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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 45 (2014) 193–206

0888-32http://d

n CorrE-m

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

Damage detection using successive parameter subsetselections and multiple modal residuals

B. Titurus a,n, M.I. Friswell b

a Department of Aerospace Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, UKb College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK

a r t i c l e i n f o

Article history:Received 6 June 2013Received in revised form21 September 2013Accepted 6 October 2013Available online 28 November 2013

Keywords:Damage detectionSubspace anglesParameter subset selectionModal residualsSymmetry

70/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.ymssp.2013.10.002

esponding author. Tel.: +441173315552.ail addresses: brano.titurus@bristol.ac.uk (B.

a b s t r a c t

The use of modal response residuals and parameterized models forms the framework forparameter subset selection based damage detection. This research explores the novelapproach in this class of methods which is characterized by the successive application ofthe homogeneous modal response residuals. The motivation behind this approach is torestrict the use of unknown weighting factors which are employed in cases with mixedresponse residuals. Particular attention is given to the parameter-effect symmetry issuesand large nonlinear changes in response residuals due to increasing damage observedacross multiple damage levels. A case study involving a real aluminum three-dimensionalframe structure with a loose joint connection is used to demonstrate the approach and itsability to localize the damaged area.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Vibration-based damage detection will form a part of future integrated health monitoring systems in spatially extendedstructures [1–3]. An attractive feature of this philosophy is its global approach to the detection of structural changes basedon the identified dynamic properties [4,5]. The changes in modal properties are linked, among other factors, with thechanges in structural properties, and these can be interpreted as possible damage instances. Friswell et al. [6] proposed adamage location method based on this assumption. This method uses Finite Element (FE) modal sensitivities to interpretidentified damage-induced changes in the corresponding modal properties. The parameterized FE models and other relatedconcepts from model updating [7] are used here to formulate a model-based inverse framework for damage identification [8].The differences in dynamic properties, called response residuals [9], identified on the reference undamaged and potentiallydamaged structure are used as damage sensitive features. The relationship between FE modal sensitivity models and measuredresponse residuals is established in a framework based on the use of angular distance measures in linear spaces [10] andparameter subset selection strategies [11]. The primary advantage inherent in this “differential” approach is based on thebelief that the interpretation capabilities of sensitivity models are more robust to potential model deficiencies than theinterpretations based on the corresponding absolute difference measures, e.g. differences between measured damaged andpredicted natural frequencies.

The original work by Friswell et al. [6] considered the use of natural frequency residuals. Titurus et al. [12] investigatedthe combined use of natural frequency and mode shape residuals. The use of combined modal residuals was also studied bySong et al. [13]. An alternative to the angular measure, called Subspace Angles, is the Multiple Damage Location Assurance

All rights reserved.

Titurus), m.i.friswell@swansea.ac.uk (M.I. Friswell).

B. Titurus, M.I. Friswell / Mechanical Systems and Signal Processing 45 (2014) 193–206194

Criterion. In this context, the modal sensitivity models along with natural frequency residuals were used by Messina et al. [14].Shi et al. [15] proposed modification of this method based on the use of mode shape residuals. More recently, this class ofmethods has been further studied by considering different types of residuals, e.g. based on the use of dynamic residual forcevectors [16], or introducing multi-stage algorithms consisting of standard or modified residual correlation measures andglobal optimization strategies [17], [18], [19].

This research investigates the complementary and successive use of natural frequency and mode shape residuals tointerpret changes in real structures with a realistic type of structural change. The framework for this research is based on theoriginal idea of parameter subset selection [6]. A part of this process is the sensitivity-based analysis of FE modelparameterization and its relationship to the topology of the structure [20–22]. The first order sensitivity representation isconsidered along with natural frequency and mode shape residuals to select the candidate damage parameter subsets. Thischoice is made because it provides a seamless link with the majority of industrially accepted FE tools. A part of this studyrepresents the analysis of the issues associated with structural symmetries [12], [23], and large parameter variations [14,24].In this respect, the specific modal response properties linked with spatially extended grid or lattice systems with structuralsymmetries are utilized and exploited as useful detection features [25–27].

The novel feature of this research compared with previously studied subset selection methods, [6,12,13], is the use ofsuccessive subset selections with homogeneous residual vectors rather than mixed weighted residuals. Natural frequencyresiduals are used to identify candidate damage regions up to structural symmetries. Mode shapes, which representdynamic responses with spatial identity, are then used to determine the actual damage location from within a small set ofcandidate symmetrically placed damage regions. An additional aspect of this approach is the use of multiple damage levelsto highlight trends in changes of the residuals and to provide an indication of the parameterization adequacy.

Structural health monitoring, or damage detection and location, can also be considered in a statistical framework. Indeedthe early work of Collins et al. [28] on model updating used a priori estimates of the measurement and initial parameterstatistics to obtain the weighting matrices in a least squares approach. Since then Bayesian methods have been extensivelyapplied to model updating [29–31] and structural health monitoring [32,33]. Indeed the detection and location of damagecan be formulated as a hypothesis test, with the location often formulated as a model selection problem [34–36].The statistical interpretation of the proposed method is not pursued in this paper, although the residuals obtained may beused in such an interpretation. Furthermore the proposed method requires no a priori knowledge of the statistics of themeasurements or the initial parameters; this is a great advantage since such information is often difficult to obtain inpractice.

Structural damage is often associated with nonlinear structural features such as cracks or loss of structural integrity, forinstance in terms of loose joints and interfaces [37–40]. Practical experience suggests that “small vibrations” and linearvibration-based methods can provide useful information about the state of structural systems. While this paper does notlook specifically at the problems of nonlinear damage a real experimental case study is chosen such that any potentialnonlinear problems can affect detection quality. An aluminum space frame with symmetric topology and a loose bolted jointis modeled, tested and further studied using the proposed approach. The damage mode studied is a loose bolted connectionbetween a frame member and a joint node.

2. Theory and modeling

2.1. Response residual-based damage detection

2.1.1. Modal response modelThis research uses parameterized FE models to introduce the parameter–response and sensitivity models which are used

later together with the measured response residuals. The considered responses are compatible with the responsesgenerated by the reference updated FE models and they include eigenvalues and eigenvectors of a system with zero ornegligible damping. FE model parameterization is considered as

ðKðpÞ�λjðpÞ MðpÞÞ φjðpÞ ¼ 0;

pAℝNP ; j¼ 1;2;…;NM ð1Þwhere M and K are the symmetric mass and stiffness matrices, λj and φj are the j-th eigenvalue and eigenvector, respectively,and λj ¼ ω 2

j . The number of degrees of freedom is ND, the parameter vector p is of size NP and the number of considered modesis NM . The model (1) is used to establish the nonlinear parameter–response map Z and its first order linearization

z¼ZðpÞ; z� z0þSðp0Þðp�p0Þ ð2Þwhere z¼ ½zi� is the vector of modal responses of size NZ and zi are selected modal responses of a single type. The sensitivitymatrix S0 ¼ Sðp0Þ, S ¼ ½ ∂ zr =∂ps � ¼ ½ sfsg �, sfsg ¼ ∂ z =∂ps, represents the linearization of Z around the chosen reference p0. Inthis paper p0 represents the updated FE model [7].

2.1.2. Subspace angles and parameter subset selectionThe incremental form of Eq. (2), Δz� S0Δp, serves as the basis for the parameter subset selection introduced for model

updating [6] and originally proposed for statistical regression [11]. In damage detection, this formula is used to explain the

B. Titurus, M.I. Friswell / Mechanical Systems and Signal Processing 45 (2014) 193–206 195

observed modal response residual vectors Δzl ¼ zm;D;l � zm;ref , where zm;ref are the reference measured responses of thehealthy system, zm;ref ¼ zm;H , and zm;D;l are the measured responses taken from the damaged structure during the l-thexperiment. It is assumed that Δzl provides sufficient spatial and damage type resolution. The sensitivity matrix columnspace cSðS0Þ ¼ ∑NP

s ¼ 1αs sfsg, αs A ℝ is the domain where this interpretation process takes place. Typically, the parametersubset selection algorithm searches the low dimensional similarity patterns between residual vectors and the individualsensitivity matrix column subspaces [12]. The associated linearized form based on (2) and the measure of similarity, theangle between two vectors, are

∑NPs ¼ 1Δps sfsg ¼Δzl; βs;l ¼∢ðsfsg;ΔzlÞw ð3Þ

where ∢ð a;b Þw is the w-weighted angle between a and b [10]. The present paper explores the use of angular measureswithout additional weighting factors, i.e. w ¼ 1.

The alternative approach is to use the sensitivity vector subsets. These subsets can be formed on the basis of spatial orconnectivity associations in the studied systems. This approach was used to parameterize joints and to provide the jointsensitivity subspaces in [24]. The joint sensitivity subspaces are defined as vector spaces spanned by the non-intersectingsubsets of sensitivity matrix columns arranged in the blocks SJj associated with the model's joint parameters. Thus

S¼ SJ1; SJ2; … ; SJNJ

h i; SJk ¼

∂ z∂ pJk1

;∂ z∂ pJk2

;…;∂ z

∂ pJknk

" #; NP ¼ ∑

NJ

k ¼ 1nk ð4Þ

where NJ is the number of joints and nk is the number of parameters in the joint parameter set Jk. Ideally, these joint vectorsare linearly independent, i.e. rankðSJkÞ ¼ nk. The maximum principal angle between two vector subspaces is one possiblemeasure which can be used to quantify the difference between two vector sets. The associated linearized form of (2)expressed in a block format, and the principle angles, are given by

∑NJ

k ¼ 1SJk Δp

Jk ¼Δzl; βk;l ¼ ∢ðSJk;ΔzlÞw: ð5Þ

In both cases, Eqs. (3) and (5), the same numerical algorithm can be used to provide the required angular measures andβk;lA ½ 0 ̂ ;90 ̂ � [10].

Typically, the response vector z consists of a combination of multiple dynamic response types. Detailed discussion aboutvarious residual types is given by Natke et al. [9]. The main motivation behind the research presented here is to explore thealternatives to this response aggregating approach due to the associated issues with the choice of weighting parameters.This configuration is written as

WðαÞSJ1;λ SJ2;λ ⋯ SJNJ ;λ

SJ1;φ SJ2;φ ⋯ SJNJ ;φ

24

35 Δp¼WðαÞ

ðΔzlÞλðΔzlÞφ

" #;

WðαÞ ¼ diagðð1�αÞ INλ; α INφ

Þ ð6ÞwhereWðαÞ is the block diagonal weighting matrix, α A ½ 0;1 � is the relative weight between two chosen response types, INλ

and INφ are the unit matrices of size Nλ and Nφ, respectively, Nλ is the number of frequencies, Nφis the number of mode shaperesponses, where Nφ þ Nλ ¼ NZ , and SJk;λ A ℝNλ � nk and SJk;φ A ℝNφ � nk are the natural frequency and mode shapesensitivity matrices corresponding to the k-th joint parameter subset. The subspaces defined by the mixed residuals andthe corresponding sensitivities depend on the parameter α. As a result, the linearized form (6) and the maximum principalangle between two vector subspaces are defined as

∑NJ

k ¼ 1SJkðαÞ Δp

Jk ¼ΔzD;lðαÞ; βk;lðαÞ ¼∢ðSJkðαÞ;ΔzD;lðαÞÞw;

SJ;Tk ðαÞ ¼ ½ð1�αÞ SJ;Tk;λ;α SJ;Tk;φ �; ΔzTD;lðαÞ ¼ ½ ð1�αÞ ðΔzlÞTλ ; α ðΔzlÞTφ�: ð7Þ

2.2. New concepts in residual-based damage detection

2.2.1. Parameter effect symmetry in response modelsThe concept of parameter-effect symmetry is used in this paper as an important qualitative feature of the response

models which influences systems with symmetries and repeated patterns such as frames, trusses or grid-based structuralsystems in general [12,24]. Damage detection response models which utilize global responses, e.g. natural frequencies, areaffected such that the changes in symmetric regions produce identical or similar changes in observed responses [23].Systems with complicated geometry or connectivity modeled by over-parameterized models, NP c NZ , are likely to beaffected by this behavior. The parameter-effect symmetry leads to multiple parameter subsets which equally well explainthe observed residuals. Mixed response residual vectors, Eq. (7), can be used to overcome this problem. This approach,however, is affected by the uncertainties in selecting the relative weighting factor α.

The main tool used in the current identification scheme is the block sensitivity matrix S¼ ½ SJ1; SJ2; … ; SJNJ� where the

blocks correspond to the regions of interest, e.g. joints. Also, an alternative block arrangement is formed on the basis of thesensitivity vector cluster analysis. This analysis highlights the existence of model symmetries in the response space [23,41].

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The clusters of parameters with similar response effects are formed using a matrix of column subspace angles

β¼ βi;j� �

; βi;j ¼∢∂ z∂ pi

;∂ z∂ pj

!ð8Þ

where β ¼ βT A ℝNP�NP , 0 rβi;j r 901 and βi;i ¼ 01, ∢ð a;b Þ is the angle between a and b. Similar parameter effects areidentified by βi;j � 00. The clustering algorithm uses NP sweeps across the columns of S to form the parameter symmetrygroups, where each group Gα is defined by the condition βα;srβ0, and α is the reference vector, β0 � 01 is the chosen smallsimilarity threshold angle and s identifies parameters in Gα.

2.2.2. Mode shape residuals in residual-based damage detectionThe use of global response residual vectors allows damage identification down to the set of symmetrically placed

candidate regions. The use of spatially distributed response residuals, such as the measured mode shapes, is proposed hereas a tool to overcome symmetry issues. The successive use of homogeneous residual types is introduced here as analternative strategy to the use of the mixed weighted residuals.

Mass normalized normal mode shapes are used in this research. Mode shapes represent a much more complexinformation resource and working with them and their residuals requires a number of correlation stages. The list of all stepsapplied in this research reflects the complex nature of mode shape residuals:

1.

Mode shape norm: negligible mass changes are assumed in the studied systems as the damage level changes.Experimental and analytical mode shape vectors are mass normalized within their own systems for all considered states.

2.

Mode shape order: correct mode shape order for all considered states is ensured by using the Modal Assurance Criterion(MAC) [7]. The known healthy state is used to provide the reference mode shape vectors for MAC studies.

3.

Mode shape vector orientation: to ensure the correct residual calculations the angles between the MAC-paired modevectors are studied. This step is completed before residual calculations Δzm;l ¼ zm;D;l � zm;ref . The angles αl;j ¼ ∢ðφD;l;j;

φref ;jÞ between the paired j-th reference and the l-th damage state modes are calculated to ensure consistent vectororientation αl;j o 901.

Further, this work assumes the existence and measurement of multiple damage states. The corresponding residualvectors Δzm;l are specified relative to the measured healthy state and the index l represents damage states. As described inthe following sections, the comparisons between the measured residuals Δzm;l and computed residuals Δza;l will be used toselect a single damage parameter from the candidate symmetry group. To ensure the correct comparison at this stage thelast check ensures,

4.

A common vector basis for all residual vectors: Steps 1–3 ensure consistency within either the experimental or analyticalmode shape data sets only. The current step establishes one universal reference vector set for all residual vectorsgenerated by the different models considered, e.g. Δzm;l and Δza;l. This step also uses the angular condition αl;j o 901

applied to the MAC paired sets of mode vectors, which, in this case, originate from the different contexts.

4. Nonlinear response residual modelThe main focus of this research is the study of the perceived complementary capabilities of the frequency and mode

shape residuals. To constrain the initial complexity of the study only a single damage site placed in the symmetric region ofthe system is considered. Further, to improve the clarity of the damage effects, the residuals collected from multiplemeasured damage states ½Δzm;1;Δzm;2; … � are considered in the individual studies.

Fig. 1 summarizes the detections steps. Initially the updated FE model is analyzed at each damage stage using frequencyresiduals only. The frequency residuals indicate damage occurrence and provide space-localized damage indicators up to theparameter-effect symmetries.

Experimental frequency residuals are specified at each test stage relative to the chosen reference healthy state.The analytical nonlinear response residual model together with the identified candidate set of symmetrically distributedelements is then used to find the parameter values for all individual measured residuals (index i) and all damage states(index j)

pD;i;j : ℛðpDÞ �ZðpDÞi�ZðpUÞi ¼ ½Δzm;j�i ð9Þ

where ½Δzm;j�i is the i-th element of the vector Δzm;j ¼ zm;j � zm;ref , index j identifies the damage state, ZðpÞi is the i-thresponse from the nonlinear response model, Eq. (2), pU and pD are the updated and calculated damage value of thecandidate parameter, pD;i;j is the candidate parameter value due to the i-th experimental residual and the j-th damage state.

The model in Eq. (9) is used to (a) form the one-to-one map between experimental and analytical frequency residualsand to determine the representative parameter value pD;j for all pD;i;j for a given j-th damage level, (b) assess the adequacyand linearity of the parameterization during increasing damage, and (c) correlate the experimental and analytical mode

Fig. 1. Data organization for response residual calculations.

B. Titurus, M.I. Friswell / Mechanical Systems and Signal Processing 45 (2014) 193–206 197

shape residuals and thus determine the unique damage location. The described concepts and associated processes areinvestigated further in the case study.

3. Case study: aluminum space frame structure

3.1. Updated finite element model and parameterization

The three-story three-dimensional aluminum frame structure, shown in Fig. 2(a), was tested and modeled with free–freeboundary conditions. The FE model of the structure was developed with the help of an in-house FE code implemented inMatlab. The model consists of 168 elements including 3D Euler–Bernoulli beams, point masses and rigid links with 708degrees-of-freedom in total. The FE model was initially parameterised for model updating purposes [42]. A summary ofthese activities is presented in [24]. The first eight global modes are located between 15 and 50 Hz, and were updated sothat the initial bias of around þ14% was reduced to a mean absolute error of approximately 0.50%, primarily by reducingjoint stiffness, as shown in Fig. 2(b) and (c). This improvement, based on the global parameterization [42], was achieved by asmall increase in the mid-sectional quadratic moments of area of all of the aluminum members and mainly by the 29%reduction of the nominal Young's modulus in all joint elements, shown in gray in Fig. 2(c).

The updated FE model was re-parameterised for damage detection. As a result all joint beam elements, for example thoseshown in Fig. 2(b) and (c), have parameterised and updated Young's modulus. Overall 16 three-, four- and five-arm joints areparameterized giving 68 parameters.

3.2. Modal analysis and damage cases

Modal analysis was performed on the three-story aluminum Meroform M12 structure, Fig. 2(a). The suspended structurewas instrumented with 33 single-axis Bruel&Kjaer accelerometers and excited with an impact hammer. The eight normalmodes were identified using an LMS system between 0 and 128 Hz using 33 measured Frequency Response Functions(FRFs). Two orthogonal accelerations were measured at each aluminum node and one corner was instrumented and exciteddiagonally to provide a point FRF.

All joints were nominally tightened to the same extent. This state was measured multiple times before and after variousdamage detection tests to produce the reference mean responses used in the updating studies [24,42]. The damageconfiguration considered in this study was a single loose bolted connection shown in Fig. 2(b). The three damage levels wereimposed by the controlled loosening of the nut linked with the aluminum node. Fig. 3 shows the point FRFs for all referencecases (gray lines) and three FRFs corresponding to the three damage levels. The damage levels were Damage 1 (D1) byunscrewing the hexagonal nut by 20 [deg], Damage 2 (D2) by incremental increase of the unscrewing level by additional 15[deg], and Damage 3 (D3) by incremental increase of the unscrewing level by additional 40 [deg].

Table 1 summarizes these tests. The reference column with the healthy frequencies used for damage detection, denotedH0, represents the last undamaged case before damage studies. The columns headed Damage 1, Damage 2 and Damage 3contain the measured responses corresponding to the damage cases. Fig. 4 provides a summary of the differences inidentified natural frequencies in [%] (horizontal axes). Ten healthy test cases are also included. While not used in thesestudies, each vertical axis shows identified modal damping ratio in [%]. Despite the applied modifications and repeatedloosening and tightening in the various parts of the structure, Fig. 4 features low variability among the healthy cases andclear distinction between the healthy and damage cases D2 and D3. Compared with the frequency spread of the healthyconfigurations, D1 produces small changes which are clearly distinguishable only when taken relative to the referencehealthy case H0 denoted by the circle markers in Fig. 4.

Fig. 4 indicates that the increased sensitivity of modes 2 and 6 and relative insensitivity of modes 3, 4 and 5 is linked totheir mode shape deformation patterns. Increased local modal curvature in the vicinity of damage regions influences

Le

lrigid

Fig. 2. Space frame structure (a) test configuration, (b) joint detail, and (c) FE joint model.

Fig. 3. Measured point frequency response functions.

Table 1Identified natural frequencies.

NF number Identified natural frequencies, [Hz] Differences, 100� (D�H)/H, [%]

Healthy (H0) Damage 1 (D1) Damage 2 (D2) Damage 3 (D3) Difference D1�H0 Difference D2�H0 Difference D3�H0

1 16.29 16.26 16.13 16.02 �0.20 �1.00 �1.642 22.37 22.21 21.83 21.41 �0.70 �2.41 �4.273 23.41 23.40 23.39 23.37 �0.02 �0.10 �0.154 29.98 29.96 29.91 29.85 �0.08 �0.25 �0.455 31.49 31.49 31.48 31.48 0.00 �0.01 �0.036 34.44 34.25 33.59 32.78 �0.53 �2.47 �4.807 36.16 35.81 35.40 35.24 �0.97 �2.11 �2.558 42.32 42.17 41.90 41.72 �0.36 �1.00 �1.42

B. Titurus, M.I. Friswell / Mechanical Systems and Signal Processing 45 (2014) 193–206198

increased modal sensitivity. For instance, mode 5 features low sensitivity due to its effectively straight longitudinalmembers.

Fig. 5 shows the relative changes in the 32 measured mode shape coordinates for all healthy and damage cases. Thesechanges are determined between all test cases and the reference H0 represented by the last healthy test case measuredbefore the introduction of damage. The differences are normalized with respect to the maximum absolute magnitude takenfrom among all mass normalized mode shapes corresponding to the case H0.

3.3. Damage detection

3.3.1. Small parameter perturbation and frequency residual analysisInitially, standard sensitivity analysis [7] for small parameter changes as well as the analysis of measured frequency

residuals is presented. The objective is to highlight the existence of symmetries and to study the changes in the measuredresiduals with increasing damage level. Fig. 6 shows joint and selected element numbers used in later analysis. Element 136contains the damage. Results from Table 1 are summarized in Fig. 7. The measured frequency residual magnitudes haveincreasing trend with increasing damage level and already the D1 case features the residual change pattern similar to D2and D3. All residuals are negative with different magnitudes indicating different sensitivities to damage. Frequency changeswith a decreasing trend leads to negative residuals. This trend is indicative of the effective stiffness reduction due to jointloosening.

Fig. 4. Identified natural frequencies (horizontal axes) and modal damping (vertical axes) Healthy: gray dots, H0: gray circle, and D1, D2, D3: black crosses.

Fig. 5. Relative changes in the measured mode shape coordinates. Healthy: gray dots, Damage 1: red diamonds, Damage 2: green squares, and Damage 3:blue circles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

B. Titurus, M.I. Friswell / Mechanical Systems and Signal Processing 45 (2014) 193–206 199

Fig. 8 shows the joint subspace angles, Eq. (5), β k, k ¼ 1; … ;16 between the measured frequency residuals ΔzD;1, ΔzD;2,ΔzD;3 where ΔzD;i A ℝ8, Eq. (3), and for all joint subspaces J k, k ¼ 1; … ;16. Individual joint subspaces consist of three, fouror five sensitivity columns depending on the topology of the structure and joint in question, as indicated in Fig. 6.

Fig. 8 indicates symmetries in the structure. Out of the three damage levels, Damage 1 gives clear and correct indicationof the four candidate damaged joints, namely 5, 7, 10 and 12. Based on Fig. 6 these joints constitute a parameter-effectsymmetry group {5,7,10,12} which is linked with the symmetrically placed joint features. This figure also indicates othersymmetry groups {1,3,14,16}, {2,4,13,15} and {6,8,9,11}, giving in total four groups, where each contains four joints.

D1 subspace angles are less than 21 [deg] due to the relatively large size of the joint subspaces compared with the size ofthe response space in this case. D2 and D3 do not provide correct damage indication. This unexpected result is caused by the

Fig. 6. FE model of the structure with joint (bold) and selected element numbers. Joint 12 and element 136 are affected by damage.

Fig. 7. Measured frequency residuals defined relative to H0.

Fig. 8. Joint subspace angles based on natural frequency residuals. Data bars from left to right: Damage 1, Damage 2, and Damage 3.

B. Titurus, M.I. Friswell / Mechanical Systems and Signal Processing 45 (2014) 193–206200

large parameter changes required to accommodate the moderate reductions in natural frequencies summarized in Table 1.These changes are not compatible with the linearity assumptions introduced in Section 2.1. The success of D1 and the failureof D2 and D3 is because the response residual model is nonlinear and large parameter changes cause significant changes tothe subspace angle; this is further studied in Section 3.3.2.

Further analysis is based on subspace angles between the measured frequency residuals and the individual sensitivitymatrix columns, Eq. (3). The angles are visualized in Fig. 9 along with the table identifying the element numbers.

Real damage is applied in the region of element 136. A low subspace angle corresponding to this element is correctlyidentified for all damage levels with minimum angle 9.2 [deg] for D1. This representation indicates the parameter-effectsymmetry group of finite elements {101,109,144,136}. The effects of near-symmetry can be also identified in this figure, e.g.element 140 and all its symmetric counterparts. This analysis isolates specific parts of the previously identified candidatejoints. The problem solved here is over-parameterized with 68 parameters and only 8 responses. Any small non-minimalsubspace angle indicates a region with some capability to explain observed frequency reductions. While being the source ofidentification ambiguity, spatial assessment of these results provides complementary information about damage type orcomponent. For instance, based on Figs. 6 and 9, the small subspace angles corresponding to elements 104, 140 and 136provide a strong indication of damage in the longitudinal rather than the transverse or diagonal members of the structure.

Fig. 9. Subspace angles using frequency residuals and element subspaces.

Fig. 10. Graphical solution of Eq. (9) using parameter–response residual maps parameter range: (a) linear scale and (b) logarithmic scale.

B. Titurus, M.I. Friswell / Mechanical Systems and Signal Processing 45 (2014) 193–206 201

3.3.2. Large parameter changes and nonlinear effectsD1 subspace angles provide correct damage indication up to symmetries while D2 and D3 feature a degraded damage

identification quality. This section studies this observation by assessing the effects due to large parameter changes. In orderto support this study the one-to-one maps between measured response residuals and the corresponding model parametersare studied.

All frequency residuals occur in the range between �0% and �5%, as shown in Fig. 7. Parameter–response residual mapsare used to find parameters pD by solving the nonlinear Eq. (9) for all NZ � 3 cases of the measured response residuals Δzm;i;j,where index j ¼ 1;2;3 represents the damage level, i ¼ 1;2; … ; 8 identifies the frequency residual and pD is the calculateddamage value for an arbitrary parameter (Young's modulus) chosen from the candidate group {101,109,144,136}.

The complete process is summarized in Fig. 10. The parameter changes required to obtain the observed frequencyresiduals on a one-to-one basis are from 0.4% to 100% of the pU value [42]. Element 101 belongs to joint 5 and is randomlyselected for this study. Eq. (9) is solved graphically for 8� 3 ¼ 24 residual cases in Fig. 10. Individual curves in this figurerepresent the response values ZðpÞ for the chosen parameter range.

A marker corresponding to D1, mode 5 is not shown due to a small positive residual (0.002%) which, when combinedwith the low sensitivity (see Fig. 4), leads to an infeasible large positive parameter change. Individual solutions of Eq. (9) arerepresented by the markers, where the triangles correspond to D1, the squares indicate D2 and the circles show D3. It isuseful for further analysis to establish unique parameter values representing damage cases D1, D2 and D3. The meanparameter values pD;j, j ¼ 1;2;3, are chosen for this purpose. Table 2 gives the summary of all calculated parameter values.

Table 2Candidate parameter value for different model states.

Model state Parameter value�1010 [Pa] 100� (col.�row)/row [%] (max.�min.)/mean�100 [%]

Min. Mean Max. D1 D2 D3

Updated – 4.971 – �38.6 �83.3 �95.6 –

Damage 1 2.784 3.051 3.394 0 �72.9 �92.9 20Damage 2 0.207 0.828 1.438 – 0 �73.8 149Damage 3 0.021 0.217 0.500 – – 0 221

Fig. 11. Sensitivity matrix with joint groups (top) and parameter-effect groups (bottom) due to structural symmetry.

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Changes in the marker spread in Fig. 10, for each damage case, indicate adequacy of the parameterization, both withregard to the involved physics and the extent of parameter changes. The D1 case features a �38.6% decrease in therepresentative parameter value with a 20% spread between the identified parameter values. This, compared with the similarparameters for D2 and D3, indicates that D1 is still within the linear range of the method using subspace angles. This is alsothe reason behind the correct damage localization indicated in Figs. 8 and 9. With increasing damage extent the responsechanges fail to comply with the linearity assumptions, Eq. (2), and the applied parameterization fails to provide narrowclusters of the solutions pD;i;j. Fig. 10 provides a measure of how well the chosen joint parameterization represents the actualdamage scenario. Both spread and overlap between the clusters are influenced by this effect. The ideal case could be studiedsimply by using simulated residuals based on the changes of the parameter used in Fig. 10. This case would provide clustersof markers perfectly aligned on the vertical line corresponding to the parameter value used for this simulated damage study.

The required reductions of more than 90% for D3 represent significant parameter variations and these map onto theresiduals representing no more than 5% reduction in the measured frequencies. Despite large parameter changes, thequalitative features of damage are predicted well. For example, the order of the most sensitive D1 residuals (i.e. 7, 2, 6) isidentical in both contexts, see Figs. 7 and 10. Moreover, modes 2 and 6 are consistently identified as the most sensitivefeatures of the residual model. These results suggest that even a physically simple parameterization using Young's moduli isable to provide correct damage indications with suitably distributed parameterized regions across the structure.

The effect of the large parameter changes on the subspace angles is studied next. The sensitivity matrix corresponding tothe healthy updated state is shown in Fig. 11. The top subplot shows this matrix with the 16 parameterized joint groupshighlighted. The bottom subplot shows the same matrix with its columns reordered to highlight the symmetries inthe model.

The low response sensitivity, i.e. the ability to detect damage, is observed in the three parameter-effect groups g4, g12and g17. These groups contain 12 parameterized joint elements corresponding to the diagonal members. The damage inthese members is not observable with the set of chosen modes. Similarities in the parameter effects are highlighted in thebottom part of Fig. 11, where, for instance, the groups g8 and g14 which correspond to the inward and outward orientedlongitudinal elements of the four 5-element joints feature similar, but not identical, sensitivity patterns.

Fig. 12 shows the changes in subspace angles in the range between 0.4% and 83% of the pU value. The solid and dashedlines represent the changes in column and joint subspace angles, respectively. These are the angles between the predictedfrequency residuals due to changes in element 101 and the reference sensitivity basis. Element and joint numbers areprovided to identify subspace angle curves βðpDÞ. Multiple identification numbers are provided next to each line due tosymmetry effects causing multiple element or joint subspaces providing identical subspace angles. Specifically, the 68 jointelements correspond to 17 element symmetry groups (solid lines) and 16 joints correspond to 4 joint symmetry groups(dashed lines). The magnitude of joint subspace angles is lower due to increased dimensionality of their subspaces. This factrepresents a trade-off between the reduced resolution capability and the increased generality to explain multiple possibledamage scenarios.

The chosen representative parameter values pD;1, pD;2 and pD;3 identified in Fig. 10 are shown as the labeled vertical linesin Fig. 12. These lines link the experimentally observed damage instances with the parametric study conveyed in this figure.

Fig. 12. Effect of large parameter changes on subspace angles, simulated damage in element 101.

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The incorrect identification in both subspace angle approaches, caused by large parameter variations pD;2 and pD;3, is clearlyshown in this figure.

3.3.3. Damage location using mode shape residualsThe previous analysis shows that the damage is located in one of the four joints {5,7,10,12} and that the candidate

elements are {101,109,144,136}. These are outward oriented elements in the 5-element joints, shown in Fig. 6. Marginallylower sensitivities are featured by the inward oriented elements in the same joints. The above studies also suggest that asthe damage level increases, see Figs. 3, 4 and 7, that this increase is reflected by the large parameter variations, Table 2, inthe residual model, Eq. (9) and Fig. 10. These changes do not agree with assumptions used in model (2) which serves as abasis for the subspace angle and subset selection calculations, Eqs. (3) and (5). Modes 2, 6 and 7 are identified as beingsensitive to damage. The previous analysis provides a small down-selected symmetry group of candidate damage elements.This section explores the use of mode shape residuals and their suitability to overcome the symmetry limitations inherentwhen using natural frequency residuals.

Reflecting on the required large parameter changes determined in the previous sections, this section studies a correlationprocess between the experimental and computational mode shape residuals to determine the unique identity of thedamaged element. Model (9) is used to generate computational residuals. Previously determined parameter values pD;j,j¼ 1;2;3 are used to position the experimental residuals in the parametric plots. The four step process outlined in Section2.2.2 is used to ensure compatibility between the residuals. The main objective of this study is to determine whetherexperimentally identified mode shape residuals can provide useful advances in damage location. For this purpose, only themost sensitive mode shape 6 is used. Similar results can be achieved with other sensitive modes, particularly mode 2.However, it is not the purpose of this study to evaluate the identifiability thresholds caused by identification uncertaintiesand experimental noise. This study motivates further extension of this approach to cover automated correlation of theresults presented in Fig. 13, as well as answering statistic, probabilistic and observability questions arising when choosingsuitable residual data for mode shape-based damage location.

Model (9) generates 32 residual lines, where each line corresponds to a single degree-of-freedom, in the parameter rangebetween 3.5% and 100% of the pU value. This study is summarized in Fig. 13. Each subplot corresponds to a parameter sweepwith one parameter selected from the candidate group {101,109,144,136}. The four rectangular floor levels use differing linestyles, the four nodes at each floor use differing markers and the two measured transverse degrees-of-freedom use blackand gray lines. This graphical encoding gives a unique identity to the lines and allows comparisons between the two sets.The match between measured and calculated response residuals implies identically placed damage. The actual damage isimplemented by loosening the nut–node connection, Fig. 2(b), and it is identified in the FE model, Fig. 6, as element 136.Results presented in Fig. 13 clearly exclude elements 109 and 144 from the candidate group as the residual lines providenon-matching trends. While elements 101 and 136 provide similar trend lines, the line style encoding identifies onlyelement 136 as the one which provides the correct match between the line shapes and their styles. The two major matchingline pairs are those which correspond to node 4 and node 8 and the transverse degrees-of-freedom measured at mode 6bending plane, see Figs. 4 and 6. Similar trends are presented by the lines corresponding to nodes 1 and 5. This behavior is

Fig. 13. Comparison between experimental and computed mode shape residuals for the four symmetrically placed candidate damage parameters.

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featured in the experimental and computational residuals lines. Similar observations can be extended to other measureddegrees-of-freedom representing the motion in the bending plane of mode 6.

3.4. Method assessment and discussion

It is shown that the successive use of parameter subset selections based on angles between various sensitivity subspaces anddynamic residuals provides correct damage location in the case of a medium complexity real structure. The characteristic featuresof this approach are (a) the use of homogeneous residual vectors at all individual selection stages to avoid relative weightinguncertainties, (b) the use of multiple damage states and nonlinear residual model (9) to assess linearity assumptions and todetermine the parameter values inducing comparable analytical residual patterns, and (c) the use of mode shape residuals toovercome the limitations inherent in the natural frequency residuals.

The one-to-one mapping study demonstrated in Fig. 10 provides useful insights not only into the parameter values whichlead to similar residual trends, but also into the appropriateness of the selected parameterization. This study shows that theparameterization representative quality degrades with increasing damage and with associated nonlinear changes in theresponse model (2).

A single location of the damage was assumed in this paper. While this is often the case, a multi-location damage casemight need to be addressed in future with schemes similar to those based on the parameter subset selection algorithmspresented in [11].

An inherent limitation of this approach is associated with the sensitivity model based on the low frequency modes asshown in Fig. 11. The extent of these limitations differs from case to case and should be of prime concern when preparing adamage detection framework for a given application. Possible ways around this limitation might lay in the addition of morelocalized higher-frequency modal data or dedicated sensors allowing observation of partially isolated structural parts.

The original motivation for this research was to avoid the use of mixed response residuals (6) and (7), [6,12,13]. In thisrespect, the approach of Titurus and Friswell [24], based on the use of mixed residuals can be seen as a counterpart to thepresent approach. They highlighted that the classical approach can work, but there are significant issues linked with theneed to determine the relative weighting parameters which are usually based on further analyses such as L-curve or cross-validation based regularization [41].

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4. Conclusion

The presented research is motivated by the perceived strengths and weaknesses of the natural frequencies when used asdamage sensitive features. On the one hand they provide relatively sensitive damage features and, on the other hand, theyare linked with the spatial resolution limitations due to structural symmetries. Moreover, the methods based on theparameter subset selection and sensitivity subspace angles can provide inconsistent results for damage cases with largeparametric variations. The determination of the relative weighting coefficients in the mixed modal residual models presentsa further complication associated with the standard approach.

A novel damage detection process is proposed here which consists of three main stages. In the first stage only the naturalfrequency residuals are used to indicate damage location down to a group of symmetrically placed candidate elements. Thesecond stage uses the local sensitivity matrix analyses and the nonlinear residual response model to provide further insightsinto the results obtained in the first stage and to direct the final third detection stage. The local sensitivity matrix analysesprovide improved understanding of system's parameter-effect symmetries. The nonlinear residual response modelinterprets experimental residuals in the context of large parameter changes and gages the limited interpretation potentialof the chosen model parameterization. The third stage consists of damage detection based purely on the measured normalmode shape residuals. The limited set of symmetrically placed candidate damage regions is exhaustively evaluated to selecta single damage location based on the mode shape residual trend line correlation. This process is successfully demonstratedin the case study involving a real three story aluminum space frame. Increasing joint loosening was used to emulateevolving damage in the structure. The use of the presented damage detection process led to the correct location of thedamage region. The analysis of measured frequency and mode shape residuals also provided relevant quantitative indicationof the evolving damage level.

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