Modal Assurance Criterion - Álvaro Rivero

DINÁMICA

Y

FATIGA

“The Modal Assurance Criterion (MAC)”

Profesor: R. Avilés

Asignatura: Dinámica y Fatiga

Alumnos: Rivero González, Álvaro

18 de Diciembre de 2009

Modal Assurance Criterion (MAC) 2

Index

Abstract 4

1. State of the Art. Antecedents. review of existing methods 5

1.1. Historical Development of MAC 5

1.1.1. Modal Vector Orthogonality 6

1.1.2. Modal Vector Consistency 8

1.1.3. Modal Assurance Criterion (MAC) Zero 10

1.1.4. Modal Assurance Criterion (MAC) Unity 11

1.1.5. MAC Presentation Formats 12

1.2. Other Similar Assurance Criteria 14

1.3. Uses of the Modal Assurance Criterion 17

1.4. Abuses of the Modal Assurance Criterion 18

1.5. Current Developments 20

2. Software 21

2.1. LMS Virtual.Lab Overview 21

2.2. LMS Virtual.Lab Correlation 25

2.2.1. Systematic Validation from the bottom up 26

2.2.2. LMS Virtual.Lab Correlation. Features and Benefits 27

2.2.3. LMS Virtual.Lab Model Updating. Features and Benefits 29


3. Uses for Aircraft Design and Testing/Certificating Companies 31

3.1. Using MSC/NASTRAN and LMS/PRETEST to find an

optimal sensor placement for modal identification and

correlation of aerospace structures 31

3.1.1. MSC/NASTRAN and LMS/PRETEST 33

3.1.2. Target Mode Selection 33

3.1.3. Sensor Placement 37

3.1.4. Shaker Positioning 41

3.2. Modal Test of L-610G Aeroplane 43

Conclusions 47

Bibliography 48


Abstract

This report firstly reviews the development of the original modal assurance criterion

(MAC) together with other related assurance criteria that have been proposed over the last twenty

years. Some of the other assurance criteria that will be discussed include the coordinate modal

assurance criterion (COMAC), the frequency response assurance criterion (FMAC), partial mode

assurance criterion (PMAC) and modal assurance criterion using reciprocal modal vectors (MACRV).

Several uses of MAC that may not be obvious to the casual observer will be identified; the common

problems with the implementation and use of modal assurance criterion computations will also be

identified. Afterward, the LMS Virtual.Lab is briefly presented, stressing the module related to the

modal correlation, the LMS Virtual.Lab Correlation. Finally, this dossier introduces some particular

uses of the model assurance criterion that could be interesting for aircraft design and

testing/certificating companies.

Este informe revisa en primer lugar el desarrollo del criterio de confianza modal (MAC)

junto con otros criterios de confianza o correlación relacionados que han sido propuestos a lo largo de

los últimos veinte años. Otros criterios de correlación que serán tratados son el criterio de correlación

modal por coordenadas (COMAC), el criterio de correlación de respuesta en frecuencia (FMAC), el

criterio de correlación de modo parcial (PMAC) y el criterio de correlación modal utilizando vectores

modales recíprocos (MACRV). Varios usos del MAC que pueden no ser obvios para el observador

casual serán identificados; los problemas habituales con la implementación y uso de computaciones

del criterio de confianza modal serán también identificados. Seguidamente, el LMS Virtual.Lab es

brevemente presentado, haciendo hincapié en el módulo relacionado con la correlación modal, el LMS

Virtual.Lab Correlation. Finalmente, este informe introduce algunos usos particulares del criterio de

correlación modal que podrían resultar interesantes para empresas de diseño y de prueba/certificación

de aeronaves.


1. State of the art. Antecedents. Review of existing methods

The development of the modal assurance criterion over twenty years ago has led to a

number of similar assurance criteria used in the area of experimental and analytical structural

dynamics. It is important to recognize the mathematical similarity of these varied criteria in order to be

certain that conclusions be correctly drawn from what is essentially a squared, linear regression

correlation coefficient. The modal assurance criterion is a statistical indicator, just like ordinary

coherence, which can be very powerful when used correctly but very misleading when used

incorrectly. This first section will first review the historical development of the modal assurance

criterion. Other similar assurance criteria will then be identified although the list is not intended to be

comprehensive. Typical uses of the modal assurance criterion will be discussed and finally, typical

abuses will be identified.

Before starting with the review of the methods, it is necessary to present the nomenclature

that will be used in the following paragraphs:

L = 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 (𝑤) = Measured frequency response function.

𝐻𝑝𝑞 (𝑤) = Synthesized frequency response function.

𝜑𝑞𝑟 = Modal coefficient for degree-of-freedom q, mode r.

𝜑𝑝𝑞𝑟 = Modal coefficient for reference p, degree-of-freedom q, mode r.

𝜑 𝑇 = Transpose of {φ}.

𝜑 𝐻 = Complex conjugate transpose (Hermitian) of {φ}.

𝜑𝑟 = Modal vector for mode r.

𝜑𝑝𝑟 = Modal vector for reference p, mode r.

1.1. Historical Development of MAC

The historical development of the modal assurance criteria originated from the need for a

quality assurance indicator for experimental modal vectors that are estimated from measured

frequency response functions. The standard of the late 1970s, when the modal assurance criterion was

developed, was the orthogonality check. The orthogonality check, however, coupled errors in the

analytical model development, the reduction of the analytical model and the estimated modal vectors

into a single indicator and was, therefore, not always the best approach. Many times, an analytical

model was not available which renders the orthogonality check impractical.


The original development of the modal assurance criterion was modeled after the

development of the ordinary coherence calculation associated with computation of the frequency

response function. It is important to recognize that this least squares based form of linear regression

analysis yields an indicator that is most sensitive to the largest difference between comparative values

(minimizing the squared error) and results in a modal assurance criterion that is insensitive to small

changes and/or small magnitudes. In the original thought process, this was considered an advantage

since small modal coefficient values are often seriously biased by frequency response function (FRF)

measurements or modal parameter estimation errors.

In the internal development of the modal assurance criterion at the University of

Cincinnati, Structural Dynamics Research Lab (UCSDRL), a little modal assurance criterion (Little

MAC), a big modal assurance criterion (Big MAC) and a multiple modal assurance criterion (Multi-

MAC) were formulated as part of the original development. Little MAC and Multi-MAC were

primarily testing methods and are not discussed further here. The modal assurance criterion that

survives today is what was originally identified as Big MAC. Since the “Big Mac” acronym was

already in use at that time, MAC is the designation that has persisted.

1.1.1. Modal Vector Orthogonality

The primary method that has historically been used to validate an experimental modal

model is the weighted orthogonality check comparing measured modal vectors and an appropriately

sized (the size of the square weighting matrix must match the length and spatial dimension of the

modal vector) analytical mass or stiffness matrix (weighting matrix). Variations of this process include

using analytical modal vectors together with experimental modal vectors and the appropriately sized

mass or stiffness matrix. This latter comparison is normally referred to as a pseudo-orthogonality

check (POC).

In the traditional orthogonality check, the experimental modal vectors are used together

with a mass matrix, normally derived from a finite element model, to evaluate orthogonality of the

experimental modal vectors. In the pseudo-orthogonality check, the experimental modal vectors are

used together with a mass matrix, normally derived from a finite element model, and the analytical

modal vectors, normally derived from the same finite element model, to evaluate orthogonality

between the experimental and analytical modal vectors. The experimental and analytical modal vectors

are scaled so that the diagonal terms of the modal mass matrix are unity. With this form of scaling, the

off-diagonal values in the modal mass matrix are expected to be less than 0.1 (10 percent of the

diagonal terms).

Theoretically, for the case of proportional damping, each modal vector of a system will be

orthogonal to all other modal vectors of that system when weighted by the mass, stiffness or damping

matrix. In practice, these matrices are made available by way of a finite element analysis and normally

the mass matrix is considered to be the most accurate. For this reason, any further discussion of


orthogonality will be made with respect to mass matrix weighting. As a result, the orthogonality

relations can be stated as follows:

For r ≠ s: 𝜑𝑟 𝑇 𝑀 𝜑𝑠 = 𝑂 (1)

For r = s: 𝜑𝑟 𝑇 𝑀 𝜑𝑠 = 𝑀𝑟 (2)

Experimentally, the result of zero for the cross orthogonality calculations (r ≠ s, Eq. 1) can

rarely be achieved but values up to one tenth of the magnitude of the generalized mass of each mode

are considered to be acceptable. It is a common procedure to form the modal vectors into a normalized

set of mode shape vectors with respect to the mass matrix weighting. The accepted criterion in the

aerospace industry, where this confidence check is made most often, is for all of the generalized mass

terms to be unity and all cross orthogonality terms to be less than 0.1. Often, even under this criterion,

an attempt is made to adjust the modal vectors so that the cross orthogonality conditions are satisfied.

Note that, in general, experimental modal vectors are not always real-valued and Eqs. 1 and 2 are

developed based upon normal or real-valued modal vectors. This complication has to be resolved by a

process of real normalization of the measured modal vectors prior to utilizing Eqs. 1 and 2 or by

applying an equivalent procedure involving the state-space form of the weighted orthogonality

relationship.

In Eqs. 1 and 2, the mass matrix must be an No×No matrix corresponding to the

measurement locations on the structure. This means that the finite element mass matrix must be

modified from whatever size and distribution of grid locations required in the finite element analysis to

the No×No square matrix corresponding to the measurement locations. This normally involves some

sort of reduction algorithm as well as interpolation of grid locations to match the measurement

situation.

When Eq. 1 is not sufficiently satisfied, one (or more) of three situations may exist. First,

the modal vectors can be invalid. This can be due to measurement errors or problems with the modal

parameter estimation algorithms. This is a very common assumption and many times contributes to the

problem. Second, the mass matrix can be invalid. Since the mass matrix does not always represent the

actual physical properties of the system when it is built or assembled, this probably contributes

significantly to the problem. Third, the reduction of the mass matrix can be invalid. This can

certainly be a realistic problem and cause severe errors. The most obvious example of this situation

would be when a relatively large amount of mass is reduced to a measurement location that is highly

flexible, such as the center of an unsupported panel. In such a situation, the measurement location is

weighted very heavily in the orthogonality calculation of Eq. 2, but may represent only incidental

motion of the overall modal vector.

In all probability, all three situations contribute to the failure of orthogonality or pseudo-

orthogonality criteria on occasion. When the orthogonality conditions are not satisfied, this result does

not indicate where the problem originates. From an experimental point of view, it is important to try to

develop methods that indicate confidence that the modal vector is, or is not, part of the problem.


1.1.2. Modal Vector Consistency

Since the frequency response function matrix contains redundant information with respect

to a modal vector, the consistency of the estimate of the modal vector under varying conditions such as

excitation locations (references) or modal parameter estimation algorithms can be a valuable

confidence factor to be utilized in the process of evaluation of experimental modal vectors. The

common approach to estimation of modal vectors from frequency response functions is to measure

several complete rows or columns of the frequency response function matrix. The estimation of modal

vectors from this frequency response function matrix will be a function of the data used in the modal

parameter estimation algorithms and the specific modal parameter estimations algorithms used. If the

modal vectors are not well represented in the frequency response function matrix, the estimation of the

modal vector will contain potential bias and variance errors. In any case, the modal vectors will

contain potential variance errors.

Frequently, different subsets of the frequency response function matrix and/or different

modal parameter estimation algorithms are utilized to estimate separate, redundant modal vectors for

comparison purposes. In these cases, if different estimates of the same modal vectors are generated,

the modal vectors can be compared and contrasted through an evaluation that consists of the

calculation of a complex modal scale factor (relating two modal vectors) and a scalar modal assurance

criterion (measuring the consistency or linearity between two modal vectors).

The function of the modal scale factor (MSF) is to provide a means of normalizing all

estimates of the same modal vector, taking into account magnitude and phase differences. Once two

different modal vector estimates are scaled similarly, elements of each vector can be averaged (with or

without weighting), differenced or sorted to provide a best estimate of the modal vector or to provide

an indication of the type of error vector superimposed on the modal vector. In terms of modern,

multiple reference modal parameter estimation algorithms, the modal scale factor is a normalized

estimate of the modal participation factor between two references for a specific mode of vibration.

The function of the modal assurance criterion (MAC) is to provide a measure of

consistency (degree of linearity) between estimates of a modal vector. This provides an additional

confidence factor in the evaluation of a modal vector from different excitation (reference) locations or

different modal parameter estimation algorithms.

The modal scale factor and the modal assurance criterion also provide a method of easily

comparing estimates of modal vectors originating from different sources. The modal vectors from a

finite element analysis can be compared and contrasted with those determined experimentally as well

as modal vectors determined by way of different experimental or modal parameter estimation

methods. In this approach, methods can be compared and contrasted in order to evaluate the mutual

consistency of different procedures rather than estimating the modal vectors specifically. If an

analytical and an experimental vector are deemed consistent or similar, the analytical modal vector,

together with the modal scale factor, can be used to complete the experimental modal vector if some

degrees of freedom could not be measured.


The modal scale factor is defined, according to this approach, as follows:

𝑀𝑆𝐹𝑐𝑑𝑟 = 𝜑𝑐𝑞𝑟 𝜑𝑑𝑞𝑟

∗𝑁0𝑞=1

𝜑𝑑𝑞𝑟 𝜑𝑑𝑞𝑟∗𝑁0

𝑞=1

(3a)

or:

𝑀𝑆𝐹𝑐𝑑𝑟 ={𝜑𝑐𝑟 }𝑇 {𝜑𝑑𝑟

∗ }

{𝜑𝑑𝑟 }𝑇 {𝜑𝑑𝑟∗ }

(3b)

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

𝑀𝑆𝐹𝑐𝑑𝑟 ={𝜑𝑑𝑟 }𝐻 {𝜑𝑐𝑟

∗ }

{𝜑𝑑𝑟 }𝐻 {𝜑𝑑𝑟∗ }

(3c)

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

compared. In the general case, modal vector c can be considered to be made of two parts. The first part

is the part correlated with modal vector d. The second part is the part that is not correlated with modal

vector d and is made up of contamination from other modal vectors and any random contribution. This

error vector is considered to be noise. The modal assurance criterion is defined as a scalar constant

relating the degree of consistency (linearity) between one modal and another reference modal vector as

follows:

𝑀𝐴𝐶𝑐𝑑𝑟 = 𝜑𝑐𝑞𝑟 𝜑𝑑𝑞𝑟

∗𝑁0𝑞=1

2

𝜑𝑐𝑞𝑟 𝜑𝑐𝑞𝑟∗𝑁0

𝑞=1 𝜑𝑑𝑞𝑟 𝜑𝑑𝑞𝑟∗𝑁0

𝑞=1

(4a)

or:

𝑀𝐴𝐶𝑐𝑑𝑟 = 𝜑𝑐𝑟 𝑇 𝜑𝑑𝑟

∗ 2

𝜑𝑐𝑟 𝑇 𝜑𝑐𝑟∗ 𝜑𝑑𝑟 𝑇 𝜑𝑑𝑟

∗ (4b)

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

𝑀𝐴𝐶𝑐𝑑𝑟 = 𝜑𝑑𝑟 𝐻 𝜑𝑐𝑟

∗ 2

𝜑𝑑𝑟 𝐻 𝜑𝑑𝑟 𝜑𝑐𝑟 𝐻 𝜑𝑐𝑟 (4c)

or:


𝑀𝐴𝐶𝑐𝑑𝑟 = 𝜑𝑑𝑟 𝐻 𝜑𝑐𝑟 𝜑𝑐𝑟 𝐻 𝜑𝑑𝑟

𝜑𝑑𝑟 𝐻 𝜑𝑑𝑟 𝜑𝑐𝑟 𝐻 𝜑𝑐𝑟 (4d)

or:

𝑀𝐴𝐶𝑐𝑑𝑟 = 𝑀𝑆𝐹𝑑𝑟 𝑀𝑆𝐹𝑑𝑟𝑐 (4e)

The modal assurance criterion takes on values from zero – representing no consistent

correspondence, to one – representing a consistent correspondence. In this manner, if the modal

vectors under consideration truly exhibit a consistent, linear relationship, the modal assurance criterion

should approach unity and the value of the modal scale factor can be considered reasonable. Note that,

unlike the orthogonality calculations, the modal assurance criterion is normalized by the magnitude of

the vectors and, thus, is bounded between zero and one.

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 the

modal assurance criterion. Invalid assumptions are normally the cause of this sort of potential error.

Even though the modal assurance criterion is unity, the assumptions involving the system or the modal

parameter estimation techniques are not necessarily correct. The assumptions may cause consistent

errors in all modal vectors under all test conditions verified by the modal assurance criterion.

1.1.3. Modal Assurance Criterion (MAC) Zero

If the modal assurance criterion has a value near zero, this is an indication that the modal

vectors are not consistent. This can be due to any of the following reasons:

The system is nonstationary. This can occur if the system is nonlinear and two data sets have

been acquired at different times or excitation levels. System nonlinearities will appear

differently in frequency response functions generated from different exciter positions or

excitation signals. The modal parameter estimation algorithms will also not handle the different

nonlinear characteristics in a consistent manner.

There is noise on the reference modal vector. This case is the same as noise on the input of a

frequency response function measurement. No amount of signal processing can remove this

type of error.

The modal parameter estimation is invalid. The frequency response function measurements

may contain no errors but the modal parameter estimation may not be consistent with the data.


The modal vectors are from linearly unrelated mode shape vectors. Hopefully, since the

different modal vector estimates are from different excitation positions, this measure of

inconsistency will imply that the modal vectors are orthogonal.

If the first four reasons can be eliminated, the modal assurance criterion can be interpreted

in a similar way as an orthogonality calculation.

1.1.4. Modal Assurance Criterion (MAC) Unity

If the modal assurance criterion has a value near unity, this is an indication that the modal

vectors are consistent. This does not necessarily mean that they are correct. The modal vectors can be

consistent for any of the following reasons:

The modal vectors have been incompletely measured. This situation can occur whenever too

few response stations have been included in the experimental determination of the modal

vector.

The modal vectors are the result of a forced excitation other than the desired input. This would

be the situation if, during the measurement of the frequency response function, a rotating piece

of equipment with an unbalance is present in the system being tested.

The modal vectors are primarily coherent noise. Since the reference modal vector may be

arbitrarily chosen, this modal vector may not be one of the true modal vectors of the system. It

could simply be a random noise vector or a vector reflecting the bias in the modal parameter

estimation algorithm. In any case, the modal assurance criterion will only reflect a consistent

(linear) relationship to the reference modal vector.

The modal vectors represent the same modal vector with different arbitrary scaling. If the two

modal vectors being compared have the same expected value when normalized, the two modal

vectors should differ only by the complex valued scale factor, which is a function of the

common modal coefficients between the rows or columns.

Therefore, if the first three reasons can be eliminated, the modal assurance criterion

indicates that the modal scale factor is the complex constant relating the modal vectors and that the

modal scale factor can be used to average, difference or sort the modal vectors.

Under the constraints mentioned previously, the modal assurance criterion can be applied

in many different ways. The modal assurance criterion can be used to verify or correlate an

experimental modal vector with respect to a theoretical modal vector (eigenvector). This can be done

by computing the modal assurance criterion between Ne modal vectors estimated from experimental

data and Na modal vectors estimated from a finite element analysis evaluated at common stations.

This process results in a Ne×Na rectangular modal assurance criterion matrix with values that


approach unity whenever an experimental modal vector and an analytical modal vector are

consistently related.

Once the modal assurance criterion establishes that two vectors represent the same

information, the vectors can be averaged, differenced or sorted to determine the best single estimate or

the potential source of contamination using the modal scale factor. Since the modal scale factor is a

complex scalar that allows two vectors to be phased the same and to the same mean value, these

vectors can be subtracted to evaluate whether the error is random or biased. If the error appears to be

random and the modal assurance criterion is high, the modal vectors can be averaged (using the modal

scale factor) to improve the estimate of a modal vector. If the error appears to be biased or skewed, the

error pattern often gives an indication that the error originates due to the location of the excitation or

due to an inadequate modal parameter estimation process. Based upon partial but overlapping

measurement of two columns of the frequency response function matrix, modal vectors can be sorted,

assuming the modal assurance function indicates consistency, into a complete estimate of each modal

vector at all measurement stations.

The modal assurance criterion can be used to evaluate modal parameter estimation

methods if a set of analytical frequency response functions with realistic levels of random and bias

errors is generated and used in common with a variety of modal parameter estimation methods. In this

way, agreement between existing methods can be established and new modal parameter estimation

methods can be checked for characteristics that are consistent with accepted procedures. Additionally,

this approach can be used to evaluate the characteristics of each modal parameter estimation method in

the presence of varying levels of random and bias error.

The concept of consistency in the estimate of modal vectors from separate testing

constraints is important considering the potential of multiple estimates of the same modal vector from

numerous input configurations and modal parameter estimation algorithms. The computation of modal

scale factor and modal assurance criterion results in a complex scalar and a correlation coefficient that

does not depend on weighting information outside the testing environment. Since the modal scale

factor and modal assurance criterion are computed analogous to the frequency response function and

coherence function, both the advantages and limitations of the computation procedure are well

understood. These characteristics, as well as others, provide a useful tool in the processing of

experimental modal vectors.

1.1.5. MAC Presentation Formats

One of the big changes in the application of the Modal Assurance Criteria over the last

twenty years is in the way the information is presented. Historically, a table of numbers was usually

presented as shown in Table 1.


Table 1.- Numerical presentation of MAC values.

Today, most computer systems routinely utilize color to present magnitude data like MAC

using a 2D or 3D plot as shown in Figures 1 and 2. It is important to remember, however, that MAC is

a discrete calculation and what appears as a color contour plot really only represents the discrete mode

to mode comparison. Nevertheless, a color plot does allow for more data to be presented in an

understandable form in a minimum space.

Figure 1.- 2-D and 3-D presentation of MAC values.


1.2. Other Similar Assurance Criteria

The following brief discussion highlights assurance criteria that utilize the same linear,

least squares computation approach to the analysis (projection) of two vector spaces as the modal

assurance criterion. The equations for each assurance criterion are not repeated unless there is a

significant computational difference that needs to be clarified or highlighted. This list is by no means

comprehensive nor is it in any particular order of importance but includes most of the frequently cited

assurance criterion found in the literature.

Weighted Modal Analysis Criterion (WMAC): A number of authors 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 weighting matrix is to recognize that MAC is not

sensitive to mass or stiffness distribution, just sensor distribution, and to adjust the modal assurance

criterion to weight the degrees-of-freedom in the modal vectors accordingly. In this case, the WMAC

becomes a unity normalized orthogonality – or pseudo-orthogonality – check where the desirable

result for a set of modal vectors would be ones along the diagonal (same modal vectors) and zeros off-

diagonal (different modal vectors) regardless of the scaling of the individual modal vectors. Note that

the weighting matrix is applied as an inner matrix product for the single numerator vector product and

both vector products in the denominator.

Partial Modal analysis Criterion (PMAC): The partial modal assurance criterion (PMAC) was

developed as a spatially limited version of the modal assurance criterion where a subset of the

complete modal vector is used in the calculation. The subset is chosen based upon the user‟s interest

and may reflect only a certain dominant sensor direction (X, Y and/or Z) or only the degrees-of-

freedom from a component of the complete modal vector.

Modal Assurance Criterion Square Root (MACSR): The square root of the modal assurance

criterion (MACSR) is developed to be more consistent with the orthogonality and pseudo-

orthogonality calculations using an identity weighting matrix. Essentially this approach utilizes the

square root of the MAC calculation, which tends to highlight the cross terms (off diagonal) that are

generally very small MAC values.

Scaled Modal Assurance Criterion (SMAC): The scaled modal assurance criterion (SMAC) is

essentially a weighted modal assurance criterion (WMAC) where the weighting matrix is chosen to

balance the scaling of translational and rotational degrees-of-freedom included in the modal vectors.

This development is needed whenever different data types (with different engineering units) are

included in the same modal vector to normalize the magnitude differences in the vectors. This is

required since the modal assurance criterion minimizes the squared error and is dominated by the

larger values.


Modal Assurance Criterion Using Reciprocal Vectors (MACRV): A reciprocal modal vector is

defined as the mathematical vector that, when transposed and premultiplied by a specific modal

vector, yields unity. When the same computation is performed with this reciprocal modal vector and

any other modal vector or any other reciprocal modal vector, the result is zero. The reciprocal modal

vector can be thought of as a product of the modal vector and the unknown weighting matrix that will

produce a perfect orthogonality result. Reciprocal modal vectors are computed directly from measured

frequency response functions and the experimental modal vectors and are, therefore, experimentally

based.

The modal assurance criterion using reciprocal modal vectors (MACRV) is the comparison of

reciprocal modal vectors with analytical modal vectors in what is very similar to a pseudo-

orthogonality check (POC). The reciprocal modal vectors are utilized in controls applications as modal

filters and the MACRV serves as a check of the mode isolation provided by each reciprocal modal

vector compared to analytical modes expected.

Modal Assurance Criterion with Frequency Scales (FMAC): Another extension of the modal

assurance criterion is the addition of frequency scaling to the modal assurance criterion. This extension

of MAC “offers a means of displaying simultaneously the mode shape correlation, the degree of

spatial aliasing and the frequency comparison in a single plot.” This development is particularly useful

in model correlation applications (model updating, assessment of parameter variation, etc.).

Coordinate Modal Assurance Criterion (COMAC): An extension of the modal assurance

criterion is the coordinate modal assurance criterion (COMAC). The COMAC attempts to identify

which measurement degrees-of-freedom contribute negatively to a low value of MAC. The COMAC

is calculated over a set of mode pairs, analytical versus analytical, experimental versus experimental or

experimental versus analytical. The two modal vectors in each mode pair represent the same modal

vector, but the set of mode pairs represents all modes of interest in a given frequency range. For two

sets of modes that are to be compared, there will be a value of COMAC computed for each

(measurement) degree-of-freedom.

The coordinate modal assurance criterion (COMAC) is calculated using the following approach,

once the mode pairs have been identified with MAC or some other approach:

𝐶𝑂𝑀𝐴𝐶𝑞 = 𝜑𝑞𝑟 𝜙𝑞𝑟

2𝐿𝑟=1

𝜑𝑞𝑟 𝜑𝑞𝑟∗ 𝜑𝑞𝑟 𝜑𝑞𝑟

∗𝐿𝑟=1

𝐿𝑟=1

(5)

Note that the above formulation assumes that there is a match for every modal vector in the two

sets and the modal vectors are renumbered accordingly so that the matching modal vectors have the

same subscript. Only those modes that match between the two sets are included in the computation.

The Enhanced Coordinate Modal Assurance Criterion (ECOMAC): One common problem

with experimental modal vectors is the potential problem of calibration scaling errors and/or sensor

orientation mistakes. The enhanced coordinate modal assurance criterion (ECOMAC) was developed

to extend the COMAC computation to be more aware of typical experimental errors that occur in


defining modal vectors such as sensor scaling mistakes and sensor orientation (plus or minus sign)

errors.

Mutual Correspondence Criterion (MCC): The mutual correspondence criterion (MCC) is the

modal assurance criterion applied to vectors that do not originate as modal vectors but as vector

measures of acoustic information (velocity, pressure, intensity, etc.). The equation in this formulation

utilizes a transpose and will only correctly apply to real valued vectors.

Modal Correlation Coefficient (MCC): One of the natural limitations of a least squares based

correlation coefficient like the modal assurance criterion is that it is relatively insensitive to small

changes in magnitude, position by position, in the vector comparisons. The modal correlation

coefficient (MCC) is a modification of MAC that attempts to provide a more sensitive indicator. This

approach is particularly important when using modal vectors in damage detection situations where the

magnitude changes of the modal vectors being measured are minimal.

Inverse Modal Assurance Criterion (IMAC): An alternative approach to increasing the

sensitivity of the modal assurance criterion to small mode shape changes is the inverse modal

assurance criterion (IMAC). This approach uses essentially the same computational scheme as MAC

but utilizes the inverse of the modal coefficients. Therefore, small modal coefficients become

significant in the least squares based correlation coefficient computation. Naturally, this computation

suffers from the possibility that a modal coefficient could be numerically zero.

Frequency Response Assurance Criterion (FRAC): Any two frequency response functions

representing the same input-output relationship can be compared using a technique known as the

frequency response assurance criterion (FRAC). The simplest example is a validation procedure that

compares the FRF data synthesized from the modal model with the measured FRF data. The basic

assumption is that the measured frequency response function and the synthesized frequency response

function should be linearly related (unity scaling coefficient) at all frequencies. Naturally, the FRFs

can be compared over the full or partial frequency range of the FRFs as long as the same discrete

frequencies are used in the comparison. This approach has been utilized in the modal parameter

estimation process for a number of years under various designations (parameter estimation correlation

coefficient, synthesis correlation coefficient and response vector assurance criterion (RVAC)). This

procedure is particularly effective as a modal parameter estimation validation procedure if the

measured data were not part of the data used to estimate the modal parameters. This serves as an

independent check of the modal parameter estimation process.

𝐹𝑅𝐴𝐶𝑝𝑞 = 𝐻𝑝𝑞 𝑤 𝐻𝑝𝑞

∗ (𝑤)𝑤 2𝑤 =𝑤 1

2

𝐻𝑝𝑞 𝑤 𝐻𝑝𝑞∗ (𝑤)

𝑤 2𝑤 =𝑤 1

𝐻𝑝𝑞 𝑤 𝐻 𝑝𝑞∗ (𝑤)

𝑤 2𝑤 =𝑤 1

(6)


Complex Correlation Coefficient (CCF): A significant variation in the frequency response

assurance criterion is the complex correlation coefficient (CCF), which is computed without squaring

the numerator term, thus yielding a complex valued coefficient. The magnitude of the coefficient is the

same as the FRAC computation but the phase describes any systematic phase lag or lead that is present

between the two FRFs. In situations where analytical and experimental FRFs are compared, the CCF

will detect the common problem of a constant phase shift that might be due to experimental signal

conditioning problems, etc.

Frequency Domain Assurance Criterion (FDAC): A similar variation in the frequency

response assurance criterion is the frequency domain assurance criterion (FDAC), which is a FRAC-

type of calculation evaluated with different frequency shifts. Since the difference in impedance (FRF)

model updating is often an FRF that is in question due to frequencies of resonances or anti-resonances,

the FDAC is formulated to identify this problem. A related criterion, the modal FRF assurance

criterion (MFAC), combines analytical modal vectors with measured frequency response functions

(FRFs) in an extension of FRAC and FDAC that weights or filters the FRF data based upon the

expected, analytical modal vectors.

Coordinate Orthogonality Check (CORTHOG): The coordinate orthogonality check

(CORTHOG) is a normalized error measure between the pseudo-orthogonality calculation, comparing

measured to analytical modal vectors, and the analytical orthogonality calculation, comparing

analytical to analytical modal vectors. Several different normalizing or scaling methods are used with

this calculation.

1.3. Uses of the Modal Assurance Criterion

Most of the potential uses of the modal assurance criterion are well known but a few may

be more subtle. A partial list of the most typical uses that have been reported in the literature are 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 estimation algorithms.

Repeated and pseudo-repeated root detection.

Structural fault/damage detection.

Quality control evaluations.

Optimal sensor placement.

1.4. Abuses of the Modal Assurance Criterion

Many of the alternate formulations of the modal assurance criterion were developed to

address some of the shortcomings of the original modal assurance criterion formulation. When users

utilize the original modal assurance criterion in these situations, a poor result will often follow. For the

purposes of this discussion, this is referred to as misuse or abuse. The misuse or abuse of the modal

assurance criterion generally results due 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 assurance 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 paragraphs.


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 spatial distribution of

the degrees-of-freedom included in the modal vectors. The modal assurance criterion does not weight

the modal vectors with a mass or stiffness matrix and, therefore, cannot compensate for situations

where a very limited number of degrees-of-freedom (sensors) have been placed on a massive sub-

structure of a mechanical system. The typical example involves the engine of an automobile. If few or

no sensors are placed on the engine and a large number are placed on the surface of the automobile

body, several modal vectors at different modal frequencies will have very high MAC numbers

indicating that the modal vectors are the same. This example indicates to the user that an incomplete

modal vector was measured and the user has violated one of the primary assumptions of experimental

modal analysis (observability).

The wrong mathematical formulation for the modal assurance criterion is used: Frequently,

users implement the modal assurance criterion, or a related similar computation, using a vector

transpose in the numerator and denominator calculations rather than an Hermitian (conjugate

transpose). This error causes no problem as long as analytical vectors or real-valued experimental

vectors are involved in the calculation. However, in the general case, where some of the vectors are

complex-valued, this does not give the correct result. The original mathematical formulation assumes

the general case but has been reported incorrectly in some literature. This innocent error often occurs

when the author is utilizing real-valued vectors and notices no problem. However, users who do not

recognize this issue are often led astray in subsequent applications involving complex-valued vectors.

The modal assurance criterion is sensitive to large values (wild points?) and insensitive to

small values: The modal assurance criterion is based upon the minimization of the squared error

between two vector spaces. This means that the degrees-of-freedom involving the largest magnitude

differences between the two modal vectors will dominate the computation while small differences will

have almost no effect. Therefore, nodal information (small modal coefficients) will generally not have

much effect on the MAC calculation and large modal coefficients will potentially have the greatest

effect. This also means that, if there have been erroneous data included in the modal vectors due to

calibration errors, modal parameter estimation mistakes, etc., these wild points may dominate the

MAC calculation.

The number of elements in the modal vectors (space) is small: Since the modal assurance

criterion is essentially a statistical computation where the number of averages comes from the number

of elements in the modal vectors, if the modal vectors have only a limited number of degrees-of-

freedom, this will skew the meaning of the numerical MAC value. This frequently happens when high

order, multiple reference modal parameter estimation algorithms estimate the stability or consistency

diagram. Modal vector stability or consistency is identified using a MAC computation where the

vectors include only the degrees-of-freedom at the reference locations, typically two to five. In these

situations, there may be great variability in the MAC computation, particularly if the modal vector is

not well excited from one or more of the reference locations. Vectors with many elements reduce the

sensitivity of MAC to this problem.


The modal vectors have been zero padded. Frequently, when modal vectors are exported from

one computational environment to another, the modal vectors include zero values when no value was

ever measured, or computed, for that degree-of-freedom. For example, in an experimental situation,

one (X) or two dimensions (X,Y) of translational response may be measured at some degrees-of-

freedom rather than three dimensions (X,Y,Z). In the commonly used Universal File Format for modal

vectors (File Format 55), this is the case since there is no designation for not measuring the

information. When the modal assurance criterion is calculated for this case, there will be 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 computation when either vector

includes a perfect zero (within computational precision) at a degree-of-freedom, but is rarely done.

1.5. Current Developments

Currently, many users are utilizing more statistical approaches to understand the meaning

and bounds of experimental modal parameters. This approach extends to the modal assurance criterion

as well. Examples are the bootstrap and jackknife approaches to the evaluation of the mean and

standard deviation of discrete sets of experimental data. These approaches remove and/or replace

portions of the computation (bootstrap uses replicative resampling, jackknife uses sequential

elimination) to evaluate the bounds or limits on the MAC values. In this way, the sensitivity of the

MAC computation can be more effectively evaluated than with the current single number indicating

the degree of linearity between two modal vectors that are being compared.


2. Software:

One of the most used computer program, and recommended by the Head of Dynamics

Programs of the CTA, is the LMS Virtual.Lab. LMS Virtual.Lab offers an integrated software suite

to simulate and optimize the performance of mechanical systems for structural integrity, noise and

vibration, system dynamics and durability. LMS Virtual.Lab covers all the process steps and required

technologies to perform an end-to-end design assessment in each key discipline. Using LMS

Virtual.Lab, engineering teams can build accurate simulation models, simulate their real-life

performance, quickly assess multiple design alternatives and optimize designs before prototype

construction.

2.1. LMS Virtual.Lab Overview

LMS Virtual.Lab Desktop: LMS Virtual.Lab Desktop provides a common environment for

multiple functional performance applications. With LMS Virtual.Lab Desktop, users have seamless

access to models and load data, geometry and simulation models from industry-standard CAD and

CAE tools as well as test data. LMS Virtual.Lab Desktop also offers a complete visualization

environment for part and assembly models, functional performance engineering data, time and

frequency functions and much more.

Figure 2.- LMS Virtual.Lab Desktop.


LMS Virtual.Lab Structures: LMS Virtual.Lab Structures offers a scalable solution for

structural modeling and analysis, integrating advanced model creation and manipulation tools to

efficiently generate component, subsystem and full-system models. LMS Virtual.Lab Structures offers

full meshing capabilities and captures the complete modeling and analysis process from CAD drawing

to multiattribute simulation results. It offers multi-solver support for Abaqus, Ansys, CATIA CAE and

Nastran (MD, MSC, NX, NEi).

Figure 3.- LMS Virtual.Lab Structures.

LMS Virtual.Lab Motion: LMS Virtual.Lab Motion offers a highly efficient, completely

integrated solution to build multibody models that simulate the full-motion behavior of complex

mechanical system designs. Users can easily create a complete and accurate system model from

scratch or import geometry models from any industry-standard CAD system. LMS Virtual.Lab Motion

applies forces and motion to simulate the actual operational behavior of the new design. The resulting

simulation is excellent input to optimize the design‟s dynamic performance. The resulting loads can

also be used for structural analysis, durability, and noise and vibration studies.

Figure 4.- LMS Virtual.Lab Motion.


LMS Virtual.Lab Acoustics: LMS Virtual.Lab Acoustics offers an integrated solution to

minimize radiated noise or optimize the sound quality in new designs. Convenient modeling

capabilities combined with efficient solvers and easy-to-interpret visualization tools enable users to

quickly gain insight to the acoustic performance of their product. LMS Virtual.Lab Acoustics

simulates both internal and external acoustic radiation and offers dedicated applications for structural

noise radiation, engine acoustics, transmission loss through panels, aero-acoustic phenomena and

much more.

Figure 5.- LMS Virtual.Lab Acoustics.

LMS Virtual.Lab Noise and Vibrations: LMS Virtual.Lab Noise and Vibration is developed to

efficiently analyze, refine and optimize the vibro-acoustic behavior of a design. It offers all the

required tools to create system-level models, build realistic load cases and simulate noise and vibration

responses. It includes a wide range of visualization and analysis tools to analyze noise and vibration

performance and accurately pinpoint the most critical contributors to noise and vibration issues.

Convenient tools enable engineers to quickly perform design modifications and assess the noise and

vibration performance of a design variant in minutes.

Figure 6.- LMS Virtual.Lab Noise and Vibrations.


LMS Virtual.Lab Correlation: LMS Virtual.Lab Correlation allows users to combine test-based

and virtual component models into system-level models for more productive simulation. It offers

direct access to standard FE and test data formats and a unique export to LMS Test.Lab. LMS

Virtual.Lab quickly compares and validates FE models to test data and identifies possible modeling

errors to systematically improve existing simulation models.

Figure 7.- LMS Virtual.Lab Correlation.

LMS Virtual.Lab Durability: LMS Virtual.Lab Durability allows engineers to design reliable

products right from the start. It predicts fatigue hotspots and system-level fatigue life by combining

dynamic component loads with stress results and fatigue material parameters. LMS Virtual.Lab

Durability provides direct feedback regarding critical fatigue areas and the root cause of fatigue

problems. This immediate insight enables engineering teams to validate more design variants for

fatigue life within ever-shorter development cycles.

Figure 8.- LMS Virtual.Lab Durability.


LMS Virtual.Lab Optimization: LMS Virtual.Lab Optimization lets design and engineering

teams automatically select the optimal design while accounting for multiple performance targets.

Users can easily identify the key variables that have the most influence on the functional performance

of a mechanical system. LMS Virtual.Lab Optimization automatically explores a multitude of design

alternatives using design of experiment and response surface modeling techniques. It also analyzes

design robustness and reliability according to Design for Six Sigma criteria.

Figure 9.- LMS Virtual.Lab Optimization.

2.2. LMS Virtual.Lab Correlation

To guarantee realistic high fidelity simulations, it is essential that simulation models meet

stringent accuracy standards. Ensuring reliable simulation results requires component, subsystem and

full-system models to be compared with experimental data, or alternatively validated models of similar

structures. Building and validating system models from the bottom up is the only way to prevent

accumulating inaccuracies. Besides more reliable what-if analyses, validated models provide a better

understanding of assumptions made regarding material properties, connections, joints and boundary

conditions.

Correlating structural characteristics: Although static physical tests serve many design

purposes, models used for vibro-acoustic simulations usually require systematic test-based validation

of dynamic properties. LMS Virtual.Lab Correlation helps correlate physical test results and prepare

structural tests. A comprehensive tool set significantly facilitates simulated and measured mode shape

comparison and operational deflection shapes and response functions. Using the original FE model as

a basis to provide optimal comparison positions, and the required number of excitation and response

points helps avoid testing errors and redundancy.


Validation-driven model updating: Deducing model improvements using validation output is

not always obvious. To facilitate model updating driven by validations, LMS Virtual.Lab Correlation

offers specialized features to identify specific locations that need improvement. For example, it runs

sensitivity analyses that efficiently retain the most influential specified parameters. Users can also

automatically update models using internal and external algorithms, such as Nastran Solution 200,

which focuses on tuning modal frequencies and response functions.

2.2.1. Systematic validation from the bottom up

Figure 10.- Systematic Validation from the bottom up.

Pre-Test: When preparing measurements for physical structures, one can use modal information

of preliminary Finite Element models to define the optimal measurement set-up. For a modal test set-

up, this means defining a set of measuring points and excitation points. LMS Virtual.Lab Correlation

provides tools to quickly carry out this pre-test analysis in a user-friendly way. The objective is to

obtain a measurement set-up that guarantees high quality measurement data.

Correlation: Once good test data for the physical model is available, LMS Virtual.Lab

Correlation allows its users to quantify the geometrical and dynamic (FRF and Modal) resemblance

between the test model and its FE equivalent model. Several correlation metrics, like MAC and

FRAC, are available to study (mode) shape or frequency response function correlation interactively.

Specialized algorithms and post-processing tools allow to localize the problem locations of bad shape

correlation and give insight in stiffness differences between the two models.

Sensitivity and updating: After the dynamic correlation between two models has been

quantified, LMS Virtual.Lab allows users to easily setup and drive Nastran Sol200 to obtain the

sensitivity of FE dynamic properties towards a set of design parameters to decide which parameters to

change to obtain better correlation results. Using Sol200 sensitivities, MAC and frequency difference


sensitivities are derived to use for modal updating. A broad range of sensitivities can also be computed

using LMS Virtual.Lab Optimization: the user can define a variety of dynamic properties to optimize

for a very broad range of design parameters. Once the set-up of design parameters (inputs) and

correlation metrics (outputs) is in place, LMS Virtual.Lab offers the possibilities to carry out Design of

Experiments, Response Surface Modeling and Updating with several local and global optimization

algorithms.

2.2.2. LMS Virtual.Lab Correlation. Features and Benefits

LMS Virtual.Lab Correlation offers tools to ensure that high-quality FE models are used in

a CAE environment and that correct sensor and excitation locations are employed in a dynamic

physical structure test environment.

For pre-test analysis, users can create an optimal test geometry from an existing FE model.

LMS Virtual.Lab Correlation interactively creates a test wireframe on top of the FE mesh and directly

quantifies its quality according to relevant mode capturing and modal excitation. In case of poor

sensor location set quality, LMS Virtual.Lab Correlation provides an easy way to analyze why the

model was off-target. Users can easily change the test geometry and directly assess new quality levels

using the MAC (Modal Assurance Criterion). The DPR (Driving Point Residue) criterion is used for

the excitation point set.

LMS Virtual.Lab Correlation also lets users easily and quickly compare the dynamic

behavior of two models and deal with incompatible meshes (test and/or FE). It helps users to

quantitatively articulate the degree of shape correlation using a MAC matrix. If the MAC values are

too low to subjectively correlate the modes, the MAC Contribution (MACCo) criterion points out the

differences to be examined. In this way, users can verify different modeling assumptions by

comparing reference or measurement data. This improves model and simulation reliability. An

orthogonality check between two models adds a degree of correlation accuracy by using the mass

matrix to compare system dynamics. For this, LMS Virtual.Lab Correlation sets up the Nastran DMIG

Solution to obtain reduced system mass matrices required for orthogonality checks between test and

FE modes. The FRAC (Frequency Response Assurance Criterion) compares transfer functions

between two models and provides information about global stiffness and mass modeling errors.

Features:

Universal access to test and FE data for models, modes and frequency spectra.

Modal Assurance Criterion (MAC) and MAC Contribution (MACCo) support error

localization.

Visual Shape correlation for side-by-side model animation (FE or Test).


Frequency Response Assurance Criterion (FRAC).

Orthogonality check for better dynamic correlation.

Driving Point Residue (DPR) for shaker location identification.

Export data to LMS Test.Lab or a universal file format.

Benefits:

Maximum test information with minimized excitation and measurement locations.

Increase measurement productivity with direct LMS Test.Lab integration.

Confirm FE simulation model validity using measurements.

Identify modeling errors or evaluate modeling strategies.

Improve simulation model reliability.

Figure 11.- LMS Virtual.Lab Correlation uses.


2.2.3. LMS Virtual.Lab Model Updating. Features and Benefits

LMS Virtual.Lab Model Updating is a model correlation and updating tool that improves

simulation model quality based on reference data. With LMS Virtual.Lab Model Updating, analysts

can make models that match reality more closely. FE models are first correlated with reference

models, which are typically test models, but can also be FE models. The next step is to compute

dynamic property sensitivity with respect to design parameter uncertainties. This can be done by

inserting a Nastran Sol200 case from LMS Virtual.Lab Desktop. In this way, users can easily define

element group properties for sensitivity analysis, including material and property data. Dynamic

targets can be the total system mass, a specific eigenfrequency that is poorly correlated, vibration

levels for unit load conditions or mode shapes. Sensitivity information is then used to update or

optimize the Nastran model to match real-life condition better. For non-Nastran users, the FE model

can still be updated or optimized, using LMS Virtual.Lab Optimization.

LMS Virtual.Lab Model Updating easily handles incompatible geometries that typically

occur when comparing test and FE models. Models can be correlated geometrically through alignment,

sizing and mapping procedures. LMS Virtual.Lab Model Updating provides numerical tools, such as

MAC (Modal Assurance Criterion), FRAC (Frequency Response Assurance Criterion) as well as tools

to check orthogonality between two models, directly driving the Nastran Guyan reduction.

The MAC combined with Nastran Sol200 sensitivities for mode shapes and

eigenfrequencies helps users to compute and study MAC and frequency difference sensitivity for

mode pair sets. These sensitivities help obtain the best dynamic match between two models. LMS

Virtual.Lab Model Updating can deal adequately with mode switching during the updating process.

This ensures that the correct FE shape is used in correlation with the reference model during the

automated updating process.

Features:

Input design parameters for material and element properties.

Targets for mass, modal frequencies and vibration levels.

Modal Assurance Criterion (MAC), Mode Pair Table and MAC Contribution (MACCo).

Frequency Response Assurance Criterion (FRAC).

Frequency difference sensitivity and MAC sensitivity.

DOE, Response Surface Modeling and several optimization algorithms.


Benefits:

Confirm FE simulation model validity using measurements.

Identify modeling errors or evaluate modeling strategies.

Improve Nastran model reliability with integrated optimization capabilities.

Figure 12.- LMS Virtual.Lab Model Updating Applications.


3. Uses for aircraft design and testing/certificating companies

In the following pages two specific examples, where the modal assurance criterion is used,

will be presented: “Using MSC/NASTRAN and LMS/PRETEST to find an optimal sensor placement

for modal identification and correlation of aerospace structures” and “Modal Test of L-610G

Aeroplane”. In both cases, the use of the MAC is essential to validate and/or check the tests performed

and the predicted FEA results.

3.1. Using MSC/NASTRAN and LMS/PRETEST to find an

optimal sensor placement for modal identification and correlation of

aerospace structures

The objective of an effective integration of finite element analysis with structural testing is

to combine the advantages of both approaches in a more valuable synergistic approach. The analytical

approach is predictive and can be used for predicting the flight loads and assessing the structural

integrity prior to the prototype production. The experimental approach, based on modal surveys on the

prototype, observes the actual behavior of the structure under controlled laboratory (ground vibration

test) or real operating conditions (in-flight testing). The benefits of such a combined approach are that:

Testing provides reliable information to cross-check predicted FEA results (Correlation

Analysis): Testing can provide reliable estimates for system damping and resonance frequencies.

Furthermore, analyzing the experimentally obtained mode shapes, and comparing them with the

results from FEA, is critical in assessing the value of the analytical model and its interpretation. After

the difficult geometry mapping (geometrical correlation) that aligns both topologies, several modal

based assessment criteria are used to validate the analytical model. In aerospace, commonly used tools

therefore are the Modal Assurance Criteria (MAC) and the Cross-Orthogonality Criteria.

The modal assurance criterion (MAC) is used to evaluate the correlation between two modes

ignoring the effects of the system mass. It is an easy criterion and has been used primarily to check the

independence of two modes.

The cross-orthogonality is used to identify the corresponding test mode that associates with an

analytical mode, including the effects of system mass. A generally accepted requirement for the cross-

orthogonality is to have all diagonal terms larger than 0.9 and all the off-diagonal terms less than 0.1.

Since the outcome of the cross-orthogonality calculation is also dependent on the quality of the

measured test modes, the orthogonality matrix of the test modes with respect to the analytical reduced

mass matrix is used to assess the quality of thereof. The test data is acceptable if the off-diagonal terms

of this orthogonality matrix are less than 0.1 when the diagonal terms are normalized to 1.0.


The requirement for modal frequencies of corresponding experimental and analytical modes is to

have a discrepancy within 5%. If both criteria, cross-orthogonality and frequency discrepancy, are met,

the analytical model is said to be test-verified.

This in-depth correlation analysis will provide understanding of the discrepancies between the

analytical results and the test results, and will teach the designer how to improve his design.

Testing results can be used to enhance the Analytical Model (FEA model Updating): The

outcome of the correlation analysis will decide if it is necessary to modify the analytical model so that

it better describes the results observed from testing. An improved analytical model is obtained by

changing analytical model parameters such that the discrepancy between test and FE resonance

frequencies is minimized. Such a structural optimization (updating) can be performed using the

MSC/NASTRAN Sol200 capability and thus are the changeable parameters shell thickness, beam

cross-sections, spring stiffnesses and such.

The FEA results can be used to better design the Test (Pretest Analysis): FEA information can

complement the Test Engineer‟s expertise in selecting optimal ways of stimulating and measuring the

dynamic behavior of the test structure. Moreover, it will make the geometry mapping of both

topologies trivial since the experimental geometry was originally created from the FEA model.

This synergistic approach consists thus of the following steps, see Figure 13:

1. FEA Modeling and Analysis, using MSC/PATRAN and MSC/NASTRAN

2. Pretest Analysis using MSC/NASTRAN and LMS/PRETEST

3. Modal Testing & Analysis using LMS CADA-X Modal

4. Correlation Analysis using LMS/Correlation and MSC/NastranForLink

5. FEM Model Updating using LMS/Updating and MSC/NASTRAN Sol200

Figure 13.- Linking test and FE.


3.1.1. MSC/NASTRAN and LMS/PRETEST

A typical pre-test analysis will usually consist of different steps, see Figure 14. Starting

from a CAD model, an analytical model is created and the dynamic behavior is calculated in terms of

resonance frequencies, mode shapes and system‟s mass and stiffness matrices. Out of all these modes,

a limited set of target modes has to be selected and sensors and shakers have to be placed such that

they efficiently capture and excite all of these target modes. Lots of techniques and methodologies

have been developed already and are still being developed and most of them are implemented by

means of user programming (DMAP) in MSC/NASTRAN.

Using both MSC/NASTRAN and LMS/PRETEST in combination offers the structural

dynamicist an additional surplus because the outcome of most of his MSC/NASTRAN dynamic

calculations becomes available for interpretation in nice displays at the same time. In addition to that,

LMS/Pretest offers some additional tools.

Figure 14.- the different steps in a pre-test analysis.

3.1.2. Target Mode Selection

A first, very important step in the pre-test analysis is the selection of the target modes,

especially since the modal density of launch vehicles and other aerospace structures within the

frequency range of interest is usually very high. It is however not necessary to „capture‟ all these

closely spaced modes during a modal survey test, because only some of them will contribute

significantly to the critical component responses. These critical responses are usually located in the

areas of hardware concern.

These important structural modes are called „target‟ modes, and their selection is critical

for the generation of a validated analytical model. A test-verified model will, by definition, have a

good correlation between the test target modes and the analytical target modes, but not necessarily for


the non-target modes. It follows that a poor selection of the target modes could result in an analytical

model, which would not accurately predict the structural responses and member loads.

Before any criteria are used to determine the target modes, all modes coming from an

analytical modal analysis should be described in detail by a „simple‟ visual inspection. This inspection

gives the fundamental insight in the modal behavior of the structure and will also serve to interpret all

used target mode selection criteria.

Generally, there are four methods or combinations thereof widely used in the aerospace

industry. These are the rigid body modal effective mass, the constraint modal effective mass, the

modal kinetic energy fraction and the modal strain energy fraction. Another method, which uses a

somewhat different approach, is the use of modal participation factors. Besides these, also other

techniques are reported already.

Rigid Body Modal and Constraint Modal Effective Mass: The rigid body effective mass

associated with each deformation mode represents the amount of system mass participating in that

mode. Therefore, a mode with a large effective mass is usually a significant contributor to the system‟s

response. These criteria are in other words used to find the important system modes. A typical

requirement for the selection of target modes is that modes with a translational effective mass equal to

or greater than 2 percent of the total mass are target modes. If the modes are calculated using mass

normalization, the formula becomes:

𝑀𝑒𝑓𝑓 = 𝜙𝑑 𝑇 𝑀𝑠 𝜙𝑟𝑏 2

Note that this is the same as the root of the mass orthogonality between the deformation modes and the

rigid body modes. The constraint modal effective mass is similar to the rigid body modal effective

mass, but the constraint modes are used instead of the rigid body modes. This formula makes more

sense if the component (e.g. payload) is over constrained.

Kinetic/Strain Energy and Kinetic/Strain Energy Fraction: Since the modal effective mass

criteria look at the structure‟s dynamic behavior on a global basis, they are usually able to identify the

important system modes but they are less useful for the determination of important local modes. To

include the significant local modes of a subsystem for improving the response prediction, the kinetic

and/or strain energy fraction of that subsystem is calculated. The kinetic energy fraction is defined as

the amount of kinetic energy in that subsystem relative to that of the whole system. The selection

criterion to consider a component mode as target mode is an energy content of 50% of the total system

energy. These target modes will be added to the target mode set if not yet been selected by the

previous criteria. If the modes are again mass normalized the formula for the kinetic energy fraction

becomes:

𝐾𝐸𝐹 =𝐷𝑖𝑎𝑔 𝜙𝑐

𝑇 𝑀𝑐 𝜙𝑐

𝐷𝑖𝑎𝑔 𝜙𝑠 𝑇 𝑀𝑠 𝜙𝑠 = 𝐷𝑖𝑎𝑔 𝜙𝑐

𝑇 𝑀𝑐 𝜙𝑐


The kinetic energy fraction for the first deformation mode of a scale model of a Boeing 747 is shows

in Figure 15 on top of the geometry. The fuselage, both wings including engines and the tail wings are

clearly visible as being the different components. Ultimately, it is possible to visualize the kinetic

energy of each element in model separately, see Figure 16.

Figure 15.- The modal kinetic energy for the first deformation

mode for several parts of Boeing 747. The lower plane is the undeformed

mode shape and each color represents a different group.

Figure 16.- The modal kinetic energy for the first deformation

mode for each element separately.

Mode Participation Factors: Although the previous methods may identify the most of the target

modes, some relevant modes critical to the payload or component responses may not be selected

because none of them takes the excitation into account. The structural integrity depends not only on

the structure‟s resonance frequencies, the mode shapes and the damping, but also on the frequency

characteristics of the excitations. Therefore, a tool that includes the excitation characteristics in the

target mode selection process will ensure the completeness of the target mode set. Useful in this

context are the mode participation factors, which are calculated during the dynamic solution

sequences, defined as (if mass normalization is used):


𝑃𝐹𝑖 = 𝜙 𝑇𝐹𝑖

𝜆𝑠2 − 𝑤2

The output is related to these participation factors by:

𝑉 = 𝑃𝐹𝑖 𝜙𝑖 𝑁

𝑖=1

Important is that these participation factors are independent of the output. The participation

factor is frequency dependant and its amplitude is determined by the structure‟s resonance behavior

(for w close to ls) and by the excitation spectra (for w far from ls), as can be seen in Figure 17, for the

PF of the first five modes. Plotting the participation factors of all modes for a certain frequency band

of interest results in the colormap diagram of Figure 18. It is now easy to investigate if some modes

are still being missed in the target mode set.

Figure 17.- The first five PF. On the x-axis the frequency

bandwidth, on the y-axis the amplitude of the PF.

Figure 18.- The PF for each mode (each vertical line

is a PF) in a colormap display.


3.1.3. Sensor Placement

Once the set of target modes has been defined, the measurement locations and their

corresponding degrees of freedom have to be chosen such that all target modes can be observed by the

modal survey test. This can be extremely though for large space structures where the target modes can

be closely spaced. Since it is not practical to instrument the test article in all degrees of freedom

corresponding to those of the analytical model, the challenge is to use a minimal number of sensors,

especially for in-flight testing, in order to sufficiently define the spatial resolution of all the target

modes. An erroneous or too limited subset of sensor locations will lead to an incomplete geometric

definition of the mode shapes, a phenomenon that is called „Spatial Aliasing‟.

To asses the correlation of the mathematical model predictions which in general do not

have dynamic degrees of freedom uniquely one to one with the modal test measurements, a reduction

(usually Guyan) to the test-analysis model (TAM) is required. Since this dynamic reduction is done

towards the measured degrees of freedom, the choice of the sensor set is also extremely important for

the outcome and the interpretation of the dynamic correlation tools.

Since the analytical model sizes of complete assemblies are way too big for a manual

selection of the sensor locations, a systematic approach in which the test engineer‟s experience is

central, see Figure 19, must thus be used. First a sensor set is searched to meet the observability

criterion. This set is then eventually modified to obtain a qualitative TAM model.

Figure 19.- Systematic approach to find the optimal sensor (and shaker) set.


Sensor set definition to meet the observability criterion: The methodology used to evaluate the

quality of a possible subset of the available analytical nodes and their corresponding degrees of

freedom, usually all three translational degrees of freedom, is a Modal Assurance Criterion (MAC)

calculation. If the off-diagonal terms of this MAC matrix are smaller than 0.1 or 0.2, the cross-

correlation between the target modes is sufficiently low and the chosen set of measurement point will

be able to observe all target modes.

If the initial group of points is not able to discriminate all target modes, a maximum

offdiagonality MAC (MODMAC) can be launched. This algorithm aims at the completion of the

initial subset with extra points/degrees of freedom that are chosen out of an additional subset such that

a resulting group of points/degrees of freedom is kept that, given a set of target modes, shows

offdiagonal MAC values below a given threshold. The initial group, the additional group and the target

modes are the only input to this algorithm.

Although MAC and MODMAC calculations are straightforward and powerful, the results and

especially the final amount of sensors still depend on the quality of the selection of initial and

additional set of possible measurement points. Different tools may assist the test engineer‟s experience

in the selection of those groups.

Master DOF Selection – Geometrical Spread: This tool constructs a group with a user-

specified number of nodes that are maximally spread out over the structure. The

spreading can be performed either on all nodes of the structure, as is illustrated in Figure

20 where 50 points are spread out over the outer shell of the X-33 Advanced Technology

Demonstrator. To avoid the risk of clustering, it is possible to ask for a minimal distance

between the chosen locations. If a lot of component target modes were selected, it is also

necessary to have a sensor distribution on these components. Figure 21 shows a spread of

nodes on the internal LO2 tanks of the X-33.

Figure 20.- 50 nodes (triax) spread of the whole

structure of the X-33: Reusable Launch Vehicle.


Figure 21.- 50 Points spread of an internal tank of the X-33. A

character line wireframe gives the position of the tank in the whole model.

Mode Shape Summation: This tools calculates the sum of a set or subset of (target)

modes and for this set of modes and within the selected nodes (assembly or component),

a user-specified number of nodes with the highest (summed) deformation will be

grouped. An example for a part of a satellite is shown in Figure 22. The summed mode

shape is shown together with the two most moving points.

Figure 22.- The summed mode shape (in color and deformation)

together with the undeformed mesh. Two points were asked as output.

Group definition tools: Of course also different tools are available such as manual

group creation and editing, creation of groups by clicking points in the geometry,

grouping all nodes that correspond with a certain element type…

A possible strategy to find an optimal set of measurement points can be starting with a

relative small number of a-priori know response locations and launching a MODMAC

with as additional group a spread of points over the structure. If the target threshold

cannot be reached, for instance because there are local component modes amongst the


target modes, a second MODMAC can be launched with an additional group that

contains a spread of points only of that component…

Sensor set definition to meet also the cross-orthogonality criterion: Once a set of possible

measurement locations is found that meets the observability criterion, one still has to check if this set

of points can be used to obtain a high quality TAM model, by performing the actual reduction in

MSC/NASTRAN. If we suppose Guyan reduction, one can check if the mass distribution by the

calculation of the orthogonality between the spatially reduced modes and the Guyan reduced mass

matrix.

𝑋𝑂𝑅 = 𝜙 𝑇 𝑀𝑇𝐴𝑀 𝜙

An example is given in Figure 23. The target put forward for this orthogonality matrix is that the

diagonal terms are larger than 0.9, and the off-diagonal terms are smaller than 0.1 Therefore, the TAM

model produced by the chosen set of measurement points in the example, is only valid for the first 13

modes (including the 6 rigid body modes).

Figure 23.- Cross-Orthogonality using original

modes and Guyan reduced mass matrix.

It is also necessary to check if the modes from the TAM model are similar to the original target

modes. The correlation between the original target modes and the reduced TAM modes and the

resonance frequency discrepancy can be investigated using a MAC calculation. Typical for Guyan

reduction is that the reduction deteriorates for higher frequencies, as can be seen also in the example of

Figure 24. It is obvious that the current set of measurement points is only valid for the first 7 target

modes, although it is possible that more target modes can be observed using this set.


Figure 24.- Correlation between the original target modes and

the reduced TAM modes and the resonance frequency discrepancy.

If the cross-orthogonality and modal assurance criteria are not met, it is possible to add some

extra measurement points to the A-set. This process is more or less trial and error. If using Guyan

reduction, it is however possible to use the Master DOF selection – Ratio M/K tool.

3.1.4. Shaker Positioning

The third stage in the pre-test analysis is the selection of the exciter locations out of the

resulting group of measurement points in order to optimally stimulate all the modes of interest. If the

structure were to be excited close to a node of a particular mode, the corresponding resonance would

be difficult to observe in the measurement data, and the experimental modal model would be hard to

identify.

The tool that is used in LMS/Pretest for the selection of excitation locations is the

calculation of the „driving point residues‟ (DPR‟s). DPR‟s are stated to be equivalent to modal

participation factors, and are a measure of how much each mode is excited, or participated in the

overall response, at the driving point. As such also the modal participation factors in all possible

measurement points can be used. The definition of the driving point residue, for mode k and node i, is:

𝐷𝑃𝑅𝑘 𝑖 =𝜙𝑖𝑘

2

2𝑚𝑘𝑤𝑘

The degrees of freedom with maximum average DPR over all mode shapes are considered to be the

best excitation dofs for the specific set of target modes. An example is given for a tail boom problem

of a helicopter. The averaged DPR for all target modes is given, and the amplitude and the direction of

the red arrow show the best position and direction to place the shaker.


Figure 25.- Averaged DPR for all possible excitation points.

Figure 26.- A typical tail boom mode together

with the undeformed mesh.


3.2. Modal Test of L-610G Aeroplane

The L-610G is a high-wing monoplane with a T-shaped tail and a pressurised cabin

powered by two General Electric turboprop engines. Maximum take-off mass with forty passengers is

15 000 kg. The test was intended to determine the effect of significant structural changes on

aeroplane's modal characteristics and to obtain data to tune up an analytical finite-element model and

also to carry out detailed investigation into flight control circuits in all failure-present conditions

tolerated by the rules.

During the test the aeroplane was standing on under-inflated tyres of the main landing

gear, the nose fuselage was elastically suspended. Illustration of L-610G model test arrangement is

shown in Figure on envelope. A total of 19 exciters were used for excitation, the response was

measured in 280 points. The flap plays, which cause disturbing shocks and deteriorate measurement

results, were eliminated by additional masses of 20 kg suspended on the trailing edges by soft rubber

bundles. The additional mass natural frequency was less than 1 Hz and so it did not have to be

considered in the aeroplane total weight. The rubber bundle stiffness was taken into account in making

calculation corrections for additional masses and stiffness of exciters and transducers and for

aeroplane suspension.

The initial part of the test consisted in identification of the aeroplane natural frequencies.

Some important parts of the test are FRFs from all 280 transducers measured as recorded in different

configurations of swept sinusoidal excitation.

The aeroplane modal parameters were investigated by the method of sinusoidal excitation

of isolated normal modes (method of appropriated forces). Relative damping and generalised mass

were measured by two techniques, the complex power method and the method of forces in quadrature.

One problem of every modal test is evaluating the quality of received natural frequencies,

generalised masses, damping and modal vectors. Factors influencing modal test results fall into three

categories:

Properties of the structure tested - modal density; linearity; damping intensity and

distribution; access to vibrating structural parts.

Effect of the experiment - suspension of the structure tested; number, position and

magnitude of exciting forces; number and position of transducers; influence of moving

parts of exciters, transducers and suspension; method used; time for experiment

available; experimenter's skill.

Technical level of experimental facilities - calibration; proper use of particular circuits.

The disturbing effect of the factors mentioned above must be minimalised. After

measuring each mode the result must be evaluated immediately so as to clear up uncertainties, if any,

or repeat the measurement. At the L-610G test several criteria were used for verification of linearity,


effects of moving parts of the test equipment and quality of natural modes isolation. Mode shapes were

checked by their graphic representation and also by verifying their orthogonality and by Modal

Assurance Criteria (MAC).

Table 2 shows the matrix of generalised masses verifying orthogonality of the L-610G

symmetric modes. Table 3 contains Auto-MAC values calculated for the same modes for all 280

measured points. In the two matrices there are following modes:

Mode f[Hz] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

SWB1 4,17 1 100

SEMVB 6,69 2 4,7 100

FVB1 8,41 3 0,1 -4,4 100

SWHB1 9,34 4 6,4 0,8 2,5 100

SWB2 10,83 5 3,1 -2,6 6,7 2,8 100

SEY 11,57 6 4,5 -2,4 -4,2 -3,2 -2,9 100

SEP 14,05 7 -2,4 3,4 -4,5 -4,7 -3,3 0,8 100

SEMHB 15,13 8 5,9 -7,0 -1,4 12,9 1,5 12,8 -5,3 100

SHTB1 15,88 9 -0,9 0,9 0,8 2,2 0,0 -0,4 -0,7 9,1 100

SWHB2 21,47 10 2,5 -0,9 -1,3 3,7 7,3 5,5 4,9 25,5 1,5 100

SWB3 22,8 11 -0,3 -3,2 -7,8 -4,2 -1,6 -0,7 7,4 0,2 1,1 23,0 100

SWT1 27,81 12 6,2 4,0 0,1 9,9 3,8 14,0 -4,9 5,2 -0,4 5,5 13,6 100

SWB4 36,78 13 -7,0 -2,9 4,1 15,7 7,5 -4,1 -7,6 3,5 1,4 -0,3 -6,3 13,4 100

SWHB3 46,1 14 -2,2 1,5 4,4 0,9 3,7 -1,6 -13,3 3,4 -0,6 -8,9 3,6 1,8 18,9 100

SWT2 46,6 15 4,8 -2,2 4,4 -8,1 -1,2 6,5 -1,5 3,3 1,3 10,6 12,1 1,0 -2,0 25,4 100

SHTHB1 54,0 16 -4,2 5,3 21,0 -1,8 -5,9 -9,5 -1,0 -5,9 -3,9 19,9 4,7 -2,0 -4,0 -4,3 -3,2 100

SHTT1 56,46 17 0,7 -0,1 7,6 0,3 -7,0 0,8 -3,8 0,7 10,2 -2,3 -2,8 0,1 0,5 0,8 0,0 25,1 100

SHTB2 60,1 18 0,6 -1,0 -4,5 -1,2 -5,5 -1,9 2,5 -0,7 -4,5 4,5 3,1 -1,0 0,8 -1,5 -1,0 -6,4 12,2 100

Table 2.- Orthogonality of L-610G aircraft symmetric mode shapes.


Mode f[Hz] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

SWB1 4,17 1 100

SEMVB 6,69 2 24,3 100

FVB1 8,41 3 0,3 0,4 100

SWHB1 9,34 4 13,9 2,1 16,8 100

SWB2 10,83 5 23,3 2,4 6,3 11,3 100

SEY 11,57 6 8,2 1,2 10,3 5,0 15,8 100

SEP 14,05 7 19,2 14,8 15,0 9,1 4,1 2,4 100

SEMHB 15,13 8 1,8 1,5 12,2 5,1 0,5 11,0 15,0 100

SHTB1 15,88 9 0,6 0,1 16,4 6,9 11,5 22,2 5,5 18,7 100

SWHB2 21,47 10 0,1 0,7 0,2 3,6 2,4 4,2 2,8 7,2 0,1 100

SWB3 22,8 11 6,7 4,2 0,1 3,7 20,4 9,5 34,8 3,1 0,3 3,2 100

SWT1 27,81 12 22,0 11,5 0,5 0,6 0,6 4,1 11,2 1,6 0,0 0,0 1,1 100

SWB4 36,78 13 3,8 2,1 0,1 0,2 0,2 1,4 4,9 0,7 0,0 0,1 0,5 16,5 100

SWHB3 46,1 14 0,4 2,0 0,0 0,3 0,3 1,4 2,4 2,9 0,3 1,1 1,5 2,1 2,8 100

SWT2 46,6 15 0,0 0,0 0,0 1,0 1,2 0,3 1,4 0,6 0,0 2,1 0,1 5,1 13,8 19,0 100

SHTHB1 54,0 16 0,0 0,1 4,3 0,2 0,9 0,6 0,4 0,0 1,9 0,5 0,3 0,0 0,4 0,7 0,0 100

SHTT1 56,46 17 0,0 0,0 0,2 0,1 0,4 0,0 0,0 0,1 0,6 0,0 0,0 0,0 0,0 0,1 0,0 30,3 100

SHTB2 60,1 18 0,0 0,0 0,6 0,2 0,1 0,0 0,0 0,4 4,1 0,0 0,0 0,0 0,0 0,1 0,0 19,5 1,2 100

Table 3.- Auto MAC of L-610G aircraft symmetric mode shapes.

SWB1 symmetric first wing bending SWHB2 symmetric second wing horizontal bending SEMVB symmetric engine mounting vertical bending SWB3 symmetric third wing bending FVB1 first fuselage vertical bending SWT1 symmetric first wing torsion SWHB1 symmetric first wing horizontal bending SWB4 symmetric forth wing bending SWB2 symmetric second wing bending SWHB3 symmetric third wing horizontal bending SEY symmetric engines yaw SWT2 symmetric second wing torsion

SEP symmetric engines pitch SHTHB1 symmetric first horizontal tailplane horizontal

bending SEMHB symmetric engine mounting horizontal bending SHTT1 symmetric first horizontal tailplane torsion SHTB1 symmetric first horizontal tailplane bending SHTB2 symmetric second horizontal tailplane bending

In the two matrices the values are given in per cent. Most non-diagonal elements are very

small under 10 per cent. Only a few values are slightly greater. Most of them are "horizontal" modes

with movement in longitudinal direction. In this direction there was limited number of measured

points on fuselage and engines. It is assumed that this is the reason why these values are rather higher.

With the Auto-MAC values the situation is similar, the elements over 10 per cent have again

"horizontal" mode shapes and in addition, mode shapes of engines. But it may be seen that higher

values may also be found with some shapes featuring good orthogonality. MAC depends to some

extent on choice of the points (and directions) on the structure whose measured displacements were


incorporated into the calculation. In analysing orthogonality and particularly the Auto-MAC values, it

was useful to assess the effect on the resulting value of motion components in the co-ordinate axes of

particular aeroplane parts.

Figure 27.- Test of L-610G Aeroplane.


Conclusions

Over the last twenty years, the modal assurance criterion has demonstrated how a simple

statistical concept can become an extremely useful tool in the field of experimental modal analysis and

structural dynamics. The use of the modal assurance criterion and the development and use of a

significant number of related criteria, has been remarkable and is most likely due to the overall

simplicity of the concept. New uses of the modal assurance criterion and new criteria will be

developed over the next years as users more fully understand the limitations of the current criteria.

Certainly in the next few years, the increased use of other statistical methods as well as further

development of singular value/vector methods are related areas that will generate useful tools in this

area.

Even so, it will always be important to recognize the origins and limitations of tools like

the modal assurance criterion to avoid misuse of the methodology. Simplistic tools like the modal

assurance criterion are limited in their meaningful application. The development of related assurance

criteria has been initiated by shortcomings, real or perceived, of the original modal assurance criterion.

Dissatisfaction often has resulted from the misuse of these tools by users, removed from the actual

development or unaware of application limitations in subsequent implementations. It is clear that users

will continue to need more feedback concerning quality assurance information relative to experimental

modal parameters and that new techniques, particularly statistical methods that utilize the redundant

information present in the measured data, will continue to be developed with strengths and

weaknesses, just like the modal assurance criterion.

In regarding to the uses for aircraft design and testing/certificating companies, this report

has tried to explain why a carefully performed pre-test analysis really is necessary to end-up with

meaningful modal survey test results. Good test results are really a sine qua non for the interpretation

of all dynamic correlation tools that are used in the analytical model verification and validation.

An overview is given of the commonly used techniques to address the target mode

selection, the sensor location placement and the positioning of the exciters. It may be clear that the use

of the available tools and the user programming capabilities of MSC/NASTRAN form a crucial aspect

for the calculation of all described tools. The synergy of MSC/NASTRAN and LMS/Pretest gives the

engineer the additional benefit that, a unique environment becomes available that guides the engineer

through the complete process, from pre-test analysis over correlation to end up eventually at the model

updating step, that the interpretation of the calculations can be visualized and that both program

communicate directly with each other.


Bibliography

[1] Allemang, R. J.: The Modal Assurance Criterion – Twenty Years of Use and Abuse. Sound and

Vibration, pp. 14-21, August 2003.

[2] Van Langenhove and T., Brughmans, M.: Using MSC/NASTRAN and LMS/PRETEST to find

an optimal sensor placement for modal identification and correlation of aerospace structures.

LMS International, Belgium, 1999.

[3] Cerný, O.: Primera Modal Analysis Systems and Software/Aircraft Modal Testing. Report

VZLÚ 3/98.

[4] LMS: LMS Virtual.Lab Introduction. LMS International (Belgium).

[5] LMS: LMS Virtual.Lab Correlation. LMS International.

[6] Braun, R., Madsen, N. and Meruane, V.: Análisis Modal Experimental de una Estructura

Aeronáutica. Universidad de Chile, Facultad de Ciencias Físicas y Matemáticas, Dpto. de

Ingeniería Mecánica, Chile, Julio de 2003.

[7] López-Díez, J., Marco-Gómez, V. and Cuerno-Rejado, C.: Modal Test Correlation and Error

Localization for Finite Elements Models of Spacecraft structures. C.A.S.A. División Espacio,

Spain, 1999.

[8] Garbayo, M. E. and Pintor, J. M.: Ajuste y validación de modelos teóricos mediante ensayos de

vibración sobre el componente. Universidad Pública de Navarra, Dpto. Ingeniería Mecánica

Energética y Materiales, Spain, 2006.

Modal Assurance Criterion - Álvaro Rivero

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