Data-driven Experimental Modal Analysis by Dynamic Mode Decomposition

This paper demonstrates that Dynamic Mode Decomposition (DMD) is an effective data-driven method for extracting accurate modal parameters from linear mechanical systems, provided measurement errors are kept relatively small, as validated through both theoretical analysis and experimental application to a cantilevered beam.

The Gist

Imagine you have a giant, complex machine, like a bridge or a skyscraper, and you want to know how it "sings." Every structure has a unique voice—a set of specific notes (frequencies) it likes to hum at, and a specific way it fades out after being hit (damping). Engineers call this Experimental Modal Analysis (EMA). It's like trying to figure out the recipe of a cake just by tasting it, but instead of a cake, it's a vibrating beam.

Traditionally, engineers have used two main ways to listen to this "song":

1. The Frequency Method (LSCF): Like using a high-tech microphone to record the sound and then using a computer to find the exact notes in the recording.
2. The Time Method (ITD): Like watching a slow-motion video of the vibration and guessing the pattern based on how the movement changes second by second.

This paper introduces a new, data-driven detective named Dynamic Mode Decomposition (DMD). Think of DMD as a super-smart AI that watches a video of a vibrating object and instantly figures out its "musical notes" and how quickly they die out, without needing to know the physics equations beforehand.

Here is the breakdown of what the researchers did and found, using some everyday analogies:

1. The New Detective (DMD) vs. The Old Detective (ITD)

The authors explain that DMD is actually a cousin to an old method called the Ibrahim Time Domain (ITD) method.

• The Analogy: Imagine you are trying to predict the next move of a dancer.
  • ITD looks at the dancer's current pose and guesses the next one based on a rigid rulebook. If the video is a little grainy (noisy), the guess gets messy.
  • DMD is like a dance instructor who watches the whole routine, ignores the tiny, shaky movements (noise), and focuses only on the big, smooth, repeating patterns. It uses a mathematical trick (called Singular Value Decomposition) to filter out the "static" and keep only the clear "signal."

2. The Test Drive: Simple Springs

First, the researchers tested DMD on simple, computer-generated springs (like a single weight bouncing on a spring, or a chain of six weights).

• The Result: When the data was perfect (no noise), DMD was incredibly accurate. It found the exact "notes" and how fast they stopped. It was just as good as the best traditional methods.
• The Catch: When they added "noise" (simulating a shaky camera or a bad sensor), DMD started to struggle.
  • The Analogy: If you are trying to hear a whisper in a quiet room, DMD works great. But if you try to hear that same whisper in a rock concert (high noise), DMD gets confused and starts hearing fake notes. Specifically, it got the "volume" of the fade-out (damping) wrong when the noise was too loud.

3. The Real-World Test: The Wobbly Beam

To see if DMD works in the real world, they took a plastic beam, clamped it to a table, and hit it with a hammer. They didn't use sensors glued to the beam; instead, they used a high-speed camera to film the whole thing.

• The Challenge: The camera footage had "pixel noise" (like digital grain).
• The Solution: They used a "filter" (Singular Value Rejection) to throw away the tiny, meaningless pixels and keep only the big, clear movements.
• The Outcome:
  • Pitch (Frequency): DMD found the notes perfectly! It matched the traditional microphone method (LSCF) and even a computer simulation (FEM) almost exactly.
  • Fade-out (Damping): This was the weak spot. DMD struggled to tell exactly how fast the vibration stopped. It was close, but not as precise as the traditional method.

The Big Takeaway

What does this mean for the future?

• The Superpower: DMD is amazing for handling massive amounts of data. Imagine a bridge with 10,000 sensors, or a video of a whole airplane wing vibrating. Traditional methods might choke on that much data, but DMD thrives on it. It can look at a whole video and say, "Here are the 5 main ways this thing vibrates."
• The Weakness: It is very sensitive to "bad data" (noise). If your measurements are shaky, DMD might get the "fade-out" speed wrong.
• The Verdict: DMD is a fantastic new tool for engineers, especially when they have huge datasets from cameras or many sensors. It can find the "shape" and "pitch" of vibrations better than ever before. However, if you need to know the exact speed at which a vibration dies out, you might still need to double-check with traditional methods or find a way to clean up the noise first.

In short: DMD is like a new, high-tech pair of glasses that lets you see the hidden "skeleton" of a vibrating object clearly, even in a crowd of data. But if the room is too dark and noisy, you might still need a flashlight (traditional methods) to get the fine details right.

Technical Summary

1. Problem Statement

Experimental Modal Analysis (EMA) is a critical technology in structural dynamics used to extract modal properties (natural frequencies, damping ratios, and mode shapes) from structures. Traditionally, these properties are extracted using:

• Frequency-domain methods: Such as the Least Squares Complex Frequency (LSCF) method, which are widely used but require input force data and Frequency Response Functions (FRFs).
• Time-domain methods: Such as the Ibrahim Time Domain (ITD) method, which are useful for low-frequency structures but often limited in handling massive datasets.

With the advent of high-speed imaging and dense sensor arrays, there is a growing need for data-driven methods capable of processing massive amounts of spatially distributed time-series data without requiring explicit input force measurements. While Dynamic Mode Decomposition (DMD) has been successful in fluid mechanics for extracting coherent structures and their temporal dynamics, its application to structural dynamics and EMA has been limited. The paper addresses the gap in applying DMD to extract modal parameters from structural vibration data and evaluates its accuracy and robustness compared to established methods.

2. Methodology

The authors propose a framework for applying DMD to linear mechanical systems for EMA. The methodology involves three main stages:

A. Theoretical Foundation

• DMD Formulation: The method treats the structural response as a discrete dynamical system. It constructs snapshot matrices $X$ (current states) and $Y$ (time-shifted states) to find a linear operator $A$ such that $Y \approx AX$.
• Relation to ITD: The paper establishes a mathematical equivalence between DMD and the classical Ibrahim Time Domain (ITD) method. However, DMD improves numerical stability by utilizing Singular Value Decomposition (SVD) to compute the pseudo-inverse, allowing for the rejection of insignificant modes (noise) via singular value truncation.
• Parameter Extraction: DMD eigenvalues ($\mu$) are converted to continuous-time eigenvalues ($s$) to derive natural frequencies ($f_n$) and damping ratios ($\zeta$). DMD modes ($\phi$) correspond to the physical mode shapes.
• Handling Standing Waves: To accurately capture standing wave behaviors typical in structural vibrations, the snapshot matrices are redefined with appended time-shifted snapshots.

B. Numerical Validation

• Single DOF System: A damped oscillator was simulated to verify the basic extraction of frequency and damping.
• Multi-DOF System: A 6-degree-of-freedom (DOF) mass-spring-damper system was analyzed.
  • Noise Sensitivity: Artificial measurement noise (Gaussian) was injected into the time histories to test robustness at different standard deviations ($\sigma$).
  • Comparison: Results were compared against the LSCF method using "pseudo-stability diagrams" (varying sampling frequencies) and "stability diagrams" (varying model orders).

C. Experimental Validation

• Setup: A polypropylene cantilever beam was subjected to an impulse hammer strike.
• Data Acquisition: A high-speed camera (480 fps) recorded the vibration. Image processing (binarization) extracted displacement time histories at 1,570 spatial points along the beam.
• Analysis: DMD was applied to the image-derived displacement field. Results were compared against:
  1. LSCF: Applied to FRFs derived from the same image data (averaged over 5 tests).
  2. FEM: A Finite Element Model (ANSYS) was created to provide a theoretical baseline.
• Noise Mitigation: Singular Value Rejection was applied to the DMD computation to filter out noise-induced eigenvalues.

3. Key Contributions

1. Theoretical Link: Demonstrated the mathematical equivalence between DMD and the ITD method, while highlighting DMD's superior numerical stability due to SVD-based pseudo-inversion.
2. Noise Sensitivity Analysis: Provided a rigorous analysis showing that DMD is highly accurate with clean data or low noise but becomes unstable in estimating damping ratios when measurement noise is significant.
3. Image-Based EMA: Successfully demonstrated the application of DMD to large-scale, image-based displacement fields, proving its capability to handle massive datasets (thousands of spatial points) without requiring input force sensors.
4. Validation: Validated the method against both numerical benchmarks and experimental data, comparing it directly with the industry-standard LSCF method.

4. Key Results

Numerical Examples (6-DOF System)

• Accuracy (No Noise): DMD extracted natural frequencies and damping ratios with negligible error (relative error $< 10^{-12}\%$ for frequency).
• Mode Shapes: DMD mode shapes matched analytic solutions and LSCF results with 100% Modal Assurance Criterion (MAC) values.
• Noise Impact:
  • Low Noise ($\sigma = 10^{-5}$): DMD maintained high accuracy for both frequency and damping.
  • High Noise ($\sigma = 10^{-2}$): DMD failed to capture higher-order natural frequencies and damping ratios accurately. The damping ratio estimation was particularly sensitive to noise, showing significant degradation.
  • Sampling Frequency: Increasing the sampling frequency in noisy conditions actually decreased prediction accuracy for damping, as the error propagation is proportional to the sampling frequency.

Experimental Example (Cantilever Beam)

• Natural Frequencies: DMD (with singular value rejection) identified the first three modes (5.91 Hz, 34.90 Hz, 97.13 Hz) with high accuracy, showing errors of less than 6% compared to LSCF and FEM results.
• Damping Ratios: DMD results for damping ratios showed poor agreement with LSCF (errors up to 48%), confirming the sensitivity to experimental noise.
• Mode Shapes:
  • DMD and LSCF mode shapes both agreed well with the FEM model (MAC > 98% for diagonal elements).
  • LSCF modes appeared slightly smoother due to the averaging of multiple FRF measurements, whereas DMD (based on a single measurement) retained some spatial noise.

5. Significance and Conclusion

The paper establishes Dynamic Mode Decomposition as a viable, powerful tool for Experimental Modal Analysis, particularly for systems where:

• Massive amounts of spatial data are available (e.g., from high-speed cameras or dense sensor arrays).
• Input force data is unavailable or difficult to measure (output-only analysis).
• High accuracy in natural frequencies and mode shapes is required.

Limitations and Future Work:
The study highlights a critical limitation: DMD is highly sensitive to measurement noise when estimating damping ratios. While singular value rejection helps filter noise, the estimation of decay rates (damping) remains challenging in noisy experimental environments compared to frequency-domain methods like LSCF. Future work should focus on developing robust noise-filtering techniques (e.g., Total Least Squares) to improve damping estimation accuracy for DMD-based EMA.

In summary, DMD offers a promising data-driven alternative to traditional EMA, capable of extracting modal parameters directly from raw time-series data with high spatial resolution, provided that noise management strategies are carefully implemented.