Modal Assurance Criterion

The modal assurance criterion is a dimensionless measure of how well two mode-shape vectors match. It is used in experimental modal analysis, finite-element correlation, rotating-machinery review and ground vibration test work to decide whether a measured mode and a predicted mode represent the same physical deformation pattern.

MAC is useful because a mode shape normally has arbitrary sign and scale. A test mode may be normalized to one accelerometer channel, while a finite-element mode may be mass-normalized or scaled for plotting. The modal assurance criterion removes that arbitrary scale and compares the direction of the vectors instead. It does not, by itself, prove that a model has the correct natural frequency, damping, mass, stiffness, boundary conditions or aeroelastic margin.

For real-valued mode-shape vectors \phi_a and \phi_b, the usual definition is:

For complex mode shapes, the transpose is replaced by the conjugate transpose:

The result lies between 0 and 1. A value close to 1 means the vectors are nearly collinear at the compared degrees of freedom. A value close to 0 means the measured and predicted shapes are poorly correlated at those degrees of freedom.

Engineering Role

MAC helps engineers pair modes, reject false matches and judge whether a model is credible for vibration or aeroelastic decisions. Frequency agreement alone is not enough. Two modes can have similar frequencies but different shapes, especially in flexible aircraft, coupled wing-store systems, control-surface modes, rotating machinery, test rigs or structures with repeated components.

The metric is insensitive to scale and sign. If one mode shape is multiplied by a constant, or by -1, the MAC value remains the same:

That invariance is useful, but it also hides some problems. The engineer still has to confirm that the compared degrees of freedom, coordinate directions, sensor locations, component choices and normalization conventions are compatible before computing the value. A high MAC calculated on the wrong coordinates is not evidence of a good model.

MAC Matrix and Mode Pairing

A single MAC value compares one pair of modes. In a real modal-correlation review, engineers usually form a matrix between test modes and model modes:

where i indexes measured modes and j indexes finite-element or analytical modes. A clean pairing matrix has high values near the intended pairings and low off-diagonal values. A matrix with several high values in the same row or column indicates repeated shapes, insufficient sensor coverage, coupled modes, coordinate mistakes or a model that cannot clearly separate the physical modes.

One practical separation check is:

This difference is not a universal certification threshold. It is a review aid. A large separation means the selected match is distinct from the alternatives at the measured coordinates. A small separation means the pairing decision is weak even when the best MAC is high.

Worked Example: Pairing a GVT Mode with a Model Mode

A ground vibration test identifies a torsion-like mode from four accelerometer channels. Two finite element modes have nearby frequencies, so the team uses MAC to decide which one matches the measured shape.

Assume all four channels have already been transformed to the same coordinate convention and that the same physical locations are present in the test vector and model vector. Without that preprocessing, the arithmetic below would not be meaningful.

Use the measured vector:

and the first candidate model vector:

The dot product is:

The squared norms are:

Therefore:

Now compare a second candidate:

The dot product is:

The second candidate squared norm is:

So:

If the release criterion requires MAC\geq0.90 for mode pairing, the first candidate is the correct match and the second is not. The pair separation against this alternative is:

Engineering comment: this result does not prove that the finite element model is fully validated. It only says that one measured shape and one predicted shape are strongly correlated at the selected degrees of freedom. The reviewer must still check frequency error, damping, sensor placement, coordinate signs, boundary conditions, mass properties, excitation quality, uncertainty and whether the mode is relevant to the intended decision.

Acceptance Evidence and Limits

MAC should be reported with the evidence that makes the comparison credible. A useful modal-correlation note states the sensor set, coordinate system, units or normalization convention, omitted degrees of freedom, frequency error, damping estimate, coherence or signal-quality checks, excitation method and the acceptance rule used for the review.

For example, a GVT-to-FEA correlation package might require:

and also require:

Those two criteria check different things. MAC checks the deformation pattern at the retained coordinates. The frequency-error criterion checks whether the dynamic stiffness and mass distribution are close enough for the intended use. A mode can pass one criterion and fail the other.

Low MAC is not automatically a bad model. It can result from missing sensors at important antinodes, weak signal-to-noise ratio, poor coherence, unmeasured rotational degrees of freedom, sensor mass loading, swapped signs, wrong coordinate transforms, inconsistent boundary conditions or comparing a test operating-deflection shape with a true normal mode. The correct response is to diagnose the measurement and coordinate chain before tuning the model.

Common Mistakes

The most serious mistake is treating MAC as a validation certificate. A high value supports a mode-pairing decision, not a full release decision. Flutter clearance, fatigue assessment, vibration-isolation design and structural model updating still need frequency, damping, mass-property, load-path, uncertainty and configuration evidence.

Another mistake is computing MAC after visually adjusting signs, removing inconvenient channels or comparing only a few coordinates that make the answer look good. The retained coordinates should be justified before seeing the result. If channels are removed because of noise or sensor failure, that exclusion belongs in the error budget or validation note.

Engineers should also avoid comparing shapes from incompatible physics. Operational deflection shapes, frequency response functions, finite-element eigenvectors and test mode shapes can all look similar on a plot, but they do not always represent the same quantity. MAC is meaningful only when the vectors represent comparable modal coordinates.

Distinction from Related Terms

Modal assurance criterion is not mean aerodynamic chord, even though both may be abbreviated MAC. Mean aerodynamic chord is an aircraft reference length used for aerodynamic moments, CG position and static margin. Modal assurance criterion is a dimensionless vector-correlation metric used in structural dynamics.

Modal assurance criterion is not natural frequency. Natural frequency is a modal frequency. MAC compares mode-shape vectors.

Modal assurance criterion is not damping ratio. Damping ratio measures decay. MAC measures shape correlation.

Modal assurance criterion is not modal analysis. Modal analysis is the broader activity of identifying and interpreting modes. MAC is one numerical check inside that workflow.

Modal assurance criterion is not proof of flutter margin. It supports model correlation, but flutter clearance still needs aerodynamic data, mass-property control, damping trends, configuration evidence and uncertainty margins.