Research ArticleStructural Damage Detection by Using Single NaturalFrequency and the Corresponding Mode Shape

Bo Zhao Zili Xu Xuanen Kan Jize Zhong and Tian Guo

State Key Laboratory for Strength and Vibration of Mechanical Structures Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Zili Xu zlxumailxjtueducn

Received 17 August 2015 Revised 29 October 2015 Accepted 16 November 2015

Academic Editor Juan P Amezquita-Sanchez

Copyright copy 2016 Bo Zhao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Damage can be identified using generalized flexibility matrix based methods by using the first natural frequency and thecorrespondingmode shape However the first mode is not always appropriate to be used in damage detectionThe contact interfaceof rod-fastened-rotor may be partially separated under bending moment which decreases the flexural stiffness of the rotor Thebending moment on the interface varies as rotating speed changes so that the first- and second-modal parameters obtained arecorresponding to different damage scenarios In this paper a structural damage detectionmethod requiring single nonfirst mode isproposed Firstly the system is updated via restricting the first few mode shapes The mass matrix stiffness matrix and modalparameters of the updated system are derived Then the generalized flexibility matrix of the updated system is obtained andits changes and sensitivity to damage are derived The changes and sensitivity are used to calculate the location and severity ofdamage Finally this method is tested through numerical means on a cantilever beam and a rod-fastened-rotor with differentdamage scenarios when only the second mode is available The results indicate that the proposed method can effectively identifysingle double and multiple damage using single nonfirst mode

1 Introduction

Damage in a structure produces variations in its geometricand physical properties which can result in changes inits natural frequencies and mode shapes In the last yearsseveral researchers have developed many damage detectionmethods based on dynamic parameters Fan andQiao [1] andJassim et al [2] presented comprehensive reviews on modalparameters-based damaged identificationmethodsThemostcommonly used methods of damage detection use changesof natural frequencies and mode shape directly Messinaet al [3] proposed a correlation coefficient termed theMultiple Damage Location Assurance Criterion (MDLAC)by introducing two methods for estimating the locationand size of defects in a structure Kim and Stubbs [4]proposed a single damage indicator (SDI) method to locateand quantify a single crack in slender structures by usingchanges in a few natural frequencies Xu et al [5] proposedan iterative algorithm to identify the locations and extent ofdamage in beams only using the changes in their first severalnatural frequencies However the natural frequency-based

methods are often ill-posed even without noise Shi et al [6]extended theMultiple Damage Location Assurance Criterion(MDLAC) by using incomplete mode shapes instead of natu-ral frequencies Pawar et al [7] proposed amethod of damagedetection using Fourier analysis of mode shapes and neuralnetworks which is limited to detecting damage of beamswithclamped-clamped boundary condition Another importantclass of damage detection methods is based on flexibilitymatrix change between damaged and undamaged structuresPandey and Biswas [8] first proposed the method based onchange in flexibility matrix to detect structural damage Yangand Liu [9] made use of the eigenparameter decompositionof structural flexibility matrix change and approached thelocation and severity of damage in a decoupled mannerBernal and Gunes [10] use the flexibility proportional matrixmethod to quantify damage without the use of a modelTomaszewska [11] investigated the effect of statistical errorson damage detection based on structural flexibility matrixand mode shape curvature Li et al [12] used the generalizedflexibility instead of original flexibility matrix to detectstructural damage which can significantly reduce the effect

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 8194549 8 pageshttpdxdoiorg10115520168194549

2 Shock and Vibration

of truncating higher-order modal parameters Masoumi etal [13] proposed a new objective function formed by usinggeneralized flexibility matrix Then imperialist competitivealgorithm was used in damage identification Yan and Ren[14] derived a closed form of the sensitivity of flexibilitybased on the algebraic eigensensitivitymethodMontazer andSeyedpoor [15] introduced a new flexibility based damageindex for damage detection of truss structures

Although the generalized flexibility matrix based damagedetection approach can precisely detect the location andseverity of damage by using only the first natural frequencyand the corresponding mode shape there are still many limi-tations in these methods One limitation lies in the damagedetection of rod-fastened-rotor of heavy duty gas turbineThe flexural stiffness of the interface decreases when somezones of the contact interface are separated with bendingmoment on the rotor [16] Flexural stiffness of interface in arod-fastened-rotor induced by bending moment is differentin first and second critical speed because bending momentdistribution varies as rotating speed changes Therefore onlythe second-modal parameters are available for the damagedetection of rod-fastened-rotor in the second critical speed

In this paper a structural damage detectionmethodbasedon changes in the flexibility matrix only using single naturalfrequency and the corresponding mode shape is presentedFirstly restricted by the first several mode shapes the systemis updated The flexibility matrix of updated system can beobtained by using non-first-modal parameters of originalsystemThen sensitivity of flexibility of the updated system todamage is derived Taking advantage of generalized flexibilitymatrix which can considerably reduce the error caused bytruncating higher-order modal parameters the location andseverity of the damage are calculated Finally two numericalexamples for a cantilever beam and a rod-fastened-rotor areused to illustrate the effectiveness of the proposed methodwhen only the second natural frequency and the correspond-ing mode shape are available

2 Structural Damage Detection Method

21 Structural System Updating Method The differentialequation governing the free vibration of a linear undampedstructural system can be expressed as

Mx + Kx = 0 (1)

where M is the global mass matrix K is the global stiffnessmatrix and x is the displacement vector When the degree offreedom for the system is 119899 the eigenvalue problem can bewritten in the form

KΦ119894

= 120582

119894MΦ119894 119894 = 1 2 119899 (2)

where 120582

119894and Φ

119894are the 119894th eigenvalue and eigenvector

respectively Restricting the system by the first 119903mode shapes

Φ119879

119894Mx = 0 119894 = 1 2 119903 (3)

Mode shape matrix mass matrix and displacement vectorcan be partitioned as

Ψ = [

Ψ119903119903Ψ119903119904

Ψ119904119903Ψ119904119904

]

M = [

M119903119903

M119903119904

M119904119903

M119904119904

]

x = [

x119903

x119904

]

(4)

where the 119894th column of Ψ is the 119894th eigenvector Φ119894

Substituting (4) into (3) yields

[Ψ119879

119903119903Ψ119879

119904119903] [

M119903119903

M119903119904

M119904119903

M119904119904

] [

x119903

x119904

] = 0 (5)

Expending (5) yields x119903

= minus(Ψ119903119903M119903119903

+ Ψ119903119904M119904119903

)

minus1(Ψ119903119903M119903119904

+

Ψ119903119904M119904119904

)x119904 then the relationship between x

119904and x is

x = Dx119904 (6)

where D = [

TRI ] in which TR = minus(Ψ

119903119903M119903119903

+

Ψ119903119904M119904119903

)

minus1(Ψ119903119903M119903119904

+Ψ119903119904M119904119904

)x119904 Substituting (6) into original

free vibration differential equation (1) yields

MDx119904

+ KDx119904

= 0 (7)

Left-multiplying (7) by D119879 yields the updated free vibra-tion differential equation

M119906x119904

+ K119906x119904

= 0 (8)

The mass and stiffness matrix of the updated system can beobtained by

M119906 = D119879MD

K119906 = D119879KD

(9)

The relationship between the updated and the original modalparameters can be described by

120582

119906

119894= 120582

119894+119903

Φ119906

119894= Dminus1Φ

119894+119903

119894 = 1 2 119899 minus 119903

(10)

where 120582

119906

119894and Φ119906

119894are the 119894th eigenvalue and eigenvector of

the updated system respectivelyDminus1 is a generalized inverseof D because D is not a square matrix Thus a new 119899 minus 119903

dimension system based on the original 119899 dimension systemis established

The complete mode shapes are difficult to obtain par-ticularly when a limited number of sensors are availableHowever incomplete mode shape data can be expanded to

Shock and Vibration 3

complete mode shapes by mode shape expansion techniqueThe expansion method in [17] is

Φ119895

=

[

[

I

minus (K119904119904

minus 120582

119895M119904119904

)

minus1

(K119904119898

minus 120582

119895M119904119898

)

]

]

Φ119898119895

(11)

where Φ119898119895

is measured degrees of mode shape Φ119895and K

119904119904

K119904119898

andM119904119904M119904119898

are submatrix of global stiffness and massmatrix respectively

22 Structural Damage Detection Based on Generalized Flex-ibility Matrix Method In this method only the decreasein structure stiffness due to damage is considered Changesin mass property are ignored The damage parameters aredenoted by 119889

119894 which stands for damage extent of the 119894th

element The decrease of global stiffness matrix ΔK canbe expressed as a sum of each elemental stiffness matrixmultiplied by damage parameters [9] that is

ΔK = K119886

minus K119889

=

119890

sum

119894=1

119889

119894K119886119894

(12)

where K119886is the global stiffness matrix of undamaged struc-

ture K119889is the global stiffness matrix of damaged structure

andK119886119894is the 119894th elemental stiffness matrix positioned within

the global matrix for undamaged structure and 119890 is thenumber of elements If the 119894th element is undamaged thevalue of 119889

119894is zero The value of 119889

119894is a nonnegative number

less than one Differentiating (12) with respect to 119889

119894leads to

120597K119889

120597119889

119894

= minusK119886119894

(13)

According to the definition of flexibility and stiffnessmatrix they satisfy the following relationship

F119906119889K119906119889

= I (14)

where F119906119889is the flexibility matrix of updated system for the

damaged structure K119906119889is the stiffness matrix of updated

system for the damaged structure and I is the identitymatrixDifferentiating (14) with respect to 119889

119894leads to

120597F119906119889

120597119889

119894

K119906119889

= minusF119906119889

120597K119906119889

120597119889

119894

(15)

Postmultiplying (15) by F119906119889yields

120597F119906119889

120597119889

119894

= minusF119906119889

120597K119906119889

120597119889

119894

F119906119889 (16)

As the damage is a small amount F119906119889

asymp F119906119886is satisfied

Substituting (9) and (13) into (16) the sensitivity of flexibilitymatrix to damage for the new system can be derived as

120597F119906

120597119889

119894

asymp minusF119906 120597K119906

120597119889

119894

F119906

= (D119879KD)

minus1

D119879K119886119894D (D119879KD)

minus1

(17)

In order to reduce the error result from truncating higher-order modes generalized flexibility matrix f119906 = F119906(M119906F119906)119897 isused [12] In this research 119897 = 2 is adopted The generalizedflexibility matrix for the updated system can be written as

f119906 = F119906M119906F119906M119906F119906 (18)

Differentiating (18) with respect to 119889

119894leads to

120597f119906

120597119889

119894

=

120597F119906

120597119889

119894

M119906F119906M119906F119906 + F119906M119906 120597F119906

120597119889

119894

M119906F119906

+ F119906M119906F119906M119906 120597F119906

120597119889

119894

(19)

Combining (17) and (19) the sensitivity of generalized flex-ibility matrix to damage can be obtained Making use ofTaylorrsquos series expansion change in generalized flexibilitymatrix can be described as

Δf119906 asymp

119890

sum

119894=1

120597f119906

120597119889

119894

119889

119894 (20)

The generalized flexibility matrix for the updated systemcan also be approximately determined by using its firstfrequency 120582

119906

1and the corresponding modeΦ119906

1 which can be

acquired by the 119903 + 1th frequency 120582

119903+1and the corresponding

mode Φ119903+1

of original system respectively Then change ingeneralized flexibility matrix can be described as

Δf119906 asymp

1

120582

3

119889119903+1

(Dminus1Φ119889119903+1

) (Dminus1Φ119889119903+1

)

119879

minus

1

120582

3

119906119903+1

(Dminus1Φ119906119903+1

) (Dminus1Φ119906119903+1

)

119879

(21)

where 120582

119889119903+1and Φ

119889119903+1are the 119903 + 1th frequency and mode

shape of the damaged structure and 120582

119886119903+1and Φ

119886119903+1are

the 119903 + 1th frequency and mode shape of the undamagedstructure respectively When first 119903 modal parameters areunavailable damage parameters can be acquired by manip-ulating (20) and (21) into a system of linear equations whichcan be solved by using the least squares method

3 Numerical Examples

In order to verify the effectiveness of the proposed methodtwo numerical examples are considered The first numericalexample is a cantilever beam and the second one is a rod-fastened-rotor considering partial separation of interface

31 Forty-Five-Element Cantilevered Beam A two-dimen-sional cantilever beam with a rectangular section as shownin Figure 1 is taken as a case study to verify the effectivenessof the proposed method The basic parameters of materialand geometrics are as follows elastic modulus 119864 = 21GPadensity 120588 = 7800 kgm3 length 119897 = 045m cross sectionarea 119860 = 16129 times 10

minus4m2 and the moment of inertia 119868 =

542 times 10

minus8m4 The total number of elements and degrees

4 Shock and Vibration

1 105 15 20 25 30 35 40 45

Figure 1 A cantilever beam

of freedom are 45 and 90 respectively The length of eachelement is 001m Two damage cases are presented here case1 element 28 is damaged with stiffness losses of 10 case 2elements 18 and 36 are damaged simultaneously with stiffnesslosses of 14 and 6 respectively

When the first mode is unavailable location and severityof damage can be obtained by using the second naturalfrequency and the corresponding mode shape with theproposed method in this paper Parameter 119903 is the numberof unavailable modes Damage parameters can be acquiredby solving (20) and (21) with 119903 = 1

The results are also compared with the results obtainedby using the method in [6] which is an extension of theMultiple Damage Location Assurance Criterion (MDLAC)Making use of the mode shape directly the damage sitescan be approximately localized as those sites with the largeMDLAC values The MDLAC value in [6] is

MDLAC (120575A)

=

1003816

1003816

1003816

1003816

1003816

ΔΦ119879

sdot 120575Φ (120575A)

1003816

1003816

1003816

1003816

1003816

2

(ΔΦ119879

sdot ΔΦ) sdot (120575Φ (120575A)

119879sdot 120575Φ (120575A))

(22)

Figure 2 shows the results calculated by the methodsproposed in this paper and [6] for damage case 1 whichrepresents the case of single damage The vertical axis ofFigures 2(a) and 2(b) is absolute damage extent and normal-ized MDLAC value by using the methods proposed in thispaper and [6] respectively Results less than zero are ignoredbecause each 119886

119894is assumed to be a nonnegative number

Damaged element can be located accurately by bothmethodswhile damage extent can also be detected by the proposedmethod in this paper

Similarly Figure 3 displays the results calculated by theproposed methods in this paper and the method in [6] fordamage case 2 (double damage)While themethod in [6] thatdirectly usesmode shape fails to detect the damage in element36 only using the second mode shape the method proposedin this paper predicts the location of the damage successfullyThe damage extent detected is 0163 and 0064 for elements 18and 36 respectively

To consider effect of themeasured noise of frequency andmode shape on accuracy of the proposed method 1 and 5random noise are added in the frequency and mode shapefor damage detection respectively [6 18] Figure 4 shows theresults calculated by the proposed method in this paper fordamage cases 1 and 2The results show that accurate locationscan also be estimated considering effect of measured noise

To investigate effect of using incomplete mode shapeson accuracy of the proposed method 80 and 60 nodaldisplacements of the complete second mode shape are usedto detect the damage The noise effect is also considered

Table 1 Location and extent of multiple damage of rod-fastened-rotor

Element number 10 11 12 13Damage extent 52 162 119 22

at the same time Figure 5 shows the results calculated bythe method proposed in this paper for damage cases 1 and2 when 80 nodal displacements of the second mode shapeare available Figure 6 displays the results calculated by themethod presented in this paper for damage cases 1 and 2whenonly 60 nodal displacements of the second mode shape areavailable

32 Simplified Rod-Fastened-Rotor The rod-fastened-rotorsare commonly used in heavy duty gas turbines and aeroengines As shown in Figure 7 the rod-fastened-rotor iscomprised of a battery of discs clamped together by tie rodsThere is contact interface between discs of the rod-fastened-rotor Some zones of the contact interface are separated andthe flexural stiffness of the rod-fastened-rotor decreaseswhenbending moment exceeds critical value [16]

Detection of partial separation of contact interface for arod-fastened-rotor in heavy duty gas turbine is investigatedin this paper The beam elements are used to model the rod-fastened-rotorThe total number of elements is 37 as shown inFigure 7The unbalance response and distribution of flexuralmoment are calculated by using the common commercialfinite element software ANSYS Figure 8 shows the flexuralmoment on each element at the second critical speedAccord-ing to the relationship between bendingmoment and flexuralstiffness in [19] damage ratio of each element is shown inTable 1

The location and extent of damage are detected with themethod proposed in this paper and the result is shown inFigure 9 The damage extent of damaged elements 10 1112 and 13 detected by our method is 59 24 14 and07 respectively Although several undamaged elements aredetected to be damaged withminor extent the result can alsohelp us to find the location of the damage

From the results obtained above the proposed methodcan effectively identify single double and multiple damagewithout the first frequency and the corresponding modeshape for numerical examples

4 Conclusions

The first-modal parameter of system plays the most impor-tant role in its flexibility matrix the original flexibilitymethod cannot be used to detect damage when the first-modal parameter is unavailable A damage detection methodbased on flexibility change by using single nonfirst modeis presented The system is updated via restricting the firstfew mode shapes The mass matrix stiffness matrix andmodal parameters of the updated system are obtained Thensensitivity of the new flexibility matrix to damage is derivedThe damage extent of each element can be calculated bysolving a linear equation This method is tested through

Shock and Vibration 5

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 2 Damage detection by only using the second-modal parameters for damage case 1 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 3 Damage detection by only using the second-modal parameters for damage case 2 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

016

012

008

004

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 4 Damage detection by only using the second-modal parameters for damage (a) case 1 and (b) case 2 when 1 and 5 random noiseare added in the frequencies and mode shapes respectively

6 Shock and Vibration

012

009

006

003

000

0 5 10 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 5 10 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 5 Damage detection by using the second natural frequency and 80 nodal displacements of the corresponding mode shape (a) case 1(b) case 2

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 6 Damage detection by using the second natural frequency and 60 nodal displacements of the corresponding mode shape (a) case1 (b) case 2

Beam elements

Contact interface

Figure 7 Schematic diagram of heavy duty gas turbine and node dividing

Shock and Vibration 7

14E6

12E6

10E6

80E5

60E5

40E5

20E5

00

0 5 10 15 20 25 30 35 40

Element number

Bending moment

Bend

ing

mom

ent (

Nm

)

Critical bending moment

Figure 8 Bending moment on each interface of the rod-fastened-rotor at the second critical speed

025

020

015

010

005

000

0 5 10 15 20 25 30 35

Dam

age e

xten

t

Element number

Figure 9 Damage detection by only using the second-modalparameter of the rod-fastened-rotor

numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the secondnatural frequency and mode shape are available The resultsof the numerical examples indicate that the proposedmethodcan effectively identify single double and multiple damage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof China (no 51275385) and Major State Basic ResearchDevelopment Program of China (no 2011CB706505)

References

[1] W Fan and P Z Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011

[2] Z A Jassim N N Ali F Mustapha and N A Abdul Jalil ldquoAreview on the vibration analysis for a damage occurrence of acantilever beamrdquo Engineering Failure Analysis vol 31 pp 442ndash461 2013

[3] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[4] J-T Kim and N Stubbs ldquoCrack detection in beam-type struc-tures using frequency datardquo Journal of Sound and Vibration vol259 no 1 pp 145ndash160 2003

[5] G Y Xu W D Zhu and B H Emory ldquoExperimental andnumerical investigation of structural damage detection usingchanges in natural frequenciesrdquo Journal of Vibration and Acous-tics vol 129 no 6 pp 686ndash700 2007

[6] Z Y Shi S S Law and L M Zhang ldquoDamage localization bydirectly using incomplete mode shapesrdquo Journal of EngineeringMechanics vol 126 no 6 pp 656ndash660 2000

[7] P M Pawar K Venkatesulu Reddy and R Ganguli ldquoDamagedetection in beams using spatial fourier analysis and neuralnetworksrdquo Journal of IntelligentMaterial Systems and Structuresvol 18 no 4 pp 347ndash359 2007

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] Q W Yang and J K Liu ldquoDamage identification by theeigenparameter decomposition of structural flexibility changerdquoInternational Journal for Numerical Methods in Engineering vol78 no 4 pp 444ndash459 2009

[10] D Bernal and B Gunes ldquoFlexibility based approach for damagecharacterization benchmark applicationrdquo Journal of Engineer-ing Mechanics vol 130 no 1 pp 61ndash70 2004

[11] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers amp Structures vol 88 no 3-4 pp 154ndash1642010

[12] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[13] M Masoumi E Jamshidi and M Bamdad ldquoApplication ofgeneralized flexibility matrix in damage identification usingImperialist Competitive Algorithmrdquo KSCE Journal of CivilEngineering vol 19 no 4 pp 994ndash1001 2015

[14] W-J Yan and W-X Ren ldquoClosed-form modal flexibility sensi-tivity and its application to structural damage detection withoutmodal truncation errorrdquo Journal of Vibration and Control vol20 no 12 pp 1816ndash1830 2014

[15] M Montazer and S M Seyedpoor ldquoA new flexibility baseddamage index for damage detection of truss structuresrdquo Shockand Vibration vol 2014 Article ID 460692 12 pages 2014

[16] J Gao Q Yuan P Li Z Feng H Zhang and Z Lv ldquoEffectsof bending moments and pretightening forces on the flexuralstiffness of contact interfaces in rod-fastened rotorsrdquo Journal ofEngineering for Gas Turbines and Power vol 134 no 10 ArticleID 102503 2012

[17] J Li Z Li H Zhong and B Wu ldquoStructural damage detectionusing generalized flexibility matrix and changes in naturalfrequenciesrdquo AIAA Journal vol 50 no 5 pp 1072ndash1078 2012

8 Shock and Vibration

[18] M R N Shirazi H Mollamahmoudi and S SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applicationsvol 160 no 3 pp 1009ndash1019 2014

[19] Q Yuan J Gao and P Li ldquoNonlinear dynamics of the rod-fastened Jeffcott rotorrdquo Journal of Vibration and Acoustics vol136 no 2 Article ID 021011 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 Shock and Vibration

of truncating higher-order modal parameters Masoumi etal [13] proposed a new objective function formed by usinggeneralized flexibility matrix Then imperialist competitivealgorithm was used in damage identification Yan and Ren[14] derived a closed form of the sensitivity of flexibilitybased on the algebraic eigensensitivitymethodMontazer andSeyedpoor [15] introduced a new flexibility based damageindex for damage detection of truss structures

Although the generalized flexibility matrix based damagedetection approach can precisely detect the location andseverity of damage by using only the first natural frequencyand the corresponding mode shape there are still many limi-tations in these methods One limitation lies in the damagedetection of rod-fastened-rotor of heavy duty gas turbineThe flexural stiffness of the interface decreases when somezones of the contact interface are separated with bendingmoment on the rotor [16] Flexural stiffness of interface in arod-fastened-rotor induced by bending moment is differentin first and second critical speed because bending momentdistribution varies as rotating speed changes Therefore onlythe second-modal parameters are available for the damagedetection of rod-fastened-rotor in the second critical speed

In this paper a structural damage detectionmethodbasedon changes in the flexibility matrix only using single naturalfrequency and the corresponding mode shape is presentedFirstly restricted by the first several mode shapes the systemis updated The flexibility matrix of updated system can beobtained by using non-first-modal parameters of originalsystemThen sensitivity of flexibility of the updated system todamage is derived Taking advantage of generalized flexibilitymatrix which can considerably reduce the error caused bytruncating higher-order modal parameters the location andseverity of the damage are calculated Finally two numericalexamples for a cantilever beam and a rod-fastened-rotor areused to illustrate the effectiveness of the proposed methodwhen only the second natural frequency and the correspond-ing mode shape are available

2 Structural Damage Detection Method

21 Structural System Updating Method The differentialequation governing the free vibration of a linear undampedstructural system can be expressed as

Mx + Kx = 0 (1)

where M is the global mass matrix K is the global stiffnessmatrix and x is the displacement vector When the degree offreedom for the system is 119899 the eigenvalue problem can bewritten in the form

KΦ119894

= 120582

119894MΦ119894 119894 = 1 2 119899 (2)

where 120582

119894and Φ

119894are the 119894th eigenvalue and eigenvector

respectively Restricting the system by the first 119903mode shapes

Φ119879

119894Mx = 0 119894 = 1 2 119903 (3)

Mode shape matrix mass matrix and displacement vectorcan be partitioned as

Ψ = [

Ψ119903119903Ψ119903119904

Ψ119904119903Ψ119904119904

]

M = [

M119903119903

M119903119904

M119904119903

M119904119904

]

x = [

x119903

x119904

]

(4)

where the 119894th column of Ψ is the 119894th eigenvector Φ119894

Substituting (4) into (3) yields

[Ψ119879

119903119903Ψ119879

119904119903] [

M119903119903

M119903119904

M119904119903

M119904119904

] [

x119903

x119904

] = 0 (5)

Expending (5) yields x119903

= minus(Ψ119903119903M119903119903

+ Ψ119903119904M119904119903

)

minus1(Ψ119903119903M119903119904

+

Ψ119903119904M119904119904

)x119904 then the relationship between x

119904and x is

x = Dx119904 (6)

where D = [

TRI ] in which TR = minus(Ψ

119903119903M119903119903

+

Ψ119903119904M119904119903

)

minus1(Ψ119903119903M119903119904

+Ψ119903119904M119904119904

)x119904 Substituting (6) into original

free vibration differential equation (1) yields

MDx119904

+ KDx119904

= 0 (7)

Left-multiplying (7) by D119879 yields the updated free vibra-tion differential equation

M119906x119904

+ K119906x119904

= 0 (8)

The mass and stiffness matrix of the updated system can beobtained by

M119906 = D119879MD

K119906 = D119879KD

(9)

The relationship between the updated and the original modalparameters can be described by

120582

119906

119894= 120582

119894+119903

Φ119906

119894= Dminus1Φ

119894+119903

119894 = 1 2 119899 minus 119903

(10)

where 120582

119906

119894and Φ119906

119894are the 119894th eigenvalue and eigenvector of

the updated system respectivelyDminus1 is a generalized inverseof D because D is not a square matrix Thus a new 119899 minus 119903

dimension system based on the original 119899 dimension systemis established

The complete mode shapes are difficult to obtain par-ticularly when a limited number of sensors are availableHowever incomplete mode shape data can be expanded to

Shock and Vibration 3

complete mode shapes by mode shape expansion techniqueThe expansion method in [17] is

Φ119895

=

[

[

I

minus (K119904119904

minus 120582

119895M119904119904

)

minus1

(K119904119898

minus 120582

119895M119904119898

)

]

]

Φ119898119895

(11)

where Φ119898119895

is measured degrees of mode shape Φ119895and K

119904119904

K119904119898

andM119904119904M119904119898

are submatrix of global stiffness and massmatrix respectively

22 Structural Damage Detection Based on Generalized Flex-ibility Matrix Method In this method only the decreasein structure stiffness due to damage is considered Changesin mass property are ignored The damage parameters aredenoted by 119889

119894 which stands for damage extent of the 119894th

element The decrease of global stiffness matrix ΔK canbe expressed as a sum of each elemental stiffness matrixmultiplied by damage parameters [9] that is

ΔK = K119886

minus K119889

=

119890

sum

119894=1

119889

119894K119886119894

(12)

where K119886is the global stiffness matrix of undamaged struc-

ture K119889is the global stiffness matrix of damaged structure

andK119886119894is the 119894th elemental stiffness matrix positioned within

the global matrix for undamaged structure and 119890 is thenumber of elements If the 119894th element is undamaged thevalue of 119889

119894is zero The value of 119889

119894is a nonnegative number

less than one Differentiating (12) with respect to 119889

119894leads to

120597K119889

120597119889

119894

= minusK119886119894

(13)

According to the definition of flexibility and stiffnessmatrix they satisfy the following relationship

F119906119889K119906119889

= I (14)

where F119906119889is the flexibility matrix of updated system for the

damaged structure K119906119889is the stiffness matrix of updated

system for the damaged structure and I is the identitymatrixDifferentiating (14) with respect to 119889

119894leads to

120597F119906119889

120597119889

119894

K119906119889

= minusF119906119889

120597K119906119889

120597119889

119894

(15)

Postmultiplying (15) by F119906119889yields

120597F119906119889

120597119889

119894

= minusF119906119889

120597K119906119889

120597119889

119894

F119906119889 (16)

As the damage is a small amount F119906119889

asymp F119906119886is satisfied

Substituting (9) and (13) into (16) the sensitivity of flexibilitymatrix to damage for the new system can be derived as

120597F119906

120597119889

119894

asymp minusF119906 120597K119906

120597119889

119894

F119906

= (D119879KD)

minus1

D119879K119886119894D (D119879KD)

minus1

(17)

In order to reduce the error result from truncating higher-order modes generalized flexibility matrix f119906 = F119906(M119906F119906)119897 isused [12] In this research 119897 = 2 is adopted The generalizedflexibility matrix for the updated system can be written as

f119906 = F119906M119906F119906M119906F119906 (18)

Differentiating (18) with respect to 119889

119894leads to

120597f119906

120597119889

119894

=

120597F119906

120597119889

119894

M119906F119906M119906F119906 + F119906M119906 120597F119906

120597119889

119894

M119906F119906

+ F119906M119906F119906M119906 120597F119906

120597119889

119894

(19)

Combining (17) and (19) the sensitivity of generalized flex-ibility matrix to damage can be obtained Making use ofTaylorrsquos series expansion change in generalized flexibilitymatrix can be described as

Δf119906 asymp

119890

sum

119894=1

120597f119906

120597119889

119894

119889

119894 (20)

The generalized flexibility matrix for the updated systemcan also be approximately determined by using its firstfrequency 120582

119906

1and the corresponding modeΦ119906

1 which can be

acquired by the 119903 + 1th frequency 120582

119903+1and the corresponding

mode Φ119903+1

of original system respectively Then change ingeneralized flexibility matrix can be described as

Δf119906 asymp

1

120582

3

119889119903+1

(Dminus1Φ119889119903+1

) (Dminus1Φ119889119903+1

)

119879

minus

1

120582

3

119906119903+1

(Dminus1Φ119906119903+1

) (Dminus1Φ119906119903+1

)

119879

(21)

where 120582

119889119903+1and Φ

119889119903+1are the 119903 + 1th frequency and mode

shape of the damaged structure and 120582

119886119903+1and Φ

119886119903+1are

the 119903 + 1th frequency and mode shape of the undamagedstructure respectively When first 119903 modal parameters areunavailable damage parameters can be acquired by manip-ulating (20) and (21) into a system of linear equations whichcan be solved by using the least squares method

3 Numerical Examples

In order to verify the effectiveness of the proposed methodtwo numerical examples are considered The first numericalexample is a cantilever beam and the second one is a rod-fastened-rotor considering partial separation of interface

31 Forty-Five-Element Cantilevered Beam A two-dimen-sional cantilever beam with a rectangular section as shownin Figure 1 is taken as a case study to verify the effectivenessof the proposed method The basic parameters of materialand geometrics are as follows elastic modulus 119864 = 21GPadensity 120588 = 7800 kgm3 length 119897 = 045m cross sectionarea 119860 = 16129 times 10

minus4m2 and the moment of inertia 119868 =

542 times 10

minus8m4 The total number of elements and degrees

4 Shock and Vibration

1 105 15 20 25 30 35 40 45

Figure 1 A cantilever beam

of freedom are 45 and 90 respectively The length of eachelement is 001m Two damage cases are presented here case1 element 28 is damaged with stiffness losses of 10 case 2elements 18 and 36 are damaged simultaneously with stiffnesslosses of 14 and 6 respectively

When the first mode is unavailable location and severityof damage can be obtained by using the second naturalfrequency and the corresponding mode shape with theproposed method in this paper Parameter 119903 is the numberof unavailable modes Damage parameters can be acquiredby solving (20) and (21) with 119903 = 1

The results are also compared with the results obtainedby using the method in [6] which is an extension of theMultiple Damage Location Assurance Criterion (MDLAC)Making use of the mode shape directly the damage sitescan be approximately localized as those sites with the largeMDLAC values The MDLAC value in [6] is

MDLAC (120575A)

=

1003816

1003816

1003816

1003816

1003816

ΔΦ119879

sdot 120575Φ (120575A)

1003816

1003816

1003816

1003816

1003816

2

(ΔΦ119879

sdot ΔΦ) sdot (120575Φ (120575A)

119879sdot 120575Φ (120575A))

(22)

Figure 2 shows the results calculated by the methodsproposed in this paper and [6] for damage case 1 whichrepresents the case of single damage The vertical axis ofFigures 2(a) and 2(b) is absolute damage extent and normal-ized MDLAC value by using the methods proposed in thispaper and [6] respectively Results less than zero are ignoredbecause each 119886

119894is assumed to be a nonnegative number

Damaged element can be located accurately by bothmethodswhile damage extent can also be detected by the proposedmethod in this paper

Similarly Figure 3 displays the results calculated by theproposed methods in this paper and the method in [6] fordamage case 2 (double damage)While themethod in [6] thatdirectly usesmode shape fails to detect the damage in element36 only using the second mode shape the method proposedin this paper predicts the location of the damage successfullyThe damage extent detected is 0163 and 0064 for elements 18and 36 respectively

To consider effect of themeasured noise of frequency andmode shape on accuracy of the proposed method 1 and 5random noise are added in the frequency and mode shapefor damage detection respectively [6 18] Figure 4 shows theresults calculated by the proposed method in this paper fordamage cases 1 and 2The results show that accurate locationscan also be estimated considering effect of measured noise

To investigate effect of using incomplete mode shapeson accuracy of the proposed method 80 and 60 nodaldisplacements of the complete second mode shape are usedto detect the damage The noise effect is also considered

Table 1 Location and extent of multiple damage of rod-fastened-rotor

Element number 10 11 12 13Damage extent 52 162 119 22

at the same time Figure 5 shows the results calculated bythe method proposed in this paper for damage cases 1 and2 when 80 nodal displacements of the second mode shapeare available Figure 6 displays the results calculated by themethod presented in this paper for damage cases 1 and 2whenonly 60 nodal displacements of the second mode shape areavailable

32 Simplified Rod-Fastened-Rotor The rod-fastened-rotorsare commonly used in heavy duty gas turbines and aeroengines As shown in Figure 7 the rod-fastened-rotor iscomprised of a battery of discs clamped together by tie rodsThere is contact interface between discs of the rod-fastened-rotor Some zones of the contact interface are separated andthe flexural stiffness of the rod-fastened-rotor decreaseswhenbending moment exceeds critical value [16]

Detection of partial separation of contact interface for arod-fastened-rotor in heavy duty gas turbine is investigatedin this paper The beam elements are used to model the rod-fastened-rotorThe total number of elements is 37 as shown inFigure 7The unbalance response and distribution of flexuralmoment are calculated by using the common commercialfinite element software ANSYS Figure 8 shows the flexuralmoment on each element at the second critical speedAccord-ing to the relationship between bendingmoment and flexuralstiffness in [19] damage ratio of each element is shown inTable 1

The location and extent of damage are detected with themethod proposed in this paper and the result is shown inFigure 9 The damage extent of damaged elements 10 1112 and 13 detected by our method is 59 24 14 and07 respectively Although several undamaged elements aredetected to be damaged withminor extent the result can alsohelp us to find the location of the damage

From the results obtained above the proposed methodcan effectively identify single double and multiple damagewithout the first frequency and the corresponding modeshape for numerical examples

4 Conclusions

The first-modal parameter of system plays the most impor-tant role in its flexibility matrix the original flexibilitymethod cannot be used to detect damage when the first-modal parameter is unavailable A damage detection methodbased on flexibility change by using single nonfirst modeis presented The system is updated via restricting the firstfew mode shapes The mass matrix stiffness matrix andmodal parameters of the updated system are obtained Thensensitivity of the new flexibility matrix to damage is derivedThe damage extent of each element can be calculated bysolving a linear equation This method is tested through

Shock and Vibration 5

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 2 Damage detection by only using the second-modal parameters for damage case 1 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 3 Damage detection by only using the second-modal parameters for damage case 2 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

016

012

008

004

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 4 Damage detection by only using the second-modal parameters for damage (a) case 1 and (b) case 2 when 1 and 5 random noiseare added in the frequencies and mode shapes respectively

6 Shock and Vibration

012

009

006

003

000

0 5 10 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 5 10 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 5 Damage detection by using the second natural frequency and 80 nodal displacements of the corresponding mode shape (a) case 1(b) case 2

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 6 Damage detection by using the second natural frequency and 60 nodal displacements of the corresponding mode shape (a) case1 (b) case 2

Beam elements

Contact interface

Figure 7 Schematic diagram of heavy duty gas turbine and node dividing

Shock and Vibration 7

14E6

12E6

10E6

80E5

60E5

40E5

20E5

00

0 5 10 15 20 25 30 35 40

Element number

Bending moment

Bend

ing

mom

ent (

Nm

)

Critical bending moment

Figure 8 Bending moment on each interface of the rod-fastened-rotor at the second critical speed

025

020

015

010

005

000

0 5 10 15 20 25 30 35

Dam

age e

xten

t

Element number

Figure 9 Damage detection by only using the second-modalparameter of the rod-fastened-rotor

numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the secondnatural frequency and mode shape are available The resultsof the numerical examples indicate that the proposedmethodcan effectively identify single double and multiple damage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof China (no 51275385) and Major State Basic ResearchDevelopment Program of China (no 2011CB706505)

References

[1] W Fan and P Z Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011

[2] Z A Jassim N N Ali F Mustapha and N A Abdul Jalil ldquoAreview on the vibration analysis for a damage occurrence of acantilever beamrdquo Engineering Failure Analysis vol 31 pp 442ndash461 2013

[3] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[4] J-T Kim and N Stubbs ldquoCrack detection in beam-type struc-tures using frequency datardquo Journal of Sound and Vibration vol259 no 1 pp 145ndash160 2003

[5] G Y Xu W D Zhu and B H Emory ldquoExperimental andnumerical investigation of structural damage detection usingchanges in natural frequenciesrdquo Journal of Vibration and Acous-tics vol 129 no 6 pp 686ndash700 2007

[6] Z Y Shi S S Law and L M Zhang ldquoDamage localization bydirectly using incomplete mode shapesrdquo Journal of EngineeringMechanics vol 126 no 6 pp 656ndash660 2000

[7] P M Pawar K Venkatesulu Reddy and R Ganguli ldquoDamagedetection in beams using spatial fourier analysis and neuralnetworksrdquo Journal of IntelligentMaterial Systems and Structuresvol 18 no 4 pp 347ndash359 2007

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] Q W Yang and J K Liu ldquoDamage identification by theeigenparameter decomposition of structural flexibility changerdquoInternational Journal for Numerical Methods in Engineering vol78 no 4 pp 444ndash459 2009

[10] D Bernal and B Gunes ldquoFlexibility based approach for damagecharacterization benchmark applicationrdquo Journal of Engineer-ing Mechanics vol 130 no 1 pp 61ndash70 2004

[11] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers amp Structures vol 88 no 3-4 pp 154ndash1642010

[12] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[13] M Masoumi E Jamshidi and M Bamdad ldquoApplication ofgeneralized flexibility matrix in damage identification usingImperialist Competitive Algorithmrdquo KSCE Journal of CivilEngineering vol 19 no 4 pp 994ndash1001 2015

[14] W-J Yan and W-X Ren ldquoClosed-form modal flexibility sensi-tivity and its application to structural damage detection withoutmodal truncation errorrdquo Journal of Vibration and Control vol20 no 12 pp 1816ndash1830 2014

[15] M Montazer and S M Seyedpoor ldquoA new flexibility baseddamage index for damage detection of truss structuresrdquo Shockand Vibration vol 2014 Article ID 460692 12 pages 2014

[16] J Gao Q Yuan P Li Z Feng H Zhang and Z Lv ldquoEffectsof bending moments and pretightening forces on the flexuralstiffness of contact interfaces in rod-fastened rotorsrdquo Journal ofEngineering for Gas Turbines and Power vol 134 no 10 ArticleID 102503 2012

[17] J Li Z Li H Zhong and B Wu ldquoStructural damage detectionusing generalized flexibility matrix and changes in naturalfrequenciesrdquo AIAA Journal vol 50 no 5 pp 1072ndash1078 2012

8 Shock and Vibration

[18] M R N Shirazi H Mollamahmoudi and S SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applicationsvol 160 no 3 pp 1009ndash1019 2014

[19] Q Yuan J Gao and P Li ldquoNonlinear dynamics of the rod-fastened Jeffcott rotorrdquo Journal of Vibration and Acoustics vol136 no 2 Article ID 021011 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 3

complete mode shapes by mode shape expansion techniqueThe expansion method in [17] is

Φ119895

=

[

[

I

minus (K119904119904

minus 120582

119895M119904119904

)

minus1

(K119904119898

minus 120582

119895M119904119898

)

]

]

Φ119898119895

(11)

where Φ119898119895

is measured degrees of mode shape Φ119895and K

119904119904

K119904119898

andM119904119904M119904119898

are submatrix of global stiffness and massmatrix respectively

22 Structural Damage Detection Based on Generalized Flex-ibility Matrix Method In this method only the decreasein structure stiffness due to damage is considered Changesin mass property are ignored The damage parameters aredenoted by 119889

119894 which stands for damage extent of the 119894th

element The decrease of global stiffness matrix ΔK canbe expressed as a sum of each elemental stiffness matrixmultiplied by damage parameters [9] that is

ΔK = K119886

minus K119889

=

119890

sum

119894=1

119889

119894K119886119894

(12)

where K119886is the global stiffness matrix of undamaged struc-

ture K119889is the global stiffness matrix of damaged structure

andK119886119894is the 119894th elemental stiffness matrix positioned within

the global matrix for undamaged structure and 119890 is thenumber of elements If the 119894th element is undamaged thevalue of 119889

119894is zero The value of 119889

119894is a nonnegative number

less than one Differentiating (12) with respect to 119889

119894leads to

120597K119889

120597119889

119894

= minusK119886119894

(13)

According to the definition of flexibility and stiffnessmatrix they satisfy the following relationship

F119906119889K119906119889

= I (14)

where F119906119889is the flexibility matrix of updated system for the

damaged structure K119906119889is the stiffness matrix of updated

system for the damaged structure and I is the identitymatrixDifferentiating (14) with respect to 119889

119894leads to

120597F119906119889

120597119889

119894

K119906119889

= minusF119906119889

120597K119906119889

120597119889

119894

(15)

Postmultiplying (15) by F119906119889yields

120597F119906119889

120597119889

119894

= minusF119906119889

120597K119906119889

120597119889

119894

F119906119889 (16)

As the damage is a small amount F119906119889

asymp F119906119886is satisfied

Substituting (9) and (13) into (16) the sensitivity of flexibilitymatrix to damage for the new system can be derived as

120597F119906

120597119889

119894

asymp minusF119906 120597K119906

120597119889

119894

F119906

= (D119879KD)

minus1

D119879K119886119894D (D119879KD)

minus1

(17)

In order to reduce the error result from truncating higher-order modes generalized flexibility matrix f119906 = F119906(M119906F119906)119897 isused [12] In this research 119897 = 2 is adopted The generalizedflexibility matrix for the updated system can be written as

f119906 = F119906M119906F119906M119906F119906 (18)

Differentiating (18) with respect to 119889

119894leads to

120597f119906

120597119889

119894

=

120597F119906

120597119889

119894

M119906F119906M119906F119906 + F119906M119906 120597F119906

120597119889

119894

M119906F119906

+ F119906M119906F119906M119906 120597F119906

120597119889

119894

(19)

Combining (17) and (19) the sensitivity of generalized flex-ibility matrix to damage can be obtained Making use ofTaylorrsquos series expansion change in generalized flexibilitymatrix can be described as

Δf119906 asymp

119890

sum

119894=1

120597f119906

120597119889

119894

119889

119894 (20)

The generalized flexibility matrix for the updated systemcan also be approximately determined by using its firstfrequency 120582

119906

1and the corresponding modeΦ119906

1 which can be

acquired by the 119903 + 1th frequency 120582

119903+1and the corresponding

mode Φ119903+1

of original system respectively Then change ingeneralized flexibility matrix can be described as

Δf119906 asymp

1

120582

3

119889119903+1

(Dminus1Φ119889119903+1

) (Dminus1Φ119889119903+1

)

119879

minus

1

120582

3

119906119903+1

(Dminus1Φ119906119903+1

) (Dminus1Φ119906119903+1

)

119879

(21)

where 120582

119889119903+1and Φ

119889119903+1are the 119903 + 1th frequency and mode

shape of the damaged structure and 120582

119886119903+1and Φ

119886119903+1are

the 119903 + 1th frequency and mode shape of the undamagedstructure respectively When first 119903 modal parameters areunavailable damage parameters can be acquired by manip-ulating (20) and (21) into a system of linear equations whichcan be solved by using the least squares method

3 Numerical Examples

In order to verify the effectiveness of the proposed methodtwo numerical examples are considered The first numericalexample is a cantilever beam and the second one is a rod-fastened-rotor considering partial separation of interface

31 Forty-Five-Element Cantilevered Beam A two-dimen-sional cantilever beam with a rectangular section as shownin Figure 1 is taken as a case study to verify the effectivenessof the proposed method The basic parameters of materialand geometrics are as follows elastic modulus 119864 = 21GPadensity 120588 = 7800 kgm3 length 119897 = 045m cross sectionarea 119860 = 16129 times 10

minus4m2 and the moment of inertia 119868 =

542 times 10

minus8m4 The total number of elements and degrees

4 Shock and Vibration

1 105 15 20 25 30 35 40 45

Figure 1 A cantilever beam

of freedom are 45 and 90 respectively The length of eachelement is 001m Two damage cases are presented here case1 element 28 is damaged with stiffness losses of 10 case 2elements 18 and 36 are damaged simultaneously with stiffnesslosses of 14 and 6 respectively

When the first mode is unavailable location and severityof damage can be obtained by using the second naturalfrequency and the corresponding mode shape with theproposed method in this paper Parameter 119903 is the numberof unavailable modes Damage parameters can be acquiredby solving (20) and (21) with 119903 = 1

The results are also compared with the results obtainedby using the method in [6] which is an extension of theMultiple Damage Location Assurance Criterion (MDLAC)Making use of the mode shape directly the damage sitescan be approximately localized as those sites with the largeMDLAC values The MDLAC value in [6] is

MDLAC (120575A)

=

1003816

1003816

1003816

1003816

1003816

ΔΦ119879

sdot 120575Φ (120575A)

1003816

1003816

1003816

1003816

1003816

2

(ΔΦ119879

sdot ΔΦ) sdot (120575Φ (120575A)

119879sdot 120575Φ (120575A))

(22)

Figure 2 shows the results calculated by the methodsproposed in this paper and [6] for damage case 1 whichrepresents the case of single damage The vertical axis ofFigures 2(a) and 2(b) is absolute damage extent and normal-ized MDLAC value by using the methods proposed in thispaper and [6] respectively Results less than zero are ignoredbecause each 119886

119894is assumed to be a nonnegative number

Damaged element can be located accurately by bothmethodswhile damage extent can also be detected by the proposedmethod in this paper

Similarly Figure 3 displays the results calculated by theproposed methods in this paper and the method in [6] fordamage case 2 (double damage)While themethod in [6] thatdirectly usesmode shape fails to detect the damage in element36 only using the second mode shape the method proposedin this paper predicts the location of the damage successfullyThe damage extent detected is 0163 and 0064 for elements 18and 36 respectively

To consider effect of themeasured noise of frequency andmode shape on accuracy of the proposed method 1 and 5random noise are added in the frequency and mode shapefor damage detection respectively [6 18] Figure 4 shows theresults calculated by the proposed method in this paper fordamage cases 1 and 2The results show that accurate locationscan also be estimated considering effect of measured noise

To investigate effect of using incomplete mode shapeson accuracy of the proposed method 80 and 60 nodaldisplacements of the complete second mode shape are usedto detect the damage The noise effect is also considered

Table 1 Location and extent of multiple damage of rod-fastened-rotor

Element number 10 11 12 13Damage extent 52 162 119 22

at the same time Figure 5 shows the results calculated bythe method proposed in this paper for damage cases 1 and2 when 80 nodal displacements of the second mode shapeare available Figure 6 displays the results calculated by themethod presented in this paper for damage cases 1 and 2whenonly 60 nodal displacements of the second mode shape areavailable

32 Simplified Rod-Fastened-Rotor The rod-fastened-rotorsare commonly used in heavy duty gas turbines and aeroengines As shown in Figure 7 the rod-fastened-rotor iscomprised of a battery of discs clamped together by tie rodsThere is contact interface between discs of the rod-fastened-rotor Some zones of the contact interface are separated andthe flexural stiffness of the rod-fastened-rotor decreaseswhenbending moment exceeds critical value [16]

Detection of partial separation of contact interface for arod-fastened-rotor in heavy duty gas turbine is investigatedin this paper The beam elements are used to model the rod-fastened-rotorThe total number of elements is 37 as shown inFigure 7The unbalance response and distribution of flexuralmoment are calculated by using the common commercialfinite element software ANSYS Figure 8 shows the flexuralmoment on each element at the second critical speedAccord-ing to the relationship between bendingmoment and flexuralstiffness in [19] damage ratio of each element is shown inTable 1

The location and extent of damage are detected with themethod proposed in this paper and the result is shown inFigure 9 The damage extent of damaged elements 10 1112 and 13 detected by our method is 59 24 14 and07 respectively Although several undamaged elements aredetected to be damaged withminor extent the result can alsohelp us to find the location of the damage

From the results obtained above the proposed methodcan effectively identify single double and multiple damagewithout the first frequency and the corresponding modeshape for numerical examples

4 Conclusions

The first-modal parameter of system plays the most impor-tant role in its flexibility matrix the original flexibilitymethod cannot be used to detect damage when the first-modal parameter is unavailable A damage detection methodbased on flexibility change by using single nonfirst modeis presented The system is updated via restricting the firstfew mode shapes The mass matrix stiffness matrix andmodal parameters of the updated system are obtained Thensensitivity of the new flexibility matrix to damage is derivedThe damage extent of each element can be calculated bysolving a linear equation This method is tested through

Shock and Vibration 5

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 2 Damage detection by only using the second-modal parameters for damage case 1 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 3 Damage detection by only using the second-modal parameters for damage case 2 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

016

012

008

004

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 4 Damage detection by only using the second-modal parameters for damage (a) case 1 and (b) case 2 when 1 and 5 random noiseare added in the frequencies and mode shapes respectively

6 Shock and Vibration

012

009

006

003

000

0 5 10 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 5 10 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 5 Damage detection by using the second natural frequency and 80 nodal displacements of the corresponding mode shape (a) case 1(b) case 2

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 6 Damage detection by using the second natural frequency and 60 nodal displacements of the corresponding mode shape (a) case1 (b) case 2

Beam elements

Contact interface

Figure 7 Schematic diagram of heavy duty gas turbine and node dividing

Shock and Vibration 7

14E6

12E6

10E6

80E5

60E5

40E5

20E5

00

0 5 10 15 20 25 30 35 40

Element number

Bending moment

Bend

ing

mom

ent (

Nm

)

Critical bending moment

Figure 8 Bending moment on each interface of the rod-fastened-rotor at the second critical speed

025

020

015

010

005

000

0 5 10 15 20 25 30 35

Dam

age e

xten

t

Element number

Figure 9 Damage detection by only using the second-modalparameter of the rod-fastened-rotor

numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the secondnatural frequency and mode shape are available The resultsof the numerical examples indicate that the proposedmethodcan effectively identify single double and multiple damage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof China (no 51275385) and Major State Basic ResearchDevelopment Program of China (no 2011CB706505)

References

[1] W Fan and P Z Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011

[2] Z A Jassim N N Ali F Mustapha and N A Abdul Jalil ldquoAreview on the vibration analysis for a damage occurrence of acantilever beamrdquo Engineering Failure Analysis vol 31 pp 442ndash461 2013

[3] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[4] J-T Kim and N Stubbs ldquoCrack detection in beam-type struc-tures using frequency datardquo Journal of Sound and Vibration vol259 no 1 pp 145ndash160 2003

[5] G Y Xu W D Zhu and B H Emory ldquoExperimental andnumerical investigation of structural damage detection usingchanges in natural frequenciesrdquo Journal of Vibration and Acous-tics vol 129 no 6 pp 686ndash700 2007

[6] Z Y Shi S S Law and L M Zhang ldquoDamage localization bydirectly using incomplete mode shapesrdquo Journal of EngineeringMechanics vol 126 no 6 pp 656ndash660 2000

[7] P M Pawar K Venkatesulu Reddy and R Ganguli ldquoDamagedetection in beams using spatial fourier analysis and neuralnetworksrdquo Journal of IntelligentMaterial Systems and Structuresvol 18 no 4 pp 347ndash359 2007

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] Q W Yang and J K Liu ldquoDamage identification by theeigenparameter decomposition of structural flexibility changerdquoInternational Journal for Numerical Methods in Engineering vol78 no 4 pp 444ndash459 2009

[10] D Bernal and B Gunes ldquoFlexibility based approach for damagecharacterization benchmark applicationrdquo Journal of Engineer-ing Mechanics vol 130 no 1 pp 61ndash70 2004

[11] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers amp Structures vol 88 no 3-4 pp 154ndash1642010

[12] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[13] M Masoumi E Jamshidi and M Bamdad ldquoApplication ofgeneralized flexibility matrix in damage identification usingImperialist Competitive Algorithmrdquo KSCE Journal of CivilEngineering vol 19 no 4 pp 994ndash1001 2015

[14] W-J Yan and W-X Ren ldquoClosed-form modal flexibility sensi-tivity and its application to structural damage detection withoutmodal truncation errorrdquo Journal of Vibration and Control vol20 no 12 pp 1816ndash1830 2014

[15] M Montazer and S M Seyedpoor ldquoA new flexibility baseddamage index for damage detection of truss structuresrdquo Shockand Vibration vol 2014 Article ID 460692 12 pages 2014

[16] J Gao Q Yuan P Li Z Feng H Zhang and Z Lv ldquoEffectsof bending moments and pretightening forces on the flexuralstiffness of contact interfaces in rod-fastened rotorsrdquo Journal ofEngineering for Gas Turbines and Power vol 134 no 10 ArticleID 102503 2012

[17] J Li Z Li H Zhong and B Wu ldquoStructural damage detectionusing generalized flexibility matrix and changes in naturalfrequenciesrdquo AIAA Journal vol 50 no 5 pp 1072ndash1078 2012

8 Shock and Vibration

[18] M R N Shirazi H Mollamahmoudi and S SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applicationsvol 160 no 3 pp 1009ndash1019 2014

[19] Q Yuan J Gao and P Li ldquoNonlinear dynamics of the rod-fastened Jeffcott rotorrdquo Journal of Vibration and Acoustics vol136 no 2 Article ID 021011 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Shock and Vibration

1 105 15 20 25 30 35 40 45

Figure 1 A cantilever beam

of freedom are 45 and 90 respectively The length of eachelement is 001m Two damage cases are presented here case1 element 28 is damaged with stiffness losses of 10 case 2elements 18 and 36 are damaged simultaneously with stiffnesslosses of 14 and 6 respectively

When the first mode is unavailable location and severityof damage can be obtained by using the second naturalfrequency and the corresponding mode shape with theproposed method in this paper Parameter 119903 is the numberof unavailable modes Damage parameters can be acquiredby solving (20) and (21) with 119903 = 1

The results are also compared with the results obtainedby using the method in [6] which is an extension of theMultiple Damage Location Assurance Criterion (MDLAC)Making use of the mode shape directly the damage sitescan be approximately localized as those sites with the largeMDLAC values The MDLAC value in [6] is

MDLAC (120575A)

=

1003816

1003816

1003816

1003816

1003816

ΔΦ119879

sdot 120575Φ (120575A)

1003816

1003816

1003816

1003816

1003816

2

(ΔΦ119879

sdot ΔΦ) sdot (120575Φ (120575A)

119879sdot 120575Φ (120575A))

(22)

Figure 2 shows the results calculated by the methodsproposed in this paper and [6] for damage case 1 whichrepresents the case of single damage The vertical axis ofFigures 2(a) and 2(b) is absolute damage extent and normal-ized MDLAC value by using the methods proposed in thispaper and [6] respectively Results less than zero are ignoredbecause each 119886

119894is assumed to be a nonnegative number

Damaged element can be located accurately by bothmethodswhile damage extent can also be detected by the proposedmethod in this paper

Similarly Figure 3 displays the results calculated by theproposed methods in this paper and the method in [6] fordamage case 2 (double damage)While themethod in [6] thatdirectly usesmode shape fails to detect the damage in element36 only using the second mode shape the method proposedin this paper predicts the location of the damage successfullyThe damage extent detected is 0163 and 0064 for elements 18and 36 respectively

To consider effect of themeasured noise of frequency andmode shape on accuracy of the proposed method 1 and 5random noise are added in the frequency and mode shapefor damage detection respectively [6 18] Figure 4 shows theresults calculated by the proposed method in this paper fordamage cases 1 and 2The results show that accurate locationscan also be estimated considering effect of measured noise

To investigate effect of using incomplete mode shapeson accuracy of the proposed method 80 and 60 nodaldisplacements of the complete second mode shape are usedto detect the damage The noise effect is also considered

Table 1 Location and extent of multiple damage of rod-fastened-rotor

Element number 10 11 12 13Damage extent 52 162 119 22

at the same time Figure 5 shows the results calculated bythe method proposed in this paper for damage cases 1 and2 when 80 nodal displacements of the second mode shapeare available Figure 6 displays the results calculated by themethod presented in this paper for damage cases 1 and 2whenonly 60 nodal displacements of the second mode shape areavailable

32 Simplified Rod-Fastened-Rotor The rod-fastened-rotorsare commonly used in heavy duty gas turbines and aeroengines As shown in Figure 7 the rod-fastened-rotor iscomprised of a battery of discs clamped together by tie rodsThere is contact interface between discs of the rod-fastened-rotor Some zones of the contact interface are separated andthe flexural stiffness of the rod-fastened-rotor decreaseswhenbending moment exceeds critical value [16]

Detection of partial separation of contact interface for arod-fastened-rotor in heavy duty gas turbine is investigatedin this paper The beam elements are used to model the rod-fastened-rotorThe total number of elements is 37 as shown inFigure 7The unbalance response and distribution of flexuralmoment are calculated by using the common commercialfinite element software ANSYS Figure 8 shows the flexuralmoment on each element at the second critical speedAccord-ing to the relationship between bendingmoment and flexuralstiffness in [19] damage ratio of each element is shown inTable 1

The location and extent of damage are detected with themethod proposed in this paper and the result is shown inFigure 9 The damage extent of damaged elements 10 1112 and 13 detected by our method is 59 24 14 and07 respectively Although several undamaged elements aredetected to be damaged withminor extent the result can alsohelp us to find the location of the damage

From the results obtained above the proposed methodcan effectively identify single double and multiple damagewithout the first frequency and the corresponding modeshape for numerical examples

4 Conclusions

The first-modal parameter of system plays the most impor-tant role in its flexibility matrix the original flexibilitymethod cannot be used to detect damage when the first-modal parameter is unavailable A damage detection methodbased on flexibility change by using single nonfirst modeis presented The system is updated via restricting the firstfew mode shapes The mass matrix stiffness matrix andmodal parameters of the updated system are obtained Thensensitivity of the new flexibility matrix to damage is derivedThe damage extent of each element can be calculated bysolving a linear equation This method is tested through

Shock and Vibration 5

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 2 Damage detection by only using the second-modal parameters for damage case 1 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 3 Damage detection by only using the second-modal parameters for damage case 2 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

016

012

008

004

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 4 Damage detection by only using the second-modal parameters for damage (a) case 1 and (b) case 2 when 1 and 5 random noiseare added in the frequencies and mode shapes respectively

6 Shock and Vibration

012

009

006

003

000

0 5 10 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 5 10 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 5 Damage detection by using the second natural frequency and 80 nodal displacements of the corresponding mode shape (a) case 1(b) case 2

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 6 Damage detection by using the second natural frequency and 60 nodal displacements of the corresponding mode shape (a) case1 (b) case 2

Beam elements

Contact interface

Figure 7 Schematic diagram of heavy duty gas turbine and node dividing

Shock and Vibration 7

14E6

12E6

10E6

80E5

60E5

40E5

20E5

00

0 5 10 15 20 25 30 35 40

Element number

Bending moment

Bend

ing

mom

ent (

Nm

)

Critical bending moment

Figure 8 Bending moment on each interface of the rod-fastened-rotor at the second critical speed

025

020

015

010

005

000

0 5 10 15 20 25 30 35

Dam

age e

xten

t

Element number

Figure 9 Damage detection by only using the second-modalparameter of the rod-fastened-rotor

numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the secondnatural frequency and mode shape are available The resultsof the numerical examples indicate that the proposedmethodcan effectively identify single double and multiple damage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof China (no 51275385) and Major State Basic ResearchDevelopment Program of China (no 2011CB706505)

References

[1] W Fan and P Z Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011

[2] Z A Jassim N N Ali F Mustapha and N A Abdul Jalil ldquoAreview on the vibration analysis for a damage occurrence of acantilever beamrdquo Engineering Failure Analysis vol 31 pp 442ndash461 2013

[3] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[4] J-T Kim and N Stubbs ldquoCrack detection in beam-type struc-tures using frequency datardquo Journal of Sound and Vibration vol259 no 1 pp 145ndash160 2003

[5] G Y Xu W D Zhu and B H Emory ldquoExperimental andnumerical investigation of structural damage detection usingchanges in natural frequenciesrdquo Journal of Vibration and Acous-tics vol 129 no 6 pp 686ndash700 2007

[6] Z Y Shi S S Law and L M Zhang ldquoDamage localization bydirectly using incomplete mode shapesrdquo Journal of EngineeringMechanics vol 126 no 6 pp 656ndash660 2000

[7] P M Pawar K Venkatesulu Reddy and R Ganguli ldquoDamagedetection in beams using spatial fourier analysis and neuralnetworksrdquo Journal of IntelligentMaterial Systems and Structuresvol 18 no 4 pp 347ndash359 2007

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] Q W Yang and J K Liu ldquoDamage identification by theeigenparameter decomposition of structural flexibility changerdquoInternational Journal for Numerical Methods in Engineering vol78 no 4 pp 444ndash459 2009

[10] D Bernal and B Gunes ldquoFlexibility based approach for damagecharacterization benchmark applicationrdquo Journal of Engineer-ing Mechanics vol 130 no 1 pp 61ndash70 2004

[11] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers amp Structures vol 88 no 3-4 pp 154ndash1642010

[12] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[13] M Masoumi E Jamshidi and M Bamdad ldquoApplication ofgeneralized flexibility matrix in damage identification usingImperialist Competitive Algorithmrdquo KSCE Journal of CivilEngineering vol 19 no 4 pp 994ndash1001 2015

[14] W-J Yan and W-X Ren ldquoClosed-form modal flexibility sensi-tivity and its application to structural damage detection withoutmodal truncation errorrdquo Journal of Vibration and Control vol20 no 12 pp 1816ndash1830 2014

[15] M Montazer and S M Seyedpoor ldquoA new flexibility baseddamage index for damage detection of truss structuresrdquo Shockand Vibration vol 2014 Article ID 460692 12 pages 2014

[16] J Gao Q Yuan P Li Z Feng H Zhang and Z Lv ldquoEffectsof bending moments and pretightening forces on the flexuralstiffness of contact interfaces in rod-fastened rotorsrdquo Journal ofEngineering for Gas Turbines and Power vol 134 no 10 ArticleID 102503 2012

[17] J Li Z Li H Zhong and B Wu ldquoStructural damage detectionusing generalized flexibility matrix and changes in naturalfrequenciesrdquo AIAA Journal vol 50 no 5 pp 1072ndash1078 2012

8 Shock and Vibration

[18] M R N Shirazi H Mollamahmoudi and S SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applicationsvol 160 no 3 pp 1009ndash1019 2014

[19] Q Yuan J Gao and P Li ldquoNonlinear dynamics of the rod-fastened Jeffcott rotorrdquo Journal of Vibration and Acoustics vol136 no 2 Article ID 021011 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 2 Damage detection by only using the second-modal parameters for damage case 1 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

10

08

06

04

02

00

0 105 15 20 25 30 35 40 45

Nor

mal

ized

MD

LAC

Element number(b)

Figure 3 Damage detection by only using the second-modal parameters for damage case 2 (a) and (b) are the results by using the proposedmethod in this paper and the method in [6] respectively

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

016

012

008

004

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 4 Damage detection by only using the second-modal parameters for damage (a) case 1 and (b) case 2 when 1 and 5 random noiseare added in the frequencies and mode shapes respectively

6 Shock and Vibration

012

009

006

003

000

0 5 10 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 5 10 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 5 Damage detection by using the second natural frequency and 80 nodal displacements of the corresponding mode shape (a) case 1(b) case 2

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 6 Damage detection by using the second natural frequency and 60 nodal displacements of the corresponding mode shape (a) case1 (b) case 2

Beam elements

Contact interface

Figure 7 Schematic diagram of heavy duty gas turbine and node dividing

Shock and Vibration 7

14E6

12E6

10E6

80E5

60E5

40E5

20E5

00

0 5 10 15 20 25 30 35 40

Element number

Bending moment

Bend

ing

mom

ent (

Nm

)

Critical bending moment

Figure 8 Bending moment on each interface of the rod-fastened-rotor at the second critical speed

025

020

015

010

005

000

0 5 10 15 20 25 30 35

Dam

age e

xten

t

Element number

Figure 9 Damage detection by only using the second-modalparameter of the rod-fastened-rotor

numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the secondnatural frequency and mode shape are available The resultsof the numerical examples indicate that the proposedmethodcan effectively identify single double and multiple damage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof China (no 51275385) and Major State Basic ResearchDevelopment Program of China (no 2011CB706505)

References

[1] W Fan and P Z Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011

[2] Z A Jassim N N Ali F Mustapha and N A Abdul Jalil ldquoAreview on the vibration analysis for a damage occurrence of acantilever beamrdquo Engineering Failure Analysis vol 31 pp 442ndash461 2013

[3] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[4] J-T Kim and N Stubbs ldquoCrack detection in beam-type struc-tures using frequency datardquo Journal of Sound and Vibration vol259 no 1 pp 145ndash160 2003

[5] G Y Xu W D Zhu and B H Emory ldquoExperimental andnumerical investigation of structural damage detection usingchanges in natural frequenciesrdquo Journal of Vibration and Acous-tics vol 129 no 6 pp 686ndash700 2007

[6] Z Y Shi S S Law and L M Zhang ldquoDamage localization bydirectly using incomplete mode shapesrdquo Journal of EngineeringMechanics vol 126 no 6 pp 656ndash660 2000

[7] P M Pawar K Venkatesulu Reddy and R Ganguli ldquoDamagedetection in beams using spatial fourier analysis and neuralnetworksrdquo Journal of IntelligentMaterial Systems and Structuresvol 18 no 4 pp 347ndash359 2007

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] Q W Yang and J K Liu ldquoDamage identification by theeigenparameter decomposition of structural flexibility changerdquoInternational Journal for Numerical Methods in Engineering vol78 no 4 pp 444ndash459 2009

[10] D Bernal and B Gunes ldquoFlexibility based approach for damagecharacterization benchmark applicationrdquo Journal of Engineer-ing Mechanics vol 130 no 1 pp 61ndash70 2004

[11] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers amp Structures vol 88 no 3-4 pp 154ndash1642010

[12] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[13] M Masoumi E Jamshidi and M Bamdad ldquoApplication ofgeneralized flexibility matrix in damage identification usingImperialist Competitive Algorithmrdquo KSCE Journal of CivilEngineering vol 19 no 4 pp 994ndash1001 2015

[14] W-J Yan and W-X Ren ldquoClosed-form modal flexibility sensi-tivity and its application to structural damage detection withoutmodal truncation errorrdquo Journal of Vibration and Control vol20 no 12 pp 1816ndash1830 2014

[15] M Montazer and S M Seyedpoor ldquoA new flexibility baseddamage index for damage detection of truss structuresrdquo Shockand Vibration vol 2014 Article ID 460692 12 pages 2014

[16] J Gao Q Yuan P Li Z Feng H Zhang and Z Lv ldquoEffectsof bending moments and pretightening forces on the flexuralstiffness of contact interfaces in rod-fastened rotorsrdquo Journal ofEngineering for Gas Turbines and Power vol 134 no 10 ArticleID 102503 2012

[17] J Li Z Li H Zhong and B Wu ldquoStructural damage detectionusing generalized flexibility matrix and changes in naturalfrequenciesrdquo AIAA Journal vol 50 no 5 pp 1072ndash1078 2012

8 Shock and Vibration

[18] M R N Shirazi H Mollamahmoudi and S SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applicationsvol 160 no 3 pp 1009ndash1019 2014

[19] Q Yuan J Gao and P Li ldquoNonlinear dynamics of the rod-fastened Jeffcott rotorrdquo Journal of Vibration and Acoustics vol136 no 2 Article ID 021011 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

012

009

006

003

000

0 5 10 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 5 10 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 5 Damage detection by using the second natural frequency and 80 nodal displacements of the corresponding mode shape (a) case 1(b) case 2

012

009

006

003

000

0 105 15 20 25 30 35 40 45

Element number

Dam

age e

xten

t

(a)

020

015

010

005

000

0 105 15 20 25 30 35 40 45

Dam

age e

xten

t

Element number(b)

Figure 6 Damage detection by using the second natural frequency and 60 nodal displacements of the corresponding mode shape (a) case1 (b) case 2

Beam elements

Contact interface

Figure 7 Schematic diagram of heavy duty gas turbine and node dividing

Shock and Vibration 7

14E6

12E6

10E6

80E5

60E5

40E5

20E5

00

0 5 10 15 20 25 30 35 40

Element number

Bending moment

Bend

ing

mom

ent (

Nm

)

Critical bending moment

Figure 8 Bending moment on each interface of the rod-fastened-rotor at the second critical speed

025

020

015

010

005

000

0 5 10 15 20 25 30 35

Dam

age e

xten

t

Element number

Figure 9 Damage detection by only using the second-modalparameter of the rod-fastened-rotor

numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the secondnatural frequency and mode shape are available The resultsof the numerical examples indicate that the proposedmethodcan effectively identify single double and multiple damage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof China (no 51275385) and Major State Basic ResearchDevelopment Program of China (no 2011CB706505)

References

[1] W Fan and P Z Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011

[2] Z A Jassim N N Ali F Mustapha and N A Abdul Jalil ldquoAreview on the vibration analysis for a damage occurrence of acantilever beamrdquo Engineering Failure Analysis vol 31 pp 442ndash461 2013

[3] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[4] J-T Kim and N Stubbs ldquoCrack detection in beam-type struc-tures using frequency datardquo Journal of Sound and Vibration vol259 no 1 pp 145ndash160 2003

[5] G Y Xu W D Zhu and B H Emory ldquoExperimental andnumerical investigation of structural damage detection usingchanges in natural frequenciesrdquo Journal of Vibration and Acous-tics vol 129 no 6 pp 686ndash700 2007

[6] Z Y Shi S S Law and L M Zhang ldquoDamage localization bydirectly using incomplete mode shapesrdquo Journal of EngineeringMechanics vol 126 no 6 pp 656ndash660 2000

[7] P M Pawar K Venkatesulu Reddy and R Ganguli ldquoDamagedetection in beams using spatial fourier analysis and neuralnetworksrdquo Journal of IntelligentMaterial Systems and Structuresvol 18 no 4 pp 347ndash359 2007

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] Q W Yang and J K Liu ldquoDamage identification by theeigenparameter decomposition of structural flexibility changerdquoInternational Journal for Numerical Methods in Engineering vol78 no 4 pp 444ndash459 2009

[10] D Bernal and B Gunes ldquoFlexibility based approach for damagecharacterization benchmark applicationrdquo Journal of Engineer-ing Mechanics vol 130 no 1 pp 61ndash70 2004

[11] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers amp Structures vol 88 no 3-4 pp 154ndash1642010

[12] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[13] M Masoumi E Jamshidi and M Bamdad ldquoApplication ofgeneralized flexibility matrix in damage identification usingImperialist Competitive Algorithmrdquo KSCE Journal of CivilEngineering vol 19 no 4 pp 994ndash1001 2015

[14] W-J Yan and W-X Ren ldquoClosed-form modal flexibility sensi-tivity and its application to structural damage detection withoutmodal truncation errorrdquo Journal of Vibration and Control vol20 no 12 pp 1816ndash1830 2014

[15] M Montazer and S M Seyedpoor ldquoA new flexibility baseddamage index for damage detection of truss structuresrdquo Shockand Vibration vol 2014 Article ID 460692 12 pages 2014

[16] J Gao Q Yuan P Li Z Feng H Zhang and Z Lv ldquoEffectsof bending moments and pretightening forces on the flexuralstiffness of contact interfaces in rod-fastened rotorsrdquo Journal ofEngineering for Gas Turbines and Power vol 134 no 10 ArticleID 102503 2012

[17] J Li Z Li H Zhong and B Wu ldquoStructural damage detectionusing generalized flexibility matrix and changes in naturalfrequenciesrdquo AIAA Journal vol 50 no 5 pp 1072ndash1078 2012

8 Shock and Vibration

[18] M R N Shirazi H Mollamahmoudi and S SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applicationsvol 160 no 3 pp 1009ndash1019 2014

[19] Q Yuan J Gao and P Li ldquoNonlinear dynamics of the rod-fastened Jeffcott rotorrdquo Journal of Vibration and Acoustics vol136 no 2 Article ID 021011 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

14E6

12E6

10E6

80E5

60E5

40E5

20E5

00

0 5 10 15 20 25 30 35 40

Element number

Bending moment

Bend

ing

mom

ent (

Nm

)

Critical bending moment

Figure 8 Bending moment on each interface of the rod-fastened-rotor at the second critical speed

025

020

015

010

005

000

0 5 10 15 20 25 30 35

Dam

age e

xten

t

Element number

Figure 9 Damage detection by only using the second-modalparameter of the rod-fastened-rotor

numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the secondnatural frequency and mode shape are available The resultsof the numerical examples indicate that the proposedmethodcan effectively identify single double and multiple damage

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof China (no 51275385) and Major State Basic ResearchDevelopment Program of China (no 2011CB706505)

References

[1] W Fan and P Z Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011

[2] Z A Jassim N N Ali F Mustapha and N A Abdul Jalil ldquoAreview on the vibration analysis for a damage occurrence of acantilever beamrdquo Engineering Failure Analysis vol 31 pp 442ndash461 2013

[3] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[4] J-T Kim and N Stubbs ldquoCrack detection in beam-type struc-tures using frequency datardquo Journal of Sound and Vibration vol259 no 1 pp 145ndash160 2003

[5] G Y Xu W D Zhu and B H Emory ldquoExperimental andnumerical investigation of structural damage detection usingchanges in natural frequenciesrdquo Journal of Vibration and Acous-tics vol 129 no 6 pp 686ndash700 2007

[6] Z Y Shi S S Law and L M Zhang ldquoDamage localization bydirectly using incomplete mode shapesrdquo Journal of EngineeringMechanics vol 126 no 6 pp 656ndash660 2000

[7] P M Pawar K Venkatesulu Reddy and R Ganguli ldquoDamagedetection in beams using spatial fourier analysis and neuralnetworksrdquo Journal of IntelligentMaterial Systems and Structuresvol 18 no 4 pp 347ndash359 2007

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] Q W Yang and J K Liu ldquoDamage identification by theeigenparameter decomposition of structural flexibility changerdquoInternational Journal for Numerical Methods in Engineering vol78 no 4 pp 444ndash459 2009

[10] D Bernal and B Gunes ldquoFlexibility based approach for damagecharacterization benchmark applicationrdquo Journal of Engineer-ing Mechanics vol 130 no 1 pp 61ndash70 2004

[11] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers amp Structures vol 88 no 3-4 pp 154ndash1642010

[12] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[13] M Masoumi E Jamshidi and M Bamdad ldquoApplication ofgeneralized flexibility matrix in damage identification usingImperialist Competitive Algorithmrdquo KSCE Journal of CivilEngineering vol 19 no 4 pp 994ndash1001 2015

[14] W-J Yan and W-X Ren ldquoClosed-form modal flexibility sensi-tivity and its application to structural damage detection withoutmodal truncation errorrdquo Journal of Vibration and Control vol20 no 12 pp 1816ndash1830 2014

[15] M Montazer and S M Seyedpoor ldquoA new flexibility baseddamage index for damage detection of truss structuresrdquo Shockand Vibration vol 2014 Article ID 460692 12 pages 2014

[16] J Gao Q Yuan P Li Z Feng H Zhang and Z Lv ldquoEffectsof bending moments and pretightening forces on the flexuralstiffness of contact interfaces in rod-fastened rotorsrdquo Journal ofEngineering for Gas Turbines and Power vol 134 no 10 ArticleID 102503 2012

[17] J Li Z Li H Zhong and B Wu ldquoStructural damage detectionusing generalized flexibility matrix and changes in naturalfrequenciesrdquo AIAA Journal vol 50 no 5 pp 1072ndash1078 2012

8 Shock and Vibration

[18] M R N Shirazi H Mollamahmoudi and S SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applicationsvol 160 no 3 pp 1009ndash1019 2014

[19] Q Yuan J Gao and P Li ldquoNonlinear dynamics of the rod-fastened Jeffcott rotorrdquo Journal of Vibration and Acoustics vol136 no 2 Article ID 021011 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

[18] M R N Shirazi H Mollamahmoudi and S SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applicationsvol 160 no 3 pp 1009ndash1019 2014

[19] Q Yuan J Gao and P Li ldquoNonlinear dynamics of the rod-fastened Jeffcott rotorrdquo Journal of Vibration and Acoustics vol136 no 2 Article ID 021011 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of