A vibrating structure — a beam, a bridge, an aircraft wing — rings at its natural frequencies. Modal analysis extracts these frequencies and their associated mode shapes from measured vibration data. The traditional approach: excite the structure with a known input (hammer strike, shaker), measure the response, compute transfer functions, fit the modes.
Dynamic Mode Decomposition (DMD) offers a data-driven alternative. Given time-series measurements of the vibration at multiple points, DMD finds the best-fit linear operator that advances the state from one time step to the next. The eigenvalues of this operator give the natural frequencies and damping ratios; the eigenvectors give the mode shapes. No input signal needed — the method works from ambient vibration or free decay alone.
The connection to classical methods is exact: DMD applied to free-decay vibration data is equivalent to the Ibrahim time-domain method, a standard technique in experimental modal analysis. The equivalence means DMD isn't a new method for this application — it's an independent derivation of a known method from a different mathematical tradition (Koopman operator theory vs. structural dynamics).
The sensitivity analysis reveals the practical limits: small measurement errors (noise, digitization, sensor misalignment) allow accurate modal parameter extraction, but larger errors degrade performance rapidly. The degradation is mode-dependent — closely spaced frequencies and lightly damped modes are more sensitive to noise than well-separated, heavily damped ones.
Old wine in new bottles, but the bottle matters. The DMD framework connects structural vibrations to Koopman operators, opening tools from dynamical systems theory to engineers testing bridges.