses EEMD method. Next, roughness-induced vibrations are extracted using second-order blind identification (SOBI) method. After these two processes the resulting signal is equivalent to the readings of mobile sensors that scan the bridge's dynamic response. Structural modal identification using mobile sensor data has been recently introduced as STRIDEX algorithm. The processed mobile sensor data is analyzed using STRIDEX to identify the modal properties of the bridge. The performance of the methods is validated on numerical case studies of a long single-span bridge monitored via a network of moving vehicles. The analyses consider three road surface roughness patterns. Results show that the proposed algorithms are successful in extracting pure bridge vibrations, and produce accurate and comprehensive modal properties of the bridge. The study shows that the proposed transfer function method can efficiently deconvolve the linear dynamics of a moving vehicle. EEMD method is able to extract vehicle dynamic response without a-priori information about the vehicle. This study is the first proposed methodology for complete bridge modal identification, including operational natural frequencies, mode shapes and damping ratios using \textit{moving vehicles sensor data}.

1 Introduction

Fixed sensors limit spatial coverage, while vehicle-carried measurements combine bridge response with suspension and road effects. The paper proposes removing these effects from moving-vehicle data before applying STRIDEX for comprehensive bridge modal identification.

1 Introduction

  • Mobile sensors can collect comprehensive spatial information with few sensors, while fixed networks constrain spatial resolution and dense arrays add cost and complexity.The introduction links mode-shape resolution to the number and arrangement of sensing nodes.Source: §1 · p. 2

1.1 Toward infrastructure vibration crowdsourcing

  • Prior mobile-sensing studies often identified only partial modal properties, motivating practical procedures for comprehensive system identification from mobile and smartphone data.The paper identifies STRIDEX as a procedure capable of complete mobile-data identification and cites an alternative with a different sensing setting.Source: §1.1 · p. 2

1.2 Review of system identification using mobile sensor networks

  • STRIDEX estimates structural parameters from dynamic sensor network data, but prior applications did not account for vehicle dynamics or road-profile effects contaminating vehicle-carried measurements.The truncated physical state-space model maps moving-sensor data to virtual probing data through the mode shape regression function.Source: §1.2 · p. 3

1.3 Bridge system identification using dynamic sensor network data

  • Earlier drive-by methods targeted individual properties or relied on special loading, smooth roads, stationary sensors, or specialized vehicles, rather than comprehensive identification under ambient excitation.For example, a frequency-domain method captured only the first natural frequency, while another mode-shape method considered a very smooth roadway.Source: §1.3 · p. 4

1.4 Problem definition

  • Vehicle-carried acceleration is contaminated by vehicle-bridge interaction and multiple sources, including bridge response, roughness displacement, and vehicle dynamics.The paper formulates the objective as extracting the bridge response for STRIDEX while neglecting engine-induced vibration.Source: §1.4 · pp. 4–5

1.5 Motivation

  • The proposed pipelines deconvolve vehicle dynamics using either the vehicle frequency response function or EEMD, then use SOBI to separate bridge vibrations from roughness before STRIDEX.The stated contribution is comprehensive modal identification from moving-vehicle acceleration, with robustness to severe roughness when the vehicle acts linearly in the frequency band considered.Source: §1.5 · pp. 5–6

2 Approaches to Extract the Bridge Response from Measurements within Moving Vehicles

The section presents two routes for removing vehicle-suspension effects, then uses SOBI to separate bridge response from roughness vibrations in the resulting mixture.

2.1 Method 1: Deconvolution using the system transfer function

  • With an accurate suspension model, Method 1 identifies the vehicle transfer function from responses at both degrees of freedom and deconvolves the vehicle input in the frequency domain.Output-only system identification estimates vehicle modal properties used to generate the transfer function.Source: §2.1 · p. 7

2.2 Deconvolution using the approximated system transfer function

  • The approximated transfer-function approach assumes known suspension mode shapes and uses only the sprung-mass channel to generate the vehicle transfer function.The paper presents this as more practical because collecting responses at the unsprung mass is difficult.Source: §2.2 · p. 7

2.3 Method 2: Deconvolution using ensemble empirical modal decomposition

  • EEMD extracts vehicle dynamics without a vehicle model by adding noise, repeating EMD, and averaging the resulting components.The paper notes that EMD components represent oscillatory trends and that EEMD addresses frequency leakage and aliasing in the original EMD.Source: §2.3 · p. 8

2.4 Source separation on the linear mixture

  • After deconvolution, SOBI separates bridge response from road-roughness effects by whitening the observations and jointly diagonalizing covariance matrices at multiple time lags.SOBI assumes uncorrelated sources and is suited to linear mixtures, especially spectrally uncorrelated sources.Source: §2.4 · p. 8

3 Generation of Vehicle Scanning Data

The simulations generate vehicle-scanning data for a 500 m bridge under three roughness patterns. Vehicle responses vary substantially with roughness, motivating the subsequent separation steps.

3.1 Bridge finite element analysis

  • The 500 m, 10K-DOF bridge model has four specified natural modes and is scanned by eight vehicles traveling 70% of the span at 2.5 m/s.Four vehicles travel in each direction; the 2D model does not estimate torsional modes.Source: §3.1 · p. 9, Figure 5 · p. 10

3.2 Simulation of the vehicle system

  • Vehicle input combines bridge displacement at each moving vehicle location with corresponding road-roughness displacement before being passed through a quarter-car model.The vehicle is modeled as a linear system; the paper notes that EEMD performs best when vehicle and bridge frequency bands do not overlap.Source: §3.1 · p. 9, §3.2 · pp. 9–11

3.1 Bridge finite element analysis

  • Three roughness profiles—sinusoidal, random white noise, and expansion joints—test robustness; the expansion-joint profile combines white noise with 0.5-meter-wide drops.The sinusoidal profile has a single PSD spike, while the other profiles have more complex spectra; the expansion-joint PSD includes high-energy fluctuations.Source: §3.1 · p. 9, Figure 7 · p. 11

3.2 Simulation of the vehicle system

  • Vehicle responses under identical bridge loads are highly sensitive to road roughness, and are then used as inputs to extract bridge modal properties.The simulated outputs include tire- and cabin-level responses.Source: §3.2 · p. 11, Figure 8 · p. 11

4 Extraction of Bridge Vibrations from Vehicle Scanning Data

The section develops two ways to remove vehicle suspension effects from moving-vehicle measurements, then separates road roughness from bridge vibrations to prepare signals for modal identification.

4.1 Method 1 - Signal deconvolution with FRF

  • Output-only identification supplies vehicle dynamics for FRF deconvolution, which reconstructs the vehicle input from its cabin response.The identified vehicle frequencies match the known frequencies of 5.14 Hz and 36.78 Hz.Source: §4.1.1 · p. 12

4.1.1 Vehicle system identification and frequency response function

  • FRF deconvolution removes vehicle-frequency spikes and roughness-shock impulses, leaving a linear mixture of bridge vibrations and roughness displacements.SOBI is then applied to separate these sources.Source: §4.1.1 · p. 12

4.2 Method 2 - Signal deconvolution with EEMD

  • EEMD identifies vehicle effects in the first IMF; subtracting two IMFs reduces vehicle energy but does not perfectly remove it, yet remains adequate for bridge modal identification.Vehicle content also appears in the second IMF for two roughness cases.Source: §4.2 · pp. 12, 14

4.3 SOBI for linear un-mixing

  • SOBI assumes the bridge and roughness sources are uncorrelated and diminishes roughness-induced peaks in outputs from both deconvolution methods.The extracted bridge vibration signal can be used for STRIDEX identification.Source: §4.3 · pp. 14–15
  • The FRF-derived PSD has higher spectral resolution than the EEMD output because its frequency range can be refined, a technique insensitive to EEMD.Source: §4.3 · p. 15

5 Bridge Modal Identification

The study applies STRIDEX to bridge-vibration signals extracted from moving vehicles and automates mode selection and superposition. Both proposed methods recover modal properties, with strong mode-shape agreement overall.

5 Bridge Modal Identification

  • Eight vehicles scan a 10K-DOF bridge model, with each one-direction trip covering 7K DOFs.STRIDEX is applied to the extracted mobile-sensor bridge-vibration signals.Source: §5 · p. 15
  • An automated algorithm selects and superposes modes, replacing the previously tedious manual selection process.The identified modal parameters are presented in Figure 18.Source: §5 · p. 15, Figure 18 · p. 16

5.1 Modal property results

  • 0.9443–0.9997 was the MAC range for all estimated modes; both methods estimated natural frequencies and mode shapes successfully, with FRF slightly outperforming EEMD on mode-shape accuracy.Damping ratios were estimated precisely in most cases, although the first FRF-identified mode was less accurate.Source: §5.1 · pp. 15, 17, Table 2 · p. 15, Figure 18 · p. 16

6 Application using Mechanical Properties of Commercial Vehicles

The commercial-vehicle case tests the methods when vehicle and bridge frequency bands are close. Modal estimates remain accurate overall, though vehicle contamination especially affects the first mode.

6 Application using Mechanical Properties of Commercial Vehicles

  • The common vehicle’s 1.64-Hz and 11.01-Hz natural frequencies create a close-band case; both methods weaken but do not fully remove vehicle content.The remaining signal is reported as suitable for bridge identification.Source: §6 · pp. 17–18
  • 0.9656–0.9987 was the MAC range for common-vehicle mode shapes, while natural frequencies and mode shapes were estimated accurately overall.The first mode’s frequency, damping ratio, and MAC were less desirable than in the previous vehicle case.Source: §6 · p. 18, Table 8 · p. 19
  • The first mode is more affected by vehicle contamination because it carries less energy, and its wider spectral peak produces very high estimated damping ratios.The authors attribute the lower accuracy in this case to close frequency bands.Source: §6 · p. 18, Table 7 · p. 19

7 System Identification using Approximated Vehicle Transfer Function

The section evaluates an approximated vehicle transfer function built from common-car properties, including a cabin-only approximation. The approximation generally preserves bridge modal identification, though first-mode damping remains a notable weakness.

7 System Identification using Approximated Vehicle Transfer Function

  • The FRF preprocessing requires identifying the sensing vehicle in advance and recording its response at two locations, which may make tire-level data impractical to obtain.The vehicle description supplies the properties needed to produce its transfer function.Source: §7 · p. 19, Table 9 · p. 21
  • Approximating the vehicle mode shape allowed its response to be characterized sufficiently using only one sensor in the vehicle cabin.The section notes that manufacturer-provided mechanical properties remain preferable when available.Source: §7 · p. 21
  • The approximated transfer function produced frequencies matching actual values, acceptably accurate damping for modes 2–4, and desirable MAC values, but first-mode damping remained far off.The approximated mode shape slightly worsened damping estimates relative to the prior case.Source: Figure 22 · p. 20, Figure 23 · p. 21, §7 · p. 21

8 Conclusion

The paper proposes FRF- and EEMD-based vehicle deconvolution, followed by SOBI extraction of roughness effects and STRIDEX analysis of the resulting mobile-sensor data. Both methods estimate bridge modes accurately in the reported numerical studies, with FRF generally more accurate and EEMD requiring no vehicle-property information.

8 Conclusion

  • FRF deconvolution with SOBI and EEMD-based deconvolution were proposed to identify bridge modal properties from vehicle-carried sensor data.The pipelines remove vehicle suspension and roughness-induced effects before STRIDEX processes the resulting mobile-sensing data.Source: §8 · pp. 21–22
  • Both methods estimated the bridge’s first four modes, with MAC values above 0.94 and frequency errors below 1.2% for FRF and 1.3% for EEMD; damping accuracy was generally on par with traditional SID.Some damping estimates were near exact, including the second mode.Source: §8 · p. 22
  • With common-vehicle properties, modal identification was not considerably affected, but estimated first-mode damping ratios changed significantly.The study reports that possible reasons for this effect were discussed.Source: §8 · p. 22
  • The FRF approach yielded more accurate bridge SID results overall, while EEMD operated without vehicle-property information.A simplified, cabin-only approximation of the vehicle transfer function produced accurate modal estimates, with frequency error at most 1.2% and MAC values above 0.97.Source: §8 · p. 22
  • The methods target bridge information from vehicle networks compatible with large-scale data collection, supporting more frequent observations and condition reports.The paper connects this potential to using everyday vehicle-based datasets for structural health monitoring.Source: §8 · p. 22

10 Appendix A: Discussion on the Generality of the Methods

The appendix tests the pipelines on a 300 m single-span bridge with ISO 8608:2016 class C road roughness and closely spaced bridge and vehicle frequencies. FRF and EEMD yield usable mode-shape estimates, though frequency proximity degrades identification, especially for EEMD’s fourth mode.

10 Appendix A: Discussion on the Generality of the Methods

  • The study applied proposed pipelines to a 300 m single-span, rigidly supported 5,000-DOF concrete beam with ISO 8608:2016 class C roughness; its first four frequencies were 0.40, 1.11, 2.18, and 3.61 Hz.The vehicle natural mode lay closely after the bridge’s fourth natural mode, adding complexity especially when applying Method 2.Source: §10 · p. 23
  • For the 300 m bridge with closely spaced vehicle and bridge frequencies, FRF deconvolution extracted a highly damped mode near 5 Hz associated with the vehicle’s fundamental mode.The FRF+SOBI mode-shape MAC values ranged from 0.9195 to 0.9718.Source: Figure 25 · p. 23, Figure 24 · p. 23, §10 · p. 23
  • EEMD-based deconvolution worked in the closely spaced-frequency case, but mode-shape MAC ranged from 0.7870 for the fourth mode to 0.9185 for the first.The fourth mode was least accurate because its frequency was close to the vehicle frequency.Source: §10 · p. 23, Figure 27 · p. 24, Figure 26 · p. 24

11 Appendix B: Modal Aggregation

The appendix describes an automated procedure for combining STRIDEX modal estimates across model orders and virtual-probing-location sets.

11 Appendix B: Modal Aggregation

  • The aggregation procedure selects and combines mode-shape estimates that pass a sine-shape dissimilarity threshold, excluding sets with damping ratios above 30%.It evaluates 11 VPL sets for each of four model orders; the sets contain eight equally spaced VPLs with varying eccentricity.Source: §11 · p. 24

12 Appendix C: Auxiliary Identifications - Road Profile and Vehicle Suspension

Appendix C uses SOBI’s non-bridge channel to estimate road roughness and compares FRF and EEMD approaches to vehicle-response separation. FRF estimates roughness more successfully, while EEMD can identify the vehicle response without prior vehicle identification.

12 Appendix C: Auxiliary Identifications - Road Profile and Vehicle Suspension

  • SOBI separates the signals into two channels: one used as bridge vibration and the other interpreted as the road roughness profile.The extracted roughness channel is compared with three predesignated profiles.Source: §12 · p. 24
  • FRF estimates roughness more successfully; EEMD does not produce an acceptable road-profile estimate for the random profile.Figure 28 compares the extracted roughness profiles with the three designated profiles.Source: Figure 28 · p. 25, §12 · p. 25
  • FRF requires prior vehicle identification, whereas EEMD extracts the vehicle effect as an IMF without that information.The EEMD-derived IMF is treated as the pure vehicle response for identifying vehicle natural frequencies.Source: §12 · p. 25
  • The EEMD-extracted vehicle-response IMF shows a spike at the vehicle fundamental mode, but the second natural frequency is unclear at the simulation’s low sampling frequency.Figure 29 presents the spatial and frequency representations for the three cases.Source: Figure 29 · p. 25, §12 · p. 25
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