Determination of Damage Severity on Rotor Shaft Due to CrackUsing Damage Index Derived from Experimental Modal DataZ.C. Ong1, A.G.A. Rahman2, Z. Ismail3
1 Mechanical Engineering Department, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia
2 Faculty of Mechanical Engineering, University Malaysia Pahang, Pahang, Malaysia
3 Civil Engineering Department, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia
After long duration of an operation, a shaft under high spin speeds and heavyloadings may develop fatigue cracks. This could lead to catastrophic failure andcould be difficult to detect. An accurate prediction of dynamic characteristicsof rotor system is fundamentally important. Hence, a practical method tonondestructively locate and estimate the severity of a crack in terms of itsdamage index by measuring the changes in natural frequencies of the rotorshaft is presented. In this study, experimental modal analysis (EMA) datawere utilized. A crack detection algorithm to locate and identify cracks inthe rotor system using the first and second natural frequencies was outlined.Subsequently, a crack-locating model was formulated by relating the fractionalchanges in modal energy to the changes in natural frequencies as a result ofcracks based on the experimentally obtained natural frequencies and modeshapes’ functions. The feasibility and practicality of the crack detection schemewere evaluated for several damage scenarios by locating and sizing cracks intest rotor shafts for which the first two natural frequencies were available. Itwas observed that crack could be confidently located with a relatively smalllocalization error. It was also observed that crack severity could be estimatedin terms of its damage index.
Introduction
One form of damage that could lead to a catastrophicfailure, if undetected, might be fatigue cracking ofthe shaft. Hence, early detection of cracks in rotorin engineering practices was of utmost importanceto the reliability and durability of large rotatingmachineries, in which a diagnosis of fault was made,and monitoring the condition of rotor shaft with crackhas to be given more attention. For the time being, theresearch on the cracked rotor is still at the theoreticalstage, and most of the previous research was onlyinvolved in detecting the crack in a rotor and notcapable of determining the location and depth of thecrack.1
A static (nonrotating) rotor with an open crackcould be considered as a simply supported beam,
and the research in relation to nonrotating structuressuch as beams and columns was useful for locatingand estimating the severity of the crack on a rotor aswell.2
A crack in a structure introduces a local flexibility,which affects the dynamic behavior of the wholestructure to a considerable degree and results in thereduction of natural frequencies and changes in thevibration of mode shapes. The effect of a crack on thedeformation of a beam has been considered as similarto that of an elastic hinge.3 The crack was modeledas a local flexibility, and the equivalent stiffnesswas computed by Dimarogonas and Papadopoulos4
using fracture mechanics methods. A finite-elementmodel was proposed using transfer matrix methodand local flexibility theorem; in order to consider thediscontinuity in deformation as a result of the crack on
Crack Detection on Rotor Shaft Z.C. Ong, A.G.A. Rahman, Z. Ismail
the beam, they adopted two different shape functionsfor two segments separated by the crack.5
The dynamic response of the structures with crackand the changes in the natural frequencies andmode shapes between the cracked and the uncrackedstructures show information about the location anddimension of the damage. Using the receptancetechnique and Taylor series expansion, it has beenshown that the ratio of the frequency changes intwo modes was only a function of the damagelocation, respectively. The crack location and depthwere estimated with a satisfactory accuracy from theresult of an analytical solution of a beam with crackand from the experimental data of the measuredamplitudes at two points of the structure vibrating atone of its natural modes and the respective vibrationfrequency.6 For a rotor removed from service, Inagakiet al.7 used natural vibration and static deflectionanalysis to find the crack size and location. Lianget al.8 proposed a two-step procedure to identifycracks in beam structures: they used the effectivestress concept coupled with Hamilton’s principle toderive a formulation relating the changes in thenatural frequencies to the changes in the stiffnessof structural elements. The elements that had crackscould be identified based on the formulation, andthen a spring damage model was used to quantifythe location and depth of the crack in each damagedelement. On the basis of the research, the dependencyof the structural eigenfrequencies on crack depthand location in contour graph form was presentedto identify the location and depth of a crack;the intersection points of the superposed contoursthat corresponded to the measured eigenfrequencyvariations caused by the presence of the crackshould be determined.5,9 A detection method wasalso presented using hybrid neuro-genetic techniqueto identify the location and depth of a crack on astructure.9
Natural frequency was one of the most commonmodal features that were used in crack detection instructures. The appealing feature was that the natu-ral frequency was relatively simple to measure andapply to structures for further use. Also, monitoringthe natural frequencies was time and cost efficientfor most structures. Research studies to nondestruc-tively detect crack location and magnitude via changesin the natural frequencies have been performed bymany researchers. Attempts have been made to relatechanges in natural frequencies to changes in beamproperties such as cracks, notches, or other geomet-rical changes10–14 and to identify crack location andmagnitude in a beam from vibration modes.14–17
Mode shape data was another parameter used inthe crack detection study. Ratcliffe18 presented atechnique for locating damage in a beam that usesa finite difference approximation of a Laplacian oper-ator on mode shape data. In the case of damage, whichis not so severe, further processing of the Laplacianoutput is necessary before damage location could bedetermined. The procedure is found to be best suitedfor the mode shape obtained from the fundamentalnatural frequency. The mode shapes obtained fromhigher natural frequencies may be used to verifythe damage location, but they are not as sensitiveas the lower modes.1,18 Yoon et al.19 expands the‘‘gapped smoothing method’’ for identifying the loca-tion of structural damage in a beam by introducinga ‘‘globally optimized smooth shape’’ with an ana-lytic mode shape function, and the procedure usesonly the mode shapes from the damaged structure.The method can detect local stiffness losses associatedwith local thickness reduction.
Instead of using mode shapes in obtaining spatialinformation about sources of vibration changes, analternative method uses the mode shape derivatives,such as curvature. It is noted that for beams, plates,and shells there is a direct relationship between curva-ture and bending strain. Pandey et al.20 demonstratedthat absolute changes in mode shape curvature canbe a good indicator of damage for the cantilever andsimply supported analytical beam structures. Stubbset al.21 presented a method based on the decrease inthe curvature of the measured mode shapes or themodal strain energy between two structural degreesof freedom. Ismail et al.22 described the determina-tion of the location of damage due to single cracks anddue to honeycombs in reinforced concrete (RC) beamsusing mode shape derivatives from modal testing.
Stubbs and Osegueda,23 Gomes and Silva,24
Narayana and Jebaraj,25 and Kim and Stubbs26 hadworked on the topic with sensitivity approach. In thesensitivity approach, damage in a structural elementwas estimated from a direct inverse solution of achange in the element stiffness if modal sensitivitywas computed from an analytical model of thestructure, and changes in natural frequencies weremeasured from the structure. Even though thesensitivity approach could produce an estimation oflocation and severity of damage at the same time,reliable output might not be expected unless both theanalytical model that should be accurate enough tocompute the modal sensitivity and a large number offrequencies that should be accurately measured wereprovided. This was because even a significant damagemight cause only very small changes in the natural
Z.C. Ong, A.G.A. Rahman, Z. Ismail Crack Detection on Rotor Shaft
frequencies, and these changes might go undetecteddue to measurement or processing errors.26
In later stages, genetic algorithms and finite ele-ments were other alternatives to identify damagein structures.27,28 Sazonov et al.29 used the geneticalgorithm to produce a sufficiently optimized ampli-tude characteristic filter to extract damage informa-tion from the strain energy mode shapes. Hassiotis30
presented an optimization algorithm which was for-mulated for the identification of damage in structures,when such damage was manifested by localizedreductions in the stiffness of structural elements. Ben-fratello et al.31 presented both numerical and exper-imental investigations to assess the capability of thenon-Gaussianity measures to detect the presence ofcrack and its position. Monte Carlo method is appliedto evaluate the higher-order statistics of a cantileverbeam modeled by finite elements in time domain.Xiang et al.32 modeled a rotor system using finiteelement method of B-spline wavelet on the interval(FEM BSWI), while the crack was detected throughlocal stiffness change. On the basis of Rayleigh beamtheory, the influences of rotatory inertia on the flex-ural vibrations of the rotor system were examinedto construct BSWI Rayleigh beam element. The newmethod could be used for the quantitative diagnosisand prognosis of crack in a rotor system.
In the study of Kim and Stubbs,26 a practicalmethodology to nondestructively localize cracks andnumerically estimate the sizes of the cracks in beam-type structures was presented. Natural frequenciesof beam were computed from theoretical modalanalysis. The crack-locating model relates frequencyratios of a few measured modes to modal sensitivityratio of the corresponding modes to identify potentialcrack locations. For each predicted location, the cracksize estimation model relates fractional changes intheoretical natural frequencies to a geometric cracksize on the basis of the theory of the linear fracturemechanics.
In this article, the methodology presented by Kimand Stubbs26 was applied to a rotor dynamic system.Natural frequencies, damping, and mode shapes of arotor shaft were determined experimentally. Theseexperimental modal data of the rotor shaft wereutilized in developing the crack detection algorithm.The algorithm aims to nondestructively localizecracks and estimate the severity of the cracks inrotor dynamic system using changes in frequenciesand mode shapes. The practicality and accuracy ofthe crack detection scheme were demonstrated bylocating and sizing cracks in test rotor shafts indifferent damage scenarios. The effects of the location
and severity of the crack were investigated anddiscussed.
A continuous model for crack identification of arotor was based on the following assumptions:
i. To avoid nonlinearity, it was assumed thatthe crack was always open. The ‘‘breathing’’phenomenon of the crack was not taken intoconsideration. Although this introduced a timevariance and nonlinearity in the equations ofmotion and spoiled the identification results, theerrors in the lowest identified eigenvectors werenot large.
ii. Only a single crack was considered along theshaft.
iii. The rotor with a single crack was assumed as anEuler–Bernoulli beam with circular cross section.
iv. The region where the crack is located wasmodeled as a local flexibility with fracturemechanics method.
Theoretical Background
A crack-detecting model was formulated by linearlyrelating the structural system’s sensitivity on modalcharacteristics to the eigenfrequency changes dueto geometrical changes. For a multiple degrees offreedom system, the damages inflicted at predefinedlocations might be predicted using the sensitivityequation as discussed by Kim and Stubbs.26
Once the eigenvalues were experimentally deter-mined, quantity Zi, which was the fractional changein the ith damped natural frequency before and afterdamage. The modal sensitivity of the ith modal stiff-ness with respect to the jth element was termed asFij. It was worthwhile to notice that the eigenvaluesand eigenvectors of both the healthy and rotor shaftswith crack needed to be determined either exper-imentally or numerically in order to complete thedetection scheme. This had limited the crack detec-tion approach where one of these parameters wasundefined.
The sensitivity equation can be described as below:
NE∑j=1
Fijαj = Zi (1)
where αj(−1 ≤ αj ≤ 0) is the damage inflicted atthe jth location (i.e., the fractional reduction in jthstiffness parameter). The term Zi is the fractionalchange in the ith eigenvalue and (by neglectingchanges in mass due to damage) is given by
Crack Detection on Rotor Shaft Z.C. Ong, A.G.A. Rahman, Z. Ismail
where δw2i (= w∗2
i − w2i ) is the change in the ith
damped natural frequency before and after damage.The term Fij is the modal sensitivity of the ith modalstiffness with respect to the jth element.
Fij = Kij/Ki (3)
where Ki is the ith modal stiffness (Ki = �Ti C�i) and
Kij is the contribution of the jth element to the ithmodal stiffness (Kij = �T
i Cj�i). In addition, �i is theith modal vector, C is the system stiffness matrix, andCj is the contribution of jth element to the systemstiffness.
The modal sensitivity of mode i and element jbetween two locations (xj, xj+1) is computed using
Fij =∫ xj+1
xj
EI{φ ′′i (x)}2 dx
Ki(4)
where Ki is computed using
Ki =∫ l
0EI{φ ′′
i (x)}2dx (5)
Once the quantity Zi is experimentally determined,Eq. 1 can be solved to locate and size damage in thesystem. However, the inverse solution is possible onlyif the number of damage parameters is close to thenumber of modes (i.e., NE ≈ NM). In the case whenNE � NM, the system becomes ill-conditioned andalternate methods to estimate damage parametersshould be sought. In an effort to overcome thisdifficulty, a sensitivity ratio concept is proposed.
Let us consider the structural system of NE elements(j = 1, 2, . . . , q, . . . , . . . , NE) and a measured set ofNM vibration modes (i = 1, . . . , m, n, . . . , NM). Eq. 1is rewritten for any two modes m and n (m �= n),respectively. On dividing Eq. 1 for mode m by theother for mode n, we obtain
Zm
Zn=
∑NEj=1 Fmjαj∑NEj=1 Fnjαj
= Fm1α1 + Fm2α2 + · · · + Fmqαq + · · · + FmNEαNE
Fn1α1 + Fn2α2 + · · · + Fnqαq + · · · + FnNEαNE
(6)
Assuming that the structure is damaged in a singleelement, such that αj �= 0 when j = q but αj = 0 whenj �= q, Eq. 6 is rewritten as
Zm
Zn= Fmq
Fnq(7)
where Zm/Zn was the measured ratio of the fractionalchanges in frequency for two modes, m and n. Fmq/Fnq
was also the ratio of the theoretically measured sen-sitivities for those modes and the element q. So the
damage inflicted at that location was defined usingEq. 1 when the left-hand side equals to the right-handside.
For all measured NM modes, Eq. 1 could be sub-stituted into
Zm∑NMk=1 Zk
= Fmq∑NMk=1 Fkq
(8)
Since Eq. 2 was true only if element q was damaged.We introduced an error index into Eq. 2 as follows:
eij = Zm∑NMk=1 Zk
− Fmq∑NMk=1 Fkq
(9)
where eij represents localization error for the ith modeand the jth location, and eij = 0 indicate that the dam-age was located at jth location using the ith modalinformation. To account for all available modes, weformed a single damage indicator (DI) for the jthmember as:
DIj =[
NM∑i=1
e2ij
]− 12
(10)
where 0 ≤ DIj < ∞ and the damage was located atelement j if DIj approaches the local maximum point.In principle, the proposed method worked if at leasttwo modes were given. But it was important to noticethat the accuracy of damage identification was up toeither the number of modes or the type of modes thatwere used in the process. Thus, damage index (DI)could serve as a good indicator to check the severityof the cracks.
Experimental Setup
In this research, Bentley Nevada RK4 Rotor Kit asshown in Fig. 1 was used. The rotor kit was 10 mmin diameter and 600 mm in length. It could closelysimulate the actual rotating machine behavior. Itsunique geometry and its ability for users to isolateand control individual machine characteristics madeit useful as both a teaching tool and as a laboratorytool for theoretical research.
Rotor shafts were prepared for open crack study andidentification purposes. Crack was simulated usingcomputer numerical control (CNC) machine with thewidth of 0.2 mm as shown in Fig. 2. The shaft wasfirst divided into 10 equal elements. The crack wasmade in different sizes and in different locations ofthe elements. Total of eight cases, which include onehealthy shaft and seven shafts with single crack, weresimulated and investigated in the crack identificationstudy. Case 0 is for healthy shaft. For cases 1, 2, 3, and5, a single crack with the depth of 2 mm was createdon elements 4, 5, 7, and 6, respectively, to simulate
Z.C. Ong, A.G.A. Rahman, Z. Ismail Crack Detection on Rotor Shaft
23
1
456
8910
7
Rotor Shaft
Balance Wheel/Disc
BearingsMotorCoupling
Figure 1 Bentley Nevada Rotor Kit (RK4).
the crack at different locations. In addition, to studythe effect of crack size using the crack identificationalgorithm, a single crack was induced at element 6with the size of 1, 2, 3, and 4 mm in cases 4, 5, 6, and7, respectively.
The research work was based on the experimentalinvestigations of the effect of crack and subsequentlydamage on the integrity of rotor, by detecting, quan-tifying, and determining its extents and locations.The dynamic characteristics of the rotor kit withhealthy and shafts with crack were determined. Thiswas achieved by obtaining the eigenfrequencies andeigenvectors of shafts from modal analysis experi-ments carried out on the shafts for all the eight cases.By using modal analysis software, the curve fittingprocess was performed on the transfer function spec-tra obtained to extract the modal parameters, that is,natural frequency, mode shape, and damping. List ofinstrumentations and test parameters are shown inTable A1.
To compute the modal sensitivity at a certainlocation of the shaft, mode shape functions for allthe corresponding modes were determined by theeigenvectors or so-called mode shapes. A computationprogram in MATLAB was used to compute the modalsensitivity at all ten locations along the shaft.
These computed modal sensitivities and experi-mental eigenfrequencies were then used to determinethe localization error and DI as in Eqs. 3 and 4. Theseindexes served as an indicator to identify the locationand the severity of the damage shaft. A damage index(DI) versus element graph as shown in Fig. 3 wasgenerated to display the computation results.
Results and Discussion
The experimental modal analysis (EMA) results interms of natural frequencies and mode shapes couldbe used to identify the location of crack and itsseverity of different damage scenarios with a crackidentification algorithm. Table 1 shows the sevendamage scenarios with the crack’s size and its location.Cases 1, 2, 3, and 5 are used to show effective locationdetection, while Cases 4, 5, 6, and 7 are used toshow effective severity detection. Case 0 is used asthe baseline.
The identification algorithm was based on modalsensitivity or local stiffness approach, and the locationof crack and its severity could be identified confidentlyusing this approach. Table 2 shows the first twonatural frequencies and mode shape functions whichwere obtained by curve fitting the experimentalmode shape data in the healthy shaft and shaft withdifferent damage scenarios. Natural frequencies wereobtained directly from experimental modal analysis,while mode shape curvatures were generated bydifferentiating the mode shape functions.
In this study, the eigenvalues and eigenvectors ofboth the healthy and rotor shafts with crack neededto be predefined either experimentally or numericallyin order to complete the detection scheme. Modalsensitivity was computed using the mode shapecurvatures where the flexural rigidity EI is assumedconstant over the shaft. The potential crack locationswere then predicted based on the fractional changesin frequencies and the sensitivity ratios in localizationerrors format. The DI of all the damage scenarios wascomputed from these error indexes.
Crack Detection on Rotor Shaft Z.C. Ong, A.G.A. Rahman, Z. Ismail
Shaft with crack size of 2mm x 0.2mm at EL4
Shaft with crack size of 2mm x 0.2mm at EL5
Shaft with crack size of 2mm x 0.2mm at EL7
Shaft with crack size of 1mm x 0.2mm at EL6
Shaft with crack size of 2mm x 0.2mm at EL6
Shaft with crack size of 3mm x 0.2mm at EL6
Shaft with crack size of 4mm x 0.2mm at EL6
Shafts with different crack locations
Figure 2 Shafts with crack simulated by CNC machines with different sizes and locations with the width of 0.2 mm.
Figures 4–10 display the crack identification resultsin DI for different crack locations and sizes. Asshown in these graphs, crack with 2 mm of depth atdifferent locations (Cases 1, 2, 3, and 5) were clearlyidentified with the DI at around the value of 20.There were some small fluctuations in DI at eachelement along the shaft. However, the fluctuations
were relatively low as compared with the DI atcrack location. These fluctuations might be because ofthe technical errors in experimental modal analysis,which will eventually affect the accuracy of thenatural frequencies and mode shape functions. Thesudden large increase of the DI shows that thereis a damage or problem on that particular element.
Z.C. Ong, A.G.A. Rahman, Z. Ismail Crack Detection on Rotor Shaft
Figure 3 Damage index graph showing
the crack location and severity of the crack
along the shaft.
Table 1 Summary of elements appointed and cracks along the shafts
Case no. Element Crack size
1 4 2 mm × 0.2 mm2 5 2 mm × 0.2 mm3 7 2 mm × 0.2 mm4 6 1 mm × 0.2 mm5 6 2 mm × 0.2 mm6 6 3 mm × 0.2 mm7 6 4 mm × 0.2 mm
Therefore, a threshold limit is set at a DI of 10.Elements with higher DI of 10 indicate problem andshould be monitored closely. Furthermore, statisticalmethod such as outlier analysis could be performed toeliminate the false alarms and measurement errors.Once an observation is identified as a potentialoutlier, root cause analysis should begin to determinewhether an assignable cause can be found for thisspurious result. If no root cause could be determined,and a retest could be justified, the potential outliershould be recorded for future evaluation as more databecome available.
The DI varies at different locations. This changedepends on how close the crack location is to themode shape nodes. It was noticed that the first modeis very significant in detecting cracks as errors inlowest eigenvectors are not large and the nonlinearityeffect at the lower modes are not taking a large effect.
Table 2 Natural frequencies and mode shape functions of different
damage scenarios
Case
no.
Mode
no.
Natural
frequency Mode shape function
0 1 37.3 (2.51x2 − 2.72x + 4.79) × 104
2 222.0 (−6.12x3 + 9.25x2 − 3.39x + 0.12) × 106
1 1 36.9 (2.73x2 − 2.93x + 4.83) × 104
2 219.0 (−7.15x3 + 11.22x2 − 4.44x + 0.26) × 106
2 1 36.7 (0.77x2 − 0.58x − 0.022) × 104
2 220.0 (−3.31x3 + 5.84x2 − 2.76x + 0.29) × 106
3 1 36.8 (0.94x2 − 1.06x + 0.18) × 104
2 220.0 (−3.59x3 + 6.43x2 − 3.16x + 0.37) × 106
4 1 36.8 (3.01x2 − 3.16x + 0.51) × 104
2 221.0 (−8.79x3 + 13.73x2 − 5.56x + 0.40) × 106
5 1 36.5 (5.53x2 − 5.73x + 1.02) × 104
2 221.0 (−8.96x3 + 13.95x2 − 5.64x + 0.40) × 106
6 1 36.4 (2.32x2 − 2.51x + 3.98) × 104
2 221.0 (−11.23x3 + 18.16x2 − 8.07x + 0.25) × 106
7 1 36.2 (4.61x2 − 4.78x + 8.42) × 104
2 220.0 (−8.13x3 + 12.62x2 − 5.06x + 0.34) × 106
In higher modes, the inertia nonlinearity, which isof the softening type, is dominant and eventuallytamper the identification results. Thus, from the firstcorresponding mode shape as shown in Fig. 11, theDI is larger if the crack is near to the antinodal pointof the mode shape. For the first mode, the antinodeis at element 6, thus the DI of the shaft with crackat this point is the most significant. This is followedby cracks at elements 7 and 5. Finally, the DI for
Crack Detection on Rotor Shaft Z.C. Ong, A.G.A. Rahman, Z. Ismail
Figure 4 Case 1–2 mm crack at element 4
with DI of 18.23.
Figure 5 Case 2–2 mm crack at element 5
with DI of 27.73.
crack at element 4 is quite small due to this particularpoint being further away from the antinode of thefirst mode shape.
Meanwhile, crack sizes of 1, 2, 3, and 4 mmat element 6 were tested (Cases 4, 5, 6, and 7).When comparing shafts with different crack sizesto a normal shaft, the change in DI of the crackedrotor monotonically increased with the increment of
the crack depth if the crack location was known inadvance. This was because larger crack size inducedlarger stiffness reduction, and therefore it causedgreater change in natural frequency and eventuallythe DI was larger. Case 7 with 4-mm crack size wasshowing the largest DI, 38.39. Subsequent to thatwere Cases 6, 5, and 4 with DI of 36.47, 29.11, and12.74, respectively.
Z.C. Ong, A.G.A. Rahman, Z. Ismail Crack Detection on Rotor Shaft
Figure 6 Case 3–2 mm crack at element 7
with DI of 28.49.
Figure 7 Case 4–1 mm crack at
element 6 with DI of 12.74.
The summary of all crack identification results in DIfor different location and size is shown in Fig. 12. Inshort, this crack detection approach could confidentlyidentify the crack locations where the DI could alsobe served as a good indicator to check the severity ofthe cracks. The DI in all of these damage scenarios
could then be used as a reference or benchmark forfuture crack identification work.
This crack detection algorithm could also be appliedto sets of shaft with different lengths and sizes as longas the dynamic characteristics of the healthy anddamage shafts are predefined. Natural frequencies
Crack Detection on Rotor Shaft Z.C. Ong, A.G.A. Rahman, Z. Ismail
Figure 8 Case 5–2 mm crack at element 6
with DI of 29.11.
Figure 9 Case 6–3 mm crack at
element 6 with DI of 36.47.
and mode shapes obtained experimentally areglobal parameters that represent the entire shafts.Meanwhile, both natural frequencies and modeshapes are greatly affected by lengths and sizes ofrotor shafts, and this would eventually give the shaftstheir own unique frequencies and shape functions.
Conclusion
By applying the crack detection approach to thetest rotor shafts, it was observed that a crack couldbe confidently located. Crack severity could also beestimated in terms of DI. However, it was importantto note that the accuracy of damage identification
Crack Detection on Rotor Shaft Z.C. Ong, A.G.A. Rahman, Z. Ismail
depended on either the number or the type of modesthat was used in the process.
Acknowledgments
The authors wish to acknowledge the financialsupport and advice given by Postgraduate ResearchFund (PV086-2011A), Advanced Shock and VibrationResearch (ASVR) Group of University of Malaya, andother project collaborators.
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Appendix
Table A1 List of instrumentation and test parameters
Instruments Description
Bently Neveda RK4 rotor kit Used as test rig to perform modal
