Using model reduction and data expansion techniques to improve SDM · 2006-04-06 · 1. Local...

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Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 1067–1089 Using model reduction and data expansion techniques to improve SDM M. Corus a, , E. Balme`s a , O. Nicolas b a E ´ cole Centrale Paris, Grande Voie des Vignes, 92295 Chaˆtenay Malabry, France b R&D E ´ lectricite´de France, 1 Avenue du Ge´ne´ral de Gaulle, 92141 Clamart Cedex, France Received 2 July 2004; received in revised form 7 January 2005; accepted 9 February 2005 Available online 27 June 2005 Abstract A method designed to predict the effects of distributed modifications of structures is proposed here. This method is an evolution of the classical formulations, and distinguishes measurements and coupling points. Based on a coarse model of the structure to be modified, the proposed methodology tackles two major difficulties: efficient predictions for distributed modifications and handling of the lack of measurement points on the coupling interface. In addition, displacements bases introduced to reconstruct unmeasured behaviour of the interface limit error propagation through the process. Moreover, two indicators are introduced to select the optimal prediction. An academic stiffened plate and an industrial application (motor pump) are used to illustrate the approach and highlight its main advantages. r 2005 Elsevier Ltd. All rights reserved. Keywords: Structural dynamics; Structural modification; Modal synthesis; Experimental modal analysis; Dynamic substructuring; Interface modes; Model reduction; Data expansion 0. Introduction Companies in charge of production processes (such as electricity, water, etc.) are not the manufacturers of their equipment. They may have no access to drawings, design studies or FE ARTICLE IN PRESS www.elsevier.com/locate/jnlabr/ymssp 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.02.012 Corresponding author. Tel.: +33 1 41 13 17 40. E-mail address: [email protected] (M. Corus).

Transcript of Using model reduction and data expansion techniques to improve SDM · 2006-04-06 · 1. Local...

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Mechanical Systemsand

Signal ProcessingMechanical Systems and Signal Processing 20 (2006) 1067–1089

0888-3270/$ -

doi:10.1016/j.

�CorresponE-mail add

www.elsevier.com/locate/jnlabr/ymssp

Using model reduction and data expansion techniques toimprove SDM

M. Corusa,�, E. Balmesa, O. Nicolasb

aEcole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay Malabry, FrancebR&D Electricite de France, 1 Avenue du General de Gaulle, 92141 Clamart Cedex, France

Received 2 July 2004; received in revised form 7 January 2005; accepted 9 February 2005

Available online 27 June 2005

Abstract

A method designed to predict the effects of distributed modifications of structures is proposed here. Thismethod is an evolution of the classical formulations, and distinguishes measurements and coupling points.

Based on a coarse model of the structure to be modified, the proposed methodology tackles two majordifficulties: efficient predictions for distributed modifications and handling of the lack of measurementpoints on the coupling interface. In addition, displacements bases introduced to reconstruct unmeasuredbehaviour of the interface limit error propagation through the process. Moreover, two indicators areintroduced to select the optimal prediction.

An academic stiffened plate and an industrial application (motor pump) are used to illustrate theapproach and highlight its main advantages.r 2005 Elsevier Ltd. All rights reserved.

Keywords: Structural dynamics; Structural modification; Modal synthesis; Experimental modal analysis; Dynamic

substructuring; Interface modes; Model reduction; Data expansion

0. Introduction

Companies in charge of production processes (such as electricity, water, etc.) are not themanufacturers of their equipment. They may have no access to drawings, design studies or FE

see front matter r 2005 Elsevier Ltd. All rights reserved.

ymssp.2005.02.012

ding author. Tel.: +33 1 41 13 17 40.

ress: [email protected] (M. Corus).

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models. When a vibration crisis occurs on a part of the installation (motor, pump, etc.), a solutionmust be proposed quickly to maintain the production capability and limit maintenance time andcosts. Due to these constraints, the effects of proposed modifications must be predicted withoutthe possibility to build a tuned FE model.Structural Dynamic modification (SDM) methods allow an estimation of the dynamic

behaviour of a structure after a modification when a test derived model of the unmodifiedstructure and a numerical model of the proposed modification are available. These methods, aspresented in [1] for example, are particularly useful when reactivity is needed, since the unmodifiedstructure can be characterised rapidly using an experimental modal test.Fig. 1 illustrates typical difficulties to be tackled with:

measurements restricted to a limited subdomain of the whole structure, � distributed modification with a continuous interface, � non-coincidence between the interface and the measurement points.

Coupling between an experimental model of a tested structure and a numerical model ofmodification is ensured by displacements continuity and virtual work of coupling efforts nullityon the interface(see Section 1.4). However, due to the experimental nature of the base structuremodel, behaviour of the interface is often unavailable, or insufficiently described. An estimationof interface behaviour, compatible with the numerical model of the modification, is then necessaryto define coupling.Few authors dealt with the problem of distributed structural modifications, mostly focusing on

rotational dof. Modal tests generally provide translational measurements, while FE model ofsimple modification also include rotational dof. Schwarz and Richardson, in [2], use bar and plateelements to model a rib stiffener. Unmeasured dof of the FE model belonging to the interface areeliminated using static condensation of the FE model of the modification on test points in themeasured direction. However, this technique is not completely correct, since some loaded dof ofthe FE model are statically condensed. A solution is provided by O’Callahan and Avitabile [3,4]with the use of a full FE model of the tested structure. The lack of rotational dof on the interface istackled with data expansion techniques. However, from the authors themselves, good results wereobtained only in cases where a properly tuned FE model was available. Ambrogio and Sestieri [5,6]mixed the use of FE model and model reduction. Limited FE models of the modified area with andwithout the modification are dynamicaly condensed over test points to account for rotational dof.Nevertheless, they all impose some of the measurement points to be on the interface, in order to

Tested structure

Measurementregion

ModificationInterface

Measurementpoints

Fig. 1. Difficulties handled by the proposed method.

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estimate the behaviour of the coupled substructures, using either impedance or modal coupling (see[1, pp. 265–302] for further details). In all cases, these methods are only applicable when modaltests have been designed for this purpose, the modification being already defined.A complete methodology allowing structural dynamic modification, is presented in this paper.

Taking advantage of non-specific tests conditions, effects of modification presenting continuousinterface with the base structure can be efficiently predicted. Local FE models, needed forinformation reconstruction on the interface, are introduced in Section 1. Observation equations,necessary to account for non-coincidence of test points and interface dof, are presented. Equationgoverning the hybrid coupled problem, deriving from test data expansion, is given. Section 2focuses on reduction basis used to perform interface estimation. Classical solutions are presentedfirst, introducing their advantages and drawbacks. A more appropriate basis, based on reducedlocal FE model mode shapes, is then presented, along with optimal interface estimation selectionindicators. Two examples are proposed to illustrate the proposed methodology. An academicaltest case is introduced in Section 3. The complete methodology is applied to a numerical test case.Prediction results are presented and their analysis widely discussed, especially interpretations ofoptimal basis selection indicators. Section 4 illustrates a true industrial case study, successfullytreated using the proposed techniques. Models quality check and practical aspects are underlined.

1. Local models for SDM

First, principles and main hypotheses will be presented. Then, since the proposed method relieson an expansion process, a model must be defined to achieve two goals: create a kinematical linkbetween test points and interface, and allow the construction of an appropriate reduction basis.The construction principles of the local FE model are detailed in the second section. Reducedbasis expansion process principles are recalled in the third section. The appropriate reductionbasis is described in the fourth section. The fifth section presents the hybrid model and theestimation of coupled behaviour.

1.1. Needs for expansion

As illustrated in Fig. 2, the fundamental difficulty is to relate test displacements fytgNt�1 andinterface dof fqI gNI�1. Since test points and interface nodes of the modification do not coincide, ageometrical description of the supporting structure, where both are defined, will be needed. This isthe first motivation for the introduction of local models. Further requirements on these modelswill be described in Section 1.2.SDM are mostly used for low frequency problems, only involving the first few normal modes of

a structure. So, both interface dof fqI gNI�1 and test displacements fytgNt�1 can be expressed aslinear combinations of these vectors. Since the normal modes cannot be known exactly, one willconsider generalised vectors ½FLg� defined at all points of the local model and assume that theresponse is a linear combination of these vectors

fqLg ¼ ½FLg�fZLg, (1)

where fZLg are the generalised coordinates depicting the behaviour of the structure.

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Fig. 2. Estimation of fqI g deriving from fytg using ½TIt�.

M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 1067–10891070

Interface motion and measurements are then computed using linear observation equations

fytg ¼ ½CtL�fqLg (2)

and

fqIg ¼ ½CIL�fqLg. (3)

as discussed in Section 1.3.The subspace ½FLg� has a dimension lower than the number of sensors, and the generalised

displacement fZLg can be estimated from measured fytg. This is done classically in modalexpansion methods (see [7]) by solving the least-squares problem

fZLg ¼ ArgMinZðkfytg � ½CtL�½FLg�fZLgk

2Þ ¼ ½CtLFLg�þfytg, (4)

where ½CtLFLg�þ denotes a pseudo-inverse. From this solution, one can build a direct linear

relation between test measurements and motion at the interface

fqIg ¼ ½TIt�fytg ¼ ½CIL�½FLg�½CtLFLg�þfytg. (5)

Using a reduced basis ½FLg� and an expansion process has the side benefit of regularising the rawtest results yt. Measurements made in industrial conditions may be subject to many perturbations,such as noise, sensors location and orientation errors. Adding biases due to the identificationprocess and modelling assumptions (linearity, reciprocity, etc.), the need for a regularising processis clear. Optimal subspace must allow a correct estimation and limit errors propagation, this willbe discussed in Section 2.

1.2. Local model

The local model is introduced with two objectives: to ensure a mechanical relationshipbetween the measurement points and the interface dof, and to generate the basis of gene-ralised vectors ½FLg�NL�Ng

. As stated in the introduction, building a tuned FE model of thestructure is typically not acceptable. It is thus proposed to build a coarse, local FE model of theinstrumented subdomain including reasonable mechanical properties. Advantages of thisapproach are:

obtain a quick design and set up of a model depicting the geometry of both structure andmodification;
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ease the construction of displacements fields defined both at measurement points and on theinterface; � ensure the continuity of displacements fields generated by the FE model; � use some a priori mechanical information; � create a basis ½FLg� that is regular with respect to the equation of motion.

Knowing that the model will be used to expand modeshapes of the modified structure, oneimportant constraint is that the proposed modification(s) must be included in the local model. Theway this inclusion is used will be detailed in Section 2.

1.3. Observation equations

The local model is often based on the test geometry. Nevertheless, it is often difficult tohave sensors fytg correspond to a subset of local model dof fqLg. For example, simple FEmodels using beam, plate or shell elements are defined with respect to the neutral fibre,while most measurements are carried out on free surfaces. Accounting for the offset is oftenimportant.The use of linear observation equations of the form (2) is a very general approach to describe

the relation between dof and measurements. Several methods exist to build ½CtL�. Applicationsshown here use a rigid link between each sensor node and the nearest FE model node, asrepresented in Fig. 3. Translational motion at test point is then the combination of translation atmaster node (Nm), and rotations at that point which are estimated by finite differences using twoother nodes (N1 and N2) defining two independent directions.More elaborate methods find the orthogonal projection of the test node on the underlying

surface and use shape function interpolations to estimate the test node motion. Robustimplementations of such methods are however difficult so that the more classical rigid linkapproach was used here.Typically, the local model is built knowing the modification so that a conform mesh can be

considered and fqIg is a subset of fqLg (that is ½CIL� is a Boolean matrix). In some cases, it can besimpler to consider a unique local model for multiple modifications. Observation matrices canthen be used to handle coupling in the absence of a coincident mesh.

Measurement pointF.E. model nodes

Nm

Nt

Linearised rigid link

N1

N2

xy

z

Fig. 3. Construction of ½CtL� using a linearised rigid link between a test point and the nearest FE model node.

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1.4. Hybrid model and coupled predictions

Given an experimentally derived normal mode model with natural frequencies ½Otest�, a possiblynon-diagonal viscous damping matrix ½Gtest� and mass normalised real mode shapes ½Ftest�, oneassumes that the interface and test motion are related by Eq. (5) and uses reciprocity to obtain theequations off motion of the base structure with loads applied on sensors

ð�o2½Id� þ jo½Gtest� þ ½O2test�ÞfZtestg ¼ ½Ftest�

T ½TIt�Tff B

I g, (6)

fytg ¼ ½Ftest�fZtestg. (7)

For the modification, the equations of motion are

½ZM �fqMg ¼ ½BM �fuMg, (8)

fyMg ¼ ½CM �fqMg, (9)

where ½BM � and ½CM � are controllability and observability matrices, respectively, accounting forthe differences between FE dof and measurement points in general cases. Dealing with a FEmodel for the modification, ½BM � ¼ ½CM � ¼ ½Id�, and inputs fuMg are assumed to be externalloads, denoted ff M

g. Assuming a partition into interface dof, denoted I, and complementary dof,denoted C, for the modification model, the structural dynamic equations become

½ZMCC � ½Z

MCI �

½ZMIC � ½Z

MII �

" #fqM

C g

fqMI g

( )¼

ff MC g

ff MI g þ ff

MI extg

( ), (10)

where ff MI extg are interface external loads, ff M

I g coupling efforts and ½ZM � the dynamic stiffness ofthe modification, usually written as

½ZM � ¼ �o2½MM � þ ½DMv � þ ½K

M �, (11)

where ½MM �, ½DMv � and ½K

M � denote, respectively, the mass, damping and stiffness matricesderiving from the FE modelling of the modification.Coupled problems must verify displacement continuity on the interface and nullity of virtual

work associated with coupling loads. Considering the reduced basis ½TIt�, displacementscontinuity is given by

fqMI g ¼ ½TIt�fytg (12)

and nullity of virtual work by

ð½TIt�fytgÞTff B

I g þ fqMI g

Tff MI g ¼ f0g. (13)

Combining Eqs. (8) and (10) and using coupling relations (12) and (13) leads to the classicaldefinition of the coupled problem

½ZMCC � ½ZM

CI �½TIt�½Ftest�

ð½TIt�½Ftest�ÞT½ZM

IC � ½Zcoupled�

" #fqM

C g

fZtestg

( )¼

ff MC g

½TIt�T ff M

Iextg

( ), (14)

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where ½Zcoupled� is the sum of mechanical impedances of tested structure and reduced FE model ofthe modification, that is,

½Zcoupled� ¼ ð½TIt�½Ftest�ÞT½ZM

II �½TIt�½Ftest�

þ ð�o2½Id� þ jo½Gtest� þ ½O2test�Þ. ð15Þ

The solutions of the eigenproblem, derived from the homogeneous form of Eq. (14), give anestimation of the coupled behaviour of the modified structure.The coupled solution is strongly dependent on the selection of ½TIt� which is related by Eq. (5)

to the selection of a set of basis vectors ½FLg� defined on the local model. The selection of thesevectors will be discussed in the following sections.

2. Optimal local model modes

The key issue is the computation of an optimal reduced basis, deriving from the local model,allowing the best prediction for the coupled substructures. Classical solutions based on constraintmodes and attachment modes are presented first.These bases are widely used, but they are not completely pertinent with the objectives, and

exhibit a major drawback. Relevant interface behaviour estimation is indeed needed, but effects ofloads applied on the interface must be taken into account to perform a proper coupling betweensubstructures. When no measurement point is available on the interface, none of these bases willrepresent the effects on the structure of loads applied on the interface, and thus, will lead toinaccurate predictions.An improved reduced basis, based on the proposed local FE model, is introduced to overcome

these issues. The interest of including the modification in this model is highlighted.

2.1. Classical static condensation/expansion

Dealing with structural dynamics and expansion process, one may use constraint modes orattachment modes of the local model associated to the sensors to define ½T �. A local FE modelexists, hence allowing to generate both mass and stiffness matrices, denoted ½ML� and ½KL�,respectively. Let us denote C the complementary subset of dof, and I interface (or kept) subset ofdof. Constraint modes correspond to operator realising the static (or Guyan) condensation ofcomplementary dof C on the subset I. Assuming the partition of Eq. (10), constraint modes ½TGI �

are given by

½TGI � ¼�½KL

CC ��1½KL

CI �

½Id�

" #. (16)

Attachment modes, ½TMN �, represent displacements of the local model fqLg for a unit force ff MNg

imposed at each interface dof I, so that

½KL�½TMN �fZMNg ¼ ff MNg. (17)

Resulting displacement is given by fqLg ¼ ½TMN �fZMNg.

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In many cases, the local FE model has rigid body modes, so that Eq. (17) cannot be solveddirectly. Some solutions, such as mass shifted stiffness matrix, or orthogonalisation of loads½BLt�fuMNg with respect to the rigid body modes of ½KL� (see [8]) are often used to overcome thisdifficulty.When ½KL� has no rigid body modes, ½TMN � is given by

½TMN � ¼½KL

CC � ½KLCI �

½KLIC � ½K

LII �

" #�1½0�NC�NI

½Id�NI�NI

" #, (18)

where NC and NI are the sizes of condensed (i.e. complementary) and kept (i.e. interface) dofsubsets, respectively.However, due to the observation matrix introduced in Section 1.3, definition of attachment

modes and constraint modes must be adapted (see [9]). For attachment modes, ½TMN � representsdisplacements of the local model for a unit force imposed at each sensor, so that, when ½KL� hasno rigid body modes,

½TMN � ¼ ½KL��1½CtL�

T . (19)

As shown in [10], the subspace spanned by static responses to loads (attachment modes) andenforced displacements are equal when ½KL� has no rigid body mode. Otherwise, inertia reliefmodes (resp. rigid body modes) have to be added to constraint modes (resp. attachment modes) toget equality. Thus, constraints modes ½TGI �, associated to sensors, verify ½CtL�½TGI � ¼ ½Id�.Assuming ½CtL� is full rank, then ½TMN � is full rank (see Eq. (17)), and then ½TGI � can be easilydefined using ½TMN � by

½TGI � ¼ ½TMN �ð½CtL�½TMN �Þ�1, (20)

ð½CtL�½TMN �Þ being a full rank square matrix.If ½CtL� is not full rank, test data can be reorganised to eliminate redundant sensors and build a

reduced full rank observability matrix.Dynamic condensation is defined considering ½ZL� at several frequencies of interest, instead

of ½KL�.

2.2. Condensation on sensors and interface

To account for loads on the interface between structure and modification, the model of themodification is included in the local FE model. A priori information for the coupled problem isthen taken into account, such as in loaded interface substructuring techniques (see [11] forexample). In order to provide pertinent a priori information on substructures interactions on theinterface, demand on the local FE model will only be a proper representation of mechanicalproperties around interface. Farther parts of the model are mainly involved in the geometricalinterpolation of test displacements, as stated in Section 1.2, so that physical properties only ensurewell suited displacements field with respect to the mechanics.A first basis, associated to attachment modes defined with respect to the sensors, derived from

Eq. (17), can be computed. However, interface loads are not explicitly represented. But the mostimportant issue is the noise sensitivity of the expansion process. The expansion basis contains as

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many vectors as the number of measurements. Thus, any error in the experimental data will beintroduced in the expansion result.The second basis considers loads on both sensors and interface. Thus, as in Eq. (19), when the

local model has no rigid body modes, ½TLIt� is directly defined by

½TLIt� ¼ ½KL��1

½CIL�

½CtL�

" #T

. (21)

Since this basis is larger than the number of sensors, the pseudo-inverse in Eq. (4) is not welldefined. One thus needs to select a subspace within the span of ½TLI

t�. In order to perform the best

choice, vectors must be classified. Dealing with low frequency problem, and considering theregularisation issues (noise, identification bias, model uncertainties, etc.), a classification withrespect to strain energy appears to be pertinent. Thus, let us define a generalised expansion basis½FLg� used to estimate interface displacements from measurements

½FLg� ¼ ½TLIt�½FI

t g�. (22)

Vectors of ½FIt g� are solutions of the eigenvalue problem associated with the reduced model defined

using ½TLIt�,

ð�o2g½TLI

t�T ½ML�½TLI

t� þ ½TLI

t�T ½KL�½TLI

t�ÞffI

t gg ¼ f0g, (23)

classified with increasing o2g.

Expansion using this basis will be referred in the following as ‘‘LMME’’ for ‘‘Local model modeexpansion’’.

2.3. Optimal truncation indicators

As presented in Section 2.2, vectors of ½FLg� are classified with respect to potential elasticenergy. ½TIt� is computed using less vectors of ½FLg� than sensors, for regularisation purposes, butthe optimal number of vectors is still to be determined. To select the appropriate basis, twoindicators are introduced. The first one is based on the comparison of displacements on theinterface for the coupled problem obtained by two different ways. The second one is based on thereversibility of the coupling process.

2.3.1. Energetic interface continuity indicators (EICI)

Let us denote ð½FC �; ½OC�Þ the solution of the undamped homogeneous eigenvalue problemderived from Eq. (14) for the coupled problem

ð�oC2½MC � þ ½KC �ÞfFCg ¼ f0g. (24)

Eigenmodes fFCg are defined for both complementary dof of the modification (fqMC g) and

generalised dof of the tested structure (fZtestg). Let us denote ½FCZ � eigenmodes defined for the

generalised subset of dof.Since displacements fyC

testg for the coupled problem are spanned by ½Ftest� (see Eq. (14)), coupledeigenmodes are given by ½FC

test� ¼ ½Ftest�½FCZ �. Thus, on the interface, one has

½FLMMEI � ¼ ½TIt�½Ftest�½FC

Z �. (25)

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To validate our confidence in this estimated interface motion, one considers a second estimategiven by the static expansion of the predicted coupled mode shapes ½Ftest�½FC

Z �. To perform thisexpansion, the FE model of the base structure, without the modification, is taken into account.The stiffness matrix associated to the local model is formally given by ½KL� ¼ ½KB� þ ½KM �, ½KB�

for the base structure, and ½KM � for the modification. Thus, the expression of the reduction basisleading to static expansion over the local model without the modification is given by

½TStat� ¼ ð½KB��1½CtL�

T Þð½CtL�½KB��1½CtL�

T Þ�1, (26)

when ½KB� has no rigid body modes.Thus, as detailed in Section 2.1, interface displacement corresponding to pseudo-test coupled

mode shapes ½FCtest� are given by

½FStatI � ¼ ½CIL�ð½CtL�½T

Stat�Þþ½FC

test�. (27)

Using estimates (25) and (27) of interface displacement for the coupled problem, an assump-tion is made that, for a proper prediction, both should provide close estimations of inter-face displacements. Thus, lack of continuity at interface is measured using the differentialdisplacements fFLMME

I g � fFStatI g for each predicted mode. An energetic measurement is defined

using the FE model of the modification to overcome the difficulty due to the differences betweendof (in plane, out of plane, rotational, etc.). The expressions of these indicators, for the jth mode,are given by

strain energy criterion:

ðDEKÞj ¼

kfFLMMEI gj � fF

StatI gjk

2KM

kfFLMMEI gjk

2KM þ kfFStat

I gjk2KM

,

kinetic energy criterion:

ðDEMÞj ¼

kfFLMMEI gj � fF

StatI gjk

2MM

kfFLMMEI gjk

2MM þ kfFStat

I gjk2MM

.

These indicators focus on the expansion process. They are not absolute criteria for predictionquality. Nevertheless, the quality in predicting eigenvalues and eigenmodes for the modifiedstructure is strongly related to the reconstruction of the interface behaviour and loads with respectto the modification. Low values for both ðDEK

Þj and ðDEMÞj would indicate close interface

behaviour estimations using methods with opposite purposes. So, results could be considered withsignificant confidence. When indicators exhibit large values, a closer inspection of intermediateresults has to be performed to evaluate prediction accuracy. Fig. 4 illustrates the link betweenEICI and the lack of continuity at the interface. Prediction results for the same mode arepresented, using expansion basis ½FLg� presenting not enough vectors (on the left), and optimalvalue (on the right). It is clear that continuity is not ensured for the prediction result presentedon the left, corresponding to large values of both indicators, when result presented on theright seems to be more accurate. A more detailed analysis of indicators behaviour is presented inSection 3.

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Fig. 4. Illustration of relationship between lack of continuity and EICI values.

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2.3.2. Coupling reversibilityThe second indicator tests the reversibility of the coupling process and its accuracy. SDM often

deal with stiffness or mass addition. Cases dealing with removals are rarely exposed. Starting fromthe prediction results previously obtained, effects of a removal of the modification will beestimated using the same methodology, adapted for this purpose. Coupling and uncouplingprocedures are done in similar ways, but using different operators and local FE models. Thus,predicted behaviour obtained from coupling and then uncoupling procedures could be comparedwith original behaviour of the tested structure to estimate the accuracy of the coupling process.

2.3.2.1. Expansion basis definition for uncoupling process. Objective of this second step is the bestestimation of behaviour after removal of the modification. Thus, considered local model does notinclude the modification. In the same manner, a priori informations on effects of the removal are

introduced. Let us denote ½Fc=u

Lg � the basis of the eigenmodes of this local model condensed on test

points and interface dof. Interface estimation operator ½Tc=u

It � is then defined using Eq. (5).

However, optimal selection of vectors of ½Fc=u

Lg � must be operated. Considering the particular issue

of the coupling/uncoupling process, simple concepts are introduced to overcome this difficulty.SDM provides good results as far as modified behaviour ½FC

test�, for a given frequency bandwith,

and initial behaviour ½Ftest� are spanned by the same subspace. Subspaces ½TIt� and ½Tc=u

It �, areintroduced to span both measurements subspaces, and add a priori information on the coupled/uncoupled behaviour through interface loading/unloading. Nevertheless, subspaces spanned by

½TIt� and ½Tc=u

It �, for arbitrary sizes of ½FLg� and ½Fc=u

Lg �, may be significantly different.Considering that on one hand, ½Ftest� and ½FC

test� are spanned by the same subspace, and on theother hand, that selection process aims to extract the smallest subspace describing interfacebehaviour from test points displacements, optimal selection is simply realised.

First, vectors of ½Fc=u

Lg � are reorganised, so that subspaces restricted to measurements points fit.

Thus, an operator ½Preorg� is introduced, verifying

½CtL�½Fc=u

Lg �½Preorg� ¼ ½CtL�½FLg�. (28)

Let us denote ½Fc=uLg � ¼ ½F

c=u

Lg �½Preorg�, and ½Tc=uIt � the interface estimation operator deriving from Eq.

(5) using ½Fc=uLg �. Thus, ½TIt� and ½T

c=uIt � provide the same representation of the measurements, while

introducing complementary informations on loads on the interface.

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A condition to ensure the uniqueness of ½Preorg�, and both ½TIt� and ½Tc=uIt � to span the same

measurement subspace, is to consider as many vectors of ½FLg� as of ½Fc=uLg �. Thus, for the optimal

number of vectors Nopt, this provides the minimal subspace spanning both ½Ftest� and ½FCtest�, since

NoptpNt.

2.3.2.2. Hybrid uncoupling model. For a given size Nn 2 ½1;Nt� of ½FLg�, hybrid coupled model isthe assembled with respect to Eq. (14). Considering ½T

c=uIt � and predicted behaviour ð½FC

test�; ½OC�Þ,

eigenvalue problem leading to initial behaviour prediction is assembled. Couples ðfFc=ug;oc=uÞ

verify

�½ZMCC � �½ZM

CI �½Tc=uIt �½F

Ctest�

�½FCtest�

T ½Tc=uIt �

T ½ZMIC � ½Zc=u�

24

35 fFc=u

C g

fFc=utestg

8<:

9=; ¼

f0g

f0g

( ), (29)

where

½Zc=u� ¼ �ðoc=uÞ2½Id� þ ½OC �2 � ð½T

c=uIt �½F

Ctest�Þ

T½ZM

II �½Tc=uIt �½F

Ctest�. (30)

Estimated behaviour derived from coupling/uncoupling is then compared to initial behaviourusing the modal assurance criterion (MAC) and relative error on the frequencies.However, eigenvalue problem presented in Eq. (29) may be ill-conditioned. Since the overall

problem, defined for both base structure and modification, is projected on a truncated basis, massand stiffness removal on the interface could be more important than projected masses andstiffnesses of the coupled structures. In this case, mass matrix is not positive definite anymore, andsome negative eigenvalues may occur. This situation may indicate that for this particular rank ofboth ½TIt� and ½T

c=uIt �, a priori information on the stiffness ratio between modification and base

structure is underestimated, so that such information is still valuable for optimal selectionpurposes.

3. Numerical case study

The first example is a numerical case study. A pseudo-test derived from a FE model is used toperform structural modification. The complete methodology is developed and the analysis ofselection indicators is amply detailed.

3.1. Test device

The FE mesh of the base structure is presented in Fig. 5. This test configuration is a simplerectangular pate, stiffened on its borders, in free–free conditions. Two orthogonal ribs areintroduced to break the symmetry of the mode shapes. The modification consists in a rib, addedon the softest semi diagonal of the base plate. Natural frequencies of the first five mode shapes arestated in Table 1.The MAC between initial and modified mode shapes is also presented in Table 1. Moreover, the

stars for modes three and four denote a switching between the mode shapes due to themodification. The proposed modification is large enough to bring noticeable variations of natural

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Fig. 5. Numerical case study: left: FE mesh with pseudo-sensors locations and measurement directions—right:

modified structure—modification shown in gray.

Table 1

Natural frequencies of the structure before and after modification

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Freq. before mod. (Hz) 32.3 118.1 152.0 181.1 203.7

Freq. after mod. (Hz) 39.7 123.8 164.8 184.9 215.1

MAC before/after 90.1 88.5 79.5� 78.7� 81.3

The star indicates that modeshapes 3 and 4 have shifted.

M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 1067–1089 1079

frequencies and still sufficiently small to cause only local mode shapes modifications. Objective isthe prediction of this first five mode shapes and natural frequencies of the stiffened structure.The first 12 flexible mode shapes of the structure are introduced in the modal model. Test

configuration is quite fine and provides observation of these modes. No measurement point islocated on the interface, and only out of plane displacements are supposed to be measured.1 Theuse of a local model, allowing an estimation of these displacements, significantly increases theprediction results.The originality of the LMME-SDM lies in the local model. Following the principles stated in

Section 1.2, the local model is a coarse FE model of the tested subdomain of the structure.Thickness and mechanical properties of the base plate are used. However, this model is not atuned FE model of the tested structure, due to the geometrical truncation. The FE mesh ispresented in Fig. 6.

3.2. Results

Table 2 summarises the selected optimal prediction results. Results for the modified FE modelare also presented for comparison. These results only derive from the analysis of indicatorsbehaviour, presented in the following, assuming that a posteriori results are unknown.Nevertheless, these predictions are in very good agreement with the FE model of the completestructure. Both frequencies and shapes are well estimated.

1A previous analysis of this case study pointed out the influence of in-plane displacements of the base structure in the

proper prediction of coupled behaviour [12].

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Table 2

Coupling results using LMME—frequencies and MAC for the first five eigenmodes

Mode 1 2 3 4 5

Base structure (Hz) 32.3 118.1 152.0 181.1 203.7

Modified structure (Hz) 39.7 123.8 164.8 184.9 215.1

LMME prediction (Hz) 38.4 122.2 163.8 184.3 212.9

MAC LMME/exact (%) 98.4 99.3 97.5 98.0 98.0

Fig. 6. Local model used for LMME-SDM application—test mesh is represented in red.

M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 1067–10891080

It is important to notice that results are considered to be valid only when predicted frequencydoes not exceed two third of the maximum frequency taken into account in ½Otest�.

3.2.1. Results analysis

As stated in Section 2.2, prediction results depend on the size of the subspace retained forexpansion. Sixteen measurement points are defined on the test device, so that the size of thereduced basis used to perform LMME cannot exceed 16 vectors. In this particular case, 13 vectorsof ½FLg� are used to compute interface estimator ½TIt�, along with the three necessary vectors toestimate rigid body modes of tested structure. Fig. 7 summarises the prediction results andselection indicators behaviour for the fifth mode of the coupled structure.Analysis to select the appropriate prediction for each mode is based on those three graphs.

Upper left quarter displays predicted frequency with respect to the size of ½FLg�. Initial frequencyis also indicated. Lower left quarter shows evolutions of EICI, while comparison of the estimatedcoupled uncoupled behaviour with initial behaviour is represented in the lower right quarter. Oneach graph, selected optimum is identified with a vertical black line. Optimal selection based onboth DEK

and DEM, along with reversibility test comparisons is explained in the following.

3.2.2. EICI analysisFirst step on the selection process is based on EICI indicators, presented in Section 2.3.1. DEM

isa mass normalised measure of the interface disjunction. This indicator is associated to a meanerror between the two expansion processes, and focuses on the regularity of the interfacedisplacements estimation. Since DEK

is a stiffness normalised measure, it puts the emphasis on the

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Fig. 7. Prediction results for mode 5.

M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 1067–1089 1081

interface displacements estimations matching. Thus, this indicator can be viewed as an estimationof the deviation of the interface disjunction. Analysis is mainly based on the following statements.

Low values would denote a good agreement between LMME and static expansion. Since twoquite different techniques provide the same interface behaviour estimation, confidence is placedin the coupled predictions, and LMME based predictions are kept. � Large values would denote noticeable differences in the two concurrent expansion processes,not prejudging of correct or erroneous results. However, leading to a low confidence in theprediction results, a close inspection of every intermediate expanded and coupled shape isrequired.

For this prediction, DEKand DEM

both decline with the growing size of ½FLg�, spelling closerinterface behaviour estimations and a good regularity. Estimation with more than nine vectors of½FLg� should thus lead to a pertinent prediction. However, frequency prediction is not reallystabilised, especially when 13 modes are introduced in ½FLg�. Reversibility tests are then used todetermine more precisely the optimal size of the expansion basis to be retained.

3.2.3. Reversibility test analysis

Coupling reversibility is described in Section 2.3.2. This process provides a finest indication onthe estimated behaviour quality, since predicted behaviour is already pertinent. Indeed, two majordrawbacks are to be tackled. The first one, mentioned in Section 2.3.2, is a bad conditioning (withrespect to a mechanical system) for the uncoupling problem. However, erroneous eigenvalues(imaginary natural frequencies) still are valuable informations, indicating a non-pertinent basis.The other drawback is a misestimation of the reversibility. For low sizes of ½FLg�, interface

behaviour is poorly estimated, so that the modification seems to have no effect on the dynamics.

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Following the same process to estimate the initial behaviour from the predictions, removal has noeffects on the behaviour. Thus, estimated behaviour after coupling/uncoupling is similar to initialbehaviour, leading to good MAC and low frequency relative errors.This point is very important, and clearly demonstrated in Fig. 7. When only the first four

vectors of ½FLg� are taken into account, C/U indicators would indicate a good prediction.However, a closer inspection of intermediate results, along with predicted frequency evolution andEICI values would contradict this conclusion.Reversibility analysis must be carried out very carefully only for cases when EICI exhibit

reasonably low values, i.e. for reasonable expansion results. In this particular case, coupling/uncoupling test is particularly efficient to determine the optimum when more than nine vectors areintroduced in ½FLg�. For 9–11 vectors, reversibility test results and initial behaviour are in goodagreement, while EICI values are low. Optimum is then determined for ten vectors, where C/Uresults are slightly better.

3.2.4. A posteriori comparisonIn most cases, optimal selection cannot be automated, since indicators and intermediate results

have to be analysed. Optimal reconstruction basis ½FLg� must realise the best compromise betweenthe following propositions:

pertinent reconstruction of interface displacements with respect to static expansion (i.e. low DEK

values),

� regular interface displacements field, compatible with low frequency motions assumptions (i.e.low DEM

values),

� correct reversibility, assuming that predicted behaviour exhibits low interface disjunction.

Fig. 8. Prediction results for mode 5—comparison with numerical results for the complete structure.

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Fig. 8 presents prediction results and the comparison with those obtained using the complete FEmodel.Indicators analysis allowed a proper basis selection, so that retained prediction is in good

agreement with expected behaviour. Results presented in Table 2 derive from the same analysisfor all the five modes.

4. Industrial case study

Example presented in Section 3 illustrates the proposed methodology and details the analysis ofindicators. Accurate predictions of effects of a modification on an academical test device areobtained. Industrial problems have also been treated successfully.The case of a motor pump is treated in this section. The motor and the pump are presented in

Fig. 9. High in-operational vibration levels were observed, occurring at the rotating frequency ofthe pump. This structure ensures the washing of drums filtering the cooling water in nuclearplants. Being critical in the electricity production process, a solution had to be quickly proposedand evaluated.This study has been fully treated using the LMME based SDM, both for reactivity and

prediction accuracy purposes. Proposed modifications have been installed on site, and predictionresults compared with in situ modal analysis after modification.

4.1. Models construction

An experimental analysis of motor/pump assembly is performed. The first five modes involvedin the in-operational behaviour are identified. High levels mainly comes from the first two modes,presented in Fig. 10. Since the motor and the pump are rotating around 25Hz, four stiffeners areproposed to reinforce the support, and set these natural frequencies away from the rotatingfrequency. Avoiding appropriation between excitation and modal response would lowervibrations levels.However, due to transmission, harmonic excitations located around 50Hz may occur. In in-

operational conditions, the torsional mode of the pump (50.4Hz) is not excited. Thus, a second

Fig. 9. Motor mounted on the support (left), pump and piping (middle) and modified support (right).

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Fig. 10. Mode shapes and natural frequencies of the first two modes—proposed modification.

M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 1067–10891084

objective is not to modify this mode, so that no appropriation exists between torsional mode andharmonic excitation around 50Hz.A modal model is derived from this experimental analysis, including the first eight modes,

ensuring valid prediction results up to 75Hz, with respect to the Rubin’s criteria (see [13]). Thesynthesised FRF presented in Fig. 11 demonstrates the quality of the modal model for the first fivemodes.Indeed, to ensure good predictions, identification leading to the modal model for the base

structure has to be carried out carefully. Model reciprocity and mode shapes scaling withrespect to the mass are the two major points to be closely inspected. Reciprocity assumptionis a key hypothesis, leading to left and right eigenvectors equality, which allows mode shapesidentification at each test and measurement points for a non-fully reciprocical test, and coupledmodel assembly presented in Eq. (15). Proper mode shapes scaling ensures good relativeimportance of substructures. For example, underestimated generalised modal masses would leadto overestimated effects of the modification. Generalised modal masses are of critical importancein the structural dynamic modification methods, and are well identified in this study.A local FE model, depicted in Fig. 11, is built, based on a crude geometrical description.

Neither precise mechanical data, nor thicknesses for the motor support and plate are available.Moreover, only the external parts are described. However, masses and stiffnesses are roughlytuned, so that deriving modal behaviour and experimental analysis fit a few for the first fourmodes.However, this model is not representative of a FE model and should not be used directly

to estimate the effects of modifications on the behaviour. Since proper physical properties

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Fig. 11. Local FE model with the proposed modification (left) and collocated FRF denoting the quality of the modal

model (right).

Table 3

Prediction results—comparison with experimental modal analysis for the modified structure

Mode n 1 2 3 4 5

FEMA ini: (Hz) 26.6 31.2 34.0 38.6 50.4

FLMME (Hz) 31.9 34.0 42.1 46.5 50.4

FEMAmod: (Hz) 32.0 34.6 41.2 43.6 50.4

MACpred:=EMA 0.93 0.92 0.87 0.92 0.98

DF (%) �0.4 �1.9 2.0 6.7 0.0

M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 1067–1089 1085

are unknown, strongly biased generalised modal masses could lead to highly erroneousprediction.

4.2. Prediction results and a posteriori comparison with modified structure

Table 3 summarises the results for the first five modes. FEMA ini:, FLMME and FEMAmod: denotethe identified frequencies for unmodified structures, predicted frequencies and identified

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frequencies for the modified structure respectively, DF concerning the relative error betweenpredicted and identified frequencies. The MACpred:=EMA indication denotes the modal assurancecriterion between predicted and identified mode shapes for the modified structure.Predicted mode shapes are presented in Fig. 12. The ability of the method to quickly and

accurately predict the effects of a distributed modification on the behaviour of a structure issuccessfully demonstrated. Modification’s objectives are successfully achieved, since motor and

Fig. 12. Predicted frequencies and mode shapes for the first five modes.

Fig. 13. Measured and predicted FRF at first control point.

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pump present no mode around 25Hz anymore, ensuring lower vibration levels. Uncouplingbetween the two structures is realised, while the torsional mode stays unmodified at 50.4Hz.The comparison between the predicted FRF and in situ measurements presented in Fig. 13

confirm the accuracy of the prediction, especially for modal masses. This information is of majorimportance when in-operational behaviour predictions are expected. In case where forceidentification would have been performed, assuming the modification would not modifyexcitation locations and spectra, in-operational vibration levels could have been accuratelypredicted.

5. Conclusion

Structural modification methods permit an estimation of the dynamic behaviour of a structureafter a modification when a behaviour model of the unmodified structure and a numerical modelof the proposed modification are available. An original approach allowing distributedmodifications is presented. The effects of many different modifications can then be estimatedusing a generic test set-up.A rough local FE model of the measurement subdomain is introduced permitting the non-

coincidence between measurement points and dof at the interface of the modification and thestructure. This model ensures kinematical links between measurement points and interface dof.Moreover, to overcome the lack of force representation on the interface, the FE model of themodifying structure is introduced in the local FE model.A smoothing expansion basis based on the eigenmodes of a reduced model derived from the

local model is computed. Tests data are interpolated using subspace based expansion process, sothat information on the interface between structure and modification is estimated. Since theestimation depends on the size of the selected subspace, two indicators are built to estimate thequality of the expansion over the interface. First estimator is based on two different evaluations ofthe behaviour of the interface for the coupled problem, leading to an energetic measurement of theinterface disjunction for the modified structure. Second estimator tests the reversibility of thewhole process. The behaviour of the initial structure is estimated considering the removal of themodification from the modified structure.Two examples are presented. A numerical academic test configuration demonstrated the

efficiency of the method, along with the analysis of indicators. Optimal selection process ispresented, leading to an accurate prediction. The other example is an industrial case study. Anon-specific modal test permitted the analysis of the high in-operation vibration levels. Asolution has been quickly proposed, precisely evaluated and installed, a posteriori testsdemonstrating the efficiency of the prediction. The ability of this method to reconstruct unknowndisplacements fields using few hypotheses is shown. Reactivity and time saving issues areillustrated.Further developments concern in-operational behaviour predictions. First step could be the

prediction of damping. No assumption is made, so that the proposed formulation could bederived to predict the effects of viscoelastic treatments or local dampers, in addition with classicalstructural modification. Second step deals with force identification procedure. Using the sameprinciples, accurate in-operational behaviour predictions could be achieved. Along with these

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M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 1067–10891088

developments, the use of strain measurements could provide a better behaviour model allowingmore precise predictions.

Appendix. Nomenclature

Matrices

½M� mass ½K � stiffness ½Dv� viscous damping ½Z� dynamic stiffness ½H� flexibility ½F� normal modes ½O� natural angular frequencies ½G� damping ratios ½T �N�NT

expansion/reduction basis

½B�N�Na

controllability

½C�Ns�N

observability

Vectors

fqgN model states (i.e. FEM dof) ff gN generalised loads ffigN ith eigenmode fZgNM

generalised dof

fugNa

inputs (i.e. physical loads)

fygNs

outputs (i.e. measurements)

Scalars

o angular frequency oi ith natural angular frequency xi ith damping ratio dij Kronecker delta

Superscripts and subscripts

B base structure M modification BþM complete structure C coupled SDM problem I interface dof C complementary dof (or condensed dof) t test quantity (measurements) g generalised quantity L local model
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