isma2010_0103

Extensions of the MAC criterion to complex modes

P. Vacher, B. Jacquier, A. BucharlesONERA, Department of Systems Control and Flight Dynamics,2, av. Edouard Belin, B.P. 4025, 31500 TOULOUSE, FRANCEe-mail: [email protected], [email protected], [email protected]

Abstract

Since its original development in the early eighties, the modal assurance criterion (MAC) has become amajor tool in modal analysis. This measure of the degree of linearity (consistency) between two modalvectors provides an invaluable means of comparing and contrasting modal vectors from different origins.However, to the authors’ knowledge, its extension to complex vectors has not been addressed thoroughly.Such situations occur when the system under study does not comply with the classical assumptions of struc-tural dynamics. This paper proposes a generalization of the MAC criterion that is applicable to any typesof vectors. It also introduces an enhanced version of this criterion that tremendously improves its ability todiscriminate modes when few measurements are available.

1 Introduction

The Modal Assurance Criterion (MAC) is an essential tool in modal analysis. Its use to perform the pairingbetween two sets of modal vectors is now widespread. The great success of the MAC has fostered theemergence of numerous derivative criteria designed to deal with more specific situations.

However a rigorous treatment of the complex case still seems to be lacking. Such situations are relativelyfrequent and occur for systems that do not fully comply with the classical assumptions of structural dynamics.The analysis of the aeroelastic behaviour of an aircraft is a typical example of such a situation. The otherdrawback of the MAC criterion resides in its limitation to discriminate efficiently the modes when only afew measurements are available. Again, the in-flight analysis of an aircraft structure is an illustration of thisproblem since the number of sensors is sometimes limited.

The development of criterions that are both appropriate to complex modes and to situations with few mea-surements constitutes the topic of this paper. It is composed of four main sections. The first one brieflyreviews the classical assumptions in structural dynamics and the definition of the MAC criterion. It alsointroduces notations used in the paper. The following section deals with the definition of a degree of com-plexity which quantifies how much a vector differs from a real-valued one. This analysis is important becauseit establishes the basic principle which shows why the MAC criterion is not quite appropriate to process com-plex vectors. The third section of the paper is devoted to an extension dubbed MACX of the MAC criterionto complex vectors. This extension is constructed from the physical interpretation of the modal contributionson the measurements of a system. Finally the last section addresses the enhancement of the MACX criterionfor situations with a limited number of measurements. This improved version named MACXP integratesthe information about the modal frequencies and dampings in order to increase the discrimination betweenmodes.

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2 Classical structural models

2.1 Conventional hypotheses

Mechanical models are based on the fundamental equations of dynamics. They are composed of a set ofsecond order differential equations on the generalized displacements of a structure. These displacements aregathered in a vector denoted q. A modeling of the damping which quantifies the dissipation of the energy inthe structure is also necessary. The most common practice is to use a viscous damping model which leads tolinear equations which have the following form:

M q + D q + K q = B U

Y = C0 q + C1 q + C2 q(1)

M, D and K designate the matrices of mass, damping and stiffness. The last equation of the model describeshow the vector of measurements Y is connected to the displacements q. The form adopted here encompassesthe various types of measurements used in structural analysis: displacements, velocities, accelerations.

Two additional hypotheses are formulated for modeling the dynamics of a structure:• the matrices M, D et K are positive symmetric matrices• they are diagonalizable in the same basis.

With such conditions, the eigenvectors of the system 1 are identical to those of the associated undampedsystem (D = 0). If moreover the damping matrix D is composed of small components, the poles of thesystems λk are complex and stable.

With these hypotheses, each pair of conjugate poles (λk, λk) shares a common eigenvector ψk and thisvector is real-valued. The mode defined by the quantities λk, λk and ψk is said to be real. When the abovehypotheses are not complied with, each pole λk is associated to a complex eigenvector ψk. The mode issaid to be complex. It is characterized by the pair of conjugate poles (λk, λk) and the pair of conjugateeigenvectors (ψk, ψk).

2.2 Mode shapes

A mode shape is a vector µk that characterizes how a pole λk affects the measurements Y . It is given by thefollowing expression:

µk =(C0 + C1 λk + C2 λ

2k

)ψk (2)

In this general formulation, the mode shape µk is not real-valued even if ψk is a real vector. However, if themeasurements performed on the system are all of the same type, the vector µk is proportional to a real-vector.In this case, only one matrix amongst C0, C1 and C2 in the model is non-zero. It is designated Ci and wehave

µk = λik (Ci ψk) (3)

Complex vectors which are equal to the product of a complex scalar by a real-valued vector are named“monophase” because the phases of their components are all equal modulo π.

2.3 MAC Criterion

The Modal Assurance Criterion (MAC) is a measure of the degree of linearity between two vectors. Giventwo vectors µ1, µ2, it is defined by

MAC(µ1, µ2) =( | µ∗1 µ2 |‖ µ1 ‖ ‖ µ2 ‖

)2

= cos2(µ1, µ2) (4)

2714 PROCEEDINGS OF ISMA2010 INCLUDING USD2010

where ∗ designate the conjugate transpose of a complex vector. The product µ∗1 µ2 is called the Hermitianinner product between two vectors.

This definition can be interpreted geometrically since the MAC criterion depends on the angle between twovectors. This criterion can be applied to both real-valued and complex-valued vectors. It is insensitive to themodulus and the phase of the vectors µ1 and µ2. For this reason, it is well-suited to the analysis of monophasevectors. On the other hand, we can notice the value of the MAC is sensitive to conjugate operations on thevectors µ1 and µ2.

As recommended by Allemang [1], this criterion is appropriate to compare two sets of modes and performpairing between these sets. One should however be aware of a few limitations. First the value of the MACis sensitive to the large components in the vectors µ1 and µ2. If this difference on the amplitude of theassociated measurements is systematic, one could possibly applied scalings on the vectors in order to equalizetheir contributions in the computation of the MAC. For instance, on aircraft, sensors installed on the wingtips produce larger values than those at the center of the fuselage.

The other defect of the MAC criterion is its reduced sensitivity when the number of components in thevectors µ1 and µ2 is small. Few spatial information is then available to differentiate mode shapes. Thisphenomenon is illustrated by the figures 1 and 2 where the modes of two aeroelastic models are comparedwith the MAC criterion. In the first case, only 18 measurements are considered. We can notice that modesquite apart exhibit significant MAC values. The situation is much clearer in figure 2 with 123 measurements.

In practice [2], two vectors are considered correlated when the MAC is greater than 0.9 which correspond toan angle lower than 18 degrees. They are judged uncorrelated when the MAC is lower than 0.6 which meansthat they are separated by an angle greater than 39 degrees.

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Figure 1: Comparison of aeroelastic models with the MAC criterion

3 Complexity criterion

In practical situations, the actual mode shapes of a physical system are not exactly monophase vectors evenif the aforementioned hypotheses for structural modeling are supposed fulfilled. It is therefore necessary todefine a criterion to quantify how distant a complex vector is from a monophase one.

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Figure 2: Comparison of aeroelastic models with the MAC criterion

In the literature, several indicators of the complexity of a vector µ are available. The thesis by Purekar[3] presents several of them. For illustrative purposes, we begin this section by a graphical approach ofcomplexity. We then concentrate on a criterion called “Modal Phase Collinearity” (MPC) developed byPappa, Elliott, and Schenk [4]. A geometrical interpretation of the complexity of a vector is also presentedand its connection with the MAC criterion is established.

3.1 Graphical approach

A simple approach to analyze the complexity of a vector consists in plotting its components in the complexplane as illustrated in figure 3. One can then easily determine if these components tend to be aligned accord-ing to a specific direction. One can also visually assess the dispersion of the components about this meandirection. This gives an indication of the degree of complexity of the vector µ. For instance, in figure 3, thefirst vector is close to a monophase vector whereas the second one is really a complex vector.

Mode 1

Real part

Imag

. p

art

Mode 6

Real part

Imag

. p

art

Figure 3: Graphical approach of vector complexity


3.2 Modal Phase Collinearity (MPC) criterion

In order to quantify the complexity of a vector µ, Pappa et al. [4] defined a criterion that makes use of thereal part µr and the imaginary part µi of µ. The variance and the covariance of these parts are computedaccording to

Sxx = µ>r µr

Syy = µ>i µi

Sxy = µ>r µi

(5)

The eigenvalues of the following covariance matrix are then calculated[Sxx SxySxy Syy

]

They are given by the expression

λ1,2 =Sxx + Syy

2± Sxy

√η2 + 1 with η =

Syy − Sxx2 Sxy

(6)

The MPC criterion is finally defined by

MPC(µ) =(λ1 − λ2

λ1 + λ2

)2

(7)

Its values ranges from 0 to 1. It is equal to 1 when µ is a monophase vector.

3.3 Geometric interpretation of the complexity

A complex eigenvector is in fact computed up to a multiplicative complex factor. So the phases of aneigenvector or of a mode shape are given up to an angle β. We can thus consider to perform a normalizationof this arbitrary phase by searching the value β of β that renders a complex vector “as real as possible”. Inmathematical terms, this phase β maximizes the norm of the real part of the vector µ e−j β

β = arg maxβ‖ υ(β) ‖ with υ(β) = Re

(µ e−j β

)This phase normalization process is illustrated in figure 4 which depicts the rotation by the angle β applied tothe components of a vector in the complex plane. The angle β maximizes the alignment of the componentswith the real axis.

It can be show that the optimum value β is equal to

β =∠(µ> µ)

2mod π (8)

where ∠(z) designates the phase of the complex scalar z and > is the transpose operation. If we denote µthe phase normalized vector µ = µ e−j β and µr et µi its real and imaginary part respectively, we have thefollowing equalities

‖ µr ‖2 =12

(‖ µ ‖2 +

∣∣µ> µ∣∣)‖ µi ‖2 =

12

(‖ µ ‖2 − ∣∣µ> µ∣∣) (9)


~

βima

g.

axis

ima

g.

axis

real axis real axis

Figure 4: Phase normalisation

If µ is a mono-phase vector we obviously have µi = 0. It is also clear that ‖ µi ‖ 6 ‖ µr ‖, the case wherethe equality holds being the most unfavorable. In this situation, we have µ> µ = 0. We can then define themeasure of the complexity of a complex vector µ by

C(µ) =

(‖ µr ‖2 − ‖ µi ‖2‖ µr ‖2 + ‖ µi ‖2

)2

(10)

This quantity lies between 0 and 1. It is equal to 1 in when µ is monophase and to 0 when ‖ µi ‖ = ‖ µr ‖.By substituting the relations 9 in the expression of C(µ), we come upon a very simple form of this criterionwhich directly depends on the vector µ

C(µ) =

(∣∣ µ> µ ∣∣µ∗ µ

)2

=

(∣∣ µ> µ ∣∣‖ µ ‖2

)2

(11)

3.4 Equivalent formulations

We are now in position to establish two remarkable equalities. First, by considering the quadratic formassociated with the matrix defined in 3.2, we can show that the criterion C(µ) is strictly equivalent to theMPC criterion defined in 7. However, the expression 11 provides a much more concise way of computingthis quantity. Moreover, it is based on simple geometric considerations.

Expression 11 can also be recast in the following form

MPC(µ) = C(µ) =( | µ∗ µ |‖ µ ‖ ‖ µ ‖

)2

(12)

where µ is the conjugate of µ. Comparing with the definition 4 of the MAC criterion, we deduce thatMPC(µ) is nothing mere than the MAC computed for the vector µ and its conjugate µ:

MPC(µ) = MAC (µ, µ) (13)

This result shows that, when the vector µ is not monophase, the angle between a complex vector and itsconjugate is non-zero.


4 Extension of the MAC to complex modes

In this section, we first consider the direct application of the MAC criterion to complex vectors which revealsthe limitations of the classical formulation of the MAC. We then proceed to the definition of an extension ofthis criterion to complex vectors that we call MACX.

4.1 Application of the MAC criterion to complex vectors

When complex modes are considered, each mode is characterized by a pair of complex-conjugate modeshapes. If we want to compute a degree of similarity between two modes using the MAC criterion, we mightwonder, as illustrated by the diagram 5, which vector in each pair of conjugate vectors, we might selectfor computing the MAC criterion. When one of the mode shapes µ1 or µ2 is monophase, this choice hasabsolutely no incidence on the value of the MAC. But if the two vectors are complex then two values are infact possible since the associations of the same color in the diagram 5 produce identical results.

?

_

2µ

2µ

_

1µ

1µ

Figure 5: MAC criterion for complex modes: pairing dilemma

This selection can lead to significant differences for the values of the criterion. This is illustrated by figure 6where the differences MAC(µ1, µ2) − MAC(µ1, µ2) between the two possibles values for the MAC areplotted for the two models considered in figure 2. We notice that these differences range from -0.64 to 0.92.

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Figure 6: Aeroelastic models: MAC(µ1, µ2)−MAC(µ1, µ2)

Which of the two cases in figure 5 should we adopt to compute the MAC Criterion?

A first solution would be to make use of the sign of the imaginary part of the poles λ1 and λ2. Theseimaginary parts represent the angular frequencies of the oscillations of the modes. The solution1 would be

1Figures 1 and 2 were calculated in this way.


to select the two vectors which are associated with poles λ1 et λ2 with imaginary parts of the same sign.

However logic this solution may be, it bears no relation with the underlying physical phenomenons. Thecontribution of a complex mode on the measurements is in fact the combination of the two complex-conjugatemode shapes associated to this mode. The value of the output vector Y (t) at time t is given by

Y (t) =nm∑k=1

(µk xk(t) + µk xk(t)

)= 2

nm∑k=1

Re(µk xk(t)

)(14)

where the scalars xk(t) are the state components in the diagonal state-space basis. So the contribution ofmode k is given by the quantity Re

(µk xk(t)

). The value of xk(t) depend on all the past of the system till

time t. We can omit the modulus of xk(t) which has no influence on the MAC criterion. We must also notforget, as mentioned before, that the phases of mode shapes are arbitrary.

Omitting the index k for simplification purposes, the contribution on the measurements of a complex modeat some time t is characterized by the real vector

υ(β) = Re(µ e−j β

)(15)

for some value of the angle β. In order to compare two complex modes, we have then to compare the vectorsof their contributions on the system outputs which are given by the real vectors υ1(β1) et υ2(β2). For givenphases β1 et β2, the similarity between these vectors depends on their correlation which is equal to the scalarproduct

υ>1 (β1) υ2(β2) (16)

If this product is equal to zero, then the contributions of the two modes are orthogonal.

If one of the mode shapes µ1 or µ2 is monophase, MAC(µ1, µ2) = 0 implies υ>1 (β1) υ2(β2) = 0 for anyangles β1 et β2. But if both mode shapes are complex, it can be shown that the scalar product will be equalto zero only if the following equality between β1 and β2 holds

β1 + β2 = ∠(µ>1 µ2)± π/2 (17)

In this situation, the MAC criterion clearly gives an underestimated value of the similarity between the twomode shapes because it indicates completely dissimilar vectors whereas the contributions of these modes willbe orthogonal only for two specific values of the sum β1 + β2 on the interval [ 0 2π [. Finally, for complexvectors, the fact that υ>1 (β1) υ2(β2) = 0 for all values of β1 and β2 is equivalent to MAC(µ1, µ2) = 0 andMAC(µ1, µ2) = 0.

This result shows that, to decide on the full orthogonality of the contribution of two complex modes, i.e. forall values of β1 and β2, we must consider simultaneously the two associations in diagram 5. This explainswhy the conventional MAC formulation is not sufficient for complex vectors.

4.2 Definition of the MACX criterion

In this section, we propose an extension of the MAC that is valid for any type of vectors. The approachadopted is quite similar to the phase normalization of complex vectors developed in subsection 3.3. Thedefinition of the new criterion is based on the correlation 16 computed for angles β1 and β2 which maximizethe absolute value of this quantity. These angles solve the following optimization problem which is illustratedin figure 7

(β1, β2) = arg maxβ1, β2

∣∣∣ υ>1 (β1) υ2(β2)∣∣∣ with

υ1(β1) = Re(µ1 e−j β1

)υ2(β2) = Re

(µ2 e−j β2

)


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ap

e c

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nts

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e c

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ima

g.

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rtim

ag

. p

art

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real part

real part

ima

g.

pa

rtim

ag

. p

art

real part

Figure 7: Maximization of the correlation between two complex vectors

It can be shown that the optimal angles are equal to

β1 =12

[∠(µ>1 µ2)− ∠(µ∗1 µ2)

]β2 =

12

[∠(µ>1 µ2) + ∠(µ∗1 µ2)

](18)

The optimum value of the scalar product is given by

| µ>r1 µr2 | = | µ∗1 µ2 |+∣∣∣ µ>1 µ2

∣∣∣ withµr1 = υ1(β1)

µr2 = υ2(β2)(19)

This quantity involves both the Hermitian product and the scalar product between µ1 and µ2. We can con-nect this result with the analysis of the previous subsection by noting that µ>r1 µr2 = 0 is equivalent toMAC(µ1, µ2) = 0 and MAC(µ1, µ2) = 0.

The criterion MACX defined below makes use of this quantity. A first idea would be to define the MACXcriterion as the MAC criterion between the two real parts µr1 and µr2 . However this choice is not acceptablebecause β1 and β2 can be quite different from the angles 2 β1 and β2 defined by equation 8 which individuallymaximize the norm of µ1(β1) and µ2(β2) as depicted in figure 7. In such a case, we might conclude to a highsimilarity between two mode shapes based on the linearity of the vectors µr1 and µr2 whereas the dominantreal parts µr1 and µr2 could be quite different or even orthogonal. Therefore we must impose that the MACXcriterion could only be equal to 1 if β1 = β1 mod π and β2 = β2 mod π.

(β1, β2) = arg maxβ1, β2

∣∣υ>1 (β1) υ2(β2)∣∣ βi = arg max

βi

‖ υi(βi) ‖

µ1 = µ1 e−j β1 = µr1 + j µi1 µ1 = µ1 e−j β1 = µr1 + j µi1

µ2 = µ2 e−j β2 = µr2 + j µi2 µ2 = µ2 e−j β2 = µr2 + j µi2

Table 1: Notations used for the definition of the MACX criterion

We thus define the MACX criterion by the following quantity:

MACX(µ1, µ2) =

(| µ>r1 µr2 |

‖ µr1 ‖ ‖ µr2 ‖‖ µr1 ‖‖ µr1 ‖

‖ µr2 ‖‖ µr2 ‖

)2

2Table 1 summarizes the various notations used in this section.


By using equations 9 and 19, we can derive a simpler expression that can be computed directly from thevectors µ1 and µ2

MACX(µ1, µ2) =

( | µ∗1 µ2 |+∣∣ µ>1 µ2

∣∣ )2(µ∗1 µ1 +

∣∣ µ>1 µ1

∣∣ ) (µ∗2 µ2 +∣∣ µ>2 µ2

∣∣ ) (20)

This formulation is quite similar to the classical MAC since we only need to perform the following substitu-tion in the expression 4 of the MAC:

| µ∗1 µ2 | −→ 12( | µ∗1 µ2 |+

∣∣∣ µ>1 µ2

∣∣∣ ) (21)

4.3 Properties of the MACX criterion

The following properties can be established for the MACX criterion:• its value ranges from 0 to 1• it is independent on the norm and the phase of µ1 et µ2.• it is insensitive to conjugate operations on its arguments• MACX(µ1, µ2) = 0 is equivalent to µ∗1 µ2 = 0 and µ>1 µ2 = 0• If µ2 = z µ1 or µ2 = z µ1 for some complex number z then MACX(µ1, µ2) = 1.

But we must notice that this is only a sufficient condition.• If one of the vectors µ1 and µ2 is monophase, then the MACX and MAC criterions are identical.• Conversely, we can find vectors µ1 and µ2 such that MAC(µ1, µ2) = 0 et MACX(µ1, µ2) = 1.

But, in this case, both vectors are“full” complex since we have MPC(µ1) = 0 and MPC(µ2) = 0.

Figure 8 shows the difference between the MACX and MAC criterions for the two models used in figure 2.It can be noticed that the MACX criterion generally gives greater values than the MAC criterion. Thedifferences between the two criterions are significant since they lie between -0.20 and 0.72.

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Figure 8: Difference MACX−MAC for the aeroelastic models


5 Enhancement of the MACX criterion by pole weighting

In subsection 2.3, we mention that the MAC is of little use when a limited number of measurements is avail-able. This is illustrated by figure 1 where two aeroelastic models are compared using only 18 measurements.The MACX criterion has the same drawback. Anyhow, in some instances, it would be very useful to be ableto discriminate modes with few measurements.

In this section, we present an improvement of the MACX criterion that takes into account the poles associatedto the mode shapes. We first present the concept of pole weighting that has already been used in the literature.Then we define a criterion named MACXP based on this concept.

5.1 The concept of pole weighting

Pole weighting consist in integrating, in the MACX criterion, the information about the poles associatedto the mode shapes for a better mode differentiation. Instead of computing the criterion on the sole modeshapes, the idea is to use the free-decay responses realted to each pole.

Given the pole λ and the associated mode shape µ, the expression of the free-decay response at time t isgiven by

d(t) = µ x(t) = x0 µ eλ t (22)

were x0 is the initial complex state associated to this pole. For each pole we can form a vector V byconcatenating the free-decay response at several equispaced instants (0, ∆t, 2 ∆t, · · · , nr ∆t) where ∆tis a constant period. Such a vector is defined by

V =

µ

µ e λ∆t

...µ enr λ∆t

(23)

The pole weighted version of the MACX criterion is nothing mere than the application of the MACX formu-lation to two of these vectors

MACXp. w.(µ1, µ2) = MACX(V1, V2) (24)

We can notice incidentally that, even if the mode shape is monophase, the vector V will not be monophaseas soon as nr > 1. Thus, even for real modes, the use of a complex version of the MAC is unavoidable.

Pole-weighted versions of the MAC have already been worked out in the literature. This approach was usedby Juang [5] at NASA to evaluate the quality of modes identified with the ERA algorithm (EigensystemRealization Algorithm). This concept was also studied more recently by Phillips and Allemang [6] whonamed it pwMAC (Pole-Weighted MAC). Anyhow the complex nature of the vectors V in equation 23 wasnot addressed.

5.2 Definition of the MACXP criterion

The MACXP criterion proposed in this paper is simply the limit of the pole-weighted version of the MACXcriterion defined in 24 when ∆t tends towards 0 and nr towards infinity:

MACXP(µ1, µ2) = lim∆t−→ 0nr −→∞

MACX(V1, V2) (25)


It can be shown that the Hermitian and scalar products between vectors V1 and V2 in the expression of theMACX criterion have the following limits

lim∆t−→ 0nr −→∞

V ∗1 V2 ∆t =∫ ∞

0d1(t)∗ d2(t) dt lim

∆t−→ 0nr −→∞

V >1 V2 ∆t =∫ ∞

0d1(t)> d2(t) dt (26)

where d1(t) and d2(t) are the free-decay responses of the poles λ1 and λ2 as defined in 22. Under theassumption that these poles are stable, the integrals in these equations are given by∫ ∞

0d1(t)∗ d2(t) dt =

−1λ1 + λ2

µ∗1 µ2

∫ ∞0

d1(t)> d2(t) dt =−1

λ1 + λ2µ∗1 µ2 (27)

Substituting for these results in the definition 25 we obtain the following formulation of the MACXP crite-rion:

MACXP(µ1, µ2) =

( | µ∗1 µ2 || λ1+λ2 | + | µ

>1 µ2 |

| λ1+λ2 |

)2

(µ∗1 µ1

2 | Re λ1 | + | µ>1 µ1 |

2 | λ1 |

) (µ∗2 µ2

2 | Re λ2 | + | µ>2 µ2 |

2 | λ2 |

) (28)

As compared to the basic MACX formulation 20, we clearly see in the above expression the influence ofpole weighting. The MACXP criterion can be interpreted in terms of the correlation function between thereal decay responses associated to two modes: it can be shown that the square root of this criterion is quiteclose to the maximum value of the modulus of this correlation function normalized by the L2-norm of thesefree-decay responses.

The application of this criterion for the comparison of the aeroelastic models with few measurements appearsin figure 9. We can see that the situation is tremendously improved as compared to figure 1. For instance,this latter criterion clearly reveals an exchange between modes 24 and 26.

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Figure 9: Application of the MACXP criterion to aeroelatic models


6 Conclusion

This paper is devoted to the study of criterions so as to compare complex modes when a small number ofmeasurements are available. This study explains why the utilization of the MAC criterion is limited to realmodes. Two extensions of this criterion are proposed.

The first one called MACX enlarges the application of the MAC criterion to complex modes. Moreover thiscriterion is strictly equivalent to the MAC when real modes are considered. The second criterion namedMACXP dramatically betters the analysis of modes when few measurements are available.

One can think of several applications in modal analysis that can take advantage of these criterions:• mode matching in stabilization diagrams• association of identified modes in various experimental situations (for instance, different flight condi-

tions for the aeroelastic testing of an aircraft)• comparison of results between several identification algorithms• mode pairing between a model of the physical system and the identification results for this system.• ...

Acknowledgements

The authors greatly thank Stephane Leroy, Aurelien Cordeau, Adrien Pavie and Adrien Berard at Airbus fortheir technical support on the aeroelastic models.

References

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[2] Etienne Balmes, Jean-Philippe Bianchi, and Jean-Michel Leclere. Structural Dynamics Toolbox - FEM-Link, User’s Guide. SDTools, Vibration Software and Consulting, version 6.1 edition, Septembre 2009.

[3] Dhanesh M. Purekar. A Study of Modal Testing Measurement Errors, Sensor Placement and ModalComplexity on the Process of FE Correlation. Master of science, University of Cincinnati, Departmentof Mechanical Engineering of the College of Engineerring, 2005.

[4] Richard S. Pappa, Kenny B. Elliott, and Axel Schenk. Consistent-mode indicator for the eigensystemrealization algorithm. Journal of Guidance, Control, and Dynamics, 16(5):852–858, Spetember–October1993.

[5] Jer-Nan Juang. Applied System Identification. PTR Prentice Hall, 1994.

[6] Allyn W. Phillips and Randall J. Allemang. Data presentation schemes for selection and identificationof modal parameters. In 23th International Modal Analysis Conference, Orlando, 2005.



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