4. Correlation metrics between test and analysis

Once a modal test has produced a set of experimental natural frequencies and mode shapes, and a finite element model has produced a corresponding set of analytical modes, the immediate engineering question is: how well do they agree? Answering that question requires quantitative correlation metrics. This chapter covers the three most widely used families of metrics in structural dynamics: the Modal Assurance Criterion (MAC), the Frequency Response Assurance Criterion (FRAC), and frequency response correlation methods based on FRF magnitude and phase. Each metric answers a slightly different question, and using them together gives a far more reliable picture of model fidelity than any single number.

Why Correlation Metrics Matter

A frequency comparison alone is insufficient. Two modes can have nearly identical natural frequencies yet completely different mode shapes — a situation that arises frequently in symmetric structures, where two orthogonal bending modes may share the same frequency. Conversely, two mode shapes can be visually similar while their frequencies differ by several percent due to stiffness or mass errors. Correlation metrics formalize these comparisons so that the pairing between test and analysis modes is objective, repeatable, and independent of visual inspection.

The correlation workflow typically proceeds in three stages: (1) pair each experimental mode with its most likely analytical counterpart, (2) quantify the shape agreement using MAC or a related criterion, and (3) validate the pairing and the frequency agreement using FRF-based metrics that operate in the frequency domain without requiring mode extraction.

The Modal Assurance Criterion (MAC)

The MAC is the most widely used mode shape correlation metric. For an experimental mode shape vector {φ}X and an analytical mode shape vector {φ}A, both sampled at the same set of measurement degrees of freedom, the MAC is defined as:

MAC({φ}_X, {φ}_A) = |{φ}_X^H {φ}_A|^2 / ( ({φ}_X^H {φ}_X) ({φ}_A^H {φ}_A) )

The result is a scalar between 0 and 1. A value of 1 indicates perfectly correlated shapes (identical up to a complex scale factor), while 0 indicates orthogonality. In practice, MAC values above 0.90 are considered excellent, 0.70–0.90 acceptable for many industrial applications, and below 0.50 indicative of a poor or incorrect pairing.

When multiple experimental and analytical modes are available, the MAC is computed for every test–analysis pair and assembled into a MAC matrix. The diagonal entries represent the correlation of paired modes; off-diagonal entries reveal cross-correlation between different modes. A well-correlated model produces a MAC matrix with diagonal values near 1 and off-diagonal values near 0. Large off-diagonal values are a warning sign: they indicate that the test and analysis mode ordering may be inconsistent, or that the structure exhibits closely spaced modes that are difficult to separate.

The Frequency Response Assurance Criterion (FRAC)

While MAC operates on extracted mode shapes, FRAC operates directly on measured and computed frequency response functions. For a given excitation–response pair, let HX(ω) be the measured FRF and HA(ω) the analytical FRF, both evaluated over the same frequency band. The FRAC is defined as:

FRAC(ω) = |H_X^H(ω) H_A(ω)|^2 / ( (H_X^H(ω) H_X(ω)) (H_A^H(ω) H_A(ω)) )

FRAC is a frequency-dependent quantity, typically plotted as a curve over the analysis band. Values near 1 indicate that the measured and analytical FRFs have the same shape (magnitude and phase relationship) at that frequency, even if their absolute amplitudes differ. This is a crucial distinction: FRAC is insensitive to overall scaling, making it useful when the absolute calibration of the test or the damping model in the analysis is uncertain.

FRAC is particularly valuable because it does not require modal parameter extraction. It can be applied directly to raw FRF data, which means it can be used even when modes are heavily damped or closely spaced and curve-fitting is unreliable. However, FRAC is also sensitive to noise in the measured FRF, so averaging and smoothing are typically applied before computing it.

Frequency Response Correlation Beyond FRAC

FRAC alone does not capture amplitude differences. For many engineering purposes, the absolute level of the FRF matters — for example, when predicting dynamic stress or acoustic radiation. In these cases, engineers supplement FRAC with:

• Magnitude ratio: |HA(ω)| / |HX(ω)|, plotted in dB. A ratio near 0 dB indicates good amplitude agreement.
• Phase difference: ∠HA(ω) − ∠HX(ω), plotted in degrees. Phase agreement is critical near resonances and anti-resonances.
• Coherence function: γ2(ω), which indicates the quality of the measured FRF itself. Low coherence (below 0.8) suggests noise, nonlinearity, or leakage, and any correlation metric computed from that FRF should be treated with caution.

A common practice is to overlay the measured and analytical FRF magnitude plots and compute FRAC, magnitude ratio, and phase difference on the same frequency axis. This gives a complete picture: FRAC tells you whether the shapes match, the magnitude ratio tells you whether the amplitudes match, and the phase difference tells you whether the resonances and anti-resonances are correctly positioned.

Interpreting Correlation Results: A Worked Example

Consider a simple cantilever beam modeled with 10 beam elements and tested with 8 accelerometers. The first three bending modes are extracted from both test and analysis. The MAC matrix is computed as follows (values are illustrative and represent a typical well-correlated model):

The diagonal values are all above 0.90, indicating strong shape correlation. The off-diagonal values are small, confirming that the mode pairing is correct. If, however, the MAC between Test Mode 2 and Analysis Mode 3 were 0.85 while the diagonal for Test Mode 2 were only 0.60, this would suggest a mispairing or a mode ordering issue — possibly due to a local stiffness error that shifts one mode more than another.

Now consider the FRAC for the same beam. Suppose the FRAC curve is near 0.95 across the first resonance but drops to 0.60 near the second resonance. This indicates that the second mode's frequency response shape is not well captured by the model, even though the MAC for that mode is 0.95. The discrepancy could be due to incorrect damping modeling, a boundary condition error, or a local mass/stiffness mismatch that affects the FRF more than the mode shape. This is exactly why MAC and FRAC should be used together: MAC validates the mode shape, FRAC validates the frequency response, and the two can disagree.

Choosing the Right Metric

The choice of correlation metric depends on the goal of the model updating exercise:

• Mode shape validation: Use MAC. It is the standard for pairing modes and detecting shape errors.
• FRF shape validation without modal extraction: Use FRAC. It works directly on measured data and is robust to damping and scaling uncertainties.
• Amplitude and phase validation: Use magnitude ratio and phase difference in addition to FRAC.
• Global model validation: Combine MAC, FRAC, and frequency error (e.g., (fA − fX) / fX) into a single dashboard. No single metric is sufficient.

Visual Summary of the Correlation Workflow

The following diagram illustrates how test and analysis data flow into the correlation metrics and how the results are interpreted. It shows the parallel paths of modal-based correlation (MAC) and FRF-based correlation (FRAC), and how they converge into a combined assessment.

The diagram shows that MAC and FRAC are complementary: MAC operates on extracted modal data, while FRAC operates on raw FRFs. Both feed into a combined assessment that also considers frequency error, magnitude ratio, phase difference, and coherence. In practice, an engineer should compute all of these metrics and examine them together before drawing conclusions about model quality.

Practical Guidelines for Reporting Correlation Results

When documenting correlation results for a model updating project, include the following:

• The MAC matrix with all test–analysis pairs, not just the diagonal.
• The FRAC curve over the full frequency band of interest, with the frequency axis clearly labeled.
• The frequency error for each paired mode: Δf / fX in percent.
• The number of measurement DOFs and their locations, since MAC depends on spatial sampling.
• The coherence function for each measured FRF, to demonstrate data quality.
• A clear statement of which modes are considered well-correlated and which are not, with thresholds justified for the specific application.

These metrics form the quantitative foundation for the model updating process. Without them, updating becomes a subjective exercise; with them, it becomes a traceable engineering activity where every adjustment to the FE model can be evaluated against objective criteria.