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Graph-Based Characterization of Vision-Derived Dynamic Modes for Structural Damage Identification

R K B M Rizmi, Khalid Mahmud Labib, Shabbir Ahmed

arXiv:2609.09666v1eess.SYcs.SY

TL;DR

The paper addresses how to identify damage-induced changes in beam dynamics from non-contact video measurements, where extracted data are high-dimensional. It models beam dynamics with delay-embedded DMD and analyzes graphs constructed from the mode matrix; graph connectivity and related measures decline as crack depth increases, with numerical analysis validating the connectivity trend observed experimentally.

  • Problem

    Video-derived vibration measurements provide rich spatiotemporal data but are high-dimensional, while selecting individual damage-sensitive DMD modes can be time-consuming and may omit useful information.

    Source: §I · p. 1

  • Method

    The study applies delay-embedded DMD to video-derived beam displacements and uses the resulting mode matrix to construct graphs whose topological features characterize structural conditions.

    Source: §I · p. 1, A. DMD-Based Prediction · p. 2, Graph Connectivity · p. 3

  • Results

    Graph connectivity, centrality, and two-star statistics decrease as crack depth increases, while null dyads increase; numerical analysis also validates the experimentally observed decreasing connectivity trend.

    Source: §IV · p. 5, §V · p. 6

  • Takeaways & Limitations

    Graph-topological measures provide indicators for identifying structural damage from vision-based measurements, with DMD-based spectral and modal measures offering complementary evidence.

    Source: §V · p. 6

Abstract

from arXiv · show

This study presents a graph theoretic framework for characterizing damage-induced changes in the dynamics of a vibrating cantilever beam from non-contact video measurements. Within this approach, dynamics of the beam under healthy and damaged conditions with crack depths of 5, 10, and 13 mm are initially modeled using delay-embedded dynamic mode decomposition (DMD) from vision-based measurements. The resulting mode matrix from DMD is used to construct the adjacency matrix of a graph for this vibrating beam system, and its topological features such as graph connectivity, centrality, two-star statistics, and dyadic configurations are evaluated as damage-sensitive measures. Additionally, different DMD-based spectral and modal measures are evaluated to support the graph theoretic damage assessment. The results show a monotonic reduction in graph connectivity, nodal accessibility, and local organization of the graph with increasing crack depth. A parametric numerical study further validates the decreasing connectivity trend observed experimentally. The results demonstrate that the topological features of the graph derived from the DMD mode matrix provide an interpretable representation of damage-induced changes in the beam dynamics and offer a promising basis for vision-based structural damage identification.

I. INTRODUCTION

Vision-based measurements provide rich spatiotemporal vibration data, but their high dimensionality calls for efficient extraction of dominant dynamics or damage-sensitive features. The study treats the DMD mode matrix as a graph whose topology can characterize structural-condition changes, alongside complementary spectral and modal measures.

I. INTRODUCTION

  • High-dimensional video-derived displacement and vibration data make efficient extraction of dominant dynamics or damage-sensitive features necessary for damage detection.The introduction contrasts this processing need with contact sensing, which requires many sensors and associated instrumentation for multiple damage locations.Source: §I · p. 1
  • Selecting individual damage-sensitive DMD modes can be time-consuming, and excluding modes may discard potentially important information.The study instead treats the entire mode matrix as an evolving graph whose connections may change as damage alters system dynamics.Source: §I · p. 1
  • Different structural conditions are represented as graph states built from their DMD mode matrices, with topology changes used to characterize changes in identified dynamics.The study also evaluates eigenspectrum eccentricity, Wasserstein, Chamfer, and Hausdorff distances, and matched-mode cosine similarity as complementary damage indicators.Source: §I · p. 1

A. DMD-Based Prediction

The method extracts beam displacement from video and arranges the measurements in delay-embedded snapshots for DMD. The resulting modes are reduced across redundant delay blocks to form a spatial mode matrix for subsequent analysis.

A. DMD-Based Prediction

  • Beam centerline displacement is measured at 120 spatial locations from video by tracking the midpoint between the detected top and bottom edges relative to equilibrium.The measurements are arranged as time-indexed displacement snapshots, with the delay-embedded representation defined by an embedding dimension d.Source: A. DMD-Based Prediction · p. 2
  • DMD seeks a best-fit linear operator to advance delay-embedded snapshots, then uses a truncated SVD and projected eigendecomposition to obtain eigenvalues, eigenvectors, and exact modes.The snapshot matrices have 120d rows and N−d columns, and the truncation retains the leading r singular triplets.Source: A. DMD-Based Prediction · p. 2
  • Averaging the d redundant blocks of DMD modes yields a reduced 120 × r mode matrix used as the graph adjacency matrix.The modes encode spatial structure and can reflect dynamics changes such as crack-induced local stiffness reduction.Source: A. DMD-Based Prediction · p. 2

B. Graph-Theoretic Measures

The framework constructs a condition-specific graph from the DMD mode matrix and analyzes its global connectivity and local organization. It also uses centrality measures to characterize node prominence and accessibility.

B. Graph-Theoretic Measures

  • The DMD mode matrix defines a weighted graph representation, with rectangular matrices embedded in a symmetric block adjacency matrix and connections thresholded into a binary graph.Rows and columns correspond to spatial locations and DMD modes, so the graph is bipartite in either matrix case.Source: B. Graph-Theoretic Measures · p. 2
  • Graph connectivity counts retained edges and normalizes them by the total possible connections, enabling comparisons across structural conditions.The resulting measure is dimensionless and describes how densely the graph nodes are connected.Source: Graph Connectivity · p. 3
  • Dyadic statistics classify each node pair as null, asymmetric, or mutual, while two-star statistics count pairs of outgoing or incoming edges sharing a node.These measures describe local graph structure and shared-source or shared-target connection patterns.Source: Dyadic and Higher-Order Structure · p. 3
  • Degree centrality measures direct outgoing connections, while closeness centrality captures how efficiently a node can reach the rest of the network through directed paths.Their network averages serve as scalar descriptors of structural conditions.Source: Centrality Measures · p. 3

C. DMD-Based Measures for Damage Detection

The section defines DMD-based spectral and modal measures for comparing damaged beam conditions with the healthy reference. The measures capture pole-distribution shape, spectral displacement, and matched-mode preservation.

C. DMD-Based Measures for Damage Detection

  • Five damage-sensitive measures comprise eigenspectrum eccentricity, Wasserstein, Chamfer, and Hausdorff distances, and matched-mode cosine similarity.The first four are spectral indicators; cosine similarity is a modal indicator.Source: C. DMD-Based Measures for Damage… · p. 3
  • Healthy-condition pole statistics set fixed scales for the real and imaginary components of continuous-time DMD poles across structural conditions.The pole components represent growth or decay rate and oscillation frequency.Source: C. DMD-Based Measures for Damage… · p. 3
  • Eigenspectrum eccentricity characterizes elongation of the retained DMD poles in the scaled spectral plane.It is defined from the covariance eigenvalues of the scaled spectral matrix.Source: C. DMD-Based Measures for Damage… · p. 4
  • Wasserstein distance uses optimal one-to-one pole assignment, Chamfer distance averages nearest-neighbor discrepancies, and Hausdorff distance captures the largest such discrepancy.These measures characterize complementary aspects of spectral change relative to the healthy reference.Source: Spectral Distances · p. 4
  • Matched-mode cosine similarity tracks modes assigned to the healthy reference using spectral proximity and mode-shape similarity; values nearer 1 indicate modal-structure preservation.Lower values reflect increased modal change.Source: Matched-Mode Cosine Similarity · p. 4

III. DATA AND METHODOLOGY

The study combines finite-element simulation and video-based experiments on a cantilever beam to evaluate graph-theoretic damage diagnosis. Both analyses use delay-embedded DMD, with distinct sampling, embedding, and model-identification settings.

III. DATA AND METHODOLOGY

  • Numerical analysis simulated a 0.8 m cantilever beam at crack depths from 0 to 14 mm in 2 mm increments, using 120 spatial locations and delay-embedded DMD with d = 2 and r = 29.The simulation was performed in-house.Source: §III · p. 4
  • Experiments used 60 fps videos of a 0.8 m polypropylene cantilever beam under healthy and 5, 10, and 13 mm crack conditions, sampled at 120 locations.The beam cross-section was 25.4 × 25.4 mm; transverse displacement came from tracked upper and lower edges.Source: §III · p. 4
  • Experimental delay-embedded DMD used d = 60 and r = 120, with the first 70% of each response for identification and parameter calculation and the remaining 30% for prediction assessment.The videos came from the open dataset of Garrido et al..Source: §III · p. 4

IV. RESULTS AND DISCUSSION

Graph connectivity and other topological measures decline as crack depth increases in both experimental and simulated beam networks. DMD spectral and modal measures also change with damage, complementing the graph-based results.

IV. RESULTS AND DISCUSSION

  • The DMD reconstruction closely matches measured tip displacement, indicating that it captures the beam’s dominant dynamics well.The displacement responses were extracted from video data and modeled using delay-embedded DMD.Source: §IV · p. 4
  • Connectivity declines across crack depths in both cases, from healthy maxima of 2×10−3 experimentally and 0.058 in simulation to minima of 0.6 × 10−3 and 0.054, respectively.Fixed thresholds were τ = 0.02 experimentally and τ = 0.0711 in simulation after network statistics showed consistent trends across the tested threshold ranges.Source: Figure 3 · p. 5, §IV · p. 5
  • Null dyads rise from 7030 to 7120 while asymmetric dyads fall from 110 to 19; 2-out-stars drop from 249 to 2 and 2-in-stars from 689 to 11.These changes indicate fewer nodes serving as common sources or receivers for multiple connections.Source: Figure 3 · p. 5, §IV · p. 5
  • Degree centrality falls from 0.9167 to 0.1750 and closeness centrality from 6.47 × 10−5 to 1.23 × 10−5 as crack depth increases.The decreases reflect losses in direct connections and accessibility through remaining network paths.Source: Figure 3 · p. 5, §IV · p. 5
  • Eigenspectrum eccentricity rises from 0.429 to 0.875, while Wasserstein, Chamfer, and Hausdorff distances rise from zero to 3.48, 3.01, and 5.64 at 13 mm.Wasserstein reflects collective spectral change, Chamfer the average nearest-neighbor distance, and Hausdorff the largest individual pole deviation.Source: Figure 4 · p. 5, §IV · pp. 5–6
  • Matched-mode cosine similarity decreases from 1.0 for the healthy reference to 0.484 at 13 mm crack, indicating increasing deviation of identified modes from the healthy condition.The reduction is relatively small at 5 and 10 mm and pronounced at 13 mm.Source: §IV · p. 6

V. CONCLUSIONS

The study uses delay-embedded DMD and graph measures to identify damage-related changes in vision-based beam measurements. Graph connectivity, centrality, and two-star statistics decrease with damage, while null dyads increase; numerical analysis also validates the connectivity trend observed experimentally.

V. CONCLUSIONS

  • The framework uses delay-embedded DMD to characterize structural damage from vision-based measurements.Graph-theoretic measures are derived from DMD mode matrices.Source: §V · p. 6
  • Increasing damage is associated with lower graph connectivity, centrality, and two-star statistics, alongside increasing null dyads.The numerical analysis further validates the decreasing connectivity trend observed experimentally.Source: §V · p. 6
  • DMD-based measures show damage-dependent changes in the eigenspectrum and modes, complementing the graph-theoretic assessment.The paper concludes that the graph measures provide indicators for identifying structural damage from vision-based measurements.Source: §V · p. 6
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