The Modal Assurance Criterion

14 SOUND AND VIBRATION/AUGUST 2003

This article reviews the development of the original modalassurance criterion (MAC) together with other related assur-ance criteria that have been proposed over the last twentyyears. Some of the other assurance criteria that will be dis-cussed include the coordinate modal assurance criterion(COMAC), the frequency response assurance criterion (FRAC),coordinate orthogonality check (CORTHOG), frequency scaledmodal assurance criterion (FMAC), partial modal assurancecriterion (PMAC), scaled modal assurance criterion (SMAC),and modal assurance criterion using reciprocal modal vectors(MACRV). In particular, the thought process that relates theoriginal MAC development to ordinary coherence and to or-thogonality computations will be explained. Several uses ofMAC that may not be obvious to the casual observer (modalparameter estimation consistency diagrams and model updat-ing are two examples) will be identified. The common prob-lems with the implementation and use of modal assurance cri-terion computations will also be identified.

The development of the modal assurance criterion1-2 overtwenty years ago has led to a number of similar assurance cri-teria used in the area of experimental and analytical structuraldynamics. It is important to recognize the mathematical simi-larity of these varied criteria in order to be certain that con-clusions be correctly drawn from what is essentially a squared,linear regression correlation coefficient. The modal assurancecriterion is a statistical indicator, just like ordinary coherence,which can be very powerful when used correctly but very mis-leading when used incorrectly. This article will first review thehistorical development of the modal assurance criterion. Othersimilar assurance criteria will then be identified although thelist is not intended to be comprehensive. Typical uses of themodal assurance criterion will be discussed and finally, typi-cal abuses will be identified.

Historical Development of the MACThe historical development of the modal assurance criteria

originated from the need for a quality assurance indicator forexperimental modal vectors that are estimated from measuredfrequency response functions. The standard of the late 1970s,when the modal assurance criterion was developed, was theorthogonality check. The orthogonality check, however,coupled errors in the analytical model development, the reduc-tion of the analytical model and the estimated modal vectorsinto a single indicator and was, therefore, not always the bestapproach. Many times, an analytical model was not availablewhich renders the orthogonality check impractical.

The original development of the modal assurance criterionwas modeled after the development of the ordinary coherencecalculation associated with computation of the frequency re-sponse function. It is important to recognize that this leastsquares based form of linear regression analysis yields an in-dicator that is most sensitive to the largest difference betweencomparative values (minimizing the squared error) and resultsin a modal assurance criterion that is insensitive to smallchanges and/or small magnitudes. In the original thought pro-cess, this was considered an advantage since small modal co-

efficient values are often seriously biased by frequency re-sponse function (FRF) measurements or modal parameter es-timation errors.

In the internal development of the modal assurance criterionat the University of Cincinnati, Structural Dynamics ResearchLab (UCSDRL), a little modal assurance criterion (Little MAC)1,a big modal assurance criterion (Big MAC) and a multiplemodal assurance criterion (Multi-MAC)3 were formulated aspart of the original development. Little MAC and Multi-MACwere primarily testing methods and are not discussed furtherhere. The modal assurance criterion that survives today is whatwas originally identified as Big MAC. Since the “Big Mac” ac-ronym was already in use at that time, MAC is the designationthat has persisted.

Modal Vector Orthogonality. The primary method that hashistorically been used to validate an experimental modal modelis the weighted orthogonality check comparing measuredmodal vectors and an appropriately sized (the size of the squareweighting matrix must match the length and spatial dimensionof the modal vector) analytical mass or stiffness matrix (weight-ing matrix). Variations of this process include using analyti-cal modal vectors together with experimental modal vectorsand the appropriately sized mass or stiffness matrix. This lat-ter comparison is normally referred to as a pseudo-orthogonal-ity check (POC).

In the traditional orthogonality check, the experimentalmodal vectors are used together with a mass matrix, normallyderived from a finite element model, to evaluate orthogonal-ity of the experimental modal vectors. In the pseudo-orthogo-nality check, the experimental modal vectors are used togetherwith a mass matrix, normally derived from a finite elementmodel, and the analytical modal vectors, normally derived fromthe same finite element model, to evaluate orthogonality be-tween the experimental and analytical modal vectors. The ex-perimental and analytical modal vectors are scaled so that thediagonal terms of the modal mass matrix are unity. With thisform of scaling, the off-diagonal values in the modal massmatrix are expected to be less than 0.1 (10 percent of the di-agonal terms).

Theoretically, for the case of proportional damping, eachmodal vector of a system will be orthogonal to all other modal

The Modal Assurance Criterion –Twenty Years of Use and AbuseRandall J. Allemang, University of Cincinnati, Cincinnati, Ohio

Based on a paper presented at IMAC-XX, the 20th International ModalAnalysis Conference, Los Angeles, CA, February 2002

NomenclatureL = Number of matching pairs of modal vectors.

A* = Complex conjugate of A.Ni = Number of inputs.No = Number of outputs (assumed to be larger than Ni).Ne = Number of experimental modal vectors.Na = Number of analytical modal vectors.

Hpq(ω) = Measured frequency response function. pq(ω)= Synthesized frequency response function.

ψqr = Modal coefficient for degree-of-freedom q, moder.

ψpqr = Modal coefficient for reference p, degree-of-free-dom q, mode r.

{ψ}T = Transpose of {ψ}.{ψ}H = Complex conjugate transpose (Hermitian) of {ψ}.{ψr} = Modal vector for mode r.

{ψpr} = Modal vector for reference p, mode r.

H

15SOUND AND VIBRATION/AUGUST 2003

vectors of that system when weighted by the mass, stiffness ordamping matrix. In practice, these matrices are made availableby way of a finite element analysis and normally the mass ma-trix is considered to be the most accurate. For this reason, anyfurther discussion of orthogonality will be made with respectto mass matrix weighting. As a result, the orthogonality rela-tions can be stated as follows:

For r ≠ s:

For r = s:

Experimentally, the result of zero for the cross orthogonal-ity calculations (r ≠ s, Eq. 1) can rarely be achieved but valuesup to one tenth of the magnitude of the generalized mass of eachmode are considered to be acceptable. It is a common proce-dure to form the modal vectors into a normalized set of modeshape vectors with respect to the mass matrix weighting. Theaccepted criterion in the aerospace industry, where this confi-dence check is made most often, is for all of the generalizedmass terms to be unity and all cross orthogonality terms to beless than 0.1. Often, even under this criteria, an attempt is madeto adjust the modal vectors so that the cross orthogonality con-ditions are satisfied.4-6 Note that, in general, experimentalmodal vectors are not always real-valued and Eqs. 1 and 2 aredeveloped based upon normal or real-valued modal vectors.This complication has to be resolved by a process of real nor-malization of the measured modal vectors prior to utilizing Eqs.1 and 2 or by applying an equivalent procedure involving thestate-space form of the weighted orthogonality relationship.

In Eqs. 1 and 2, the mass matrix must be an No×No matrixcorresponding to the measurement locations on the structure.This means that the finite element mass matrix must be modi-fied from whatever size and distribution of grid locations re-quired in the finite element analysis to the No×No square ma-trix corresponding to the measurement locations. Thisnormally involves some sort of reduction algorithm as well asinterpolation of grid locations to match the measurement situ-ation.7-13

When Eq. 1 is not sufficiently satisfied, one (or more) of threesituations may exist. First, the modal vectors can be invalid.This can be due to measurement errors or problems with themodal parameter estimation algorithms. This is a very commonassumption and many times contributes to the problem. Sec-ond, the mass matrix can be invalid. Since the mass matrixdoes not always represent the actual physical properties of thesystem when it is built or assembled, this probably contributessignificantly to the problem. Third, the reduction of the massmatrix can be invalid.7-13 This can certainly be a realistic prob-lem and cause severe errors. The most obvious example of thissituation would be when a relatively large amount of mass isreduced to a measurement location that is highly flexible, suchas the center of an unsupported panel. In such a situation, themeasurement location is weighted very heavily in the orthogo-nality calculation of Eq. 2, but may represent only incidentalmotion of the overall modal vector.

In all probability, all three situations contribute to the fail-ure of orthogonality or pseudo-orthogonality criteria on occa-sion. When the orthogonality conditions are not satisfied, thisresult does not indicate where the problem originates. From anexperimental point of view, it is important to try to developmethods that indicate confidence that the modal vector is, oris not, part of the problem.

Modal Vector Consistency. Since the frequency responsefunction matrix contains redundant information with respectto a modal vector, the consistency of the estimate of the modalvector under varying conditions such as excitation locations(references) or modal parameter estimation algorithms can bea valuable confidence factor to be utilized in the process ofevaluation of experimental modal vectors.

The common approach to estimation of modal vectors fromfrequency response functions is to measure several complete

rows or columns of the frequency response function matrix.The estimation of modal vectors from this frequency responsefunction matrix will be a function of the data used in the modalparameter estimation algorithms and the specific modal param-eter estimations algorithms used. If the modal vectors are notwell represented in the frequency response function matrix, theestimation of the modal vector will contain potential bias andvariance errors. In any case, the modal vectors will containpotential variance errors.

Frequently, different subsets of the frequency response func-tion matrix and/or different modal parameter estimation algo-rithms are utilized to estimate separate, redundant modal vec-tors for comparison purposes. In these cases, if differentestimates of the same modal vectors are generated, the modalvectors can be compared and contrasted through an evaluationthat consists of the calculation of a complex modal scale fac-tor (relating two modal vectors) and a scalar modal assurancecriterion (measuring the consistency or linearity between twomodal vectors).

The function of the modal scale factor (MSF) is to providea means of normalizing all estimates of the same modal vec-tor, taking into account magnitude and phase differences. Oncetwo different modal vector estimates are scaled similarly, ele-ments of each vector can be averaged (with or without weight-ing), differenced or sorted to provide a best estimate of themodal vector or to provide an indication of the type of errorvector superimposed on the modal vector. In terms of modern,multiple reference modal parameter estimation algorithms, themodal scale factor is a normalized estimate of the modal par-ticipation factor between two references for a specific mode ofvibration.

The function of the modal assurance criterion (MAC) is toprovide a measure of consistency (degree of linearity) betweenestimates of a modal vector. This provides an additional con-fidence factor in the evaluation of a modal vector from differ-ent excitation (reference) locations or different modal param-eter estimation algorithms.

The modal scale factor and the modal assurance criterion alsoprovide a method of easily comparing estimates of modal vec-tors originating from different sources. The modal vectors froma finite element analysis can be compared and contrasted withthose determined experimentally as well as modal vectors de-termined by way of different experimental or modal parameterestimation methods. In this approach, methods can be com-pared and contrasted in order to evaluate the mutual consis-tency of different procedures rather than estimating the modalvectors specifically. If an analytical and an experimental vec-tor are deemed consistent or similar, the analytical modal vec-tor, together with the modal scale factor, can be used to com-plete the experimental modal vector if some degrees of freedomcould not be measured.

The modal scale factor is defined, according to this ap-proach, as follows:

or:

Since the modal scale factor is a complex-valued scalar, this isalso equivalent to:

Eq. 3 implies that the modal vector d is the reference to which

(1)

(2)

{ } [ ]{ }y yrT

sM = 0

{ } [ ]{ }y yrT

s rM M=

MSFcdr

cqr dqrq

N

dqr dqrq

N

o

o= =

=

∑

∑

y y

y y

*

*

1

1

MSFcdrcr

Tdr

drT

dr

= { } { }

{ } { }

*

*y yy y

MSFcdrdr

Hcr

drH

dr

= { } { }

{ } { }

*

*y yy y

(3a)

(3b)

(3c)


the modal vector c is compared. In the general case, modalvector c can be considered to be made of two parts. The firstpart is the part correlated with modal vector d. The second partis the part that is not correlated with modal vector d and ismade up of contamination from other modal vectors and anyrandom contribution. This error vector is considered to benoise. The modal assurance criterion is defined as a scalarconstant relating the degree of consistency (linearity) betweenone modal and another reference modal vector as follows:

or:

Since the modal assurance criterion is a real-valued scalar, thisis also equivalent to:

or:

or:

The modal assurance criterion takes on values from zero –representing no consistent correspondence, to one – represent-ing a consistent correspondence. In this manner, if the modalvectors under consideration truly exhibit a consistent, linearrelationship, the modal assurance criterion should approachunity and the value of the modal scale factor can be consid-ered reasonable. Note that, unlike the orthogonality calcula-tions, the modal assurance criterion is normalized by the mag-nitude of the vectors and, thus, is bounded between zero andone.

The modal assurance criterion can only indicate consistency,not validity or orthogonality. If the same errors, random or bias,exist in all modal vector estimates, this is not delineated by themodal assurance criterion. Invalid assumptions are normallythe cause of this sort of potential error. Even though the modalassurance criterion is unity, the assumptions involving thesystem or the modal parameter estimation techniques are notnecessarily correct. The assumptions may cause consistenterrors in all modal vectors under all test conditions verifiedby the modal assurance criterion.

Modal Assurance Criterion (MAC) Zero. If the modal assur-ance criterion has a value near zero, this is an indication thatthe modal vectors are not consistent. This can be due to any ofthe following reasons:• The system is nonstationary. This can occur if the system is

nonlinear and two data sets have been acquired at differenttimes or excitation levels. System nonlinearities will appeardifferently in frequency response functions generated fromdifferent exciter positions or excitation signals. The modalparameter estimation algorithms will also not handle the dif-ferent nonlinear characteristics in a consistent manner.

• There is noise on the reference modal vector. This case is thesame as noise on the input of a frequency response functionmeasurement. No amount of signal processing can removethis type of error.

• The modal parameter estimation is invalid. The frequencyresponse function measurements may contain no errors butthe modal parameter estimation may not be consistent withthe data.

• The modal vectors are from linearly unrelated mode shapevectors. Hopefully, since the different modal vector estimatesare from different excitation positions, this measure of in-consistency will imply that the modal vectors are orthogo-nal.If the first four reasons can be eliminated, the modal assur-

ance criterion can be interpreted in a similar way as an orthogo-nality calculation.

Modal Assurance Criterion (MAC) Unity. If the modal as-surance criterion has a value near unity, this is an indicationthat the modal vectors are consistent. This does not necessar-ily mean that they are correct. The modal vectors can be con-sistent for any of the following reasons:• The modal vectors have been incompletely measured. This

situation can occur whenever too few response stations havebeen included in the experimental determination of themodal vector.

• The modal vectors are the result of a forced excitation otherthan the desired input. This would be the situation if, dur-ing the measurement of the frequency response function, arotating piece of equipment with an unbalance is present inthe system being tested.

• The modal vectors are primarily coherent noise. Since thereference modal vector may be arbitrarily chosen, this modalvector may not be one of the true modal vectors of the sys-tem. It could simply be a random noise vector or a vectorreflecting the bias in the modal parameter estimation algo-rithm. In any case, the modal assurance criterion will onlyreflect a consistent (linear) relationship to the referencemodal vector.

• The modal vectors represent the same modal vector with dif-ferent arbitrary scaling. If the two modal vectors being com-pared have the same expected value when normalized, thetwo modal vectors should differ only by the complex valuedscale factor, which is a function of the common modal coef-ficients between the rows or columns.

MACcdr

cqr dqrq

N

cqr cqrq

N

dqr dqrq

N

o

o o= =

= =

∑

∑ ∑

y y

y y y y

*

* *

1

2

1 1

MACcdrcr

Tdr

crT

cr drT

dr

={ } { }

{ } { }{ } { }

*

* *

y y

y y y y

2

MACcdrdr

Hcr

drH

dr crH

cr

={ } { }

{ } { }{ } { }

y y

y y y y

2

MACcdrdr

Hcr cr

Hdr

drH

dr crH

cr

= { } { }{ } { }

{ } { }{ } { }

y y y yy y y y

MAC MSF MSFdrccdr dr=

(4a)

(4b)

(4c)

(4d)

(4e)

Mode

1234567891011121314

1

1.0000.0000.0010.0110.0210.0120.0080.0020.0290.005 0.001 0.1180.2750.007

2

0.0001.0000.0120.0060.0100.0010.0090.0030.0070.0060.0290.2730.1260.002

3

0.0010.0121.0000.0020.0070.0170.0010.0010.0060.0220.0190.0600.0450.394

4

0.0110.0060.0021.0000.9930.0110.0070.0010.0140.0130.0030.0020.0020.010

5

0.0210.0100.0070.9931.0000.0130.0060.0120.0150.0140.0040.0020.0030.003

6

0.0120.0010.0170.0110.0131.0000.0000.0050.0040.0070.0110.0120.0090.006

7

0.0080.0090.0010.0070.0060.0001.0000.0000.0220.0050.0020.0080.0090.007

8

0.0020.0030.0010.0010.0120.0050.0001.0000.0000.0200.0190.0110.0070.006

9

0.0290.0070.0060.0140.0150.0040.0220.0001.0000.0150.0080.0050.0190.006

10

0.0050.0060.0220.0130.0140.0070.0050.0200.0151.0000.0930.0020.0030.025

11

0.0010.0290.0190.0030.0040.0110.0020.0190.0080.0931.0000.0260.0170.023

12

0.1180.2730.0600.0020.0020.0120.0080.0110.0050.0020.0261.0000.7550.067

13

0.2750.1260.0450.0020.0030.0090.0090.0070.0190.0030.0170.7551.0000.067

14

0.0070.0020.3940.0100.0030.0060.0070.0060.0060.0250.0230.0670.0671.000

Table 1. Numerical presentation of MAC values.


Therefore, if the first three reasons can be eliminated, themodal assurance criterion indicates that the modal scale fac-tor is the complex constant relating the modal vectors andthat the modal scale factor can be used to average, differenceor sort the modal vectors.

Under the constraints mentioned previously, the modal as-surance criterion can be applied in many different ways. Themodal assurance criterion can be used to verify or correlatean experimental modal vector with respect to a theoreticalmodal vector (eigenvector). This can be done by computingthe modal assurance criterion between Ne modal vectors es-timated from experimental data and Na modal vectors esti-mated from a finite element analysis evaluated at commonstations. This process results in a Ne×Na rectangular modalassurance criterion matrix with values that approach unitywhenever an experimental modal vector and an analyticalmodal vector are consistently related.

Once the modal assurance criterion establishes that twovectors represent the same information, the vectors can beaveraged, differenced or sorted to determine the best singleestimate or the potential source of contamination using themodal scale factor. Since the modal scale factor is a complexscalar that allows two vectors to be phased the same and tothe same mean value, these vectors can be subtracted toevaluate whether the error is random or biased. If the errorappears to be random and the modal assurance criterion ishigh, the modal vectors can be averaged (using the modalscale factor) to improve the estimate of a modal vector. If theerror appears to be biased or skewed, the error pattern oftengives an indication that the error originates due to the loca-tion of the excitation or due to an inadequate modal param-eter estimation process. Based upon partial but overlappingmeasurement of two columns of the frequency response func-tion matrix, modal vectors can be sorted, assuming the modalassurance function indicates consistency, into a complete es-timate of each modal vector at all measurement stations.

The modal assurance criterion can be used to evaluatemodal parameter estimation methods if a set of analyticalfrequency response functions with realistic levels of randomand bias errors is generated and used in common with a va-riety of modal parameter estimation methods. In this way,agreement between existing methods can be established andnew modal parameter estimation methods can be checked forcharacteristics that are consistent with accepted procedures.Additionally, this approach can be used to evaluate the char-acteristics of each modal parameter estimation method in thepresence of varying levels of random and bias error.

The concept of consistency in the estimate of modal vec-tors from separate testing constraints is important consider-

ing the potential of multiple estimates of the same modal vec-tor from numerous input configurations and modal parameterestimation algorithms. The computation of modal scale factorand modal assurance criterion results in a complex scalar anda correlation coefficient that does not depend on weighting in-formation outside the testing environment. Since the modalscale factor and modal assurance criterion are computed analo-gous to the frequency response function and coherence func-tion, both the advantages and limitations of the computationprocedure are well understood. These characteristics, as wellas others, provide a useful tool in the processing of experimen-tal modal vectors.

MAC Presentation Formats. One of the big changes in theapplication of the Modal Assurance Criteria over the lasttwenty years is in the way the information is presented. His-torically, a table of numbers was usually presented as shownin Table 1.

Today, most computer systems routinely utilize color topresent magnitude data like MAC using a 2D or 3D plot asshown in Figures 1 and 2. It is important to remember, how-ever, that MAC is a discrete calculation and what appears as acolor contour plot really only represents the discrete mode tomode comparison. Nevertheless, a color plot does allow formore data to be presented in an understandable form in a mini-mum space.

Other Similar Assurance CriteriaThe following brief discussion highlights assurance criteria

that utilize the same linear, least squares computation approachto the analysis (projection) of two vector spaces as the modalassurance criterion. The equations for each assurance criterionare not repeated unless there is a significant computationaldifference that needs to be clarified or highlighted. This list isby no means comprehensive nor is it in any particular order ofimportance but includes most of the frequently cited assurancecriterion found in the literature.

Weighted Modal Analysis Criterion (WMAC). A number ofauthors have utilized a weighted modal assurance criterion(WMAC) without developing a special designation for this case.WMAC is proposed for these cases. The purpose of the weight-ing matrix is to recognize that MAC is not sensitive to mass orstiffness distribution, just sensor distribution, and to adjust themodal assurance criterion to weight the degrees-of-freedom inthe modal vectors accordingly. In this case, the WMAC becomesa unity normalized orthogonality – or psuedo-orthogonality –check where the desirable result for a set of modal vectorswould be ones along the diagonal (same modal vectors) andzeros off-diagonal (different modal vectors) regardless of thescaling of the individual modal vectors. Note that the weight-

Figure 1. 2-D presentation of MAC Values. Figure 2. 3-D presentation of MAC Values.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Mode Number

Mod

e N

umbe

r

MA

C V

alue

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MA

C V

alue

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14

12

34

56

78

910

1112

1314

Mode Number

Mode N

umber


ing matrix is applied as an inner matrix product for the singlenumerator vector product and both vector products in the de-nominator.

Partial Modal Analysis Criterion (PMAC). The partial modalassurance criterion (PMAC)14 was developed as a spatially lim-ited version of the modal assurance criterion where a subsetof the complete modal vector is used in the calculation. Thesubset is chosen based upon the user’s interest and may reflectonly a certain dominant sensor direction (X, Y and/or Z) or onlythe degrees-of-freedom from a component of the completemodal vector.

Modal Assurance Criterion Square Root (MACSR). Thesquare root of the modal assurance criterion (MACSR)15 is de-veloped to be more consistent with the orthogonality andpsuedo-orthogonality calculations using an identity weightingmatrix. Essentially this approach utilizes the square root of theMAC calculation, which tends to highlight the cross terms (offdiagonal) that are generally very small MAC values.

Scaled Modal Assurance Criterion (SMAC). The scaledmodal assurance criterion (SMAC)16 is essentially a weightedmodal assurance criteria (WMAC) where the weighting matrixis chosen to balance the scaling of translational and rotationaldegrees-of-freedom included in the modal vectors. This devel-opment is needed whenever different data types (with differ-ent engineering units) are included in the same modal vectorto normalize the magnitude differences in the vectors. This isrequired since the modal assurance criterion minimizes thesquared error and is dominated by the larger values.

Modal Assurance Criterion Using Reciprocal Vectors(MACRV). A reciprocal modal vector is defined as the math-ematical vector that, when transposed and premultiplied by aspecific modal vector, yields unity. When the same computa-tion is performed with this reciprocal modal vector and anyother modal vector or any other reciprocal modal vector, theresult is zero. The reciprocal modal vector can be thought ofas a product of the modal vector and the unknown weightingmatrix that will produce a perfect orthogonality result. Recip-rocal modal vectors are computed directly from measured fre-quency response functions and the experimental modal vec-tors and are, therefore, experimentally based.

The modal assurance criterion using reciprocal modal vec-tors (MACRV)17 is the comparison of reciprocal modal vectorswith analytical modal vectors in what is very similar to apsuedo-orthogonality check (POC). The reciprocal modal vec-tors are utilized in controls applications as modal filters andthe MACRV serves as a check of the mode isolation providedby each reciprocal modal vector compared to analytical modesexpected.

Modal Assurance Criterion with Frequency Scales (FMAC).Another extension of the modal assurance criterion is the ad-dition of frequency scaling to the modal assurance criterion.18-

19 This extension of MAC “offers a means of displaying simul-taneously the mode shape correlation, the degree of spatialaliasing and the frequency comparison in a single plot.” Thisdevelopment is particularly useful in model correlation appli-cations (model updating, assessment of parameter variation,etc.).

Coordinate Modal Assurance Criterion (COMAC). An exten-sion of the modal assurance criterion is the coordinate modalassurance criterion (COMAC).20 The COMAC attempts to iden-tify which measurement degrees-of-freedom contribute nega-tively to a low value of MAC. The COMAC is calculated over aset of mode pairs, analytical versus analytical, experimentalversus experimental or experimental versus analytical. The twomodal vectors in each mode pair represent the same modalvector, but the set of mode pairs represents all modes of inter-est in a given frequency range. For two sets of modes that areto be compared, there will be a value of COMAC computed foreach (measurement) degree-of-freedom.

The coordinate modal assurance criterion (COMAC) is cal-culated using the following approach, once the mode pairs havebeen identified with MAC or some other approach:

Note that the above formulation assumes that there is a matchfor every modal vector in the two sets and the modal vectorsare renumbered accordingly so that the matching modal vec-tors have the same subscript. Only those modes that match be-tween the two sets are included in the computation.

The Enhanced Coordinate Modal Assurance Criterion(ECOMAC). One common problem with experimental modalvectors is the potential problem of calibration scaling errorsand/or sensor orientation mistakes. The enhanced coordinatemodal assurance criterion (ECOMAC)21 was developed to ex-tend the COMAC computation to be more aware of typical ex-perimental errors that occur in defining modal vectors such assensor scaling mistakes and sensor orientation (plus or minussign) errors.

Mutual Correspondence Criterion (MCC). The mutual cor-respondence criterion (MCC)22 is the modal assurance criterionapplied to vectors that do not originate as modal vectors butas vector measures of acoustic information (velocity, pressure,intensity, etc.). The equation in this formulation utilizes atranspose and will only correctly apply to real valued vectors.

Modal Correlation Coefficient (MCC). One of the naturallimitations of a least squares based correlation coefficient likethe modal assurance criterion is that it is relatively insensitiveto small changes in magnitude, position by position, in the vec-tor comparisons. The modal correlation coefficient (MCC)23-24

is a modification of MAC that attempts to provide a more sen-sitive indicator. This approach is particularly important whenusing modal vectors in damage detection situations where themagnitude changes of the modal vectors being measured areminimal.

Inverse Modal Assurance Criterion (IMAC). An alternativeapproach to increasing the sensitivity of the modal assurancecriterion to small mode shape changes is the inverse modal as-surance criterion (IMAC).25 This approach uses essentially thesame computational scheme as MAC but utilizes the inverseof the modal coefficients. Therefore, small modal coefficientsbecome significant in the least squares based correlation coef-ficient computation. Naturally, this computation suffers fromthe possibility that a modal coefficient could be numericallyzero.

Frequency Response Assurance Criterion (FRAC). Any twofrequency response functions representing the same input-out-put relationship can be compared using a technique known asthe frequency response assurance criterion (FRAC).26-28 Thesimplest example is a validation procedure that compares theFRF data synthesized from the modal model with the measuredFRF data. The basic assumption is that the measured frequencyresponse function and the synthesized frequency responsefunction should be linearly related (unity scaling coefficient)at all frequencies. Naturally, the FRFs can be compared overthe full or partial frequency range of the FRFs as long as thesame discrete frequencies are used in the comparison. Thisapproach has been utilized in the modal parameter estimationprocess for a number of years under various designations (pa-rameter estimation correlation coefficient29, synthesis correla-tion coefficient30 and response vector assurance criterion(RVAC)31). This procedure is particularly effective as a modalparameter estimation validation procedure if the measured datawere not part of the data used to estimate the modal param-eters. This serves as an independent check of the modal param-eter estimation process.

COMACq

qr qrr

L

qr qrr

L

qr qrr

L= =

= =

∑

∑ ∑

y f

y y y y

2

1

1 1

* *

(5)

(6)FRAC

H H

H H Hpq

pq pq

pq pq pq

= =

= =

∑

∑

( ) ( )

( ) ( ) ( )

*

*

w w

w w w

w w

w

w w

w

w

1

2

1

2

2

ww

ww

1

2

∑ ˆ ( )*Hpq


Complex Correlation Coefficient (CCF). A significant varia-tion in the frequency response assurance criterion is the com-plex correlation coefficient (CCF)31, which is computed with-out squaring the numerator term, thus yielding a complexvalued coefficient. The magnitude of the coefficient is the sameas the FRAC computation but the phase describes any system-atic phase lag or lead that is present between the two FRFs. Insituations where analytical and experimental FRFs are com-pared, the CCF will detect the common problem of a constantphase shift that might be due to experimental signal condition-ing problems, etc.

Frequency Domain Assurance Criterion (FDAC). A similarvariation in the frequency response assurance criterion is thefrequency domain assurance criterion (FDAC)32, which is aFRAC-type of calculation evaluated with different frequencyshifts. Since the difference in impedance (FRF) model updat-ing is often an FRF that is in question due to frequencies ofresonances or anti-resonances, the FDAC is formulated to iden-tify this problem. A related criterion, the modal FRF assurancecriterion (MFAC)18, combines analytical modal vectors withmeasured frequency response functions (FRFs) in an extensionof FRAC and FDAC that weights or filters the FRF data basedupon the expected, analytical modal vectors.

Coordinate Orthogonality Check (CORTHOG). The coordi-nate orthogonality check (CORTHOG)33 is a normalized errormeasure between the pseudo-orthogonality calculation, com-paring measured to analytical modal vectors, and the analyti-cal orthogonality calculation, comparing analytical to analyti-cal modal vectors. Several different normalizing or scalingmethods are used with this calculation.

Uses of the Modal Assurance CriterionMost of the potential uses of the modal assurance criterion

are well known but a few may be more subtle. A partial list ofthe most typical uses that have been reported in the literatureare as follows:• Validation of experimental modal models• Correlation with analytical modal models (mode pairing)• Correlation with operating response vectors• Mapping matrix between analytical and experimental modal

models• Modal vector error analysis• Modal vector averaging• Experimental modal vector completion and/or expansion• Weighting for model updating algorithms• Modal vector consistency/stability in modal parameter esti-

mation algorithms• Repeated and psuedo-repeated root detection• Structural fault/damage detection• Quality control evaluations• Optimal sensor placement

Abuses of the Modal Assurance CriterionMany of the alternate formulations of the modal assurance

criterion were developed to address some of the shortcomingsof the original modal assurance criterion formulation. Whenusers utilize the original modal assurance criterion in thesesituations, a poor result will often follow. For the purposes ofthis discussion, this is referred to as misuse or abuse. The mis-use or abuse of the modal assurance criterion generally resultsdue to one of five issues. These issues can be summarized as:• The modal analysis criterion is not an orthogonality check.• The wrong mathematical formulation for the modal assur-

ance criterion is used.• The modal assurance criterion is sensitive to large values

(wild points?) and insensitive to small values.• The number of elements in the modal vectors (space) is small.• The modal vectors have been zero padded.These issues can be further explained in the following para-graphs.

The modal analysis criterion is not an orthogonality check.It is important to recognize that the modal assurance criterion

effectively weights the computation based upon the spatialdistribution of the degrees-of-freedom included in the modalvectors. The modal assurance criterion does not weight themodal vectors with a mass or stiffness matrix and, therefore,cannot compensate for situations where a very limited num-ber of degrees-of-freedom (sensors) have been placed on amassive sub-structure of a mechanical system. The typicalexample involves the engine of an automobile. If few or nosensors are placed on the engine and a large number are placedon the surface of the automobile body, several modal vectorsat different modal frequencies will have very high MAC num-bers indicating that the modal vectors are the same. This ex-ample indicates to the user that an incomplete modal vectorwas measured and the user has violated one of the primaryassumptions of experimental modal analysis (observability).

The wrong mathematical formulation for the modal assur-ance criterion is used. Frequently, users implement the modalassurance criterion, or a related similar computation, using avector transpose in the numerator and denominator calcula-tions rather than an Hermitian (conjugate transpose). This er-ror causes no problem as long as analytical vectors or real-val-ued experimental vectors are involved in the calculation.However, in the general case, where some of the vectors arecomplex-valued, this does not give the correct result. The origi-nal mathematical formulation assumes the general case but hasbeen reported incorrectly in some literature. This innocenterror often occurs when the author is utilizing real-valued vec-tors and notices no problem. However, users who do not rec-ognize this issue are often led astray in subsequent applicationsinvolving complex-valued vectors.

The modal assurance criterion is sensitive to large values(wild points?) and insensitive to small values. The modal as-surance criterion is based upon the minimization of the squarederror between two vector spaces. This means that the degrees-of-freedom involving the largest magnitude differences be-tween the two modal vectors will dominate the computationwhile small differences will have almost no effect. Therefore,nodal information (small modal coefficients) will generally nothave much effect on the MAC calculation and large modal co-efficients will potentially have the greatest effect. This alsomeans that, if there have been erroneous data included in themodal vectors due to calibration errors, modal parameter esti-mation mistakes, etc., these wild points may dominate the MACcalculation.

The number of elements in the modal vectors (space) issmall. Since the modal assurance criterion is essentially a sta-tistical computation where the number of averages comes fromthe number of elements in the modal vectors, if the modal vec-tors have only a limited number of degrees-of-freedom, this willskew the meaning of the numerical MAC value. This frequentlyhappens when high order, multiple reference modal parameterestimation algorithms estimate the stability or consistency dia-gram. Modal vector stability or consistency is identified usinga MAC computation where the vectors include only the de-grees-of-freedom at the reference locations, typically two tofive. In these situations, there may be great variability in theMAC computation, particularly if the modal vector is not wellexcited from one or more of the reference locations. Vectorswith many elements reduce the sensitivity of MAC to this prob-lem.

The modal vectors have been zero padded. Frequently,when modal vectors are exported from one computational en-vironment to another, the modal vectors include zero valueswhen no value was ever measured, or computed, for that de-gree-of-freedom. For example, in an experimental situation, one(X) or two dimensions (X,Y) of translational response may bemeasured at some degrees-of-freedom rather than three dimen-sions (X,Y,Z). In the commonly used Universal File Format formodal vectors (File Format 55), this is the case since there isno designation for not measuring the information. When themodal assurance criterion is calculated for this case, there willbe a problem if some other vectors, with nonzero information


at these degrees-of-freedom, are included in the computation.This can be avoided if information is dropped from the com-putation when either vector includes a perfect zero (withincomputational precision) at a degree-of-freedom, but is rarelydone.

Current DevelopmentsCurrently, many users are utilizing more statistical ap-

proaches to understand the meaning and bounds of experimen-tal modal parameters.34-39 This approach extends to the modalassurance criterion as well. Examples are the bootstrap andjackknife approaches40-42 to the evaluation of the mean andstandard deviation of discrete sets of experimental data. Theseapproaches remove and/or replace portions of the computation(bootstrap uses replicative resampling, jackknife uses sequen-tial elimination) to evaluate the bounds or limits on the MACvalues. In this way, the sensitivity of the MAC computation canbe more effectively evaluated than with the current single num-ber indicating the degree of linearity between two modal vec-tors that are being compared.

ConclusionsOver the last twenty years, the modal assurance criterion has

demonstrated how a simple statistical concept can become anextremely useful tool in the field of experimental modal analy-sis and structural dynamics. The use of the modal assurancecriterion and the development and use of a significant num-ber of related criteria, has been remarkable and is most likelydue to the overall simplicity of the concept. New uses of themodal assurance criterion and new criteria will be developedover the next years as users more fully understand the limita-tions of the current criteria. Certainly in the next few years, theincreased use of other statistical methods as well as further de-velopment of singular value/vector methods are related areasthat will generate useful tools in this area.

Even so, it will always be important to recognize the originsand limitations of tools like the modal assurance criterion toavoid misuse of the methodology. Simplistic tools like themodal assurance criterion are limited in their meaningful ap-plication. The development of related assurance criteria hasbeen initiated by shortcomings, real or perceived, of the origi-nal modal assurance criterion. Dissatisfaction often has re-sulted from the misuse of these tools by users, removed fromthe actual development or unaware of application limitationsin subsequent implementations. It is clear that users will con-tinue to need more feedback concerning quality assurance in-formation relative to experimental modal parameters and thatnew techniques, particularly statistical methods that utilize theredundant information present in the measured data, will con-tinue to be developed with strengths and weaknesses, just likethe modal assurance criterion.

AcknowledgementsThe author would like to acknowledge the number of authors

cited as developing similar assurance criterion and apologizeto those who were not identified due to space limitations. Mostof all, the author would like to acknowledge the contributionof Ray Zimmerman in the original development of the modalassurance criterion. Ray contributed to many of the develop-ments of the UCSDRL during the 1970s and 1980s by provid-ing interesting ideas and thoughtful discussion. We miss hav-ing the opportunity to bounce ideas off Ray.

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The author can be contacted at: [email protected].

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The Modal Assurance Criterion