Library / Structural dynamics and vibration testing / SDY-36
Structural dynamics and vibration testing
Modal assurance criterion
Correlation between two mode shapes measured at the same coordinates: the squared cosine of the angle between them.
- Ref
- SDY-36
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- MAC
The modal assurance criterion MAC = |aᴴb|²/[(aᴴa)(bᴴb)] is the squared cosine of the angle between two mode shapes measured at the same coordinates. It is 1 for shapes that differ only by a scale factor (including a sign flip) and 0 for orthogonal shapes. It is insensitive to overall scale but cannot tell closely spaced modes apart. This entry uses three real coordinates.
Correlation between two mode shapes measured at the same coordinates: the squared cosine of the angle between them. It is 1 for identical shapes up to scale and 0 for orthogonal ones. This entry uses three real coordinates.
Calculator
- Dot product a·b4
- Length of a, √(a·a)2.236
- Angle between the shapes36.87 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A MAC of 0.64 means the shapes are only partly correlated; values above about 0.9 are normally taken as a good match.
- The angle between them is 36.9°; MAC = cos² of that angle.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| a2 Shape a, coordinate 2 | −3 to 3 | 0 | 0.9941 | 0.9941 | +0.12 | −3.46 % |
| b1 Shape b, coordinate 1 | −3 to 3 | 0 | 0.9941 | 0.9941 | +0.12 | −3.46 % |
| a1 Shape a, coordinate 1 | −3 to 3 | 0 | 0.9846 | 0.9846 | +0.12 | +6.92 % |
| b2 Shape b, coordinate 2 | −3 to 3 | 0 | 0.9846 | 0.9846 | +0.12 | +6.92 % |
| a3 Shape a, coordinate 3 | −3 to 3 | 0.2286 | 0.64 | 0.4114 | +0.12 | −0.287 % |
| b3 Shape b, coordinate 3 | −3 to 3 | 0.2286 | 0.64 | 0.4114 | +0.12 | −0.287 % |
Sweep of Shape b, coordinate 1
| Shape b, coordinate 1 | Modal assurance criterion |
|---|---|
| −3 | 0.02 |
| −2.25 | 0.002062 |
| −1.5 | 0.01538 |
| −0.75 | 0.2 |
| 0 | 0.8 |
| 0.75 | 0.968 |
| 1.5 | 0.7538 |
| 2.25 | 0.5959 |
| 3 | 0.5 |
Sweep one variable
Variables and how each one contributes
Modal assurance criterion
What the equation gives you. Everything else feeds it.
Shape a, coordinate 1
The ordinate of shape a at coordinate 1 (dimensionless). Changes how well shape a lines up with shape b. Raising it by 0.1 (to 1.1) moves the result by +5.81 %, from 0.64 to 0.6772.
Shape a, coordinate 2
The ordinate of shape a at coordinate 2 (dimensionless). Changes how well shape a lines up with shape b. Raising it by 0.1 (to 2.1) moves the result by −2.9 %, from 0.64 to 0.6214.
Shape a, coordinate 3
The ordinate of shape a at coordinate 3 (dimensionless). Zero if only two coordinates are used. Changes how well shape a lines up with shape b. Raising it by 0.1 (to 0.1) moves the result by −0.2 %, from 0.64 to 0.6387.
Shape b, coordinate 1
The ordinate of shape b at coordinate 1 (dimensionless). Changes how well shape b lines up with shape a. Raising it by 0.1 (to 2.1) moves the result by −2.9 %, from 0.64 to 0.6214.
Shape b, coordinate 2
The ordinate of shape b at coordinate 2 (dimensionless). Changes how well shape b lines up with shape a. Raising it by 0.1 (to 1.1) moves the result by +5.81 %, from 0.64 to 0.6772.
Shape b, coordinate 3
The ordinate of shape b at coordinate 3 (dimensionless). Zero if only two coordinates are used. Changes how well shape b lines up with shape a. Raising it by 0.1 (to 0.1) moves the result by −0.2 %, from 0.64 to 0.6387.
Worked example with the values above
- Dot product: a·b = 4.
- Lengths squared: a·a = 5, b·b = 5.
- MAC = 4² / (5 × 5) = 0.64.
Assumptions and limits
- Both shapes are measured or computed at the same coordinates and directions, in the same order.
- Real shapes are used here; for complex modes use the conjugate transpose.
Common mistakes
- Comparing shapes that do not share the same coordinates or sign convention.
- Trusting a high MAC alone when two modes have close frequencies, since closely spaced modes can have similar shapes.
- Using too few coordinates, so that different shapes look alike.
Statistics and uncertainty
A correlation measure from a handful of noisy coordinates is itself uncertain. Measurement noise and too few sensors both lower the apparent agreement, so a threshold such as 0.9 should be judged against the test quality.
Where it is used
- Matching test modes to finite-element modes and tracking modes between test runs.
- Scaling a computed shape to a measured one before updating a model.
Maths you need
- Dot product: the sum of products of two vectors
- Vector norm: the length of a vector
- Least squares: the best fit in the sum-of-squares sense