Using model reduction and data expansion techniques to improve SDM

Transcription

1 Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) Using model reduction and data expansion techniques to improve SDM M. Corus a,, E. Balme` s a, O. Nicolas b a École Centrale Paris, Grande Voie des Vignes, Châtenay Malabry, France b R&D Électricité de France, 1 Avenue du Général de Gaulle, 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 Corresponding author. Tel.: address: corus@mssmat.ecp.fr (M. Corus) /$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi: /j.ymssp

2 1068 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) models. When a vibration crisis occurs on a part of the installation (motor, pump, etc.), a solution must be proposed quickly to maintain the production capability and limit maintenance time and costs. Due to these constraints, the effects of proposed modifications must be predicted without the 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 unmodified structure and a numerical model of the proposed modification are available. These methods, as presented in [1] for example, are particularly useful when reactivity is needed, since the unmodified structure 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 of modification is ensured by displacements continuity and virtual work of coupling efforts nullity on the interface(see Section 1.4). However, due to the experimental nature of the base structure model, behaviour of the interface is often unavailable, or insufficiently described. An estimation of interface behaviour, compatible with the numerical model of the modification, is then necessary to 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 of simple modification also include rotational dof. Schwarz and Richardson, in [2], use bar and plate elements to model a rib stiffener. Unmeasured dof of the FE model belonging to the interface are eliminated using static condensation of the FE model of the modification on test points in the measured direction. However, this technique is not completely correct, since some loaded dof of the 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 is tackled with data expansion techniques. However, from the authors themselves, good results were obtained 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 and without 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 Interface Modification Measurement points Measurement region Tested structure Fig. 1. Difficulties handled by the proposed method.

3 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) estimate the behaviour of the coupled substructures, using either impedance or modal coupling (see [1, pp ] for further details). In all cases, these methods are only applicable when modal tests 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 continuous interface with the base structure can be efficiently predicted. Local FE models, needed for information 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. Equation governing the hybrid coupled problem, deriving from test data expansion, is given. Section 2 focuses on reduction basis used to perform interface estimation. Classical solutions are presented first, introducing their advantages and drawbacks. A more appropriate basis, based on reduced local FE model mode shapes, is then presented, along with optimal interface estimation selection indicators. Two examples are proposed to illustrate the proposed methodology. An academical test 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 of optimal basis selection indicators. Section 4 illustrates a true industrial case study, successfully treated 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 relies on an expansion process, a model must be defined to achieve two goals: create a kinematical link between 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. Reduced basis expansion process principles are recalled in the third section. The appropriate reduction basis is described in the fourth section. The fifth section presents the hybrid model and the estimation of coupled behaviour Needs for expansion As illustrated in Fig. 2, the fundamental difficulty is to relate test displacements fy t g Nt 1 and interface dof fq I g NI 1. Since test points and interface nodes of the modification do not coincide, a geometrical description of the supporting structure, where both are defined, will be needed. This is the first motivation for the introduction of local models. Further requirements on these models will 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 fq I g NI 1 and test displacements fy t g Nt 1 can be expressed as linear combinations of these vectors. Since the normal modes cannot be known exactly, one will consider generalised vectors ½F Lg Š defined at all points of the local model and assume that the response is a linear combination of these vectors fq L g¼½f Lg ŠfZ L g, (1) where fz L g are the generalised coordinates depicting the behaviour of the structure.

4 1070 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) Fig. 2. Estimation of fq I g deriving from fy t g using ½T It Š. Interface motion and measurements are then computed using linear observation equations fy t g¼½c tl Šfq L g (2) and fq I g¼½c IL Šfq L g. (3) as discussed in Section 1.3. The subspace ½F Lg Š has a dimension lower than the number of sensors, and the generalised displacement fz L g can be estimated from measured fy t g. This is done classically in modal expansion methods (see [7]) by solving the least-squares problem fz L g¼argminðkfy t g ½C tl Š½F Lg ŠfZ L gk 2 Þ¼½C tl F Lg Š þ fy t g, (4) Z where ½C tl F Lg Š þ denotes a pseudo-inverse. From this solution, one can build a direct linear relation between test measurements and motion at the interface fq I g¼½t It Šfy t g¼½c IL Š½F Lg Š½C tl F Lg Š þ fy t g. (5) Using a reduced basis ½F Lg Š and an expansion process has the side benefit of regularising the raw test results y t. Measurements made in industrial conditions may be subject to many perturbations, such as noise, sensors location and orientation errors. Adding biases due to the identification process and modelling assumptions (linearity, reciprocity, etc.), the need for a regularising process is clear. Optimal subspace must allow a correct estimation and limit errors propagation, this will be discussed in Section Local model The local model is introduced with two objectives: to ensure a mechanical relationship between the measurement points and the interface dof, and to generate the basis of generalised vectors ½F Lg Š NL N g. As stated in the introduction, building a tuned FE model of the structure is typically not acceptable. It is thus proposed to build a coarse, local FE model of the instrumented subdomain including reasonable mechanical properties. Advantages of this approach are: obtain a quick design and set up of a model depicting the geometry of both structure and modification;

5 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) ease the construction of displacements fields defined both at measurement points and on the interface; ensure the continuity of displacements fields generated by the FE model; use some a priori mechanical information; create a basis ½F Lg Š that is regular with respect to the equation of motion. Knowing that the model will be used to expand modeshapes of the modified structure, one important constraint is that the proposed modification(s) must be included in the local model. The way this inclusion is used will be detailed in Section Observation equations The local model is often based on the test geometry. Nevertheless, it is often difficult to have sensors fy t g correspond to a subset of local model dof fq L g. For example, simple FE models 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 often important. 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 ½C tl Š. Applications shown here use a rigid link between each sensor node and the nearest FE model node, as represented in Fig. 3. Translational motion at test point is then the combination of translation at master node (Nm), and rotations at that point which are estimated by finite differences using two other 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. Robust implementations of such methods are however difficult so that the more classical rigid link approach was used here. Typically, the local model is built knowing the modification so that a conform mesh can be considered and fq I g is a subset of fq L g (that is ½C IL Š is a Boolean matrix). In some cases, it can be simpler to consider a unique local model for multiple modifications. Observation matrices can then be used to handle coupling in the absence of a coincident mesh. Linearised rigid link Nt Nm N1 N2 y z x F.E. model nodes Measurement point Fig. 3. Construction of ½C tl Š using a linearised rigid link between a test point and the nearest FE model node.

6 1072 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) Hybrid model and coupled predictions Given an experimentally derived normal mode model with natural frequencies ½O test Š, a possibly non-diagonal viscous damping matrix ½G test Š and mass normalised real mode shapes ½F test Š, one assumes that the interface and test motion are related by Eq. (5) and uses reciprocity to obtain the equations off motion of the base structure with loads applied on sensors ð o 2 ½IdŠþjo½G test Šþ½O 2 test ŠÞfZ testg ¼½F test Š T ½T It Š T ff B I g, (6) fy t g¼½f test ŠfZ test g. (7) For the modification, the equations of motion are ½Z M Šfq M g¼½b M Šfu M g, (8) fy M g¼½c M Šfq M g, (9) where ½B M Š and ½C M Š are controllability and observability matrices, respectively, accounting for the differences between FE dof and measurement points in general cases. Dealing with a FE model for the modification, ½B M Š¼½C M Š¼½IdŠ, and inputs fu M g are assumed to be external loads, 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 " ½Z M CC Š ½ZM CI Š #( ) ( fq M C g ff M C ½Z M IC Š ½ZM II Š fq M I g ¼ g ) ff M I gþff M, (10) I ext g where ff M I ext g are interface external loads, ff M I g coupling efforts and ½ZM Š the dynamic stiffness of the modification, usually written as ½Z M Š¼ o 2 ½M M Šþ½D M v Šþ½KM Š, (11) where ½M M Š, ½D M v Š and ½KM Š denote, respectively, the mass, damping and stiffness matrices deriving 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 ½T It Š, displacements continuity is given by fq M I g¼½t ItŠfy t g and nullity of virtual work by ð½t It Šfy t gþ T ff B I gþfqm I gt ff M I g¼f0g. (13) Combining Eqs. (8) and (10) and using coupling relations (12) and (13) leads to the classical definition of the coupled problem " ½Z M CC Š ½ZM CI Š½T #( ) ( ItŠ½F test Š fq M C g ff M C ð½t It Š½F test ŠÞ T ½Z M IC Š ½Z ¼ g ) coupledš fz test g ½T It Š T ff M, (14) I ext g (12)

7 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) where ½Z coupled Š is the sum of mechanical impedances of tested structure and reduced FE model of the modification, that is, ½Z coupled Š¼ð½T It Š½F test ŠÞ T ½Z M II Š½T ItŠ½F test Š þð o 2 ½IdŠþjo½G test Šþ½O 2 test ŠÞ. The solutions of the eigenproblem, derived from the homogeneous form of Eq. (14), give an estimation of the coupled behaviour of the modified structure. The coupled solution is strongly dependent on the selection of ½T It Š which is related by Eq. (5) to the selection of a set of basis vectors ½F Lg Š defined on the local model. The selection of these vectors will be discussed in the following sections. ð15þ 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 constraint modes 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 of loads applied on the interface must be taken into account to perform a proper coupling between substructures. When no measurement point is available on the interface, none of these bases will represent the effects on the structure of loads applied on the interface, and thus, will lead to inaccurate 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 Classical static condensation/expansion Dealing with structural dynamics and expansion process, one may use constraint modes or attachment modes of the local model associated to the sensors to define ½TŠ. A local FE model exists, hence allowing to generate both mass and stiffness matrices, denoted ½M L Š and ½K L Š, respectively. Let us denote C the complementary subset of dof, and I interface (or kept) subset of dof. Constraint modes correspond to operator realising the static (or Guyan) condensation of complementary dof C on the subset I. Assuming the partition of Eq. (10), constraint modes ½T GI Š are given by " ½T GI Š¼ ½KL CC Š 1 ½K L CI Š #. (16) ½IdŠ Attachment modes, ½T MN Š, represent displacements of the local model fq L g for a unit force ff MN g imposed at each interface dof I, so that ½K L Š½T MN ŠfZ MN g¼ff MN g. (17) Resulting displacement is given by fq L g¼½t MN ŠfZ MN g.

8 1074 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) In many cases, the local FE model has rigid body modes, so that Eq. (17) cannot be solved directly. Some solutions, such as mass shifted stiffness matrix, or orthogonalisation of loads ½B Lt Šfu MN g with respect to the rigid body modes of ½K L Š (see [8]) are often used to overcome this difficulty. When ½K L Š has no rigid body modes, ½T MN Š is given by " # 1 " # ½0Š NC N I, (18) ½IdŠ NI N I ½T MN Š¼ ½KL CC Š ½KL CI Š ½K L IC Š ½KL II Š where N C and N I are the sizes of condensed (i.e. complementary) and kept (i.e. interface) dof subsets, 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, ½T MN Š represents displacements of the local model for a unit force imposed at each sensor, so that, when ½K L Š has no rigid body modes, ½T MN Š¼½K L Š 1 ½C tl Š T. (19) As shown in [10], the subspace spanned by static responses to loads (attachment modes) and enforced displacements are equal when ½K L Š has no rigid body mode. Otherwise, inertia relief modes (resp. rigid body modes) have to be added to constraint modes (resp. attachment modes) to get equality. Thus, constraints modes ½T GI Š, associated to sensors, verify ½C tl Š½T GI Š¼½IdŠ. Assuming ½C tl Š is full rank, then ½T MN Š is full rank (see Eq. (17)), and then ½T GI Š can be easily defined using ½T MN Š by ½T GI Š¼½T MN Šð½C tl Š½T MN ŠÞ 1, (20) ð½c tl Š½T MN ŠÞ being a full rank square matrix. If ½C tl Š 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 ½Z L Š at several frequencies of interest, instead of ½K L Š Condensation on sensors and interface To account for loads on the interface between structure and modification, the model of the modification is included in the local FE model. A priori information for the coupled problem is then taken into account, such as in loaded interface substructuring techniques (see [11] for example). In order to provide pertinent a priori information on substructures interactions on the interface, demand on the local FE model will only be a proper representation of mechanical properties around interface. Farther parts of the model are mainly involved in the geometrical interpolation of test displacements, as stated in Section 1.2, so that physical properties only ensure well 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 most important issue is the noise sensitivity of the expansion process. The expansion basis contains as

9 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) many vectors as the number of measurements. Thus, any error in the experimental data will be introduced 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, ½T L I t Š is directly defined by " # T ½C ½T L I t Š¼½K L Š 1 IL Š. (21) ½C tl Š Since this basis is larger than the number of sensors, the pseudo-inverse in Eq. (4) is not well defined. One thus needs to select a subspace within the span of ½T L I t Š. In order to perform the best choice, vectors must be classified. Dealing with low frequency problem, and considering the regularisation issues (noise, identification bias, model uncertainties, etc.), a classification with respect to strain energy appears to be pertinent. Thus, let us define a generalised expansion basis ½F Lg Š used to estimate interface displacements from measurements ½F Lg Š¼½T L I Š½FI t t gš. (22) Vectors of ½FI t gš are solutions of the eigenvalue problem associated with the reduced model defined using ½T L I t Š, ð o 2 g ½T L I ŠT ½M t L Š½T L I t Šþ½T L I t Š T ½K L Š½T L I ŠÞffI t t gg¼f0g, (23) classified with increasing o 2 g. Expansion using this basis will be referred in the following as LMME for Local model mode expansion Optimal truncation indicators As presented in Section 2.2, vectors of ½F Lg Š are classified with respect to potential elastic energy. ½T It Š is computed using less vectors of ½F Lg Š than sensors, for regularisation purposes, but the optimal number of vectors is still to be determined. To select the appropriate basis, two indicators are introduced. The first one is based on the comparison of displacements on the interface for the coupled problem obtained by two different ways. The second one is based on the reversibility of the coupling process Energetic interface continuity indicators (EICI) Let us denote ð½f C Š; ½O C ŠÞ the solution of the undamped homogeneous eigenvalue problem derived from Eq. (14) for the coupled problem ð o C2 ½M C Šþ½K C ŠÞfF C g¼f0g. (24) Eigenmodes ff C g are defined for both complementary dof of the modification (fq M C g) and generalised dof of the tested structure (fz test g). Let us denote ½F C Z Š eigenmodes defined for the generalised subset of dof. Since displacements fy C test g for the coupled problem are spanned by ½F testš (see Eq. (14)), coupled eigenmodes are given by ½F C test Š¼½F testš½f C Z Š. Thus, on the interface, one has ½F LMME I Š¼½T It Š½F test Š½F C Z Š. (25)

10 1076 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) To validate our confidence in this estimated interface motion, one considers a second estimate given by the static expansion of the predicted coupled mode shapes ½F test Š½F C Z Š. To perform this expansion, 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 ½K L Š¼½K B Šþ½K M Š, ½K B Š for the base structure, and ½K M Š for the modification. Thus, the expression of the reduction basis leading to static expansion over the local model without the modification is given by ½T Stat Š¼ð½K B Š 1 ½C tl Š T Þð½C tl Š½K B Š 1 ½C tl Š T Þ 1, (26) when ½K B Š has no rigid body modes. Thus, as detailed in Section 2.1, interface displacement corresponding to pseudo-test coupled mode shapes ½F C testš are given by ½F Stat I Š¼½C IL Šð½C tl Š½T Stat ŠÞ þ ½F C testš. (27) Using estimates (25) and (27) of interface displacement for the coupled problem, an assumption is made that, for a proper prediction, both should provide close estimations of interface displacements. Thus, lack of continuity at interface is measured using the differential displacements ff LMME I g ff Stat I 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 between dof (in plane, out of plane, rotational, etc.). The expressions of these indicators, for the jth mode, are given by strain energy criterion: ðd EK Þ j ¼ kfflmme I kff LMME I kinetic energy criterion: ðd EM Þ j ¼ kfflmme I kff LMME I g j ff Stat I g j k 2 K M g j k 2 þkff Stat K M I g j k 2, K M g j ff Stat I g j k 2 M M g j k 2 þkff Stat M M I g j k 2. M M These indicators focus on the expansion process. They are not absolute criteria for prediction quality. Nevertheless, the quality in predicting eigenvalues and eigenmodes for the modified structure is strongly related to the reconstruction of the interface behaviour and loads with respect to the modification. Low values for both ðd EK Þ j and ðd EM Þ j would indicate close interface behaviour estimations using methods with opposite purposes. So, results could be considered with significant confidence. When indicators exhibit large values, a closer inspection of intermediate results has to be performed to evaluate prediction accuracy. Fig. 4 illustrates the link between EICI and the lack of continuity at the interface. Prediction results for the same mode are presented, using expansion basis ½F Lg Š presenting not enough vectors (on the left), and optimal value (on the right). It is clear that continuity is not ensured for the prediction result presented on the left, corresponding to large values of both indicators, when result presented on the right seems to be more accurate. A more detailed analysis of indicators behaviour is presented in Section 3.

11 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) Fig. 4. Illustration of relationship between lack of continuity and EICI values Coupling reversibility The 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 from the prediction results previously obtained, effects of a removal of the modification will be estimated using the same methodology, adapted for this purpose. Coupling and uncoupling procedures 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 compared with original behaviour of the tested structure to estimate the accuracy of the coupling process Expansion basis definition for uncoupling process. Objective of this second step is the best estimation of behaviour after removal of the modification. Thus, considered local model does not include the modification. In the same manner, a priori informations on effects of the removal are introduced. Let us denote ½ ^F c=u Lg Š the basis of the eigenmodes of this local model condensed on test points and interface dof. Interface estimation operator ½ ^T c=u It Š is then defined using Eq. (5). However, optimal selection of vectors of ½ ^F c=u Lg Š 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 ½F C testš, for a given frequency bandwith, and initial behaviour ½F test Š are spanned by the same subspace. Subspaces ½T It Š and ½ ^T c=u It Š, are introduced to span both measurements subspaces, and add a priori information on the coupled/ uncoupled behaviour through interface loading/unloading. Nevertheless, subspaces spanned by ½T It Š and ½ ^T c=u It Š, for arbitrary sizes of ½F Lg Š and ½ ^F c=u Lg Š, may be significantly different. Considering that on one hand, ½F test Š and ½F C testš are spanned by the same subspace, and on the other hand, that selection process aims to extract the smallest subspace describing interface behaviour from test points displacements, optimal selection is simply realised. First, vectors of ½ ^F c=u Lg Š are reorganised, so that subspaces restricted to measurements points fit. Thus, an operator ½P reorg Š is introduced, verifying ½C tl Š½ ^F c=u Lg Š½P reorgš ¼½C tl Š½F Lg Š. (28) Let us denote ½F c=u c=u Lg Š¼½^F Lg Š½P reorgš, and ½T c=u It Š the interface estimation operator deriving from Eq. (5) using ½F c=u Lg Š. Thus, ½T ItŠ and ½T c=u It Š provide the same representation of the measurements, while introducing complementary informations on loads on the interface.

12 1078 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) A condition to ensure the uniqueness of ½P reorg Š, and both ½T It Š and ½T c=u It Š to span the same measurement subspace, is to consider as many vectors of ½F Lg Š as of ½F c=u Lg Š. Thus, for the optimal number of vectors N opt, this provides the minimal subspace spanning both ½F test Š and ½F C testš, since N opt pn t Hybrid uncoupling model. For a given size N n 2½1; N t Š of ½F Lg Š, hybrid coupled model is the assembled with respect to Eq. (14). Considering ½T c=u It Š and predicted behaviour ð½f C test Š; ½OC ŠÞ, eigenvalue problem leading to initial behaviour prediction is assembled. Couples ðff c=u g; o c=u Þ verify ½Z M CC Š ½ZM c=u CIŠ½T It Š½F C test 4 Š 5 ffc=u C g ( < = ½F C test ŠT ½T c=u It Š T ½Z M IC Š ½Zc=u Š : ff c=u testg ; ¼ f0g ), (29) f0g where ½Z c=u Š¼ ðo c=u Þ 2 ½IdŠþ½O C Š 2 ð½t c=u It Š½F C test ŠÞT ½Z M c=u II Š½T It Š½F C testš. (30) Estimated behaviour derived from coupling/uncoupling is then compared to initial behaviour using 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, mass and stiffness removal on the interface could be more important than projected masses and stiffnesses of the coupled structures. In this case, mass matrix is not positive definite anymore, and some negative eigenvalues may occur. This situation may indicate that for this particular rank of both ½T It Š and ½T c=u It Š, a priori information on the stiffness ratio between modification and base structure is underestimated, so that such information is still valuable for optimal selection purposes. 3. Numerical case study The first example is a numerical case study. A pseudo-test derived from a FE model is used to perform structural modification. The complete methodology is developed and the analysis of selection indicators is amply detailed Test device The FE mesh of the base structure is presented in Fig. 5. This test configuration is a simple rectangular pate, stiffened on its borders, in free free conditions. Two orthogonal ribs are introduced to break the symmetry of the mode shapes. The modification consists in a rib, added on the softest semi diagonal of the base plate. Natural frequencies of the first five mode shapes are stated 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 the modification. The proposed modification is large enough to bring noticeable variations of natural

13 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 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) Freq. after mod. (Hz) MAC before/after The star indicates that modeshapes 3 and 4 have shifted. frequencies and still sufficiently small to cause only local mode shapes modifications. Objective is the 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 is located on the interface, and only out of plane displacements are supposed to be measured. 1 The use of a local model, allowing an estimation of these displacements, significantly increases the prediction 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 a tuned FE model of the tested structure, due to the geometrical truncation. The FE mesh is presented in Fig Results Table 2 summarises the selected optimal prediction results. Results for the modified FE model are also presented for comparison. These results only derive from the analysis of indicators behaviour, 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 complete structure. Both frequencies and shapes are well estimated. 1 A 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].

14 1080 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) Fig. 6. Local model used for LMME-SDM application test mesh is represented in red. Table 2 Coupling results using LMME frequencies and MAC for the first five eigenmodes Mode Base structure (Hz) Modified structure (Hz) LMME prediction (Hz) MAC LMME/exact (%) It is important to notice that results are considered to be valid only when predicted frequency does not exceed two third of the maximum frequency taken into account in ½O test Š Results analysis As stated in Section 2.2, prediction results depend on the size of the subspace retained for expansion. Sixteen measurement points are defined on the test device, so that the size of the reduced basis used to perform LMME cannot exceed 16 vectors. In this particular case, 13 vectors of ½F Lg Š are used to compute interface estimator ½T It Š, along with the three necessary vectors to estimate rigid body modes of tested structure. Fig. 7 summarises the prediction results and selection 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 ½F Lg Š. Initial frequency is also indicated. Lower left quarter shows evolutions of EICI, while comparison of the estimated coupled uncoupled behaviour with initial behaviour is represented in the lower right quarter. On each graph, selected optimum is identified with a vertical black line. Optimal selection based on both D EK and D EM, along with reversibility test comparisons is explained in the following EICI analysis First step on the selection process is based on EICI indicators, presented in Section D EM is a mass normalised measure of the interface disjunction. This indicator is associated to a mean error between the two expansion processes, and focuses on the regularity of the interface displacements estimation. Since D EK is a stiffness normalised measure, it puts the emphasis on the

15 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) Fig. 7. Prediction results for mode 5. interface displacements estimations matching. Thus, this indicator can be viewed as an estimation of 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 two quite different techniques provide the same interface behaviour estimation, confidence is placed in 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 the prediction results, a close inspection of every intermediate expanded and coupled shape is required. For this prediction, D EK and D EM both decline with the growing size of ½F Lg Š, spelling closer interface behaviour estimations and a good regularity. Estimation with more than nine vectors of ½F Lg Š should thus lead to a pertinent prediction. However, frequency prediction is not really stabilised, especially when 13 modes are introduced in ½F Lg Š. Reversibility tests are then used to determine more precisely the optimal size of the expansion basis to be retained Reversibility test analysis Coupling reversibility is described in Section This process provides a finest indication on the estimated behaviour quality, since predicted behaviour is already pertinent. Indeed, two major drawbacks are to be tackled. The first one, mentioned in Section 2.3.2, is a bad conditioning (with respect 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 ½F Lg Š, interface behaviour is poorly estimated, so that the modification seems to have no effect on the dynamics.

16 1082 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) Following the same process to estimate the initial behaviour from the predictions, removal has no effects on the behaviour. Thus, estimated behaviour after coupling/uncoupling is similar to initial behaviour, 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 ½F Lg Š are taken into account, C/U indicators would indicate a good prediction. However, a closer inspection of intermediate results, along with predicted frequency evolution and EICI 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 are introduced in ½F Lg Š. For 9 11 vectors, reversibility test results and initial behaviour are in good agreement, while EICI values are low. Optimum is then determined for ten vectors, where C/U results are slightly better A posteriori comparison In most cases, optimal selection cannot be automated, since indicators and intermediate results have to be analysed. Optimal reconstruction basis ½F Lg Š must realise the best compromise between the following propositions: pertinent reconstruction of interface displacements with respect to static expansion (i.e. low D EK values), regular interface displacements field, compatible with low frequency motions assumptions (i.e. low D EM 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.

17 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) Fig. 8 presents prediction results and the comparison with those obtained using the complete FE model.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 analysis for all the five modes. 4. Industrial case study Example presented in Section 3 illustrates the proposed methodology and details the analysis of indicators. Accurate predictions of effects of a modification on an academical test device are obtained. 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 of the pump. This structure ensures the washing of drums filtering the cooling water in nuclear plants. Being critical in the electricity production process, a solution had to be quickly proposed and 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 prediction results compared with in situ modal analysis after modification Models construction An experimental analysis of motor/pump assembly is performed. The first five modes involved in 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 25 Hz, four stiffeners are proposed to reinforce the support, and set these natural frequencies away from the rotating frequency. Avoiding appropriation between excitation and modal response would lower vibrations levels. However, due to transmission, harmonic excitations located around 50 Hz may occur. In inoperational conditions, the torsional mode of the pump (50.4 Hz) is not excited. Thus, a second Fig. 9. Motor mounted on the support (left), pump and piping (middle) and modified support (right).

18 1084 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) Fig. 10. Mode shapes and natural frequencies of the first two modes proposed modification. objective is not to modify this mode, so that no appropriation exists between torsional mode and harmonic excitation around 50 Hz. A modal model is derived from this experimental analysis, including the first eight modes, ensuring valid prediction results up to 75 Hz, with respect to the Rubin s criteria (see [13]). The synthesised FRF presented in Fig. 11 demonstrates the quality of the modal model for the first five modes. 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 with respect to the mass are the two major points to be closely inspected. Reciprocity assumption is a key hypothesis, leading to left and right eigenvectors equality, which allows mode shapes identification at each test and measurement points for a non-fully reciprocical test, and coupled model assembly presented in Eq. (15). Proper mode shapes scaling ensures good relative importance of substructures. For example, underestimated generalised modal masses would lead to overestimated effects of the modification. Generalised modal masses are of critical importance in 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 roughly tuned, so that deriving modal behaviour and experimental analysis fit a few for the first four modes. 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

19 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) 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 F EMA ini: (Hz) F LMME (Hz) F EMA mod: (Hz) MAC pred:=ema D F (%) are unknown, strongly biased generalised modal masses could lead to highly erroneous prediction Prediction results and a posteriori comparison with modified structure Table 3 summarises the results for the first five modes. F EMA ini:, F LMME and F EMA mod: denote the identified frequencies for unmodified structures, predicted frequencies and identified

20 1086 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) frequencies for the modified structure respectively, D F concerning the relative error between predicted and identified frequencies. The MAC pred:=ema indication denotes the modal assurance criterion 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 is successfully 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.

21 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) pump present no mode around 25 Hz anymore, ensuring lower vibration levels. Uncoupling between the two structures is realised, while the torsional mode stays unmodified at 50.4 Hz. 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 major importance when in-operational behaviour predictions are expected. In case where force identification would have been performed, assuming the modification would not modify excitation locations and spectra, in-operational vibration levels could have been accurately predicted. 5. Conclusion Structural modification methods permit an estimation of the dynamic behaviour of a structure after a modification when a behaviour model of the unmodified structure and a numerical model of the proposed modification are available. An original approach allowing distributed modifications is presented. The effects of many different modifications can then be estimated using a generic test set-up. A rough local FE model of the measurement subdomain is introduced permitting the noncoincidence between measurement points and dof at the interface of the modification and the structure. 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 the modifying 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, so that information on the interface between structure and modification is estimated. Since the estimation depends on the size of the selected subspace, two indicators are built to estimate the quality of the expansion over the interface. First estimator is based on two different evaluations of the behaviour of the interface for the coupled problem, leading to an energetic measurement of the interface disjunction for the modified structure. Second estimator tests the reversibility of the whole process. The behaviour of the initial structure is estimated considering the removal of the modification 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 is presented, leading to an accurate prediction. The other example is an industrial case study. A non-specific modal test permitted the analysis of the high in-operation vibration levels. A solution has been quickly proposed, precisely evaluated and installed, a posteriori tests demonstrating the efficiency of the prediction. The ability of this method to reconstruct unknown displacements fields using few hypotheses is shown. Reactivity and time saving issues are illustrated. 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 be derived to predict the effects of viscoelastic treatments or local dampers, in addition with classical structural modification. Second step deals with force identification procedure. Using the same principles, accurate in-operational behaviour predictions could be achieved. Along with these

22 1088 M. Corus et al. / Mechanical Systems and Signal Processing 20 (2006) developments, the use of strain measurements could provide a better behaviour model allowing more precise predictions. Appendix. Nomenclature Matrices ½MŠ ½KŠ ½D v Š ½ZŠ ½HŠ ½FŠ ½OŠ ½GŠ ½TŠ NNT ½BŠ NNa ½CŠ Ns N Vectors fqg N ff g N ff i g N fzg NM fug Na fyg Ns Scalars o o i x i d ij mass stiffness viscous damping dynamic stiffness flexibility normal modes natural angular frequencies damping ratios expansion/reduction basis controllability observability model states (i.e. FEM dof) generalised loads ith eigenmode generalised dof inputs (i.e. physical loads) outputs (i.e. measurements) angular frequency ith natural angular frequency ith damping ratio Kronecker delta Superscripts and subscripts B base structure M modification B þ M 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