The Modal Assurance Criterion Analysis (MAC) analysis is used to determine the similarity of two mode shapes:
If a mode shape was compared to itself, the Modal Assurance Criterion value should be one or 100%.
For modes with different shapes, the MAC is less than 1. Shapes that are very different will have a value close to zero as shown in Figure 2.
Mode shapes that are used in the comparison can originate from a Finite Element Analysis or from an experimental modal analysis.
In a typical MAC analysis, one might make a ‘MAC Matrix’. A ‘MAC Matrix’ is a series of bar graphs of MAC values, that each range from 0 to 100% as shown in Figure 3.
In the case of Figure 3, this is a mode set compared to itself. The mode set contains nine different individual modes, so 81 different MAC values are being calculated. About half the values are redundant –e.g., the MAC value between mode 1 and 3 is the same as between mode 3 and 1.
In Figure 3, the first mode shape at 133 Hz is identical to itself, hence a single red bar of a value of 1. Along the diagonal, every mode is identical to itself, 1 to 1 (133 Hz), 2 to 2 (135 Hz), 3 to 3 (304 Hz), etc.
Off of the diagonal, the MAC values are very low. Ideally, each mode should be uniquely observed and have a different shape than the other modes. This is the case for this mode set. The highest off diagonal mode pair is mode 2 compared to mode 9 (and vice versa 9 to 2) with a MAC value of 20%. All the other off-diagonal mode pairs are below 20%.
Modal Assurance Criterion Equation
The MAC value between two modes is essentially the normalized dot product of the complex modal vector at each common nodes (i.e., points), as shown in Equation 1. It can also be thought of as the square of correlation between two modal vectors φr and φs.
If a linear relationship exists (i.e., the vectors move the same way) between the two complex vectors, the MAC value will be near to one. If they are linearly independent, the MAC value will be small (near zero).
A complex vector simply includes both amplitude and phase, whereas a real vector is real part only. In Equation 1, it is also clear that the MAC is not sensitive to scaling, so if all mode shape components are multiplied with the same factor, the MAC will not be affected.
If an experimental modal analysis had 20 different nodes where measurements were made, the mode shape components at all 20 nodes are taken into account to calculate the MAC value, but more importance will be attributed to the higher amplitude node locations.
A Modal Assurance Criterion (or MAC) analysis can be used in several different ways:
A MAC analysis is only looking at the mode shape, it does not compare the frequency value.
Experimental Modal Analysis Application Example
When performing an experimental modal analysis, the test operator must decide the number of points (i.e., nodes) to be measured. Determining the proper number is critical to the success of the test. If not enough points are measured, then the mode shape will not be identified properly.
Consider the case of an experimental modal analysis performed on a rectangular plate, suspended with free-free boundary conditions. Frequency Response Function (FRF) data was acquired at 6 locations on the plate. The FRF data was analyzed and a mode set extracted. After performing a MAC on the resulting mode set, not all the off-diagonal MAC values are close to zero (Figure 5).
Upon closer inspection of modes 3 and 7, the shapes themselves are unexpected (Figure 6). They appear to be rigid body modes. Rigid body modes on a free-free suspended structure are normally around 0 Hz. There are 6 of them: translation in X, Y, Z and rotation in X, Y, and Z.
At frequencies like 385 Hz and 764 Hz, flexible modes of the structure are expected. In this case, only measuring at 6 locations on the structure leads to “spatial aliasing”. There are not enough points to capture the modes correctly.
Acquiring an additional 9 points leads to better results. With 15 total points, the mode shapes look completely different (Figure 7).
When viewing modes 3 and 7 again, but with 15 measured points, one can better appreciate how measuring only 6 points created the spatial aliasing error (Figure 6 versus Figure 7).
When comparing Figure 6 (6 point modal) versus Figure 7 (15 point modal):
The MAC analysis of the plate structure experimental modal analysis of the 15 point modal analysis is much improved as shown in Figure 8. The off-diagonal MAC values are much closer to zero.
In experimental modal analysis, the data measured in the 6 point modal analysis is not "wrong". The FRF measurements at these nodes were no different in the 6 point modal versus the 15 point, since the physical structure being tested did not change. There was simply not enough measurement points to determine the complete mode shape. This is different than a Finite Element modal analysis were the number of nodes does determine the dynamic behavior.
In this case, a Modal Assurance Criterion (MAC) analysis flagged a problem with an inadequate set of measurement points. Because the off-diagonal MAC values in the MAC matrix were not low, the error was easy to find.
FEA-Test Application Example
An experimental modal analysis was done on an exhaust system and compared to a finite element modal analysis of the same exhaust (Figure 9).
After collecting Frequency Response Functions (FRFs) on the exhaust system, a MAC analysis was done between the first thirteen experimental test modes and the first thirteen finite element analysis modes. The results are shown in Figure 10.
Looking at the diagonal of the MAC matrix:
In this case, the MAC analysis indicates that there is room for improvement in the correlation of the test and FEA.
Using a variant of the MAC analysis called a ‘MAC Contribution Analysis’, the nodal points which most reduced the average MAC below 100% can be identified. Performing this analysis, it was found that nodal points of the Y pipe part of the exhaust were most responsible for the decrease.
Using this information, a visual inspection of the actual exhaust found that welds were present at the joint of the Y pipe (Figure 10). These welds were not physically represented in the finite element model, which was created from a CAD geometry. This can happen because when weld locations are only indicated on CAD drawings, but not physically present.
After introducing appropriate elements at the Y joint, the finite element model was recalculated and compared to the test results (Figure 11).
While not perfect, the following improvements were observed:
Using the MAC and ‘MAC Contribution Analysis’ as guidance, the Finite Element model was significantly improved. This process can be continued until the results match within desired limits.
The most important part of this process is the actual “detective” work. Figuring out that welds need to be added to the FEA model (which are not in the base CAD parts) is an important lesson for future modelling projects. The MAC and ‘MAC Contribution Analysis’ are tools to be used in this process.
Calculating MAC in LMS Test.Lab
To calculate a Modal Assurance Criterion in LMS Test.Lab, use the ‘Modal Validation’ worksheet of LMS Test.Lab Modal Analysis (Figure 12).
A MAC analysis can be performed:
To calculate a MAC analysis within a single mode set, select the “Processing” set that contains the modes in the upper left of the ‘Modal Validation’ worksheet (Figure 13).
Then press the “Auto-MAC” button in the middle left of the screen. After pressing the “Auto-MAC” button, a table of MAC values is created in the upper right.
The default view of the table is called “Table/Geometry”. Rather than looking at a table, a set of bars can be viewed by selecting “Matrix/Geometry” in the upper right (Figure 14).
In the resulting Matrix view, one can click on the bars to see the MAC value for any mode pair. The corresponding mode shapes are automatically displayed below (Figure 15).
A MAC analysis can also be performed between two different modes sets. This is done by selecting two different “Processing” sets (Figure 16).
To do a comparison between two different mode sets, each needs to be identified in a separate location on the left side of the ‘Modal Validation’ worksheet:
Press the “MAC” button to perform the analysis.
A Modal Assurance Criterion (or MAC) analysis can be used for FEA-Test, FEA-FEA and Test-Test comparisons of modes. By analyzing a MAC matrix, an engineer can improve the quality of an experimental modal test, verify finite element models, and update FEA models with test data.
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