Glossary term

Frequency-Domain Decomposition

Output-only modal identification method that estimates structural modes from singular values and vectors of response spectral density matrices.

Branch: Mechanical Engineering
Glossary type: method
Content: Glossary term
Updated: Jun 24, 2026
Revision: v1.0.0 · reviewed

Definition

method

Frequency-domain decomposition is an output-only modal identification method that uses the singular values and singular vectors of response spectral density matrices to estimate modal frequencies and relative mode shapes.

In structural dynamics, frequency-domain decomposition, often abbreviated FDD, decomposes the measured response spectral density matrix at each frequency line. When one lightly damped mode dominates a narrow band, the largest singular value indicates the modal peak and the corresponding singular vector approximates the relative mode shape. FDD is widely used in operational modal analysis because it does not require measured input force.

Frequency-domain decomposition (FDD) identifies structural modes from output-only response data by decomposing the response spectral density matrix at each frequency. It is a frequency-domain method used in operational modal analysis when input forces are unknown or impractical to measure.

For a response vector $\mathbf{y}(t)$ measured at several locations, estimate the output spectral density matrix:

\mathbf{S}_{yy}(f)

At each frequency line, FDD applies a singular value decomposition:

\mathbf{S}_{yy}(f)=\mathbf{U}(f)\boldsymbol{\Sigma}(f)\mathbf{U}^{H}(f)

where $\boldsymbol{\Sigma}(f)$ contains singular values and $\mathbf{U}(f)$ contains singular vectors. For a well-separated mode, the first singular value $\sigma_1(f)$ has a peak near the modal frequency, and the first singular vector approximates the relative mode shape.

Engineering Role

FDD is useful when a structure can be instrumented with several response sensors but cannot be excited with a measured force. Common applications include bridges under traffic, floors under occupancy, towers under wind, offshore structures under waves, ship structures under service excitation and large machinery foundations during operation.

The method gives engineers a visual and numerical way to separate structural modes from multi-channel response data. Instead of reviewing many single-channel spectra independently, the engineer reviews the singular-value spectrum and the spatial consistency of the singular vectors.

Typical FDD outputs include:

candidate modal frequencies from singular-value peaks;
relative mode shapes from dominant singular vectors;
singular-value separation between dominant and secondary response patterns;
mode-shape consistency checks across frequency bins;
input for modal assurance criterion comparisons with finite-element modes or previous tests.

FDD is most defensible for lightly damped, approximately linear structures with adequate sensor coverage, broad enough excitation and enough record length to resolve close modes.

Worked Example: Interpret a Singular-Value Peak

An engineer records ambient vibration from four accelerometers on a pedestrian bridge for:

T=900\ \text{s}

The frequency resolution of one full record is approximately:

\displaystyle \Delta f=\frac{1}{T}=\frac{1}{900}=0.00111\ \text{Hz}

At a candidate modal peak near:

f=3.20\ \text{Hz}

the singular values of the acceleration spectral density matrix are:

\sigma_1=1.80\times10^{-4}

\sigma_2=2.50\times10^{-5}

\sigma_3=1.10\times10^{-5}

\sigma_4=0.80\times10^{-5}

The dominance ratio of the first singular value over the second is:

\displaystyle R=\frac{\sigma_1}{\sigma_2}=\frac{1.80\times10^{-4}}{2.50\times10^{-5}}=7.2

In decibels:

20\log_{10}(7.2)=17.1\ \text{dB}

Engineering comment: a 17.1 dB separation suggests that one response pattern dominates this frequency line, so the first singular vector is a plausible mode-shape estimate. It is not final proof. The engineer still checks neighbouring frequency lines, repeatability across records, sensor layout, harmonic forcing and whether the same shape correlates with a finite-element or baseline mode.

Suppose the first singular vector at this frequency is approximately:

\mathbf{u}_1=\begin{bmatrix}0.18 & -0.52 & -0.68 & -0.48\end{bmatrix}^{T}

For reporting a relative mode shape, normalize by the largest absolute component:

\displaystyle \boldsymbol{\phi}\approx\frac{\mathbf{u}_1}{0.68}=\begin{bmatrix}0.26 & -0.76 & -1.00 & -0.71\end{bmatrix}^{T}

The sign is arbitrary. Multiplying every component by $-1$ gives the same physical mode shape. The result also remains relative because output-only FDD does not produce a force-normalized or mass-normalized mode unless additional information is used.

Frequency-domain decomposition is not operational modal analysis as a whole. OMA is the broader output-only identification discipline; FDD is one frequency-domain method inside it.

Frequency-domain decomposition is not a frequency response function. An FRF requires measured input and output. FDD uses output spectral density matrices when input forces are unknown.

Frequency-domain decomposition is not magnitude-squared coherence. Coherence is a two-signal quality metric. FDD is a multi-channel modal identification method based on matrix decomposition.

Frequency-domain decomposition is not modal assurance criterion. MAC compares mode-shape vectors. FDD may produce candidate mode-shape vectors that are later compared with MAC.

Frequency-domain decomposition is not simple peak picking. Peak picking may select spectral peaks from one channel. FDD uses the spatial structure of multi-channel response data and singular vectors to support modal interpretation.

Validation and Common Mistakes

A defensible FDD result states sensor locations, response quantity, calibration, sampling rate, anti-alias filtering, record length, windowing, averaging, frequency resolution, singular-value plots, selected frequency bins, mode-shape normalization, environmental conditions, operating conditions and uncertainty.

Common mistakes include:

accepting a singular-value peak without checking whether the singular vector is spatially plausible;
confusing a harmonic operating order with a structural mode;
using too few sensors to distinguish bending, torsion or local modes;
over-interpreting damping from basic FDD without an appropriate enhanced FDD or curve-fitting step;
comparing FDD mode shapes with finite-element modes in mismatched coordinates;
ignoring boundary-condition, temperature, traffic, machinery or wave-state changes;
assuming that output-only FDD can recover absolute force-normalized response functions.

REF

Disciplines

Frequency-Domain Decomposition

Definition

Engineering Role

Worked Example: Interpret a Singular-Value Peak

Validation and Common Mistakes

See also

Engineering Role

Worked Example: Interpret a Singular-Value Peak

Distinction from Related Terms

Validation and Common Mistakes