1 INTRODUCTION

The presence of a crack in a repeating structure such as turbomachinery blade set, has a signifi-cant effect on the modal characteristics obtained. Cracks near the blade root generally affect natural frequencies more than those near the blade tip. Rehman et al. (2010) indicated that when a crack propagates into a blade beyond a certain depth, mode swapping occurs in repeat-ing structures due to the occurrence of a new mode, referred to as cracked blade mode. This is also indicated by the sudden drop in the frequencies for the particular modes where swapping has occurred. For the accurate functioning of a system, cracks need to be detected primarily during crack initiation or in the early propagation stage, before ultimate failure occurs. Pitkanen and Hakkarainen (2001) explained different NDT techniques, which were used for detecting cracks in repeating structures. These included dye penetration, eddy current inspection, radiog-raphy, ultrasonic methods, shearography and thermography. Natural frequencies and modes shapes have also been used extensively in the past as an indicator for the detection of damage in a system. Salawu (1997) reviewed in detail these attempts for the possibility of utilising modal properties as a damage indicator. The mode shapes of cyclic symmetric structures are very sensitive to small changes in the geometric as well as material properties of constituent parts (blades). These minute perturba-tions, known as mistuning, are inevitable due to manufacturing constraints. Hou (2002) ex-amined that any crack developed inside the bladed system increases the level of mistuning, making it difficult to understand the underlying mechanisms governing the blade dynamic re-sponse. Saito et al. (2009) showed that mistuning can cause localisation of the vibration about a few blades resulting in concentration of vibration energy Where the modal properties are to be utilised for damage detection, modal assurance crite-rion (MAC) provides a method to compare modeshapes obtained. An alternative method, dem-onstrated by Fotsch and Ewins, involves a MAC plot, an auto-MAC plot and a natural fre-quency comparison for the model correlation. These three parameters can be viewed simultaneously using the frequency-scaled modal assurance criterion (FMAC). In this paper, experimental investigation of MAC based crack detection technique is carried out for a bladed disc. Besides that, the effect of different mistuning patterns and varying crack depths on the modal characteristics is also explained numerically and compared with experimental results.

Experimental investigation of the MAC based crack detection technique for repeating structures

A. U. Rehman, K. Worden and J. A. Rongong Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, S1 3JD, United Kingdom.

ABSTRACT: This study is an extension of Modal Assurance Criteria (MAC) based crack de-tection technique for detecting cracks in the repeating structures by the author. Previously this technique was explored numerically but in this work possibility of obtaining a damage index based on the MAC is investigated experimentally. A bladed disc is selected as a representative base model for the complex real turbomachinery blade-rotor system. A test rig is established and the modal properties of the disc are identified by obtaining an impulse response from dif-ferent blades. Modal characteristics of the blades are quite sensitive to the level of mistuning present inside the structure. Therefore, various mistuning patterns are considered and the results of the split of vibration modes of the perfectly tuned bladed disc are obtained. Damage indices based on differences in the MAC that give a measure of the change in the mode shapes are in-troduced. The damage index is obtained from the Frobenius norm of MAC matrix subtracted from the AutoMAC of perfectly tuned model. A clear correlation between the experimental and numerical investigation of MAC based crack detection technique is shown for tuned and mis-tuned bladed disc.

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2 FINITE ELEMENT MODEL OF BLADED DISC

A blade row in a turbomachine is a cyclic symmetric structure which is quite complex in its geometry and operation. Structures made up of nearly identical parts can be used as a base model for actual complex repeating structures like rotors or stators in gas turbines. Such base models represent the behavior of actual models, ignoring intricate details. A two dimensional circular disc type structure was considered as a representative base model for actual turbomachinery bladed system. It comprised eight blades mounted on a thick circular base. The interface between blade and the circular base was modeled as a rigid single piece to represent the geometry of a blisk. The geometry of the circular disc model along with its finite element mesh is shown in Figure 1.

Figure 1 : Finite element model of bladed disc.

To keep the finite element model simple, the bladed disc was meshed using 2D plane stress

(eight node quadrilateral) elements with mid-side nodes. Out of plane modeshapes were not considered and the material assigned to this FE model was steel (E = 200GPa, υ = 0.3). The bladed disc model was constrained radially at the centre, indicating a rigid connection between rotor and the connecting shaft.

3 CRACK DEPTH AND LOCATION

In order to investigate the effect of crack initiation and propagation on the modal characteristics of the disc model, cracks of various depths were considered at different crack locations along the length of the blade. Cracks were initiated and propagated in Blade-1 of the bladed disc. The crack depth was defined as the ratio of crack length (a) to width of the blade (d). This crack depth ratio (a/d) was then varied from 0.01 (very small crack) to 0.9 (large crack). The location of the crack on the blade was varied as the ratio of distance of crack edge from the base (b) to the total length of the blade (l). Three different crack locations were considered: near the blade root with b/l = 0.14, in the middle of blade with b/l = 0.5 and near the blade tip at b/l = 0.8. Crack depths and locations are shown in Figure 2.

Figure 2 : Crack depth and location parameters.

Constraint Location Blades

1

2

3

4

5

6

7

8

3

4 BLADE MISTUNING PATTERENS

Cyclic symmetric structures, such as blisks in turbomchinery, are analyzed under the assump-tion that the whole structure is composed of perfectly identical blades attached to the rotor. In fact, these repeating structures have small variations in their geometric and material properties. These small differences in different blade sectors are referred to as mistuning in the system. Tuned and mistuned turbomachinery bladed systems behave in a significantly different fashion. Presence of mistuning in blisk sectors results in changed dynamics, especially modal characte-ristics of the whole system. In order to investigate the effect of different mistuning patterns, in the presence of a crack, on the modal characteristics of the whole bladed disc, the following mistuning patterns were considered:

4.1 No mistuning present in the system

In this scenario, the bladed disc was assumed to be a perfectly tuned system with all the blisk sectors perfectly identical to each other. This case provided a reference for comparing the be-havior of the mistuned system with the tuned one in the presence of crack in one of the blades of bladed disc.

4.2 Arbitrarily mistuned system (low mistuning level)---Model-1

In this case, bladed disc was arbitrarily mistuned by attaching an equal quantity of small lumped mass (0.1g) on blade 1, 4 and 6. The purpose was to distinguish between the effects of an alternately mistuned system and perfectly tuned model.

4.3 Arbitrarily mistuned system (high mistuning level)---Model-2

The bladed disc model was once again arbitrarily mistuned but this time, the level of mistuning was high in comparison to the previous case. Only blades 1, 3, 4, 6, 7 and 8 were mistuned by attaching 0.2g, 0.1g, 0.2g, 0.1g, 0.1g and 0.5g respectively. The objective of this particular case was to elaborate the comparison between arbitrarily low level mistuned bladed disc with arbi-trarily high level mistuned model along with the alternately mistuned system and perfectly tuned bladed disc model.

4.4 Alternately mistuned system---Model-2

In alternately mistuned system, each alternate blade sector was equally mistuned. There were

eight blades in bladed disc and blades 1, 3, 5 & 7 were mistuned by attaching 0.1g of mass.

All these three mistuning patterns considered for the bladed disc are elaborated in Figure 2.

a- Arbitrarily (low level)

mistuned system

b- Arbitrarily (high level)

mistuned system

c- Alternately mistuned

system

-- 0.1g mass -- 0.2g mass -- 0.5g mass

Figure 2 : Bladed disc mistuning patterns.

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5 MAC AND AUTOMAC PLOTS OF BLADED DISC

As far as model selection for MAC plots considering bladed disc is concerned, AutoMAC plots of complete model, blades only and model skeleton were obtained. The similarity of the Auto-MAC plots for the complete set and the skeleton suggested that the modal vector based on the skeleton is a simpler representation. The model skeleton of the complete model is shown in Figure 3.

i- Complete Model ii- Model Skeleton

Figure 3 : Model skeleton out of complete model.

For detailed explanation of workability of MAC and AutoMAC plots, two cases of bladed

disc model were considered. Case-1 is the tuned system without cracks whereas in Case-2, the

tuned bladed disc had a crack of depth ratio 0.1 at the first crack location. The first nine mode

shapes were obtained for both the cases and are shown in Figure 4. It can be seen that among

various other correlations, Mode-2 of Case-1 and Mode-3 of Case-2 are nearly identical and the

matching of these two modes is indicated by a light grey box in MAC plot (Figure 5c). Auto-

MAC plots of Case-1 and Case-2 are also shown in Figure 5a and 5b respectively. It is obvious

that on both the AutoMAC plots, all the modes can be easily distinguished with each other and

there is no indication of spatial aliasing present. This means that the correlation between Mode-

2 of Case-1 and Mode-3 of Case-2 is a genuine correlation. Similarly, Mode-7 of Case-1 and

Mode-8 of Case-2 are approximately similar, indicated by a dark gray box in MAC plot.

Figure 4 : Mode shapes for cases 1 and 2.

5

a- AutoMAC plot Case-1 b- AutoMAC plot Case-2 c- MAC plot between Case-1 & 2

Figure 5 : MAC and AutoMAC plots for Case-1 and Case-2.

While obtaining MAC plots of bladed disc for the comparison of different scenarios, the ef-

fect of rotational symmetry caused some confusion while comparing mode shapes. This issue

was resolved according to the guidelines described by Rehman et al. (2010).

6 MAC BASED CRACK DETECTION

For each of the crack depth ratios (a/d = 0.01-0.9) at different crack locations (b/l = 0.14, 0.5 and 0.8), displacement vectors were obtained for twenty modes. MAC matrices were then gen-erated comparing each case with a defect with the reference (model without crack). The mode shape matrix obtained for each case was then compared with itself, to produce the AutoMAC, and also with other cases, to produce MAC matrices. All mode shape matrices were obtained using data acquired from nodes at the same geometric locations. These MAC matrices were subtracted from the AutoMAC matrix of the reference case. The resulting matrix represents the MAC change and gives a measure of disorder due to the presence of the crack. A single scalar value for this matrix was obtained by calculating its Frobenius norm which therefore provides a measure of the extent of that disorder. The Frobenius norm was calculated for each crack depth and location and plotted against crack depth ratios giving three curves, each representing differ-ent crack locations. In this way the extent of damage (damage index) due to crack initiation and propagation inside the blade was estimated.

Damage index verses crack depth ratio plot for perfectly tuned circular disc is shown in Fig-

ure 6a. Magnitude of damage index for first crack location represented by b/l = 0.14 came out

to be higher than second (b/l = 0.5) and third (b/l = 0.8) crack locations in circular disc. First

crack location, b/l = 0.14 gave steep damage index curve up to crack depth ratio of 0.25 and

then followed a continuous horizontal pattern in the magnitude over a/d > 0.25. Whereas the

second crack location near the middle of the blade, b/l = 0.5, and third crack location near the

blade tip, b/l = 0.8, showed an approximately smooth increase rate in the magnitude of damage

index considering crack initiation and then propagation. It was found that a very minute dam-

age in one of the blades of circular disc model resulted in sudden increase in the magnitude of

Frobenius norm and this was indicated by a steep rise in the damage index curve. The extend of

this sharp increment in damage index is more for a crack of certain depth near the blade root as

compared to the crack in the middle of the blade and near the blade tip. Due to the limitation of

finite element modeling of crack depth ratios less than 0.01, damage index values for early

crack initiation stage (from no crack to a/d = 0.01), were not computed.

a- Tuned Model b- Mistuned Model

Figure 6 : Damage index curves for bladed disc.

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Damage index curves, considering different mistuning patterns and varying crack depths, are shown in Figure 6b. It can be seen that the presence of arbitrarily low and high level of mistun-ing in circular disc resulted in almost same values for damage index for the entire range of crack depth ratios considered. Whereas the alternately mistuned system indicated lower magni-tude of damage index. Mistuning causes each mode to be highly localized, corresponding pri-marily to the oscillation of a single blade. Besides this, vibrating energy is no more distributed along all the blades but confined to limited set of blades. As damage index plots are strictly de-pendent on the mode shapes obtained, higher magnitudes of damage index can be attributed to-wards presence of localized modes due to mistuning.

7 BLADED DISC SPECIMEN AND EXPERIMENTAL RESULTS

The bladed disc specimen is shown in Figure 7. The dimensions of the specimen were nominal-

ly the same as the FE model, except for inner diameter of disc which was reduced from 40mm

to 8mm. This was done to reduce the interblade coupling ratio. Besides this, the thickness of the

bladed disc specimen was selected to be 40mm. Purpose of making the specimen a bit thick was

to make it consistent with the FE model of bladed disc by avoiding out of plane modes. The ma-

terial selected for the bladed disc specimen was medium Carbon steel.

Figure 7 : Experimental setup for bladed disc.

Experimental natural frequencies and mode shapes of the bladed disc specimen were ob-

tained by carrying out the modal analysis using LMS test lab equipment. Impulse excitation us-

ing an instrumented hammer was used to excite each blade (1-8) at six locations (top right, top

center, top left, middle right, middle center and middle left) and response was obtained on blade

3 at top right location (Figure 8a).

a- Response locations on blade b- Bladed disc model in LMS

Figure 8 : Response locations and bladed disc model.

Impulse

Hammer

Bladed

Disc Laser Acceler-

ometer

Loc-1(Top Right)

Loc-4(Middle Right)

Loc-5(Middle Centre)

Loc-2(Top Centre)

Loc-6(Middle Left)

Loc-3(Top Left)

7

The geometry of the bladed disc specimen was modeled in LMS to visualize the mode shapes

(Figure 8b). The excitation locations on the specimen were then correlated to the specimen

model in LMS. The frequency band width selected for this test was 0-4096 Hz and 8192 spec-

tral lines were specified in this range to obtain frequency resolution of 0.5 Hz. For each test, a

total of 48 data sets (six per blade) were obtained. The first nine natural frequencies and mode

shapes were obtained experimentally. Experimental modes shapes are compared with numerical

mode shapes in Figure 9. The bladed disc specimen was slightly mistuned due to the presence

of small variability in its sector components, which may be caused by manufacturing tolerance

or material degradation, resulted in different modes shapes as compared to the perfectly tuned

numerical bladed disc.

Experimental

Numerical

Mode-1

Mode-2 Mode-3

Experimental

Numerical

Mode-4

Mode-5 Mode-6

Experimental

Numerical

Mode-7 Mode-8 Mode-9

Figure 9 : Experimental and numerical mode shapes of bladed disc.

In order to obtain the damage index for experimental bladed disc, another 3D FE model of

bladed disc was created and only six nodes per blades along the thickness (corresponding to

experimental response locations on each blade) were considered to obtain displacement vectors

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in ‘x’ and ‘y’ axes. Resultant modal vector was then obtained. Only first nine modes were con-

sidered to form a modal displacement matrix in a similar fashion as described in previous sec-

tion of this paper. Experimental modal displacement matrix comprising of nine modal vectors

was also assembled. MAC matrix was then obtained by comparing the mistuned experimental

modal matrix with numerical modal displacement matrix of perfectly tuned model. This MAC

matrix was then subtracted from AutoMAC of tuned numerical model of bladed disc and dam-

age index for this subtracted matrix was obtained by calculating its Frobenius norm. This came

out to be 1.6291. Quantification of the level of mistuning in the bladed disc specimen can pro-

vide a threshold for a certain crack depth that can be highlighted in a mistuned environment.

This can be achieved by plotting the mistuning level on the damage index curve obtained for

bladed disc specimen having cracks of various depths. Figure 10 shows the details of Figure 6

with mistuning threshold plotted as a horizontal line. It can be seen that the experimental bladed

disc was very slightly mistuned, indicated by the smaller magnitude of frobenius norm, as com-

pared to the mistuning levels considered numerically (Figure 10b). In addition to that, with the

current mistuning level in experimental bladed disc, a crack of even smaller depth can be hig-

hlighted (Figure 10a).

a- Tuned Model b- Mistuned Model

Figure 10 : Damage index curves with mistuning threshold.

8 CONCLUSIONS In this paper, experimental investigation of damage index curves utilizing MAC is carried out along with the comparison with numerical results. Modal characteristics were altered due to the presence of crack and mistuning inside bladed disc model. Presence of mistuning caused mode localisation and non-uniform distribution of vibration energy, which resulted in higher magni-tudes of damage index. Besides this, highlighting a crack in a mistuned environment depends on the pattern and level of mistuning. A threshold to identify a crack in mistuned environment can be obtained by mistuning quantification and plotting it on damage index curves. REFERENCES Ewins, D. J. and Fotsch, D., Further applications of the FMAC, Proceedings of the International Modal

Analysis Conference - IMAC 1, pp. 635-639.

Ewins, D. J. and Fotsch, D., Application of Mac in Frequency Domain, Proceedings of SPIE - The Inter-

national Society for Optical Engineering, 4062, pp.

Hou, J., 2002, Cracking-Induced Mistuning in Bladed Disks, AIAA Journal, 44 (11), pp. 2542-2546.

Pitkanen, J. and Hakkarainen, T., 2001, NDT methods for revealing anomalies and defects in gas turbine

blades, Insight: Non-Destructive Testing and Condition Monitoring, 43 (9), pp 601-604.

Rehman, A.U., Rongong, J. A. and Worden, K., 2010, Detection of Damage in Repeating Structures, Pro-

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bration, Journal of Vibration and Acoustics, Vol. 131.

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