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§ Tool 04 — Operational Modal Analysis

FDD Explorer

An output-only modal identification sandbox using Frequency Domain Decomposition. Simulate a shear-building under ambient white-noise excitation (or upload multi-channel vibration data), estimate the cross power spectral density matrix by Welch's method, and take its singular value decomposition at every frequency line. Peaks of the first singular value give the natural frequencies; the leading singular vector gives the mode shape.

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Data

A lumped-mass shear building (uniform floor mass, uniform inter-story stiffness) is driven by independent Gaussian white-noise forces at every floor. The response is built by modal superposition with light damping, then contaminated with measurement noise. FDD never sees the modes directly: it recovers them from the output channels alone.
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Welch / FDD settings

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Channel time histories

First seconds of each output channel, normalized and vertically offset for readability. These output-only signals are the entire input to FDD: no excitation measurement is used.
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Singular value spectrum

Singular values of the cross power spectral density matrix versus frequency, in decibels relative to the maximum of SV1. Peaks of SV1 (markers, dashed lines) are the identified natural frequencies. For synthetic data the true frequencies are shown as dotted vertical lines.
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Mode shapes

Mode shape at the selected peak: the first singular vector of the CPSD matrix at that frequency line, with the base fixed at zero. For synthetic data the true mass-normalized mode shape is overlaid (sign and scale aligned).
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Modal Assurance Criterion

MAC between each identified mode and each true mode (synthetic data only). A near-unity diagonal means the identified shapes match the true shapes; off-diagonal values near zero indicate well-separated, cleanly identified modes.
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Method

Frequency Domain Decomposition (Brincker, Zhang and Andersen, 2001) is an output-only method: it identifies modal properties from measured responses alone, with no knowledge of the ambient excitation, assumed to be broadband and close to white noise.

From the n output channels, the n × n cross power spectral density matrix Gyy(f) is estimated by Welch's method: the records are split into overlapping segments, each segment is windowed (Hann) and Fourier transformed, and the cross-periodograms Yi(f) Yj(f)* are averaged over all segments. The diagonal holds the auto-spectra and the off-diagonal the cross-spectra, so Gyy(f) is Hermitian and positive semi-definite at every frequency line.

At each frequency the matrix is decomposed, Gyy(f) = U S UH. Because Gyy is Hermitian positive semi-definite, this singular value decomposition equals its eigendecomposition: the singular values are the eigenvalues and the singular vectors are the eigenvectors. Here the Hermitian eigenproblem is solved by embedding A + iB into the real symmetric matrix [[A, -B], [B, A]] and running a cyclic Jacobi rotation, whose eigenvalues come in identical pairs.

Near an isolated, lightly damped mode the response is dominated by that single mode, so Gyy(f) is nearly rank one: SV1 peaks at the natural frequency and the first singular vector estimates the mode shape. Detected and true mode shapes are compared by the Modal Assurance Criterion, MAC = |φaHφb|2 / ((φaHφa)(φbHφb)).

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