⚡ electrical engineering

Vision-Based Structural Damage Identification in Vibrating Beams via Dynamic Mode Decomposition

This study proposes and validates a non-contact, data-driven framework using Dynamic Mode Decomposition (DMD) on high-speed video data to extract modal features and identify structural damage in vibrating beams through both numerical simulations and experimental tests.

Original authors: R K B M Rizmi, Shabbir Ahmed

Published 2026-05-05

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: R K B M Rizmi, Shabbir Ahmed

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a long, flexible ruler clamped to a table. If you flick the free end, it wiggles back and forth. This is a vibrating beam. Engineers need to know if this ruler is healthy or if it has a hidden crack, because a crack changes how it wiggles.

Traditionally, to check for cracks, engineers would glue tiny sensors (like sticky notes with wires) all over the ruler. This is messy, expensive, and only checks the spots where the sensors are. If a crack is between sensors, they might miss it.

This paper proposes a smarter way: watch the ruler with a high-speed camera.

Here is the simple breakdown of what the researchers did and found:

1. The "Magic Eye" (The Camera)

Instead of sticking sensors on the beam, they filmed it vibrating with a super-fast camera (1,000 frames per second).

The Analogy: Imagine looking at a crowd of people waving their hands. Instead of asking every single person to hold a sign, you just watch the whole crowd move. The camera captures the movement of every single pixel on the screen, creating a "full-field" view of the vibration.

2. The "Pattern Finder" (Dynamic Mode Decomposition - DMD)

The video creates a massive amount of data (millions of pixels changing over time). Humans can't process that. The researchers used a mathematical tool called Dynamic Mode Decomposition (DMD).

The Analogy: Think of a song. A song is just a bunch of sound waves mixed together. DMD is like a smart music app that listens to the song and says, "Okay, I hear a bass note at 50Hz, a guitar note at 200Hz, and a drum beat."
In this paper, DMD looks at the video and says, "I see the beam bending in this specific shape (Mode 1), and it's wobbling at this specific speed." It breaks the complex video down into simple, understandable "shapes" and "speeds."

3. The "Damage Detective" (Finding the Crack)

When a beam is healthy, it bends in smooth, predictable curves. When it has a crack, that part gets softer, and the curve gets weird or "kinked" right at the damage spot.

The Analogy: Imagine a smooth, rolling hill. If someone digs a hole in the middle of the hill, the shape changes. DMD finds these "holes" in the shape.
The researchers created a "Damage Index" (a score).
- Healthy Beam: The score is zero. The shapes look perfect.
- Damaged Beam: The score goes up. The math detects that the "hill" is no longer smooth.

4. The Experiment: Computer vs. Reality

They tested this in two ways:

Computer Simulation: They built a virtual beam in a computer program (ANSYS) and "broke" it virtually. The camera (simulated) filmed it, and DMD found the damage.
Real Life: They took a real plastic beam, clamped it, and filmed it with a high-speed camera. They cut tiny notches (cracks) into it.
- The Result: In both cases, the method worked. It could tell the difference between a healthy beam and a damaged one just by watching the video. It even noticed that a bigger crack caused a bigger change in the "Damage Index" score.

5. What They Found (The Takeaway)

It works without touching: You don't need to glue sensors to the beam. Just film it.
It sees the invisible: The math can spot tiny changes in how the beam bends that a human eye might miss in the video.
It's fast and clean: They didn't need to do heavy, messy calculations to figure out how much the beam moved; they just fed the raw video into the DMD tool.

What They Didn't Do (Important Limitations)

The paper is very clear about what this tool can and cannot do right now:

It can tell you that there is damage. (The score went up).
It can tell you how bad the damage is. (The score went up more for bigger cracks).
It CANNOT tell you exactly where the damage is yet. The authors admit this is a limitation. They know the beam is broken, but they haven't perfected the math to point a finger at the exact spot of the crack yet. That is what they plan to work on next.

In summary: This paper shows that you can use a high-speed camera and a smart math tool to "listen" to the vibrations of a structure and tell if it's sick, without ever touching it. It's like diagnosing a patient just by watching them walk, rather than sticking needles in them.

Technical Summary: Vision-Based Structural Damage Identification in Vibrating Beams via Dynamic Mode Decomposition

Problem Statement
Structural health monitoring (SHM) traditionally relies on contact sensors (e.g., accelerometers, strain gauges) which are intrusive, costly to install, and offer limited spatial coverage. While non-contact methods like laser vibrometry exist, they often remain point-wise or expensive for full-field analysis. Video-based monitoring offers a scalable, non-contact alternative; however, extracting physically meaningful dynamic information from high-dimensional visual data remains challenging. Existing approaches often require intensive preprocessing to reconstruct displacement fields or rely on specialized variants of Dynamic Mode Decomposition (DMD) with multi-stage post-processing to isolate damage signatures. This study addresses the need for a streamlined, data-driven framework that extracts diagnostic features directly from raw video data or simulation snapshots without extensive heuristic preprocessing.

Methodology
The proposed framework utilizes Dynamic Mode Decomposition (DMD) to approximate the underlying linear dynamics of a vibrating cantilever beam directly from spatiotemporal data. The methodology is divided into two parallel investigations: a numerical simulation using ANSYS and an experimental study using high-speed video recordings.

Data Acquisition and Preprocessing:
- Numerical: Transient finite element analyses generate displacement fields for a polypropylene cantilever beam under healthy and two damaged conditions (cracks at varying depths and locations).
- Experimental: High-speed video (1000 fps) captures the free vibration of a similar beam. The video frames are processed into spatiotemporal snapshot matrices.
- State Representation: The study constructs data matrices $X$ and $X'$ from sequential snapshots. While delay embedding (Hankel matrices) was investigated to enrich state representation, the final experimental analysis utilized a truncation rank of 150 without delay embedding to balance fidelity and complexity.
Dynamic Mode Decomposition (DMD):
- DMD identifies a best-fit linear operator $A$ that advances the system state in time ( $X' \approx AX$ ).
- Through Singular Value Decomposition (SVD) and projection onto a reduced subspace, the algorithm extracts DMD eigenvalues (characterizing temporal growth/decay and frequency) and DMD modes (spatial coherent structures).
- These modes serve as a substitute for classical experimental mode shapes.
Damage Detection Algorithm:
- Feature Extraction: From each DMD mode profile $\phi_k(x)$ , four spatial feature vectors are derived: Mode Shape (MS), Mode Shape Slope (MSS), Mode Shape Curvature (MSC), and Mode Shape Curvature Square (MSCS).
- Baseline Construction: A reference matrix $P$ is constructed from the healthy state using a variance-weighted outer-product accumulation of the healthy feature vectors.
- Damage Index ( $Q$ ): A novel damage index is formulated based on the Mahalanobis-type quadratic form. It measures the deviation of the current (potentially damaged) feature vector from the healthy reference, weighted by the inverse of the reference covariance structure. A global index $Q$ is obtained by averaging over selected modes.

Key Results

Reconstruction and Prediction: DMD accurately reconstructed and predicted the tip displacement for both healthy and damaged configurations. The Root-Mean-Square Error (RMSE) was lowest for the healthy case ( $1.12 \times 10^{-3}$ ) and slightly higher for damaged cases ( $\approx 2.1 \times 10^{-3}$ ), indicating that damage introduces dynamical complexity that is slightly more challenging for the linear DMD framework to capture, though errors remain low.
Modal Analysis: Extracted DMD modes showed smooth, physically consistent shapes for the healthy beam. Damaged configurations exhibited subtle but systematic deviations in spatial structure, particularly in localized curvature, reflecting changes in stiffness distribution.
Eigenvalue Analysis: Damaged configurations exhibited an inward shift of eigenvalues toward the origin in the complex plane compared to the healthy state. This indicates increased damping or faster attenuation of dynamics, suggesting enhanced energy dissipation due to structural degradation.
Damage Index Performance:
- The proposed index $Q$ successfully distinguished between healthy and damaged states, with $Q=0$ for the healthy baseline and significantly higher values for damaged cases.
- The index showed consistent sensitivity to damage severity, with values increasing from Damage Case 1 to Damage Case 2.
- Among the features, the MSCS (Mode Shape Curvature Square) based index demonstrated the highest sensitivity to damage severity, showing the largest proportional change between damage levels. This is attributed to the amplification of curvature variations when squared.

Significance and Claims
The paper claims that DMD provides a robust, interpretable, and computationally efficient tool for non-contact damage detection using video data. Key contributions include:

Direct Feature Extraction: The ability to extract damage-sensitive signatures directly from raw modal manifolds (DMD modes) without the need for explicit displacement reconstruction or extensive preprocessing.
Novel Damage Index: The formulation of a characteristic quantity and damage index ( $Q$ ) derived exclusively from DMD-extracted mode shapes and eigenvalues, which facilitates efficient damage detection and identification.
Interpretability: The framework offers a physically interpretable representation of system response, linking visual data to modal properties (frequencies, damping, and spatial shapes) that are known to be sensitive to stiffness degradation.
Validation: The methodology is validated across both high-fidelity simulations and real-world experimental video data, demonstrating consistent patterns between healthy and damaged states.

Limitations and Future Work
The authors note that the current framework is limited to damage detection and the quantification of damage severity; it does not currently perform damage localization. Future work aims to extend the framework to enable localization and to validate the methodology on more complex, higher-dimensional structural systems, including those exhibiting nonlinear behavior.