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A new scenario-based approach to damage detection using operational modal parameter estimates
Mechanical Systems and Signal Processing 94 (2017) 359–373
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A new scenario-based approach to damage detection using operational modal parameter estimates J.B. Hansen a,⇑, R. Brincker b, M. López-Aenlle c, C.F. Overgaard a, K. Kloborg a a
Department of Engineering, Aarhus University, Denmark Department of Civil Engineering, Technical University of Denmark, Denmark c Department of Construction and Manufacturing Engineering, University of Oviedo, Spain b
a r t i c l e
i n f o
Article history: Received 26 May 2016 Received in revised form 3 March 2017 Accepted 4 March 2017
Keywords: Damage detection Model updating Operational modal analysis Modal sensitivities
a b s t r a c t In this paper a vibration-based damage localization and quantification method, based on natural frequencies and mode shapes, is presented. The proposed technique is inspired by a damage assessment methodology based solely on the sensitivity of mass-normalized experimental determined mode shapes. The present method differs by being based on modal data extracted by means of Operational Modal Analysis (OMA) combined with a reasonable Finite Element (FE) representation of the test structure and implemented in a scenario-based framework. Besides a review of the basic methodology this paper addresses fundamental theoretical as well as practical considerations which are crucial to the applicability of a given vibrationbased damage assessment configuration. Lastly, the technique is demonstrated on an experimental test case using automated OMA. Both the numerical study as well as the experimental test case presented in this paper are restricted to perturbations concerning mass change. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction In the academic as well as the industrial world the interest in the ability to monitor and detect damage(s) in a civil or mechanical structure is immense. For decades researchers and practitioners in the Structural Health Monitoring (SHM) community have been developing an extensive amount of methods based on a wide range of physically interpretable structural features. At this point in time there is no universal monitoring system hence the term SHM refers to the implementation of one or more damage identification strategies [1]. In vibration-based damage detection the underlying premise is that the modal parameters are governed by the physical properties of the structure, hence an observed shift in the modal parameters indicate changes in the inherent properties of the monitored structure (e.g. structural degradation). Even though the area of vibration-based damage evaluation is merely a branch of SHM, the literature produced in this area alone is extensive [2,3]. The capabilities of a given damage technique has through many years been classified in reference to Rytters four levels of damage identification [4]. The four levels are as follows: 1. 2. 3. 4.
Detection - Is the structure damaged? Localization - Where is the damage? Quantification - How severe is the damage? Prognosis - What is the remaining service life of the structure?
⇑ Corresponding author. E-mail address: [email protected] (J.B. Hansen). http://dx.doi.org/10.1016/j.ymssp.2017.03.007 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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The vibration-based methods range from simple tracking of natural frequencies, fundamental relations between the modal parameters and the system matrices, to modal-based Finite Element (FE) model updating. Methods utilizing only the natural frequencies usually fall into to the level 1 category whereas level 2–3 techniques also require the corresponding mode shapes. Since, the modal parameters, traditionally, have been extracted by means of Experimental Modal Analysis (EMA) many techniques capable of localizing damage, without depending on a parametric model, require scaled or mass normalized mode shapes. This fact rules out the immediate application of most vibration-based localization methods to structures which cannot be subjected to a controlled vibration test (i.e. where the input force is known). Over the last decades the concept of OMA has matured into a well-established discipline within the area of modal testing [5]. The output of OMA is an estimate of the modal model for the whole system (i.e. the loading filter and the physical structure) which presents the challenge of separating the dynamic characteristics of the loading from the modal properties of the structure. The loads/excitation does not change the physical modes of the system [6], so if one can separate the noise modes from physical modes the accuracy of an OMA estimate is comparable to an EMA estimate. Furthermore, OMA lacks the ability to produce mass normalized mode shapes, however, both experimental techniques [7] and FE-based methods [8] have with success eliminated this shortcoming. The concept of automated OMA (i.e. OMA without any user interaction) enables the constant extraction modal features from any structure to be utilized in an SHM context. This makes OMA the obvious modal extraction technique within vibration-based SHM. The scope of the present method is to localize and quantify damage, restricted to a limited number of candidate scenarios by merging operational modal parameter estimates with the system matrices of an FE model. Conceptually, the methodology is similar to a level 2–3 technique based on the sensitivity of experimentally obtained mass normalized mode shapes [9]. The differences are found in; 1) the current technique implements, via a parametric model, the idea of limiting the number of possible damage locations [10] and 2) the current method utilizes OMA to extract the modal properties. Both modifications increase the robustness and applicability to civil and larger mechanical structures, however, they also prompt the necessity of having an FE model of the monitored structure for mass normalization and prediction purposes. Damage detection techniques which use experimental modal data in combination with a parametric representation of the structure, falls into the category of model-based (or parametric) methods [11]. The experimental modal parameters are easily related to the properties of a parametric model, however the implementation of the FE model as well as the utilization of the model data vary substantially across the model-based techniques. Some methods synthesize a experimental measure for a change in flexibility to derive a forcing configuration to be subjected to an parametric model [12,13]. Methods based on FE model updating attempts to tune an FE model to the measured modal data using a predefined subset of adjustable parameters which are sensitive to damage [14–16]. The main drawback of these FE updating methods is the inverse-problem which often is ill-posed. This has prompted FE based methods which bypass the ill-condition issue by introducing statistical frameworks [17,18]. The damage identification technique presented in this paper is very closely related to the FE model updating methods listed above and features the same fundamental algebraic operation. Therefore the methods inherits some of the same challenges when dealing with limited, noisy data and a wide range of potential damage mechanisms. The current method does not attempt to fit a parametric model to test data but merely utilize FE model information in combination with measured modal data in the computation of the modal sensitivities - this is the most distinct feature of the present method in relation to other published model-based techniques. Since the reference data set and the modal sensitivities are based on measured data, the gap between the reference and ‘‘damaged” data can be assumed limited, hence no iterative process is necessary. Ultimately the proposed technique is able to locate and quantify damage given the structural modification is small and corresponds to a predefined scenario in one calculation step. A successful application of the current method relies on the modal parameters being sensitive towards the predefined scenarios of structural change. The fundamental relations which found the basis of the method in combination with the implementation of the scenario idea enables an examination of the method usability. Therefore, this paper features a section which describes the preliminary considerations necessary to determine whether or not the current technique is feasible, with respect to a given structure and configuration of scenarios. On a more general level one might argue that the usability of using any modal-based technique to detect and locate specific candidate structural modifications is disclosed. Lastly, the current method is demonstrated on a experimental test case where the structural modifications consist of local mass perturbations. Hence, the technique is demonstrated on damage cases in the form of mass increase only. In the test case the modal parameters are extracted with no user interaction and by means of a time domain identification technique. 2. Scenario-based approach using modal sensitivities The mode shape and frequency sensitivity equations in the form presented by Heylen [19] founds the basis for the present technique. The undamped sensitivity or derivative of mode shape i with respect to a local change parameter u in the mass and stiffness matrix is given by
Nm X @bi 1 T @M 1 1 T @K 2 @M bi bi þ þ bi br bi b x ¼ i 2mi @u @u @u @u x2i x2r mr r r¼1;r–i
ð1Þ
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where b is a (N dof 1) mode shape vector of the unperturbed system, M is the mass matrix, K is the stiffness matrix, m is the modal scaling factor, x is the angular frequency, Nm is the number of modes and the subscript r denotes the remaining modes of the system. In this context N dof is the number of experimental Degrees Of Freedom (DOF) Likewise, the sensitivity of angular frequency i due to local change parameter u is given by
@ xi 1 T @M 1 @K bi þ bi xi ¼ 2mi @u xi @u @u
ð2Þ
For a finite change scenario j consisting of small alterations in the mass and stiffness matrix yields the following approximations of (1) and (2), respectively
Dbij
Dxij
Nm X 1 T 1 1 T bi DMj bi bi þ br x2i DMj þ DKj bi br 2 2 m 2mi x x r r i r¼1;r–i
1 T 1 bi xi DMj þ DKj bi 2mi xi
ð3Þ
ð4Þ
In general, the mode shapes and angular frequencies are experimentally determined quantities whereas the modal scaling factors and perturbed matrices are based on FE data. The methodology is to compute the approximated changes in mode shapes and natural frequencies using (3) and (4) in relation to a series of predefined structural alterations or scenarios. Obviously, the accuracy of the aforementioned estimates depend on the quality of the initial FE model and the engineers ability to parametrize the structural modification. In Fig. 1, four arbitrary scenario examples on an FE shell representation of a Tshaped structure is shown. The approximated mode shape and frequency changes of mode i due to all predefined scenarios of change in structural stiffness and/or mass are arranged in the matrix SDbi and vector sDxi respectively, thus
SDbi ¼ ½Dbi1 Dbi2 DbiNsc sDxi ¼ ½Dxi1 Dxi2 DxiNsc
ð5Þ
where SDbi is a (N dof N sc ) mode shape sensitivity matrix, sDxi is a (1 N sc ) frequency sensitivity vector and N sc is the total number of predefined scenarios. If the tructural system is subjected to a small change, the measured mode shape and angular frequency change of mode i can be defined as
Dbi ¼ ai bi
Dxbi ¼ xai xbi
ð6Þ
where a and subscript a denotes modal parameters from the perturbed system Given that the structural change prompting the measured changes in (6) correspond to one, or to a linear combination, of the predefined damage scenarios, collected in (5), the following equations for mode i can be written
Dbi ¼ SDbi DuDbi
Dxbi ¼ sDxbi DuDxbi
ð7Þ
where uDbi and DuDxbi can be regarded as a (N sc 1) selection vector. The equations in (7) can be extended to include all recorded modes of the system. In relation to the mode shape changes one can write
Db ¼ SDb DuDb
8 9 2 3 SDb1 Db1 > > > > > > > 6 SDb 7 < Db2 > = 6 2 7 7 where Db ¼ . and SDb ¼ 6 6 .. 7 . > > > > 4 . 5 . > > > > : ; DbNm SDbNm
Analogously, for frequency changes we get
Fig. 1. Scenario examples, the red contour represent modified elements.
ð8Þ
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Dxb ¼ SDxb DuDxb
8 9 3 2 sDxb1 Dxb1 > > > > > > > 6 sDx 7 < Dxb2 > = 6 b2 7 7 where Dxb ¼ . and SDxb ¼ 6 6 .. 7 . > > > > 5 4 . . > > > > : ; DxbNm sDxbNm
ð9Þ
By combining (8) and (9) a final set of equations reached, hence
Dc ¼ SDu where Dc ¼
Db Dx b
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SDb
SDxb
ð10Þ
S is a (ðN dof þ 1ÞN m N sc ) sensitivity matrix and Dc is a (ðN dof þ 1ÞN m 1) measured change vector and Du remains a (N sc 1) selection vector. Solving (10) for Du yields to following estimate
^ ¼ S þ Dc Du
ð11Þ
where þ denotes the pseudo-inverse. Because the approximations due to either stiffness or mass alterations in (3) and (4) are proportional to the perturbed matrices DM or DK the selection vector Du not only single out present scenario but also produce an estimate of the extent in reference to parametrization of the particular candidate mechanism. As mentioned, the approximated changes is a combination of experimental modal parameters and FE matrices which means that there will be a discrepancy in number of the experimental and the FE DOFs. There are two possible solutions to this issue, either FE matrix reduction or experimental mode shape expansion. Further comments on this issue can be fund in Section 6.
3. Weighting sensitivity matrix components The sensitivity matrix, defined in (10), holds the approximated changes of both frequency and mode shape. The numerical size of the two entities, the frequency change and mode shape coordinate change approximation, are very different (the frequency change approximation being the larger). This means that the evaluation of the damage vector by means of the expression stated in (11) will to a certain extend be dominated by the frequency changes. Furthermore, although less important, both the individual frequency and mode shape estimates which found the basis of sensitivity computation may not have been identified with equal confidence. The considerations stated above suggests that some sort of weighting of the individual sensitivities is necessary to avoid a skewed sensitivity matrix. Since, the mathematical operation in (11) is a ‘‘best fit” (least squares) solution to the system of equations, one can turn to a weighted least squares approach [20]. The procedure is to pre-multiply both sides of (10) with weight matrix W, hence
WDc ¼ WSDu where W ¼ diagðw1 w2 . . . wn Þ
ð12Þ
diagðÞ denotes a diagonal matrix. By pre-multiplying (12) with the transpose of the sensitivity matrix S and then isolating for Du one gets
1 Du ¼ ST WS ST WDc
ð13Þ
The immediate selection of the weight components w1 ; w2 ; . . . ; wn is rather difficult if no preliminary knowledge of the importance or statistical properties of the individual terms are available. However, an initial selection in reference to (10) could be 1
W ¼ ½diagðb1;1 b1;2 . . . bNm;Ndof x1 x2 . . . xNm Þ
ð14Þ
where b is a mode shape coordinate. This selection of weight components eliminates the difference in numerical values of the modal quantities [21]. Hence, both the approximated and measured change are made relative to the corresponding unperturbed modal parameter. However, in a monitoring context numerous estimates of the modal parameters are expected to be available, thus also the statistical properties of each individual modal quantity, i.e. standard deviation and variance. This enables the statistically founded choice of scaling the individual terms by means of the corresponding variance [22], by selecting the following weight components
W ¼ ½diagðr2b1;1 where
r2b1;2 . . . r2bNm;Ndof r2x1 r2x2 . . . r2xNm Þ
1
r2 is the variance of the modal parameter estimate.
ð15Þ
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4. The condition of the system of equations The procedure of weighting the sensitivity matrix does not add any additional information nor does it change the fact that (13) remains a noisy system of equations and potentially ill-conditioned [23]. In relation to the current method the problem itself can arise from one or several circumstances such as noised and limited modal estimates, poor FE model, misalignment between actual and modeled damage etc. In this section it is assumed that the FE model is a perfect representation of the monitored structure. This enables focus on the general factors affecting the overall condition of the system of equations, these factors are itemized below. 1. The ‘‘modes-scenario-ratio” It is crucial that the extent of available information surpass the number of unknowns to keep the problem overdetermined. In reference to the present technique the information are the recorded modes and the unknowns are the predefined scenarios (e.g. a configuration containing 5 scenarios requires at least 5 modes). Using OMA one is subject to the in-operation response of the structure which basically means that the number of identified modes will be governed by the nature of the loading filter, i.e. the number of modes will be limited. So, to meet this requirement of over-determination the end user must restrict the number of damage scenarios relatively to the amount of identified modes. This prompts the need for preliminary knowledge on where the monitored structure is most likely be subjected to alterations. 2. Sensitivity matrix rank To ensure full rank of the sensitivity matrix the predefined scenarios must differ across the columns in terms of impact to at least some of the available modal parameters. In other words the approximated changes in one column of the sensitivity matrix must be unique or distinguishable in relation to the remaining columns. 3. Parametrized versus actual scenario For (10) to be valid the nature of the perturbation of the monitored structure must correspond to one, or to a linear combination, of the predefined scenarios. Furthermore, the content of the sensitivity matrix are frequency and mode shape gradients defined in (3) and (4), which means that the validity of these approximations depend on the size of the actual structural perturbation. Naturally, the occurring mechanism must be large enough to penetrate the noise floor of the measurements, but on the other hand small enough to match the approximations. 4. Approximated versus measured change Given that the actual occurring scenario coincides with one of the predefined alteration. There will most probably be a discrepancy between the approximated change and the actual measured change due to noised and limited modal data and erroneous damage parametrization. From a practical point of view there is little that can be done to improve the third and fourth item listed above. The first two items, however, has to do with the construction and content of the sensitivity matrix which can be examined and potentially improved by the end user. An advantageous approach to avoid a poorly conditioned system of equations is to examine the contents of the sensitivity matrix, i.e. the estimated changes in both frequency and mode shape due to the predefined scenarios. Lastly, the Singular Value Decomposition (SVD) is a useful tool to determine the rank of the sensitivity matrix [24].
5. Simulation study Before applying the current technique on a physical structure a preliminary simulation study on a parametric model is to be conducted. In continuation of the previous section the purpose is to assess the overall condition of the sensitivity matrix
(a) Scenario 1
(b) Scenario 2
(c) Scenario 3
(d) Scenario 4
(e) Scenario 5
Fig. 2. Scenario 1–5 – the red dot marks the added mass. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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based on particular set of predefined scenario and an available set of modal parameters. The investigation is similar to an FE model sensitivity analysis conducted to determine whether or not a preselected model parameter could be the source of error between model and test. Here, the purpose is to determined if the available information (i.e. the modes of the system) is sufficient, sensitive towards the scenarios of interest and distinguishable in terms of impact. On a more general level, one might argue that this investigation discloses the feasibility, of using vibration-based monitoring to detect the candidate structural alterations on the specific structure. To illustrate this concept an FE based simulation using a beam element model of a steel T-structure is reviewed. The FE model consist of 126 beam elements connected by means of 127 nodes and is fixed/clamped at the base, ultimately constituting a 756 DOF system. The task is to investigate the ability to detect mass changes in five locations on the structure. The structure and mass change scenarios are sketched in Fig. 2. As mentioned, nearly all contributory causes to the potential ill-condition issue present in the current technique is to some extent related to the range of available information. So to keep the premise of having limited information only the first eight modes are included in this investigation. The size of the mass change is, in all scenarios, set to 0.5% of the total weight of the FE model. In the following subsections the contents of the sensitivity matrix due to the predefined mass change scenarios is examined, i.e. both the frequency and mode shape change approximations, in relation to the ill-condition considerations.
5.1. The frequency change approximations Each column in frequency sensitivity matrix, SDxb defined in (9), holds the approximated frequency changes of all the recorded modes due to a specific scenario. So, by plotting the columns of the aforementioned matrix a preliminary graphical examination of the impact of the various perturbations is possible. In Fig. 3 absolute relative approximated frequency changes caused by scenario 1–5 are shown. In reference to the ill-condition problem there is a lot to be learned from the bar plots in Fig. 3. The first observation to be made is that the approximated changes due to scenario 3, as a whole, seem unique compare to the other scenarios but only at relatively few modes are influenced and some to a very limited extent. Secondly, the approximated frequency changes due to scenario 4 and 5 are identical. Obviously, this is due to the structural symmetry combined with the fact that the scenarios 4 and 5 are added mass in opposite positions of the structure. In relation to the frequency sensitivity matrix, however, this means that the 4th and 5th column are identical thus the matrix does not have full rank. In other words the application of the current technique using only the natural frequencies would make scenario 4 and 5 indistinguishable. Furthermore, there seem to be a rather large difference in the general level of change when comparing scenario 1, 2 and 3 to scenario
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4 and 5. This presents the fact that these scenarios are more sensitive than scenario 1–3. Naturally, this is due to cantilever nature of the structure and the nature of the perturbation (i.e. mass change). 5.2. The mode shape change approximations It is obvious that the estimated mode shapes need to be affected by the predefined structural modification scenarios. However, it is less obvious that the approximated change of a specific mode shape includes contributions from the remaining mode shapes of the system. This emphasizes the need for a ‘‘sufficient” number of modes to get a proper approximation of change. Since, (3) consist of several inner products it can be simplified by defining scalar values tij and trij , thus Nm X
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1 1 T and trij ¼ 2 b x2i DMj þ DKj bi xi x2r mr r
1 T where tij ¼ b DMj bi 2mi i
In terms of the mode shape derivatives it would be rather hard to construct a meaningful plot of the approximated mode shape changes. The changes themselves, being vectors and furthermore a combination of contributions from the mode shape vector space, are difficult to interpret. The scalar values tij and t rij defined in (16) supply the information on how much and which mode shapes contribute to the change approximation of a given mode shape due to a particular scenario. So instead of attempting to dissect a vector representation of the mode shape changes one can examine t-scalars in a stacked bar plot. Fig. 4 is a graphical representation of the aforementioned scalar values prompted by scenario 1–5. Each bar represents the composition of a mode shape change approximation where the color of the individual boxes refer to the contribution from a particular mode. For instance, in Fig. 4a the change approximation of mode 6 consists, besides a contribution from mode 6 itself, of contributions from mode 1 and 4. The general pattern of the frequency change plots in Fig. 3 is more or less the same in the stacked bar plots in Fig. 4. Since, the nature of the perturbation is local one expects a correlation between the approximated frequency and mode shape changes. Likewise, the magnitude of the scalars based on scenario 1–3 are a factor of ten smaller than the ones based on scenario 4 and 5 emphasizing that these scenarios are more sensitivity than scenario 1–3. But contrary to the frequency change approximation produced by scenario 4 and 5 the composition of mode shape contributions due to these scenario differ which means that approximated mode shape changes will differ as well. So, in this particular simulation case by adding the mode shape sensitivities to the system of equations the ability to separate scenario 4 and 5 is ensured. As previously noticed the change approximation of a mode shape does rely on contributions from the remaining modes which potentially imply an incomplete approximation. If we briefly suspend the underlying assumption of having only 8
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mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 mode 7 mode 8 mode 9 mode 10
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modes available and include mode 9 and 10 in the t-scalar computation we can assess the impact of the missing contributions from these modes by extending the bar plot in Fig. 4. In Fig. 5 the aforementioned contributions has been added to the the plots of scenario 2 and 5. From Fig. 5 it is evident that the consequence of not having mode 9 and 10 result in a truncation of some of the shape change estimates. Especially, in Fig. 5a the change estimates in relation to modes 5, 6 and 8 contain non-negligible contributions from modes which originally was said to be outside the information pool. The impact, or the lack of it, cannot be directly assessed based on the t-value alone since the nature of the corresponding mode shape also is part of the equation. Furthermore, one cannot know if there is major contributions from the higher modes. Ultimately, the limited number of modes adds to the uncertainty of the approximation of (3) which contribute to the ill-condition of the system of equations. Based on the experimental data alone, one cannot know if there is major contributions from the higher modes this question has to be answered by an FE simulation.
5.3. The combined system of equations By intuition it seems advantageous to combined the available frequency and mode shape information, as it is done in (10). In the current framework both the frequency and mode shape change approximations have their advantages and disadvantages. An important fact is that frequency change approximation does not, according to (4), depend on any contributions from the remaining modes. In other words the accuracy of a particular frequency change approximation is not influenced by a limited modal data set. On the other hand the frequency measure itself does not contain any spatial information. This prompts the necessity of including mode shape change estimates in the system of equations when dealing with structures featuring some sort of symmetry. For instance, in the simulation example, reviewed above, the uniqueness of the columns of the sensitivity matrix is achieved by adding the mode shape change approximations. The examination of the sensitivity matrix content is an initial investigation of the feasibility of the current technique. Every structure is different and the available modal information, the number of scenarios etc. varies from case to case which make the presented exercises a natural first step. Even though the various modal parameters are sensitive and wellseparated in terms of scenario impact, it does not rule out that the equation system is ill-conditioned.
5.4. The influence of missing modes The investigations presented in the subSections 5.1 and 5.2 examines of the sensitivity of the modal parameters in relation to the considered perturbation scenarios, and furthermore the corresponding patterns of approximated change. Al tough these investigations indicate that the predefined damage cases impact the modal parameters and furthermore in a unique manner, they do not disclose whether or not the amount of available modes is sufficient. So to highlight the influence of ^ utilizing a growing missing modes on the output of the present method, a repeated computation of the selection vector Du number of modes has been done. In Fig. 6 the selection vector estimate as a function of the number of modes (initially only using 1 mode up to using 8 modes) is shown. In the calculation of the selection vector both mode shapes and frequencies are utilized as stated in (11). The result of the computation results in Fig. 6 show that a correct indication of all the predefined mass perturbation cases become reliable when at least 5 modes are used. Naturally, it should be noted that the results are based on noise-free modal parameters along with a complete set of DOFs and using a ‘‘perfect” FE representation, hence in an experimental test case the need for additional modes must be expected.
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(e) Selection vectors, scenario 5
Fig. 6. 3d bar charts of the selection vectors computed using 1–8 modes vectors colored in green indicate correct location, vectors colored in red indicate wrong location. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
6. Experimental test cases The presented technique has been tested on a physical version of the FE model which founded the basis of the examinations reviewed in the previous sections. The structure is made up of two 40 40 2 mm quadratic hollow steel tubes welded together to constitute a T-shape structure with a total height and width of 1,5 and 1 m, respectively. A 10 mm steel plate has been welded to the base to enable clamped boundary conditions. The total weight of the structure, without the steel plate, is approximately 7.5 kg. The vibrations of the structure were recorded by means of 10 uni-axial accelerometers with a sensitivity of 1 V/g. A picture of the test structure including position and orientation of the sensors is shown Fig. 7. Since, the vibrational characteristics of the structure were to be identified by means of OMA, the excitation had to be unknown and random. In order to meet these fundamental conditions an ‘‘excitation system” consisting of 3 highpressure air nozzles was used. The nozzles were placed strategically to ensure sufficient turbulence around the structure. The dynamic characteristics were extracted automatically by means of the Time Domain Poly Reference (TDPR) [5] in combination with a DOF-condensation technique [25]. In terms of ‘‘damage” parametrization and repeatability, mass perturbations are the most convenient scenarios for proving the validity of the presented technique. Therefore, the selected scenario definition and candidate mass locations in the test cases were coincident with the scenarios shown in Fig. 2. The five test cases are listed in Table 1 below. In the measurement campaign the T-structure has been monitored in six configurations, i.e. the reference state and the five perturbed states. To minimize noise 1200 individual modal tests were conducted, 200 for each state. Hence, the modal parameters which were utilized in the algorithm were averaged natural frequencies and mode shapes. In all tests the initial 8 modes were identified consistently. The averaged natural frequencies and corresponding mode shapes of these modes (reference state) are shown in Table 2. Besides the measured modal parameters the calculation of the gradients require, according to (3) and (4), the modal masses and mass change matrices of the scenarios. As mentioned, the FE model presented in Section 5 is a parametric version of the physical lab model and therefore suitable to constitute the required FE information. The Modal Assurance Criterion (MAC) values between the initial FE and test modes ranged from 95.6% to 99.9%, however, major differences in the corresponding natural frequencies were observed. The boundary conditions of the FE model were initially defined as fixed, which to an unknown extent deviate from the actual boundaries of the experimental model. Therefore, the boundary conditions of the FE model were changed from fixed to spring supported (rotational springs). Using the reference modal parameters as target responses the stiffness of the rotational springs were updated using FEMtools [26]. The correlation between the test case and the updated FE model is displayed in Table 3 In order to enable the computation of modal scaling factors, or modal masses, the size of the mass matrix and the experimental mode shapes must match. In the current test cases the system matrices of the FE model are condensed by means of
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S10 S08 S09
S07 S05 S06 S03 S04
S01 S02
(a) Test structure
(b) Sensor positions and orientations
Fig. 7. The test structure together with sensor positions and orientations.
Table 1 Test case definitions. Test case no.
Added mass size (g)
1 2 3 4 5
150 150 150 150 150
Mass location corresponds to Scenario Scenario Scenario Scenario Scenario
1 2 3 4 5
Table 2 The experimental mode shapes and the corresponding natural frequencies of the structure in the reference state. Mode 1 1st in plane bending mode
Mode 2 1st out of plane bending mode
Mode 3 1st Torsion mode
Mode 4 2nd in plane bending mode
f 1 ¼ 6:9819 Hz
f 2 ¼ 8:4228 Hz
f 3 ¼ 22:939 Hz
f 4 ¼ 43:538 Hz
Mode 5 2nd out of plane bending mode
Mode 6 2rd in plane bending mode
Mode 7 Local flange bending mode
Mode 8 3rd out of plane bending mode
f 5 ¼ 76:118 Hz
f 6 ¼ 106:22 Hz
f 7 ¼ 144:82 Hz
f 8 ¼ 204:37 Hz
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J.B. Hansen et al. / Mechanical Systems and Signal Processing 94 (2017) 359–373 Table 3 Mode pairs between updated FE model and test. Pair
FEA freq. (Hz)
EMA freq. (Hz)
Diff. (%)
MAC (%)
1 2 3 4 5 6 7 8
6.9593 8.4223 23.130 45.157 75.909 111.42 151.43 206.85
6.9819 8.4228 22.939 43.538 76.118 106.22 144.82 204.37
0.32 0.01 0.83 3.72 0.27 4.90 4.56 1.21
99.8 99.8 99.9 99.8 96.3 98.3 99.9 97.1
Fig. 8. The interconnections of the data processing steps related to the experimental test cases, where B and A are the mode shape matrices of the reference and perturbed structural state, respectively.
0.1
0.05
0
0.15
Mass (kg)
0.15
Mass (kg)
Mass (kg)
0.15
0.1
0.05
0 1
2
3
4
Scenario no.
(a) Selection vector
5
0.1
0.05
0 1
2
3
4
5
Scenario no.
(b) Selection vector
1
2
3
4
5
Scenario no.
(c) Selection vector
Fig. 9. Selection vectors, test case No. 1.
Guyan reduction [27] to match the experimental DOFs. The mass change matrices are produced by; 1) adding a point mass to the FE structure in accordance to one of the predefined specific scenario, 2) reducing the system matrices and 3) subtracting the perturbed mass matrix from the reference mass matrix. The size of the added mass in the FE scenario definition is set to 1 kg, this means the output, i.e. selection vector entries, will be relative to this ‘‘unit” perturbation. In other words if the FE
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0.1
0.05
0
0.15
Mass (kg)
0.15
Mass (kg)
Mass (kg)
0.15
0.1
0.05
0 1
2
3
4
5
0.1
0.05
0 1
Scenario no.
2
3
4
5
1
Scenario no.
(b) Selection vector
(a) Selection vector
2
3
4
5
Scenario no.
(c) Selection vector
Fig. 10. Selection vectors, test case No. 2.
0.1
0.05
0
1
2
3
4
0.1
0.05
0
5
0.15
Mass (kg)
0.15
Mass (kg)
Mass (kg)
0.15
1
Scenario no.
2
3
4
0.1
0.05
0
5
1
Scenario no.
(b) Selection vector
(a) Selection vector
2
3
4
5
Scenario no.
(c) Selection vector
Fig. 11. Selection vectors, test case No. 3.
0.1
0.05
0
1
2
3
4
0.1
0.05
0
5
0.15
Mass (kg)
0.15
Mass (kg)
Mass (kg)
0.15
1
Scenario no.
2
3
4
0.1
0.05
0
5
1
Scenario no.
(b) Selection vector
(a) Selection vector
2
3
4
5
Scenario no.
(c) Selection vector
Fig. 12. Selection vectors, test case No. 4.
0.1
0.05
0
1
2
3
4
Scenario no.
(a) Selection vector
5
0.15
Mass (kg)
0.15
Mass (kg)
Mass (kg)
0.15
0.1
0.05
0
1
2
3
4
5
Scenario no.
(b) Selection vector Fig. 13. Selection vectors, test case No. 5.
0.1
0.05
0
1
2
3
4
Scenario no.
(c) Selection vector
5
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Sc. 5
Sc. 3 Sc. 4
Sc. 2 Pos. 1
50
50
40
40
Mass (g)
Mass (g)
Pos. 2
30 20 10
30 20 10
0
Sc. 1
0 1
2
3
4
5
Scenario no.
(b) Selection vector mass added in position 1
1
2
3
4
5
Scenario no.
(c) Selection vector mass added in position 2
(a) Scenario definitions (red spheres) and mass perturbation positions (blue spheres) Fig. 14. Perturbation cases in between scenario locations.
perturbation was set to 150 g, a precise quantification of an actual mass change of 150 g, in the correct selection vector entry, would read 1. All data processing steps related to the current test cases are shown in Fig. 8. The Figs. 9–13 displays the results of the experimental test cases using 8 modes, which proved to be the minimum number of modes required for a reliable output for all test cases. Each figure contain 3 bar plots; the plot on the left is the estimated selection vector using only mode shape information in accordance with (8), the middle plot is the estimated selection vector using frequency information in accordance with (9) and the outermost right plot is the estimated selection vector is the combined information using (13) and (15). In all plots only the indication of positive mass change are shown and a horizontal red line has been added to mark the level of actual mass perturbation. Fig. 9 shows the selection vectors from the test case 1. All three versions of the selection vector indicate the correct occurring scenario and furthermore a rather accurate estimate of the mass size. In Fig. 10 the computed results from the test case 2 is displayed. Also in this test case the selection vectors are accurate, both in terms of location and quantification of the mass perturbation. In test case 3, a point mass is attached at the center top of the test structure. The selection vectors of this perturbation case is shown in Fig. 11. Here two initial selection vectors indicate the correct location, however, the size of the mass change is underestimated. From Figs. 3c and 4c it is apparent the this mass change scenario has a relatively large impact on the two lowest bending modes due to the structure being cantilevered. Therefore, underestimation suggests that the actual change no longer is proportional the linear approximations of the modal parameter changes. The outermost right plot of Fig. 11 is the selection vector computed by means of the combined mode shape and frequency sensitivity matrix. In this plot small mass changes in a false and the correct location is indicated. This result seems strange due to fact that the selection vectors using mode shapes and frequency individually prompt more convincing results. Therefore, the explanation for the poor result lies in the actual mass size and the chosen weighting. Due to the geometry and boundary conditions of the T-stucture, the mass change scenarios 4 and 5 are the most sensitive scenarios in terms of size of the mass perturbation. These locations are not only cantilevered with respect to the boundary conditions but also with respect to the vertical centroidal line of the structure. This is why the approximated modal changes in these locations, with the same mass size, are around 10 times larger than the ones cause by scenario 1 and 2. Furthermore, adding mass in these locations impacts all of the estimated modes, as seen in Figs. approxrelfreqd and 4d. The results from test case 4 and 5 are shown in Figs. 12 and 13. The selection vectors computed using only the mode shapes are shown in Figs. 12a and 13a. In both cases the correct scenario is indicated however, the underestimation of the actual mass size is pronounced. As expected to selections vectors based on the frequency sensitivities prompt poor results, see Figs. 12b and 13b (the numerical values of the scenario 4 bar in both figures are 0.93 and 0.71, respectively). Naturally, this is due to the structural symmetry which makes scenario 4 and 5 indistinguishable. Lastly, the selection vectors based on the combined frequency and mode shape change approximation is shown in Figs. 12c and 13c. Once again, these results are influenced by the fact that the actual mass size is too large, however, both test cases the correct position is indicated. In general more accurate mass change estimates would have been achieved by using smaller masses in test case 3, 4 and 5.
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7. Discussion An important aspect of the scenario-based approach is the risk of a mismatch between the actual perturbation and the predefined scenarios. This discrepancy can arise from the following; the perturbation occurring in a location which is unaccounted for, an erroneous parametrization of the damage mechanism or a combination of both. The occurrence of a ‘‘missing” scenario which has no relation to the predefined group of damage scenarios will undoubtedly prompt a false alarm, i.e. an incorrect estimate of both damage location and extent. However, if the actual structural perturbation corresponds to a linear combination of the predefined scenarios, for instance a mass perturbation somewhere in between two mass change scenarios, the output of the scenario-based method is still valid. To illustrate this situation we return to the FE model of the simulation study presented in Section 5 and define two mass perturbation cases (30 grams added) in between the mass scenarios. Both the predefined scenarios and the two mass perturbation positions along with the corresponding output of the current method are shown in Fig. 14. The selection vector output in these cases indicate a mass change in the neighboring scenarios which either suggest that mass has been added in the scenario locations or somewhere in between. Intuitively the values of the bars in the selection vector plots should of equal value, however, the explanation for the difference of the bar sizes is found in the difference in the sensitivity of the scenarios in question. In the present paper both the simulation study and experimental test cases have only treated structural perturbations concerning mass change. As mentioned earlier this has been convenient in relation to the perturbation parametrizations and the repeatability of the experimental test cases. However, a change in stiffness is a more common consequence of damage which in terms of the parametrization of the structural perturbation is more complex. The scenario-based method, in the present framework, require that the unperturbed stiffness matrix can be subtracted from the stiffness matrix of the modeled damaged state. Hence, perturbed FE models which consist a discontinuity in the FE mesh, e.g. a crack modeled by disconnecting adjacent FE nodes, are incompatible with the present framework of the scenario-based technique. This means that damage parametrization is limited to an equivalent modeling approach, i.e. the reduction in stiffness of the selected subset of FE elements. Many times, this approach is sufficient to identify local damage using low frequency vibration measurements [23]. Whether or not the equivalent approach is physically meaningful, and prompt a reliable estimate of model parameter change, is highly depended on the FE mesh, the nature of the scenario damage mechanism and the structural geometry and material. Since, the presented method is closely related to the damage detection techniques based on FE model updating the application in terms of test cases is expected to be coincident. Naturally, the present method requires knowledge related to where and how the structural modifications occurs. An example of a structural perturbation that is well known, is ice accretion on wind turbines. Ice accretion on the rotor blades of a wind turbine leads, among other things, to added loads, safety issues and diminished aerodynamic performance of the airfoil. This type of perturbation constitute an added mass and occurs frequently in northern regions. The presented technique could be implemented directly to localize and quantify ice accretion. 8. Conclusions In this paper a scenario-based damage assessment methodology using modal parameters estimated by means of automated OMA techniques was presented. Besides a review of the theoretical aspects of the methodology, this paper has introduced a series of considerations related to the feasibility of the current as well as other modal driven techniques. Lastly, the method was tested on the series of perturbation scenarios on a experimental test case proving the validity and demonstrating the limitations. References [1] K. Worden, C.R. Farrar, G. Manson, G. Park, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 463, The Royal Society, 2007, pp. 1639–1664. [2] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review, Technical Report, Los Alamos National Lab., NM (United States), 1996. [3] H. Sohn, C.R. Farrar, F.M. Hemez, D.D. Shunk, D.W. Stinemates, B.R. Nadler, J.J. Czarnecki, A review of structural health monitoring literature: 1996– 2001, 2004. [4] A. Rytter, Vibration based inspection of civil structures, Ph.D. thesis, Dept. of Building Technology and structural engineering, Aalborg University, Aalborg, Denmark, 1993. [5] R. Brincker, C. Ventura, Introduction to Operational Modal Analysis, John Wiley & Sons, 2015. [6] S. Ibrahim, J. Asmussen, R. Brincker, Modal parameter identification from responses of general unknown random inputs, Technical Report, Dept. of Building Technology and Structural Engineering, Aalborg University, 1995. [7] D. Bernal, J. Eng. Mech. 130 (2004) 1083–1088. [8] M. Aenlle, R. Brincker, Int. J. Mech. Sci. 76 (2013) 86–101. [9] E. Parloo, P. Guillaume, M. Van Overmeire, Mech. Syst. Signal Process. 17 (2003) 499–518. [10] T. Lauwagie, E. Dascotte, Topics in Modal Analysis II, vol. 6, Springer, 2012, pp. 295–303. [11] W. Fan, P. Qiao, Struct. Health Monit. 10 (2011) 83–111. [12] D. Bernal, J. Eng. Mech. 128 (2002) 7–14. [13] E. Reynders, G. De Roeck, J. Sound Vib. 329 (2010) 2367–2383. [14] J.M. Brownjohn, P.-Q. Xia, H. Hao, Y. Xia, Finite Elem. Anal. Des. 37 (2001) 761–775. [15] M. Link, M. Weiland, Mech. Syst. Signal Process. 23 (2009) 1734–1746. [16] B. Titurus, M. Friswell, Mech. Syst. Signal Process. 45 (2014) 193–206. [17] M. Basseville, L. Mevel, M. Goursat, J. Sound Vib. 275 (2004) 769–794.
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M.K. Hovgaard, J.B. Hansen, A. Skafte, P. Olsen, R. Brincker, Struct. Health Monit. 2015 (2015). W. Heylen, P. Sas, Modal Analysis Theory and Testing, Katholieke Universteit Leuven, Departement Werktuigkunde, 2006. M. Friswell, J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, vol. 38, Springe Science & Business Mediar, 1995. J.E. Mottershead, M. Link, M.I. Friswell, Mech. Syst. Signal Process. 25 (2011) 2275–2296. J.L. Crassidis, J.L. Junkins, Optimal Estimation of Dynamic Systems, CRC Press, 2011. A. Morassi, F. Vestroni, Dynamic Methods for Damage Detection in Structures, Springer Science & Business Media, 2008. P.C. Hansen, Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion, vol. 4, Siam, 1998. R. Brincker, A note on optimal dof condensation, Unpublished, 2015. Dynamic Design Solutions (DDS) N.V., FEMtools 3.8.2, 2016.. R.J. Guyan, AIAA J. 3 (1965), 380–380.
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
A new scenario-based approach to damage detection using operational modal parameter estimates J.B. Hansen a,⇑, R. Brincker b, M. López-Aenlle c, C.F. Overgaard a, K. Kloborg a a
Department of Engineering, Aarhus University, Denmark Department of Civil Engineering, Technical University of Denmark, Denmark c Department of Construction and Manufacturing Engineering, University of Oviedo, Spain b
a r t i c l e
i n f o
Article history: Received 26 May 2016 Received in revised form 3 March 2017 Accepted 4 March 2017
Keywords: Damage detection Model updating Operational modal analysis Modal sensitivities
a b s t r a c t In this paper a vibration-based damage localization and quantification method, based on natural frequencies and mode shapes, is presented. The proposed technique is inspired by a damage assessment methodology based solely on the sensitivity of mass-normalized experimental determined mode shapes. The present method differs by being based on modal data extracted by means of Operational Modal Analysis (OMA) combined with a reasonable Finite Element (FE) representation of the test structure and implemented in a scenario-based framework. Besides a review of the basic methodology this paper addresses fundamental theoretical as well as practical considerations which are crucial to the applicability of a given vibrationbased damage assessment configuration. Lastly, the technique is demonstrated on an experimental test case using automated OMA. Both the numerical study as well as the experimental test case presented in this paper are restricted to perturbations concerning mass change. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction In the academic as well as the industrial world the interest in the ability to monitor and detect damage(s) in a civil or mechanical structure is immense. For decades researchers and practitioners in the Structural Health Monitoring (SHM) community have been developing an extensive amount of methods based on a wide range of physically interpretable structural features. At this point in time there is no universal monitoring system hence the term SHM refers to the implementation of one or more damage identification strategies [1]. In vibration-based damage detection the underlying premise is that the modal parameters are governed by the physical properties of the structure, hence an observed shift in the modal parameters indicate changes in the inherent properties of the monitored structure (e.g. structural degradation). Even though the area of vibration-based damage evaluation is merely a branch of SHM, the literature produced in this area alone is extensive [2,3]. The capabilities of a given damage technique has through many years been classified in reference to Rytters four levels of damage identification [4]. The four levels are as follows: 1. 2. 3. 4.
Detection - Is the structure damaged? Localization - Where is the damage? Quantification - How severe is the damage? Prognosis - What is the remaining service life of the structure?
⇑ Corresponding author. E-mail address: [email protected] (J.B. Hansen). http://dx.doi.org/10.1016/j.ymssp.2017.03.007 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.
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The vibration-based methods range from simple tracking of natural frequencies, fundamental relations between the modal parameters and the system matrices, to modal-based Finite Element (FE) model updating. Methods utilizing only the natural frequencies usually fall into to the level 1 category whereas level 2–3 techniques also require the corresponding mode shapes. Since, the modal parameters, traditionally, have been extracted by means of Experimental Modal Analysis (EMA) many techniques capable of localizing damage, without depending on a parametric model, require scaled or mass normalized mode shapes. This fact rules out the immediate application of most vibration-based localization methods to structures which cannot be subjected to a controlled vibration test (i.e. where the input force is known). Over the last decades the concept of OMA has matured into a well-established discipline within the area of modal testing [5]. The output of OMA is an estimate of the modal model for the whole system (i.e. the loading filter and the physical structure) which presents the challenge of separating the dynamic characteristics of the loading from the modal properties of the structure. The loads/excitation does not change the physical modes of the system [6], so if one can separate the noise modes from physical modes the accuracy of an OMA estimate is comparable to an EMA estimate. Furthermore, OMA lacks the ability to produce mass normalized mode shapes, however, both experimental techniques [7] and FE-based methods [8] have with success eliminated this shortcoming. The concept of automated OMA (i.e. OMA without any user interaction) enables the constant extraction modal features from any structure to be utilized in an SHM context. This makes OMA the obvious modal extraction technique within vibration-based SHM. The scope of the present method is to localize and quantify damage, restricted to a limited number of candidate scenarios by merging operational modal parameter estimates with the system matrices of an FE model. Conceptually, the methodology is similar to a level 2–3 technique based on the sensitivity of experimentally obtained mass normalized mode shapes [9]. The differences are found in; 1) the current technique implements, via a parametric model, the idea of limiting the number of possible damage locations [10] and 2) the current method utilizes OMA to extract the modal properties. Both modifications increase the robustness and applicability to civil and larger mechanical structures, however, they also prompt the necessity of having an FE model of the monitored structure for mass normalization and prediction purposes. Damage detection techniques which use experimental modal data in combination with a parametric representation of the structure, falls into the category of model-based (or parametric) methods [11]. The experimental modal parameters are easily related to the properties of a parametric model, however the implementation of the FE model as well as the utilization of the model data vary substantially across the model-based techniques. Some methods synthesize a experimental measure for a change in flexibility to derive a forcing configuration to be subjected to an parametric model [12,13]. Methods based on FE model updating attempts to tune an FE model to the measured modal data using a predefined subset of adjustable parameters which are sensitive to damage [14–16]. The main drawback of these FE updating methods is the inverse-problem which often is ill-posed. This has prompted FE based methods which bypass the ill-condition issue by introducing statistical frameworks [17,18]. The damage identification technique presented in this paper is very closely related to the FE model updating methods listed above and features the same fundamental algebraic operation. Therefore the methods inherits some of the same challenges when dealing with limited, noisy data and a wide range of potential damage mechanisms. The current method does not attempt to fit a parametric model to test data but merely utilize FE model information in combination with measured modal data in the computation of the modal sensitivities - this is the most distinct feature of the present method in relation to other published model-based techniques. Since the reference data set and the modal sensitivities are based on measured data, the gap between the reference and ‘‘damaged” data can be assumed limited, hence no iterative process is necessary. Ultimately the proposed technique is able to locate and quantify damage given the structural modification is small and corresponds to a predefined scenario in one calculation step. A successful application of the current method relies on the modal parameters being sensitive towards the predefined scenarios of structural change. The fundamental relations which found the basis of the method in combination with the implementation of the scenario idea enables an examination of the method usability. Therefore, this paper features a section which describes the preliminary considerations necessary to determine whether or not the current technique is feasible, with respect to a given structure and configuration of scenarios. On a more general level one might argue that the usability of using any modal-based technique to detect and locate specific candidate structural modifications is disclosed. Lastly, the current method is demonstrated on a experimental test case where the structural modifications consist of local mass perturbations. Hence, the technique is demonstrated on damage cases in the form of mass increase only. In the test case the modal parameters are extracted with no user interaction and by means of a time domain identification technique. 2. Scenario-based approach using modal sensitivities The mode shape and frequency sensitivity equations in the form presented by Heylen [19] founds the basis for the present technique. The undamped sensitivity or derivative of mode shape i with respect to a local change parameter u in the mass and stiffness matrix is given by
Nm X @bi 1 T @M 1 1 T @K 2 @M bi bi þ þ bi br bi b x ¼ i 2mi @u @u @u @u x2i x2r mr r r¼1;r–i
ð1Þ
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where b is a (N dof 1) mode shape vector of the unperturbed system, M is the mass matrix, K is the stiffness matrix, m is the modal scaling factor, x is the angular frequency, Nm is the number of modes and the subscript r denotes the remaining modes of the system. In this context N dof is the number of experimental Degrees Of Freedom (DOF) Likewise, the sensitivity of angular frequency i due to local change parameter u is given by
@ xi 1 T @M 1 @K bi þ bi xi ¼ 2mi @u xi @u @u
ð2Þ
For a finite change scenario j consisting of small alterations in the mass and stiffness matrix yields the following approximations of (1) and (2), respectively
Dbij
Dxij
Nm X 1 T 1 1 T bi DMj bi bi þ br x2i DMj þ DKj bi br 2 2 m 2mi x x r r i r¼1;r–i
1 T 1 bi xi DMj þ DKj bi 2mi xi
ð3Þ
ð4Þ
In general, the mode shapes and angular frequencies are experimentally determined quantities whereas the modal scaling factors and perturbed matrices are based on FE data. The methodology is to compute the approximated changes in mode shapes and natural frequencies using (3) and (4) in relation to a series of predefined structural alterations or scenarios. Obviously, the accuracy of the aforementioned estimates depend on the quality of the initial FE model and the engineers ability to parametrize the structural modification. In Fig. 1, four arbitrary scenario examples on an FE shell representation of a Tshaped structure is shown. The approximated mode shape and frequency changes of mode i due to all predefined scenarios of change in structural stiffness and/or mass are arranged in the matrix SDbi and vector sDxi respectively, thus
SDbi ¼ ½Dbi1 Dbi2 DbiNsc sDxi ¼ ½Dxi1 Dxi2 DxiNsc
ð5Þ
where SDbi is a (N dof N sc ) mode shape sensitivity matrix, sDxi is a (1 N sc ) frequency sensitivity vector and N sc is the total number of predefined scenarios. If the tructural system is subjected to a small change, the measured mode shape and angular frequency change of mode i can be defined as
Dbi ¼ ai bi
Dxbi ¼ xai xbi
ð6Þ
where a and subscript a denotes modal parameters from the perturbed system Given that the structural change prompting the measured changes in (6) correspond to one, or to a linear combination, of the predefined damage scenarios, collected in (5), the following equations for mode i can be written
Dbi ¼ SDbi DuDbi
Dxbi ¼ sDxbi DuDxbi
ð7Þ
where uDbi and DuDxbi can be regarded as a (N sc 1) selection vector. The equations in (7) can be extended to include all recorded modes of the system. In relation to the mode shape changes one can write
Db ¼ SDb DuDb
8 9 2 3 SDb1 Db1 > > > > > > > 6 SDb 7 < Db2 > = 6 2 7 7 where Db ¼ . and SDb ¼ 6 6 .. 7 . > > > > 4 . 5 . > > > > : ; DbNm SDbNm
Analogously, for frequency changes we get
Fig. 1. Scenario examples, the red contour represent modified elements.
ð8Þ
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Dxb ¼ SDxb DuDxb
8 9 3 2 sDxb1 Dxb1 > > > > > > > 6 sDx 7 < Dxb2 > = 6 b2 7 7 where Dxb ¼ . and SDxb ¼ 6 6 .. 7 . > > > > 5 4 . . > > > > : ; DxbNm sDxbNm
ð9Þ
By combining (8) and (9) a final set of equations reached, hence
Dc ¼ SDu where Dc ¼
Db Dx b
and S ¼
SDb
SDxb
ð10Þ
S is a (ðN dof þ 1ÞN m N sc ) sensitivity matrix and Dc is a (ðN dof þ 1ÞN m 1) measured change vector and Du remains a (N sc 1) selection vector. Solving (10) for Du yields to following estimate
^ ¼ S þ Dc Du
ð11Þ
where þ denotes the pseudo-inverse. Because the approximations due to either stiffness or mass alterations in (3) and (4) are proportional to the perturbed matrices DM or DK the selection vector Du not only single out present scenario but also produce an estimate of the extent in reference to parametrization of the particular candidate mechanism. As mentioned, the approximated changes is a combination of experimental modal parameters and FE matrices which means that there will be a discrepancy in number of the experimental and the FE DOFs. There are two possible solutions to this issue, either FE matrix reduction or experimental mode shape expansion. Further comments on this issue can be fund in Section 6.
3. Weighting sensitivity matrix components The sensitivity matrix, defined in (10), holds the approximated changes of both frequency and mode shape. The numerical size of the two entities, the frequency change and mode shape coordinate change approximation, are very different (the frequency change approximation being the larger). This means that the evaluation of the damage vector by means of the expression stated in (11) will to a certain extend be dominated by the frequency changes. Furthermore, although less important, both the individual frequency and mode shape estimates which found the basis of sensitivity computation may not have been identified with equal confidence. The considerations stated above suggests that some sort of weighting of the individual sensitivities is necessary to avoid a skewed sensitivity matrix. Since, the mathematical operation in (11) is a ‘‘best fit” (least squares) solution to the system of equations, one can turn to a weighted least squares approach [20]. The procedure is to pre-multiply both sides of (10) with weight matrix W, hence
WDc ¼ WSDu where W ¼ diagðw1 w2 . . . wn Þ
ð12Þ
diagðÞ denotes a diagonal matrix. By pre-multiplying (12) with the transpose of the sensitivity matrix S and then isolating for Du one gets
1 Du ¼ ST WS ST WDc
ð13Þ
The immediate selection of the weight components w1 ; w2 ; . . . ; wn is rather difficult if no preliminary knowledge of the importance or statistical properties of the individual terms are available. However, an initial selection in reference to (10) could be 1
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ð14Þ
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ð15Þ
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4. The condition of the system of equations The procedure of weighting the sensitivity matrix does not add any additional information nor does it change the fact that (13) remains a noisy system of equations and potentially ill-conditioned [23]. In relation to the current method the problem itself can arise from one or several circumstances such as noised and limited modal estimates, poor FE model, misalignment between actual and modeled damage etc. In this section it is assumed that the FE model is a perfect representation of the monitored structure. This enables focus on the general factors affecting the overall condition of the system of equations, these factors are itemized below. 1. The ‘‘modes-scenario-ratio” It is crucial that the extent of available information surpass the number of unknowns to keep the problem overdetermined. In reference to the present technique the information are the recorded modes and the unknowns are the predefined scenarios (e.g. a configuration containing 5 scenarios requires at least 5 modes). Using OMA one is subject to the in-operation response of the structure which basically means that the number of identified modes will be governed by the nature of the loading filter, i.e. the number of modes will be limited. So, to meet this requirement of over-determination the end user must restrict the number of damage scenarios relatively to the amount of identified modes. This prompts the need for preliminary knowledge on where the monitored structure is most likely be subjected to alterations. 2. Sensitivity matrix rank To ensure full rank of the sensitivity matrix the predefined scenarios must differ across the columns in terms of impact to at least some of the available modal parameters. In other words the approximated changes in one column of the sensitivity matrix must be unique or distinguishable in relation to the remaining columns. 3. Parametrized versus actual scenario For (10) to be valid the nature of the perturbation of the monitored structure must correspond to one, or to a linear combination, of the predefined scenarios. Furthermore, the content of the sensitivity matrix are frequency and mode shape gradients defined in (3) and (4), which means that the validity of these approximations depend on the size of the actual structural perturbation. Naturally, the occurring mechanism must be large enough to penetrate the noise floor of the measurements, but on the other hand small enough to match the approximations. 4. Approximated versus measured change Given that the actual occurring scenario coincides with one of the predefined alteration. There will most probably be a discrepancy between the approximated change and the actual measured change due to noised and limited modal data and erroneous damage parametrization. From a practical point of view there is little that can be done to improve the third and fourth item listed above. The first two items, however, has to do with the construction and content of the sensitivity matrix which can be examined and potentially improved by the end user. An advantageous approach to avoid a poorly conditioned system of equations is to examine the contents of the sensitivity matrix, i.e. the estimated changes in both frequency and mode shape due to the predefined scenarios. Lastly, the Singular Value Decomposition (SVD) is a useful tool to determine the rank of the sensitivity matrix [24].
5. Simulation study Before applying the current technique on a physical structure a preliminary simulation study on a parametric model is to be conducted. In continuation of the previous section the purpose is to assess the overall condition of the sensitivity matrix
(a) Scenario 1
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Fig. 2. Scenario 1–5 – the red dot marks the added mass. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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based on particular set of predefined scenario and an available set of modal parameters. The investigation is similar to an FE model sensitivity analysis conducted to determine whether or not a preselected model parameter could be the source of error between model and test. Here, the purpose is to determined if the available information (i.e. the modes of the system) is sufficient, sensitive towards the scenarios of interest and distinguishable in terms of impact. On a more general level, one might argue that this investigation discloses the feasibility, of using vibration-based monitoring to detect the candidate structural alterations on the specific structure. To illustrate this concept an FE based simulation using a beam element model of a steel T-structure is reviewed. The FE model consist of 126 beam elements connected by means of 127 nodes and is fixed/clamped at the base, ultimately constituting a 756 DOF system. The task is to investigate the ability to detect mass changes in five locations on the structure. The structure and mass change scenarios are sketched in Fig. 2. As mentioned, nearly all contributory causes to the potential ill-condition issue present in the current technique is to some extent related to the range of available information. So to keep the premise of having limited information only the first eight modes are included in this investigation. The size of the mass change is, in all scenarios, set to 0.5% of the total weight of the FE model. In the following subsections the contents of the sensitivity matrix due to the predefined mass change scenarios is examined, i.e. both the frequency and mode shape change approximations, in relation to the ill-condition considerations.
5.1. The frequency change approximations Each column in frequency sensitivity matrix, SDxb defined in (9), holds the approximated frequency changes of all the recorded modes due to a specific scenario. So, by plotting the columns of the aforementioned matrix a preliminary graphical examination of the impact of the various perturbations is possible. In Fig. 3 absolute relative approximated frequency changes caused by scenario 1–5 are shown. In reference to the ill-condition problem there is a lot to be learned from the bar plots in Fig. 3. The first observation to be made is that the approximated changes due to scenario 3, as a whole, seem unique compare to the other scenarios but only at relatively few modes are influenced and some to a very limited extent. Secondly, the approximated frequency changes due to scenario 4 and 5 are identical. Obviously, this is due to the structural symmetry combined with the fact that the scenarios 4 and 5 are added mass in opposite positions of the structure. In relation to the frequency sensitivity matrix, however, this means that the 4th and 5th column are identical thus the matrix does not have full rank. In other words the application of the current technique using only the natural frequencies would make scenario 4 and 5 indistinguishable. Furthermore, there seem to be a rather large difference in the general level of change when comparing scenario 1, 2 and 3 to scenario
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4 and 5. This presents the fact that these scenarios are more sensitive than scenario 1–3. Naturally, this is due to cantilever nature of the structure and the nature of the perturbation (i.e. mass change). 5.2. The mode shape change approximations It is obvious that the estimated mode shapes need to be affected by the predefined structural modification scenarios. However, it is less obvious that the approximated change of a specific mode shape includes contributions from the remaining mode shapes of the system. This emphasizes the need for a ‘‘sufficient” number of modes to get a proper approximation of change. Since, (3) consist of several inner products it can be simplified by defining scalar values tij and trij , thus Nm X
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In terms of the mode shape derivatives it would be rather hard to construct a meaningful plot of the approximated mode shape changes. The changes themselves, being vectors and furthermore a combination of contributions from the mode shape vector space, are difficult to interpret. The scalar values tij and t rij defined in (16) supply the information on how much and which mode shapes contribute to the change approximation of a given mode shape due to a particular scenario. So instead of attempting to dissect a vector representation of the mode shape changes one can examine t-scalars in a stacked bar plot. Fig. 4 is a graphical representation of the aforementioned scalar values prompted by scenario 1–5. Each bar represents the composition of a mode shape change approximation where the color of the individual boxes refer to the contribution from a particular mode. For instance, in Fig. 4a the change approximation of mode 6 consists, besides a contribution from mode 6 itself, of contributions from mode 1 and 4. The general pattern of the frequency change plots in Fig. 3 is more or less the same in the stacked bar plots in Fig. 4. Since, the nature of the perturbation is local one expects a correlation between the approximated frequency and mode shape changes. Likewise, the magnitude of the scalars based on scenario 1–3 are a factor of ten smaller than the ones based on scenario 4 and 5 emphasizing that these scenarios are more sensitivity than scenario 1–3. But contrary to the frequency change approximation produced by scenario 4 and 5 the composition of mode shape contributions due to these scenario differ which means that approximated mode shape changes will differ as well. So, in this particular simulation case by adding the mode shape sensitivities to the system of equations the ability to separate scenario 4 and 5 is ensured. As previously noticed the change approximation of a mode shape does rely on contributions from the remaining modes which potentially imply an incomplete approximation. If we briefly suspend the underlying assumption of having only 8
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modes available and include mode 9 and 10 in the t-scalar computation we can assess the impact of the missing contributions from these modes by extending the bar plot in Fig. 4. In Fig. 5 the aforementioned contributions has been added to the the plots of scenario 2 and 5. From Fig. 5 it is evident that the consequence of not having mode 9 and 10 result in a truncation of some of the shape change estimates. Especially, in Fig. 5a the change estimates in relation to modes 5, 6 and 8 contain non-negligible contributions from modes which originally was said to be outside the information pool. The impact, or the lack of it, cannot be directly assessed based on the t-value alone since the nature of the corresponding mode shape also is part of the equation. Furthermore, one cannot know if there is major contributions from the higher modes. Ultimately, the limited number of modes adds to the uncertainty of the approximation of (3) which contribute to the ill-condition of the system of equations. Based on the experimental data alone, one cannot know if there is major contributions from the higher modes this question has to be answered by an FE simulation.
5.3. The combined system of equations By intuition it seems advantageous to combined the available frequency and mode shape information, as it is done in (10). In the current framework both the frequency and mode shape change approximations have their advantages and disadvantages. An important fact is that frequency change approximation does not, according to (4), depend on any contributions from the remaining modes. In other words the accuracy of a particular frequency change approximation is not influenced by a limited modal data set. On the other hand the frequency measure itself does not contain any spatial information. This prompts the necessity of including mode shape change estimates in the system of equations when dealing with structures featuring some sort of symmetry. For instance, in the simulation example, reviewed above, the uniqueness of the columns of the sensitivity matrix is achieved by adding the mode shape change approximations. The examination of the sensitivity matrix content is an initial investigation of the feasibility of the current technique. Every structure is different and the available modal information, the number of scenarios etc. varies from case to case which make the presented exercises a natural first step. Even though the various modal parameters are sensitive and wellseparated in terms of scenario impact, it does not rule out that the equation system is ill-conditioned.
5.4. The influence of missing modes The investigations presented in the subSections 5.1 and 5.2 examines of the sensitivity of the modal parameters in relation to the considered perturbation scenarios, and furthermore the corresponding patterns of approximated change. Al tough these investigations indicate that the predefined damage cases impact the modal parameters and furthermore in a unique manner, they do not disclose whether or not the amount of available modes is sufficient. So to highlight the influence of ^ utilizing a growing missing modes on the output of the present method, a repeated computation of the selection vector Du number of modes has been done. In Fig. 6 the selection vector estimate as a function of the number of modes (initially only using 1 mode up to using 8 modes) is shown. In the calculation of the selection vector both mode shapes and frequencies are utilized as stated in (11). The result of the computation results in Fig. 6 show that a correct indication of all the predefined mass perturbation cases become reliable when at least 5 modes are used. Naturally, it should be noted that the results are based on noise-free modal parameters along with a complete set of DOFs and using a ‘‘perfect” FE representation, hence in an experimental test case the need for additional modes must be expected.
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6. Experimental test cases The presented technique has been tested on a physical version of the FE model which founded the basis of the examinations reviewed in the previous sections. The structure is made up of two 40 40 2 mm quadratic hollow steel tubes welded together to constitute a T-shape structure with a total height and width of 1,5 and 1 m, respectively. A 10 mm steel plate has been welded to the base to enable clamped boundary conditions. The total weight of the structure, without the steel plate, is approximately 7.5 kg. The vibrations of the structure were recorded by means of 10 uni-axial accelerometers with a sensitivity of 1 V/g. A picture of the test structure including position and orientation of the sensors is shown Fig. 7. Since, the vibrational characteristics of the structure were to be identified by means of OMA, the excitation had to be unknown and random. In order to meet these fundamental conditions an ‘‘excitation system” consisting of 3 highpressure air nozzles was used. The nozzles were placed strategically to ensure sufficient turbulence around the structure. The dynamic characteristics were extracted automatically by means of the Time Domain Poly Reference (TDPR) [5] in combination with a DOF-condensation technique [25]. In terms of ‘‘damage” parametrization and repeatability, mass perturbations are the most convenient scenarios for proving the validity of the presented technique. Therefore, the selected scenario definition and candidate mass locations in the test cases were coincident with the scenarios shown in Fig. 2. The five test cases are listed in Table 1 below. In the measurement campaign the T-structure has been monitored in six configurations, i.e. the reference state and the five perturbed states. To minimize noise 1200 individual modal tests were conducted, 200 for each state. Hence, the modal parameters which were utilized in the algorithm were averaged natural frequencies and mode shapes. In all tests the initial 8 modes were identified consistently. The averaged natural frequencies and corresponding mode shapes of these modes (reference state) are shown in Table 2. Besides the measured modal parameters the calculation of the gradients require, according to (3) and (4), the modal masses and mass change matrices of the scenarios. As mentioned, the FE model presented in Section 5 is a parametric version of the physical lab model and therefore suitable to constitute the required FE information. The Modal Assurance Criterion (MAC) values between the initial FE and test modes ranged from 95.6% to 99.9%, however, major differences in the corresponding natural frequencies were observed. The boundary conditions of the FE model were initially defined as fixed, which to an unknown extent deviate from the actual boundaries of the experimental model. Therefore, the boundary conditions of the FE model were changed from fixed to spring supported (rotational springs). Using the reference modal parameters as target responses the stiffness of the rotational springs were updated using FEMtools [26]. The correlation between the test case and the updated FE model is displayed in Table 3 In order to enable the computation of modal scaling factors, or modal masses, the size of the mass matrix and the experimental mode shapes must match. In the current test cases the system matrices of the FE model are condensed by means of
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S10 S08 S09
S07 S05 S06 S03 S04
S01 S02
(a) Test structure
(b) Sensor positions and orientations
Fig. 7. The test structure together with sensor positions and orientations.
Table 1 Test case definitions. Test case no.
Added mass size (g)
1 2 3 4 5
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Mass location corresponds to Scenario Scenario Scenario Scenario Scenario
1 2 3 4 5
Table 2 The experimental mode shapes and the corresponding natural frequencies of the structure in the reference state. Mode 1 1st in plane bending mode
Mode 2 1st out of plane bending mode
Mode 3 1st Torsion mode
Mode 4 2nd in plane bending mode
f 1 ¼ 6:9819 Hz
f 2 ¼ 8:4228 Hz
f 3 ¼ 22:939 Hz
f 4 ¼ 43:538 Hz
Mode 5 2nd out of plane bending mode
Mode 6 2rd in plane bending mode
Mode 7 Local flange bending mode
Mode 8 3rd out of plane bending mode
f 5 ¼ 76:118 Hz
f 6 ¼ 106:22 Hz
f 7 ¼ 144:82 Hz
f 8 ¼ 204:37 Hz
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Fig. 8. The interconnections of the data processing steps related to the experimental test cases, where B and A are the mode shape matrices of the reference and perturbed structural state, respectively.
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Guyan reduction [27] to match the experimental DOFs. The mass change matrices are produced by; 1) adding a point mass to the FE structure in accordance to one of the predefined specific scenario, 2) reducing the system matrices and 3) subtracting the perturbed mass matrix from the reference mass matrix. The size of the added mass in the FE scenario definition is set to 1 kg, this means the output, i.e. selection vector entries, will be relative to this ‘‘unit” perturbation. In other words if the FE
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perturbation was set to 150 g, a precise quantification of an actual mass change of 150 g, in the correct selection vector entry, would read 1. All data processing steps related to the current test cases are shown in Fig. 8. The Figs. 9–13 displays the results of the experimental test cases using 8 modes, which proved to be the minimum number of modes required for a reliable output for all test cases. Each figure contain 3 bar plots; the plot on the left is the estimated selection vector using only mode shape information in accordance with (8), the middle plot is the estimated selection vector using frequency information in accordance with (9) and the outermost right plot is the estimated selection vector is the combined information using (13) and (15). In all plots only the indication of positive mass change are shown and a horizontal red line has been added to mark the level of actual mass perturbation. Fig. 9 shows the selection vectors from the test case 1. All three versions of the selection vector indicate the correct occurring scenario and furthermore a rather accurate estimate of the mass size. In Fig. 10 the computed results from the test case 2 is displayed. Also in this test case the selection vectors are accurate, both in terms of location and quantification of the mass perturbation. In test case 3, a point mass is attached at the center top of the test structure. The selection vectors of this perturbation case is shown in Fig. 11. Here two initial selection vectors indicate the correct location, however, the size of the mass change is underestimated. From Figs. 3c and 4c it is apparent the this mass change scenario has a relatively large impact on the two lowest bending modes due to the structure being cantilevered. Therefore, underestimation suggests that the actual change no longer is proportional the linear approximations of the modal parameter changes. The outermost right plot of Fig. 11 is the selection vector computed by means of the combined mode shape and frequency sensitivity matrix. In this plot small mass changes in a false and the correct location is indicated. This result seems strange due to fact that the selection vectors using mode shapes and frequency individually prompt more convincing results. Therefore, the explanation for the poor result lies in the actual mass size and the chosen weighting. Due to the geometry and boundary conditions of the T-stucture, the mass change scenarios 4 and 5 are the most sensitive scenarios in terms of size of the mass perturbation. These locations are not only cantilevered with respect to the boundary conditions but also with respect to the vertical centroidal line of the structure. This is why the approximated modal changes in these locations, with the same mass size, are around 10 times larger than the ones cause by scenario 1 and 2. Furthermore, adding mass in these locations impacts all of the estimated modes, as seen in Figs. approxrelfreqd and 4d. The results from test case 4 and 5 are shown in Figs. 12 and 13. The selection vectors computed using only the mode shapes are shown in Figs. 12a and 13a. In both cases the correct scenario is indicated however, the underestimation of the actual mass size is pronounced. As expected to selections vectors based on the frequency sensitivities prompt poor results, see Figs. 12b and 13b (the numerical values of the scenario 4 bar in both figures are 0.93 and 0.71, respectively). Naturally, this is due to the structural symmetry which makes scenario 4 and 5 indistinguishable. Lastly, the selection vectors based on the combined frequency and mode shape change approximation is shown in Figs. 12c and 13c. Once again, these results are influenced by the fact that the actual mass size is too large, however, both test cases the correct position is indicated. In general more accurate mass change estimates would have been achieved by using smaller masses in test case 3, 4 and 5.
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7. Discussion An important aspect of the scenario-based approach is the risk of a mismatch between the actual perturbation and the predefined scenarios. This discrepancy can arise from the following; the perturbation occurring in a location which is unaccounted for, an erroneous parametrization of the damage mechanism or a combination of both. The occurrence of a ‘‘missing” scenario which has no relation to the predefined group of damage scenarios will undoubtedly prompt a false alarm, i.e. an incorrect estimate of both damage location and extent. However, if the actual structural perturbation corresponds to a linear combination of the predefined scenarios, for instance a mass perturbation somewhere in between two mass change scenarios, the output of the scenario-based method is still valid. To illustrate this situation we return to the FE model of the simulation study presented in Section 5 and define two mass perturbation cases (30 grams added) in between the mass scenarios. Both the predefined scenarios and the two mass perturbation positions along with the corresponding output of the current method are shown in Fig. 14. The selection vector output in these cases indicate a mass change in the neighboring scenarios which either suggest that mass has been added in the scenario locations or somewhere in between. Intuitively the values of the bars in the selection vector plots should of equal value, however, the explanation for the difference of the bar sizes is found in the difference in the sensitivity of the scenarios in question. In the present paper both the simulation study and experimental test cases have only treated structural perturbations concerning mass change. As mentioned earlier this has been convenient in relation to the perturbation parametrizations and the repeatability of the experimental test cases. However, a change in stiffness is a more common consequence of damage which in terms of the parametrization of the structural perturbation is more complex. The scenario-based method, in the present framework, require that the unperturbed stiffness matrix can be subtracted from the stiffness matrix of the modeled damaged state. Hence, perturbed FE models which consist a discontinuity in the FE mesh, e.g. a crack modeled by disconnecting adjacent FE nodes, are incompatible with the present framework of the scenario-based technique. This means that damage parametrization is limited to an equivalent modeling approach, i.e. the reduction in stiffness of the selected subset of FE elements. Many times, this approach is sufficient to identify local damage using low frequency vibration measurements [23]. Whether or not the equivalent approach is physically meaningful, and prompt a reliable estimate of model parameter change, is highly depended on the FE mesh, the nature of the scenario damage mechanism and the structural geometry and material. Since, the presented method is closely related to the damage detection techniques based on FE model updating the application in terms of test cases is expected to be coincident. Naturally, the present method requires knowledge related to where and how the structural modifications occurs. An example of a structural perturbation that is well known, is ice accretion on wind turbines. Ice accretion on the rotor blades of a wind turbine leads, among other things, to added loads, safety issues and diminished aerodynamic performance of the airfoil. This type of perturbation constitute an added mass and occurs frequently in northern regions. The presented technique could be implemented directly to localize and quantify ice accretion. 8. Conclusions In this paper a scenario-based damage assessment methodology using modal parameters estimated by means of automated OMA techniques was presented. Besides a review of the theoretical aspects of the methodology, this paper has introduced a series of considerations related to the feasibility of the current as well as other modal driven techniques. Lastly, the method was tested on the series of perturbation scenarios on a experimental test case proving the validity and demonstrating the limitations. References [1] K. Worden, C.R. Farrar, G. Manson, G. Park, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 463, The Royal Society, 2007, pp. 1639–1664. [2] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review, Technical Report, Los Alamos National Lab., NM (United States), 1996. 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