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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 45 (2014) 80–90

0888-32http://d

n Tel.:E-m

journal homepage: www.elsevier.com/locate/ymssp

Complex eigenvector scaling from mass perturbations

Dionisio Bernal n

Civil and Environmental Engineering Department, Center for Digital Signal Processing, Northeastern University,405 Cushing Hall, 102 The Fenway, Boston, MA 02115, United States

a r t i c l e i n f o

Article history:Received 18 April 2012Received in revised form15 October 2013Accepted 23 October 2013Available online 2 December 2013

Keywords:Mode normalizationModal scalingMass perturbationSystem identification

70/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.ymssp.2013.10.019

þ1 617 373 4417; fax: þ1 617 373 4419.ail address: bernal@neu.edu

a b s t r a c t

This paper presents an approach to normalize experimentally extracted complex eigen-vectors so that their outer product gives transfer function residues. The approach, animplementation of the mass perturbation strategy, is exact for arbitrary perturbationmagnitudes and number of sensors when the modal space is complete and is robustagainst modal truncation. It is shown that improvements over a sensitivity solution aresignificant when the relation between the eigenvalue and the perturbation magnitude isstrongly nonlinear.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Results from output only modal identification cannot be used to establish input–output relations because theinformation to properly scale the mode shapes is unavailable. Since a number of applications in experimental dynamicsand diagnostics are based on input–output maps, procedures to determine the modal scaling constants for output onlysettings are of practical interest. A strategy that has received notable attention computes these constants by equating theeigenvalue sensitivity (with respect to a given perturbation) to an experimental estimate, the simplest being a finitedifference approximation from the results of two tests [1]. In theory the type of perturbation is arbitrary but in practice theaddition of masses is typically easiest to implement and has thus been most commonly considered [2–7].

Scaling constants estimated using the mass perturbation scheme are random variables with a bias and a variance thatdepend on the perturbation magnitude and on the specifics of how the information is used to extract the constants.For conciseness in the discussion we introduce some notation from the outset, namely, we parameterize the massperturbation as ΔM¼ βM1, where β is a scalar and note that as β-0 the information vanishes and thus s2-1, where s2 isthe variance of the scaling constant estimate. It is evident, therefore, that to realize a reasonable variance sufficiently largeperturbations are needed. As β gets larger, however, the bias of a finite difference estimate of the sensitivity tends to increasebecause the relation between the eigenvalue λ and the perturbation magnitude β is generally nonlinear. Two different pathsto minimize the bias issue can be pursued. The first one is to select distributions of the perturbation that minimizenonlinearity in λðβÞ (or in a predetermined function of this function) [3,4] and the second is to formulate the problem in sucha way that the solution depends on the total changes in the eigenvalues, instead of the derivatives. Belonging to the firstalternative one can mention the use of distributionsM1 that more or less mimic the actual mass, as these lead to quasi-linearrelations between the inverse of the (undamped) eigenvalue and the perturbation magnitude. The limiting case, however,

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D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–90 81

being redundant since in this instance the modes have to be measured “everywhere” and knowledge of the massdistribution allows normalization using the orthogonality relationship. Optimization of the positioning of a few masses toreduce nonlinearity in (some) of the λðβÞ relations is analytically feasible but this requires use of a model and is thusincompatible with the data-driven constraints that apply to the present problem.

Formulations that work with the total changes in the eigenvalues (instead of the derivatives) do not impose constraintson the number, or on the spatial distribution of the perturbation, and this is the framework pursued in this paper. At firstglance it may appear that working with total changes would remove the perturbation magnitude bias altogether but this isnot generally the case because finite perturbations lead to modal coupling and bias enters the problem through the modallytruncated space. A formulation for the scaling constants based on total eigenvalue changes, however, can offer significantreductions gains over a sensitivity scheme because the bias in this case (as shall be shown) is only weakly dependent on β.The first formulation that used total changes to estimate the scaling constants traced the nonlinear λðβÞ relation usingsensitivities for a set of assumed constants and obtained the solution by minimizing discrepancy between predictions andidentified results [2]. Albeit straightforward conceptually, the outlined optimization framework is computationallycumbersome compared to a direct solution and, perhaps for this reason, has not received much attention. Methods thatoffer direct solutions include the Projection Approach (PA) [5], derived for the normal mode model on the premise that theperturbed eigenvectors lay on the basis of the unperturbed modes and the Receptance Based Normalization technique (RBN)[7], where the constants are obtained from an over-determined linear system of equations.

Implicit throughout the previous discussions is the fact that the modes that are to be normalized are the real modes ofthe normal mode model. While the normal mode model is sufficiently accurate in the large majority of cases, there areinstances, typically due to the existence of a set of modes with small eigenvalue gaps (see Appendix A) where irreducibleeigenvector complexity is significant and the more general first order modal model must be used to avoid undue error [8].This paper presents an extension of the RBN scheme to the normalization of the complex eigenvectors of the first orderformulation. The paper presents the theory, examines performance in a stochastic setting and clarifies the conditions werethe gains with respect to a sensitivity solution are important. The contrast between the complex form, designated as CRBN,and the approach that applies in the case of normal modes is highlighted in Appendix B for convenience.

2. Preliminaries

Consider an arbitrarily viscously damped linear time invariant model, accepting finite dimensionality the equations ofdynamic equilibrium can be written as

M €qðtÞþC _qðtÞþKqðtÞ ¼ f ðtÞ ð1Þwhere M;C;KARN�N are the mass damping and stiffness matrices, N is the number of degree of freedom and f(t) are theapplied loads. Taking a Laplace transform writes

½Ms2þCsþK�qðsÞ ¼ f ðsÞ ð2ÞSolutions are possible for f(s)¼0 if the matrix in the parenthesis is rank deficient. The values of s for which this matrix

loses rank are the poles, or complex eigenvalues, λ, and the vectors in the null space are the latent vectors ψ [9,10]. It iscustomary to refer to the latent vectors as complex eigenvectors, or simply eigenvector. For any value of s, other than a pole,the matrix in Eq. (2) can be inverted and one has

qðsÞ ¼ GðsÞf ðsÞ ð3Þwhere G(s) is known as the Receptance. The Receptance can be expressed as [11]

GðsÞ ¼ ∑2N

j ¼ 1

ψ jψTj ρj

s�λjð4Þ

taking

κj ¼ffiffiffiffiρj

p ð5Þ

and defining the normalized complex eigenvectors as

φj ¼ κjψ j ð6Þ

the Receptance writes

GðsÞ ¼ ∑2N

j ¼ 1

φjφTj

s�λjð7Þ

For the eigenvectors in Eq. (7) one has, among other relations [10]

ΦTCΦþΛΦTMΦþΦTMΦΛ¼ I ð8Þ

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–9082

and

ΦTΦ¼ 0 ð9Þwhere Φ¼ ½φ1 φ2 ⋯ φ2N� and Λ¼ diag½λ1 λ2 ⋯ λ2N �. Inspection of the diagonal in Eq. (8) shows that

2λjφTMφjþφTj Cφj ¼ 1 ð10Þ

Note that while the constraints in Eq. (9) are sufficient to relate all the complex eigenvectors to within a single scalar, thisequation is seldom useful for normalization because one does not have “all the modes” nor are “all the DOF” measured.

2.1. A motivating example

To illustrate a situation where adoption of the normal mode model leads to undue error consider the 2-DOF system inFig. 1. This systemwas shown in [12] to have repeated poles when the location of the dashpot c1 is α¼�0.69. Fig. 2 plots theabsolute value of the three independent entries of the transfer matrix; the top row depicts results for α¼0.65, which leadsto an eigenvalue gap of 6.2% of the magnitude of the first eigenvalue and the bottom for α¼�0.65, in which case the gap is2.4%. For each entry of the transfer matrix we plot the exact result (E) and the result from the normal mode model when theproperties are estimated from the damped solution in the usual manner (NMC). Namely, the frequency is the magnitude ofthe pole, the damping the negative of the ratio of the real part divided by the magnitude, the mode shapes are obtained bynormalizing so the largest amplitude is real and discarding the imaginary parts. As can be seen, in the case of the smalleigenvalue gap the normal mode estimates are very poor.

k1

c1 c2

m1m2

L

αL

0.062L

k2

1.0 0.31

-1.62

#1 #2ω1= 1.55 rad/sec

ω2= 1.45 rad/sec

m1 = 1m2 = 2c1 = 0.0778c2 = 0.1018k1 = 2.6759k2 = 4.2907

Fig. 1. Two-DOF system used in the example.

g11 g12

g12g11

g22

g22

0 1 20

2

4

6

8

10

0 1 20

1

2

3

4

0 1 20

0.5

1

1.5

2

2.5

0 1 20

5

10

15

20

25

0 1 20

2

4

6

8

0 1 20

2

4

6

8E

NMC

ω(rad/sec)

Fig. 2. Transfer matrix entries for the system of Fig. 1; top row α¼0.65, bottom row α¼�0.65 (E¼exact result, NMC¼using the normal mode withproperties estimated from the complex eigensolution).

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–90 83

3. The Receptance approach to complex mode normalization

Let ΔM be the mass perturbation and λj and ψ j be the pole and the arbitrarily scaled complex eigenvector in the massperturbed condition for the jth mode, with the symbols without the supra-bar being the values for the unperturbed system.In the derivation is convenient to treat the eigenvectors as if they are available at all the coordinates, although it will beapparent at the end that only the measured coordinates are needed. From the polynomial eigenvalue problem in theperturbed condition one has

½ðMþΔMÞλ2j þCλjþK�ψ j ¼ 0 ð11Þ

from where it follows that

½ðMþΔMÞλ2j þCλjþK��1ΔMλ2j ψ j ¼ �ψ j ð12Þ

Inspection of Eq. (12) shows that the inverted matrix is the Receptance of the unperturbed system evaluated at s¼ λj,therefore

∑2N

ℓ ¼ 1

ψℓψTℓρℓ

λj�λℓ

!ΔMλ

2j ψ j ¼ �ψ j ð13Þ

which, after some simple rearranging, can be written as

∑2N

ℓ ¼ 1

ψℓψTℓρℓ

λj�λℓ

!ρj ¼ ψ j ð14Þ

Eq. (14) contains m equations, where m is the number of sensors, in the 2N modal constants formulated using the jthpolynomial eigenvalue equation of the perturbed system. By combining the equations for each perturbed mode a linearsystem (albeit a complex valued one) is obtained. In practice there are only 2n identified modes so the summation′s upperlimit in Eq. (14) has to be taken as 2n and, after evaluating it at each of the identified modes, the coefficient matrixdimension is (m�2n)�2n, showing that the system is, for more than one sensor, over-determined. As is evident from thederivation the CRBN formulation uses all the information and, except for approximation resulting from modal truncation,is exact.

3.1. Obtaining the coefficient matrices

The linear system of equations implicit in Eq. (14) can be conveniently formed as follows: let Ψ ¼ ½ψ ⋯ ψ2n� be the setof identified complex eigenvectors at whatever coordinates are measured and Ψ ¼ ½ψ ⋯ ψ2n� the set in the mass modifiedstate. The system of equations writes

TρI ¼ vecðΨ Þ ð15Þwhere ρI ¼ ρ1 … ρ2ngT

�, and

T ¼Ψϒ1

⋮Ψϒ2n

264

375 ð16Þ

with ϒ j ¼ diag½b1j ⋯ b2nj � and

bij ¼λ2j

λi�λj

0@

1AψT

i ΔMψ j ð17Þ

As can be seen from Eq. (15), the constants are coefficients in a projection of the perturbed complex modes on the basisof the original system. An important property is the fact that results from the formulation presented do not depend on theordering of eigenvalues. This property can be readily verified by direct substitution and is important because establishing aone to one map between the unperturbed and perturbed states in the case of closely spaced poles, which is likely when thecomplex modal model is needed, is difficult.

3.2. Initial scaling

It is understood that what is unique is the normalized complex eigenvector of Eq. (6) and not the modal constant. For agiven set of modes in the unperturbed state, however, the exact constants are unique and what we want to note here is thatthe approximate solution in the truncated case has some dependence on the relative scaling of the modes of the perturbedcondition. One can appreciate this by noting that if Eq. (15) (for whatever scaling was selected) is treated as the referencethen all other possible normalizations of the perturbed state are realized by pre-multiplying Eq. (15) by a diagonal matrix E

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–9084

so, in general

ETρI ¼ EvecðΨ ÞþEη ð18Þwhere η is included so the equality holds in the truncated case. From Eq. (18) one can see that the least square solution for ρIdepends on E, except when rank(T)¼rankð½T vecðΨ Þ�Þ. The effect of the scaling in the solution, however, is small (forreasonable scaling) because, as can be seen from Eq. (18), the matrix E plays the role of weights in the least square solution.

3.3. Truncation error

To illustrate how modal truncation introduces approximation we write Eq. (15) as

T11 T12

T21 T22

" #ρI1ρI2

( )¼ Ψ1

Ψ2

( )ð19Þ

where the subscript “1” refers to identified modes and “2” to the non-identified set. The exact normalization constants forthe identified modal space are thus

ρI1 ¼ T �n

11 Ψ1�T �n

11 T12ρI2 ð20Þwhere �n is used to designate pseudo-inversion. Since the second term in Eq. (20) is not generally zero the truncatedsolution contains approximation. As shall be shown, however, the bias introduced by modal truncation is small.

4. A measure of performance

In this section we derive a scalar measure of performance that can be used to examine how perturbation magnitudeaffects results. For this purpose let the difference between the magnitude of the estimated and the magnitude of the exactscaling constant for any mode, normalized by the magnitude of the exact result be θm and the error in the phase angle θp.Let the probability density functions of these errors be pðθjÞ and the cost of errorWjðθjÞ for j¼m and p. A reasonable measureof performance is then

J ¼Z 1

�1WmðθmÞpðθmÞdθmþτ

Z 1

�1WpðθpÞpðθpÞdθp ð21Þ

where τ is a weight. Numerical examinations show that the error in the phase is relatively small so, to simplify we take τ¼0and, assuming the density is Gaussian and the cost function quadratic, namely WðθÞ ¼ αθ2 one gets

J ¼ α

smffiffiffiffiffiffi2π

pZ 1

�1θ2 e θ�θmð Þ= ffiffi2p

smð Þ2 dθm ð22Þ

which can be integrated and gives

J ¼ αffiffiffi2

p θ2mþ s2m2

� �ð23Þ

Eq. (23) shows that a reasonable measure of performance is proportional to the square of the bias plus one half of thevariance of the distribution of the error in amplitude. In the numerical section we use

ℑ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðθ2mþ0:5s2mÞ

q½κ� � 100 ð24Þ

which can be loosely interpreted as percent error.

5. The sensitivity approach

The sensitivity approach will be used in the numerical section to provide contrast to CRBN so we derive the relevantexpression for the complex eigenvector normalization. The polynomial eigenvalue problem writes

½ðMþβM1Þλ2þCλþK�ψ ¼ 0 ð25ÞDifferentiating with respect to β, pre-multiplying by ψT and evaluating the result at β¼0 gives

ð2λψTMψþψTCψÞλ′þψTM1ψλ2 ¼ 0 ð26Þ

where the prime notation indicates derivative with respect to β. Taking the eigenvector that is to be scaled so that ρ¼1 andusing the relation in Eq. (10) one gets

λ′þψTM1ψλ2 ¼ 0 ð27Þ

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–90 85

from where it follows, adding the subscript j for specificity, that

ρj ¼�λ′j

ψ jTM1ψ jλ

2 ð28Þ

Estimating the derivative with the forward difference one has

ρj ¼λj�λj

ψ jTM1ψ jλ

2 ð29Þ

which is the sensitivity formula for the complex mode normalization constant.

5.1. Bias of SA

An expression for the bias of the sensitivity estimate can be derived as follows: accepting that the eigenvalues areanalytic functions of the mass perturbation magnitude one has

λðβÞ ¼ λð0Þþλ′βþ0:5λ″β2þOðβ3Þ ð30ÞFrom where it follows that the error of the forward difference estimation, normalized to the true derivative is

NE¼ 0:5λ″λ′β ð31Þ

An index indicating whether or not the perturbation is likely to introduce significant bias is thus the absolute value of theratio of the second to the first derivative of the eigenvalue at β ¼0.

6. Numerical examinations

We consider a 16-DOF chain system having masses and stiffness m¼[1, 2, 1,…] and k¼[5000, 5000…] in a consistent setof units. The system has a non-classical damping distribution obtained by first specifying 2% of critical in each mode andsubsequently scaling the (1,1) and (2,2) entries of the damping matrix by a factor of 10. Sensors are located at coordinates{4, 8, 13 and 16} and the mass perturbation consists of three equal masses at coordinates {4, 8 and 13} (with a magnitudethat is varied). Table 1 lists the poles, the first and second derivatives, the weighted modal co-linearity index (mpcw) [13],the absolute value of the ratio of the second derivative to the first derivative, and the (nominal) damping ratio, computed asthe real part of the pole divided by the magnitude. In light of Eq. (31) one gathers that the modes where SA is expected tohave the poorest performance are the 5th and the 8th.

6.1. Deterministic examination

Figs. 3 and 4 plot the percent error in the magnitude of the scaling constants for the first 8 modes and the error in thephase vs. the perturbation magnitude. To exemplify the robustness of CRBN with respect to truncation, results when only 4modes are identified are also shown. As anticipated, the difference between CRBN and the sensitivity estimate is large in the5th and the 8th modes, where the eigenvalue vs. perturbation is strongly nonlinear. To gauge the magnitude of theperturbation note that for β¼1 the sum of the perturbation masses is 1/8 of the total mass of the system.

6.2. Stochastic examination

The deterministic results in Figs. 3 and 4 illustrate how the bias of the error behaves with perturbation magnitude. It isevident that as β increases the variance will decrease towards an asymptote that is fixed by the variability in theidentification and the issue in any case is whether the perturbation magnitude can be taken as large as practical constraintsallow or whether increases in the bias limit the attainable accuracy. From examination of the results in Fig. 3 one expectsthat bias will not play a role in CRBN and that the situation in SA depends on which mode is being normalized, e.g., in the

Table 1Some parameters of the 16-DOF system used in the numerical investigations.

Mode #(pair) λ λ′ λ″ λ00λ0�� �� % Damping ratio mpcw

1 �0.184þ5.443i 0.009–0.145i �0.002þ0.025i 0.173 3.38 99.942 �0.932þ16.374i 0.056–0.502i �0.009þ0.083i 0.166 5.68 97.163 �1.997þ27.227i 0.117–0.683i 0.005þ0.001i 0.008 7.32 84.304 �3.236þ37.482i 1.071–0.878i �0.559þ0.749i 0.675 8.60 60.535 �3.995þ46.342i �1.018–0.474i 0.320þ1.463i 1.333 8.59 42.446 �3.289þ54.558i �0.164–2.641i 0.456þ0.691i 0.313 6.02 48.277 �2.346þ62.442i 0.622–1.614i �0.051þ0.635i 0.395 3.75 72.448 �1.686þ68.397i 0.362–2.193i �0.465þ3.156i 1.435 2.46 92.00

0 0.5 1-6

-4

-2

0

2

4

6

0 0.5 1-6

-4

-2

0

2

4

6

0 0.5 1-6

-4

-2

0

2

4

6

0 0.5 1-6

-4

-2

0

2

4

6

0 0.5 1-30

-20

-10

0

10

0 0.5 1-30

-20

-10

0

10

0 0.5 1-30

-20

-10

0

10

0 0.5 1-30

-20

-10

0

10

mode1 mode2 mode3 mode4

mode5 mode6 mode7 mode8

SACRBN (8modes) CRBN (4modes)

ββββ

Fig. 3. Percent error in the amplitude of the scaling constant vs. perturbation magnitude.

SACRBN (8modes) CRBN (4modes)

0 0.5 1-15

-10

-5

0

0 0.5 1-15

-10

-5

0

0 0.5 1-15

-10

-5

0

0 0.5 1-15

-10

-5

0

0 0.5 1-15

-10

-5

0

0 0.5 1-15

-10

-5

0

0 0.5 1-15

-10

-5

0

0 0.5 1-15

-10

-5

0

mode1 mode2 mode3 mode4

mode5 mode6 mode7 mode8

ββββ

Fig. 4. Difference in phase (in degrees) between the exact and the estimated scaling constant vs. perturbation magnitude.

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–9086

first mode the bias may not be critical but the attainable accuracy in the 5th or the 8th modes may be bias limited.To illustrate we simulate the uncertainty in the identification by assigning normal distributions to the poles such that thecoefficients of variation are 0.0025 and 0.1 for frequency and damping respectively and perform 500 simulations for each of3 values of the perturbation magnitude, namely β¼0.2, 0.4 and 0.6. The resulting histograms for the 1st and the 8th mode

-60 -40 -20 0 20 40 600

40

80

120

-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 600

40

80

120

0

40

80

120

0

40

80

120

0

40

80

120

0

40

80

120

-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60

Top Row = CRBN Bottom Row = SA

β = 0.6β = 0.4β = 0.2

β = 0.6β = 0.4β = 0.2

Fig. 5. Histograms of the percent error in the magnitude of the normalization constant for the 1st mode (500 simulations with pole variability Gaussian).

-60 -40 -20 0 20 40 60

40

80

120

-60 -40 -20 0 20 40 600

-60 -40 -20 0 20 40 60

40

80

120

0

40

80

120

0

40

80

120

0

40

80

120

0

40

80

120

0-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60

Top Row = CRBN Bottom Row = SA

β = 0.6β = 0.4β = 0.2

β = 0.6β = 0.4β = 0.2

Fig. 6. Histograms of the percent error in the magnitude of the normalization constant for the 8th mode (500 simulations with pole variability Gaussian).

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–90 87

are depicted in Figs. 5 and 6. The reduction in the variance as the perturbation magnitude increases is clear and, as can beseen, is essentially the same for CRBN and SA. The consistency of the bias in the SA predictions with the results in Fig. 3 iseasily checked.

6.3. Performance

A format where the bias-variance tradeoff is clearly illustrated is a plot of the index of Eq. (24) vs. perturbationmagnitude. Fig. 7 plots this index for the 1st the 5th and the 8th modes. Each data point obtained from 500 simulationsusing the variability of the poles described in connection with Figs. 5 and 6. The fact that CRBN allows the perturbationmagnitude to be large without the bias error outweighing the variance reduction is clear.

0 0.5 10

5

10

15

20

25

30

35

40

0 0.5 10

5

10

15

20

25

30

35

40

0 0.5 10

5

10

15

20

25

30

35

40

Mode 1 Mode 5 Mode 8

β ββ

Á

Fig. 7. Index of Eq. (24) vs. perturbation magnitude.

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–9088

7. Conclusions

Normalization of complex eigenvectors using the mass perturbation strategy shows that a Receptance based scheme canoffer significant improvements in accuracy over a sensitivity solution in modes where eigenvalue vs. perturbationmagnitude is significantly nonlinear. The fact that CRBN is invariant with respect to the relative order of eigenvalues is auseful feature since closely spaced poles are common when modal complexity is high and eigenvalue pairing can, in thesecases, be difficult.

Acknowledgment

Part of the research reported in this paper was carried with support from NSF under grant CMMI-1000391, this support isgratefully acknowledged. The author also thanks the Department of Structural Engineering and Mechanics of the Universityof Trento, Italy, which provided support during a visit in which this work was initiated.

Appendix A. Modal complexity and the eigenvalue gap

It is shown here that for small, but arbitrarily distributed damping, the error in the frequency response functions of anormal mode model are accurate if the frequencies are well separated. The need for the complex modal model in lowdamped systems arises, therefore, when there are closely spaced poles.

A.1. Derivation

Let the mass normalized undamped mode shapes be collected in the matrix Φand let q¼ΦY . Substituting into Eq. (1) andpre-multiplying by ΦTgives

€Yþd _YþΛY ¼ΦT f ðtÞ�d0 _Y ðA:1Þwhere ½d� is diagonal with dj ¼ 2ωjξj, ½Λ� is diagonal with the square of the undamped frequencies, ωj, and ½d0� contains theoff-diagonal terms of the product ΦTCΦ. Designating the normal mode approximation (i.e., ½d0� ¼ 0) as YnmðtÞ one hasYðtÞ ¼ YnmðtÞþeðtÞ and it follows that

€eðtÞþd_eðtÞþΛeðtÞ ¼ �d0 _Y ðA:2Þ

so for the jth mode

€ejðtÞþdj _ejðtÞþω2j ejðtÞ ¼ �dT0;j _Y ðA:3Þ

where dT0;j is the jth row of ½d0�. Taking a Fourier transform of Eq. (A.3) writes

εjðωÞ ¼ejðωÞYjðωÞ

¼ gjðωÞdT0;jηðωÞ ðA:4Þ

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–90 89

where

gjðωÞ ¼�ωi

ω2j �ω2þ2ωjωξji

ðA:5Þ

and

ηðωÞ ¼ YðωÞYjðωÞ

ðA:6Þ

Eq. (A.4) is thus the normalized error in the FRF resulting from neglecting the off diagonal terms of the damping matrix.Evaluating Eq. (A.4) at ω¼ωj and taking

d0;j ¼ jd0;jjmaxrTj ðA:7Þ

writes

εjðωÞ ¼�12ωjξj

d0;jjmaxrTj ηðωÞ

��� ðA:8Þ

where rTj ¼ … 1 … 0 vi …f g has a zero at the jth position (by definition) and the entries vi have absolute value lessthan one. Assuming that the off-diagonal terms of ΦTCΦ are no larger than the diagonals it follows that

jεjðω¼ωjÞjrrT jηðω¼ωjÞj ðA:9Þ

Eq. (A.9) shows that the normalized error at resonance is no larger than the sum of the ratio of the modal amplitudesnormalized to the jth one, for all modes other than j. The entries in η can be estimated as follows: for the resonant mode onehas

YjffiϕTj f ðωÞ

2ω2j ξji

ðA:10Þ

In the non-resonant modes the damping term can be neglected and one has

YkffiϕTk f ðωÞ

ω2j �ω2

k

ðA:11Þ

therefore

ηjffiℓk;j2ω2

j ξj

ðω2j �ω2

k Þi ðA:12Þ

where

ℓk;jffiϕTk f ðωÞ

ϕTj f ðωÞ

ðA:13Þ

For frequencies ωk that are sufficiently smaller than ωj Eq. (A.12) simplifies to a¼ 2ℓj;kξji and for those that are sufficientlylarger to b¼ �2ℓj;kξjiðωj=ωkÞ2. These results show that the contribution of the off-diagonal terms of the damping are likelyto have more effect in the higher modes than in the lower ones since the “a” terms are larger than the “b” terms. More to thepoint of this derivation, however, is the fact that if the damping ratios are small then the “a” and “b” terms are small andsince they do not add monotonically (given the form of r) the normalized error of Eq. (A.8) is small. Needless to say, if thereis any frequency ωk that is close to ωj then the corresponding term in Eq. (A.12) can be large and, as a result, the error in thenormal mode estimate of the FRF.

Appendix B. RBN vs. CRBN

To clarify the distinction between RBN and CRBN we note that in the case of the normal mode model one has

∑N

ℓ ¼ 1

ϕℓðϕTℓΔMϕjÞλjλℓ�λj

γℓ ¼ ϕj ðB:1Þ

where ϕ is the arbitrary scaled real mode, λ is the square of the undamped natural frequency and γ is the (square of the)normalization constant, with the supra-bar indicating the perturbed state. In Eq. (B.1) the constant is such that the realmode multiplied by ffiffiffi

γp is the mass normalized mode while in CRBN the complex mode shape multiplied by

ffiffiffiρ

pis a vector

that satisfies Eq. (10) and can thus be used to compute the residues in Eq. (7). Note that while Eq. (B.1) has the same form asEq. (14), the two are not related by a one to one change from real to the complex case and a switch from N to 2N.

D. Bernal / Mechanical Systems and Signal Processing 45 (2014) 80–9090

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