[American Institute of Aeronautics and Astronautics 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit - Long Beach,CA,U.S.A. (20 April

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

A98-25334 AIAA^8-2100

AN EIGENSTRUCTURE ASSIGNMENT TECHNIQUE FORDAMAGE DETECTION IN ROTATING STRUCTURES *

Jason Kiddy *and Darryll Pines *Alfred Gessow Rotorcraft Center

University of Maryland, College Park, MD 20742-3015

Abstract

In this work, the detection and identification of damagein a helicopter rotor blade is considered. An eigenstruc-ture assignment technique is developed using measuredmodal test data and a finite element model of the bladeto detect and characterize the extent of damage in thesystem. Previous eigenstructure assignment techniqueshave not attempted to consider the centrifugal and gy-roscopic forces found in rotating systems. Additionalelemental stiffness matrix terms are generated for theinboard elements due to the centrifugal force causedby the mass of the outboard elements. This force cre-ates an enhanced sensitivity of the eigenstructure tomass changes thereby leading to an improved damagedetection capability. Feasibility of the methodology isdemonstrated analytically on a finite element model ofthe TH-55A main rotor blade.

Nomenclature

[C] =n x n Proportional Damping MatrixEIX =Structural Stiffness in Lag DirectionEIZ ^Structural Stiffness in Flap Direction[F] =n x p Control Influence MatrixFz =Distributed Force in z direction

=n x nSkew-Symmetric Gyroscopic Matrix=n x n Global Stiffness Matrix=Stiffness Matrix for Element i=n x n Centrifugal Stiffness Component=n x n Non-rotating Stiffness Component=n x n Global Mass Matrix=Mass Matrix for Element i

[Qij] =Combined Mass Influence MatrixR =Beam Radius[Rj] =jth Modal Residuala =Best Achievable Eigenvector AngleAj =jth Eigenvalue{({>}j =j*ft Mode Shape{(j)}j =jth Damaged Mode Shapeujj =jth Natural Frequencyft ^Rotation Rate

[K]CF

[K]NR

[M]

a, ^Damage in Stiffness of Element ibi =Damage in Mass of Element in =Number of Degrees of Freedomp =Number of Sensorstf, =Thickness of the Blade{u} =p x 1 Vector of Control Forcesv =Lag Displacementw —Flap Displacement{x} —n x 1 Physical Displacement Vector[Aij] =Stiffness Influence Matrix[ 5 ] f c =Mass Influence Matrix

Introduction

'Copyright ©1998 by Jason Kiddy and Darryll Pines. Pub-lished by the American Institute of Aeronautics and Astronau-tics, Inc. with permission

t Graduate Fellow, Aerospace Engineering Department, Stu-dent Member AIAA

^Assistant Professor, Aerospace Engineering Department, Se-nior Member AIAA

Motivation

The development of Health and Usage Monitoring Sys-tems (HUMS) for rotorcraft has been driven by a de-sire to reduce the operating costs associated with bothcivilian and military helicopters. Several recent stud-ies illustrate that installation of HUMS on civilian ormilitary helicopters may significantly reduce insuranceand maintenance costs while improving vehicle relia-bility and safety.1'2 The goal of HUMS is to providereal-time information regarding the health of criticalflight components.

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State of the Art

Interest in health monitoring and damage detection ofnon-rotating structures using modal test data has re-ceived a considerable amount of attention over the pasttwo decades with a significant increase in published re-ports appearing in the literature in the last 10 years.The basic idea is that modal parameters, notably fre-quencies, mode shapes, and modal damping are a func-tion of the physical properties of the structure (mass,stiffness, damping and boundary conditions). Withthis in mind, many researchers have developed severalpromising modal damage detection methods3"9 whichuse damaged eigenvectors and eigenvalues with an ini-tial undamaged model to detect and locate damage ina structure. A comprehensive literature review of thesemethods is given by'Doebling et al.10

Application to Rotating Structures

Most of the previously developed modal based damagedetection techniques have been applied to non-rotatingstructures such as ofT-shore oil platforms, bridges, andtrusses. There has been very little work reported inthe literature11"16 on the detection of damage in ro-tating structures using model-based damage detectionmethods. These structures possess unique characteris-tics such as centrifugal loading and Coriolis couplings.Thus, the dynamics are quite different than non-rotatingstructures. One of the earliest attempts at determin-ing the effect of damage in rotorcraft flight structureswas studied by Azzam11 using simulated math modelsof the rotor system. Faults were simulated and char-acterized to help determine their effect on the rotor-craft system response. However, no attempt was madeto automate the fault classification process. Followingthis modeling approach, Ganguli et al.12 applied finiteelement analysis to develop comprehensive mathemat-ical models of coupled rotor dynamics to simulate var-ious faults and damaged states in a rotor system. Thedamaged conditions that were simulated include pitchlink failures, moisture absorption, and trim mass loss.The effects of these damaged conditions on rotorcraftsystem loads were plotted as a function of harmonicsof the input RPM. Later, Ganguli et al.13 trained aneural network to map the input-output response sothat damage classification could be performed on thevarious simulated damage conditions. Although thismethod has many merits, it is limited to detecting onlyforms of damage that have been trained by the neuralnetwork model. In addition, a large amount of data isrequired to train the neural network to learn the sys-

tem model.

This paper develops an extension to previously de-veloped eigenstructure assignment based damage de-tection methods to account for rotational effects. Theextension developed in this paper attempts to accountfor mass damage which subsequently affects both thesystem's mass and stiffness matrices due to the cen-trifugal forces. Previous application of eigenstructureassignment methods to the damage detection problemdid not attempt to consider the effect of a mass dam-age on a structure's stiffness. This papers builds theanalytical framework for detecting damage in rotatingstructures using an eigenstructure assignment method-ology.

Sensors

, Sensor Data

Cockpit Warning

Diagnosis

Figure 1: Pictorial of a Real-Time Main Rotor BladeHealth Monitoring System

Development of EigenstructureAssignment Methodology

Model Development

Consider the continuum mechanics description of thecoupled flap-lag bending partial differential equationsfor a rotating beam.

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Flap bending:

d? I' d?w\ .. d (dw f1

-TV EIX -TV + mw ~ T -j- /dr2 V dr2 1 dr \ dr Jr

= FZ

mil

(I)

Lag bending:

-T-Tdr2.. rt-^r } + mv - mfdr2 J dr \^dr Jr

( dv di> dw d/u^\dr dr dr dr

m£l2pdp

A Closer inspection of these two equations revealsthat the first two terms of each equation are from thestandard (non-rotational) Bernoulli-Euler beam equa-tions. The second set of terms, which are proportionalto Q2, are centrifugal terms and the remaining termsare due to the Coriolis forces. Note that the Cori-olis terms generate gyroscopic coupling between theflap and lag degrees of freedom which then affects theoverall damping of the system. By linearizing and dis-cretizing the above equations into finite elements, theequation of motion for an 'n' DOF rotating structuralsystem is given by:

(3)

Figure 2: Finite Element Model (Flap DOF's Only)

Due to the impracticality of measuring the displace-ments of a rotating blade and the greater sensitivity ofstrain mode shapes to damage, a strain based formu-lation of Equation 4 is preferred for damage detection.The above model can be converted into a formulationwhich uses strain-based modal parameters. This alsoenables the use of smaller and less costly strain sensorsas transducers. To do this, the finite element modelof the system is converted from a displacement-basedmodel to a strain-based model.

By applying the partial differential operator -j^ onEquation 4 the FEM model is converted into a strainformulation.

where [M] is the n x n system mass matrix. [C] and[G(fl)j are the system proportional damping and skew-symmetric gyroscopic matrices. [K]NR and [K(£l)]CF

are the n x n system non-rotating and centrifugal stiff-ness matrices respectively, {x} is a physical displace-ment vector, {u} is a p x 1 vector of control forces and[F] is an n x p control influence matrix.

Neglecting the proportional and gyroscopic dampingterms permits the equation to be rewritten as:

[M] {x} + ((K}NR + (K(tl)]CF) {x} = (F}{u}(4)

((K] NR = tb

(5)

The mass and stiffness matrices are removed from thedifferential operators becuase they are not a function ofspatial position due to their discretization. It shouldalso be noted that the boundary conditions are nowstrain boundary conditions such that e = e' = 0 at thefree edge. At the rotor hub, the boundary conditionscan vary depending on what type of rotor blade systemis being modeled. An articulated or hinged rotor (e =0, e" = 0) or a hingeless/bearingless rotor (e = e' = 0)can be modeled.

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The eigenvalue problem for the rotating blade dynam- Substituting for the first order approximation to theics can be represented as: damaged matrices given in Equations 7-11 leads to the

following expansion of Equation 12.

((K] NR = 0 (6)

where w? = \j and <j>j are the jih eigenvalue and eigen-vector of the undamaged rotating system respectively.

Damage Approximation

N N

[K]

t=l k=i (13)

For a damaged blade structure, we can approximatemass and stiffness damage in terms of a first order Tay- By collecting undamaged and damaged terms, Equa-lor series expansion given by: tjon 13 can be rearranged as

N[M], =

Nrv*\NR _[K\d =

CF2

(7)

(8)

(9)

(10)

N

i=l

k=i

(14)

To simplify the above expression, define the matrixR as

(11)i=l k=i

, . , , ,where the scalars at and o, denote a percent damage. , , . , , , , . „ , , , ,, .41,associated with the stiffness and mass of the i™ struc-tural element. It is important to note here that thecentrifugal portion of the stiffness matrix has been di-vided into two sections. [.K"^)]^1 which is affected bya change of mass of a given element (bi), and [K(£l)]CF2

which is dependent on both the mass of that element,i, and each of the outboard elements (6*>j).

Damage Location

Formulating the eigenvalue problem for the damagedstructure leads to

- [K]) -i (15)

([K]d-X«[M}d) (12)

Now the structural stiffness influence matrix, [Aj,-l, and, , . a x - m i uthe mass influence matrix, .0,,-U, become

(16)

(17)

(18)

(19)

where [Aij] corresponds to the effect of damage associ-ated with the structural stiffness in element i to modej and [Bij\k corresponds to damage associated withmass of element k affecting element i and mode j.

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By substituting the relations defined in Equations 15through 19, Equation 14 can be rewritten as

{<(>}<* = [Amj}am{^ + (Qmj]bm{<t>}t (26)

N

(20)

Expanding the summation of Equation 20 out for allelements and collecting terms yields

([BIJ]N (21)

This equation can be written in matrix form as

f [Amj] [Qmj] ] { HS! 1 =L J I 0m\<PSj ) (27)

isThis equation can only be true if the vector {spanned by the matrix [Lmj\ = [ [Amj] [Qmj] ] . Eig-enstructure assignment shows that the eigenvector whichbest approximates the damaged eigenvector can be ob-tained from [Lmj] and its psuedo-inverse leading to

achieve _ rr 1+fj ~ [Lmj\ l (28)

For convenience, the following matrices can be dennedas

[Qij] = -[Bij]i (22)(23)

+ (B2j]N + ... + [BNj]N)(24)

where [Qij] refers to a mass damage in element i af-fecting mode j.

Equation 21 can now be written more compactly as

(25)

It is now necessary to determine how closely this 'bestachievable eigenvector' matches the damaged eigenvec-tor. To do this, the angle between the best achievableeigenvector and the damaged eigenvector can be com-puted.

a = cos"1.achieve . {(f)}d

(29)

If the best achievable eigenvector lies parallel to thedamaged eigenvector, this angle will be zero. If theangle is not zero, the damaged eigenvector does not liein the subspace spanned by [Ly] and hence that par-ticular element is not damaged. Through this method,it is possible to determine the damaged element by lo-cating the element, i, from which [Ly-] produces thesmallest angle a. Furthermore, the process can be re-peated for every mode, j. Ideally, a will be zero forthe damaged element for every mode.

Extent of Damage

Up to this point, there has been no restrictions made asto the number of damaged elements. However, if onlyone element, m, is considered damaged (i.e. a, = bi = 0for i 7^ m), the above equation becomes

Once the damaged element is determined, the extent ofdamage in that element can be calculated by rewritingEquation 27 in a slightly different form.

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(30)

Upon taking the pseudo-inverse of the first matrix, wecan obtain the extent of damage for a given element asthe following:

ambm (31)

To improve the conditioning of the pseudo-inverse, theabove equation can be rewritten to include the effect oftwo modes in the calculation of the extent of damage.The average of various combinations of modes can thenbe used to improve the estimate of am and bm.

il

the eigenvalues and eigenvectors of the 'damaged' bladeare calculated. These 'damaged' eigenvalues and eigen-vectors are used with the 'undamaged' finite elementmodel and the eigenstructure assignment approach tolocate and characterize the damage.

Tables 1 and 2 show the properties of the rotorblade used for this study. These properties correspondto the main rotor blade of a Hughes TH-55A as shownin Figures 3 and 4. Three different scenarios are consid-ered. The first is the perfect case where all degrees offreedom of the finite element model are measured andno noise is present. The second case examines the effectof the modal expansion process on the damage detec-tion algorithm. In this case, only the strain degreesof freedom are retained from the eigenvalue analysis ofthe 'damaged' finite element model. The reduced modeshapes are then expanded to generate the full degree offreedom mode shapes. The expanded mode shapes areused to characterize the damage. The final scenario isusing noisy data with reduced mode shapes. Randomnoise is simulated on the 'measured' eigenvalues andeigenvectors.

RadiusChord (c/R)

n1.848 m.0722 m

113.8 r ad /sec

Modal Expansion Table 1: Blade Reference Parameters

The above analysis assumes that all degrees of freedomare available to apply the eigenstructure assignmentdamage detection method. However, the internal shearforce degrees of freedom are not easily measured, espe-cially in a rotating environment. For this reason, it isnecessary to expand the measured portion of the modeshapes to include all degrees of freedom. In this paper,the modal expansion technique developed by Williamsand Green17 is employed to recover the internal shearforce degrees of freedom.

Element123456

Length (m).535.610.610.610.610.610

Mass (kg)3.4672.4592.4592.4592.4593.994

Stiffness (Nm2)299929992999299929992999

Table 2: Analytical Blade Elemental Properties

Simulation Results

To analytically validate the feasibility of this ex-tension to eigenstructure assignment for damage de-tection in rotating structures, a detailed finite elementmodel of a scaled rotor blade is considered. To simu-late damage, the mass and/or stiffness of one element isreduced. The finite element model is reassembled and

Figure 3: TH-55A Main Rotor Blade

For the following examples, the simulated damage isa 10% loss of mass and stiffness in the third element.

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Figure 4: Cross-Section of the TH-55A's NACA 0015Airfoil and Spar

The first four 'damaged' modes were used to calculatethe angle and extent of damage to each element. Thesemodes were used in pairs of two resulting in six differentcombinations (i.e. modes 1&2, 1&3, 1&4, 2&3, 2&4,and 3&4).

Case 1: Fully Populated, Clean ModeShapes

Non- Rotating

rupted with noise. The corresponding natural frequen-cies are listed in Table 4 and the results of the damagedetection are included in Table 5. Here, it can alsobe seen that the methodology correctly predicts thata = 0 for the third element. The algorithm also prop-erly determines the extent of damage for both massand stiffness. However, the algorithm determines thata = 0 for every element outboard of the third ele-ment. This effect is due to the [Qij] term. For thenon-rotating case, [Qij] simplifies to [By]j. [Qij] con-tains [Bij]i but it also contains extra terms due to thecentrifugal forces. For higher rotation rates, the cen-trifugal terms dominate thereby making [Qij] for aninboard element linearly dependant on the [Qij] for ev-ery element further outboard. As the distance from therotation axis increases, the rank of [Qij] also increases.Thus, there are more basis vectors to generate the 'bestachievable' eigenvector. If an inboard element is ableto produce a 'best achievable' eigenvector, so too arethe elements further outboard for a rotating blade.

Before validating the extension to the eigenstructureassignment methodology on a rotating blade, the methodwas first tested for a non-rotating blade. In this case,the extension to eigenstructure assignment condensesinto the form given by Lim and Kashangaki.8 Theaverage values of the angle and extent of damage areshown in Table 3. It can be seen that the location of thedamage was properly identified as the third element asit is the only element with an a = 0. Furthermore, theextent is properly identified by all six combinations ofmodes.

Elementa

StiffnessChange (%)

MassChange (%)

1.011

31.1

0.95

2.010

-8.18

-9.87

3.000

-10.0

-10.0

4.011

7.58

8.27

5.012

5.46

5.37

6.013

-0.69

14.9

Table 3: Damage Detection Results for 0 RPM andFully Populated, Clean Mode Shapes

Rotating

The next test of the methodology is for the same dam-age case but with the rotational effects included. Again,the mode shapes are fully populated and are not cor-

Mode No.1234

Undamaged15.60653.58992.315136.017

Damaged15.75853.51793.311136.573

Table 4: Undamaged and Damaged Natural Frequen-cies

ElementAngle

StiffnessChange (%)

MassChange (%)

1.017

43.1

-0.03

2.005

360

1.15

3.000

-10.0

-10.0

4.000

-0.36

1.32

5.000

61.4

12.4

6.000

9.38

-1.26

Table 5: Damage Detection Results with Rotation andFully Populated, Clean Mode Shapes

For this 'ideal' case, the presence of the additional zeroangles does not pose a problem as the element whichfirst shows a = 0 denotes the damaged element. How-ever, it will be shown that this is not viable for morerealistic situations when the modal data is lacking theinternal shear force degrees of freedom or is corruptedwith sensor noise.

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Case 2; Reduced, Clean Mode Shapes

The third test of the methodology is done by 'mea-suring' only the strain degrees of freedom of the modeshape. Information about the shear force degrees offreedom are recovered using modal expansion. Again,the average results are tabulated in Table 6

the damaged element will give a standard deviation ofzero. Realistically, the element with the lowest stan-dard deviation represents the damaged element. Forthe Case 2 data presented above, the standard devia-tions to the changes in mass and stiffness are shown inTable 7. Element 3 clearly shows the lowest standarddeviation and is hence the damaged element.

ElementAngle

StiffnessChange (%)

MassChange (%)

1.015

45.4

-3.23

2.007

401

1.59

3.001

-10.4

-10.0

4.000

-5.48

1.11

5.000

61.4

12.5

6.000

9.51

-1.31

Table 6: Damage Detection Results with Rotation andReduced, Clean Mode Shapes

ElementStiffness

DeviationMass

Deviation

1

3.90

.081

2

9.54

.284

3

.001

.000

4

3.65

.277

5

.719

.167

6

.2968

.0767

Table 7: Improved Damage Location Methodology Us-ing Standard Deviations

It is shown that the extent of damage in the third el-ement is still predicted at 10% with very little error.However, the angle criteria no longer gives a clear indi-cation of which element is actually damaged. In fact,the angles for elements 4, 5 and 6 are lower than theangle for element 3. Again this is due to the additionalrank of the [Qij] matrix for the outboard elements.Since the angle of the third element is no longer identi-cally zero due to errors in the modal expansion process,the outer elements are able to create a more accurate'best achievable' eigenvector and hence a lower angle.For this reason, the angle between the 'best achievable'eigenvector and the damaged eigenvector is not a goodindicator of the damage location for a rotating system.

Improved Damage Location Methodology

To compensate for this problem, a second method oflocating damage is developed using a statistical ap-proach. An examination of Equation 32 shows that it isvalid for any combination of modes (j,k) if the correctelement (m) is assumed to be damaged. Thus, Equa-tion 32 will yield the same extent of damage regardlessof the modes used in the calculation of am and bm.However, this will not be true if the incorrect elementis assumed to be damaged. In this case, each combi-nation of modes will provide a different solution. Thisfact can be used to provide information on the locationof damage. By considering the various combination ofmeasured modes, the standard deviation can be com-puted for the extent of damage values (am,bm). Ideally,

Case 3: Reduced, Noisy Mode Shapes

The final test is the case where the reduced mode shapesare contaminated with noise. For this simulation, 3%(la) Gaussian noise is applied to the 'damaged' modeshapes and natural frequencies. The result of this sim-ulation is shown in Table 8.

ElementStiffness

DeviationStiffness

Change(%)Mass

DeviationMass

Change(%)

1

4.49

83.3

.088

-2.74

2

15.6

24.3

.237

-18.2

3

.185

8.35

.002

-9.76

4

4.25

-43.1

.247

-1.24

5

.659

65.9

.188

11.3

6

.284

6.57

.083

0.13

Table 8: Damage Detection Results for Reduced, NoisyMode Shapes

It is clear from both standard deviations that the thirdelement is the damaged element. Furthermore, theaverage mass change properly identifies the extent ofdamage. However, the stiffness damage is shown tobe approximately an 8% increase instead of a 10% de-crease. This inaccuracy can partially be explained bythe insignificance of structural stiffness to the over-all stiffness of the blade at high rotation rates. Thecentrifugal stiffening effect contributes far more to the

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overall blade stiffness than the structural stiffness. There-fore, the mass damage, which decreases the centrifu-gal forces, is more accurately detected at high rotationrates.

Conclusions

A damage detection and characterization algorithmis developed for rotating blades. The algorithm is anextension to an eigenstructure assignment based algo-rithm which was developed for non-rotating structures.This extension accounts for the large centrifugal forcesfound in the rotating environment. By including thiseffect, an enhanced sensitivity to mass damage is ob-tained at the expense of a reduced sensitivity to struc-tural stiffness change.

The algorithm is validated using analytical data withencouraging results. However, the analytical study de-monstrated the need for an improved damage locationmethodology which is also developed. Furthermore,the methodology is demonstrated to be effective forboth reduced and noisy modes. The extension to eigen-structure assignment for damage detection in rotatingstructures appears to be a fast and reliable method foruse in helicopter main rotor blades.

Acknowledgements

This work was supported in part by the U.S. ArmyResearch Office under the Smart Structures UniversityResearch Initiative, contract no. DAAL03-92-G0121,with Dr. Gary Anderson serving as contract moni-tor, and in part by the National Rotorcraft Technol-ogy Center (NRTC), contract no. NCC2944, with Dr.Yung Yu serving as contract monitor

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[8] T.W. Lim, and T.A.L. Kashangaki, "StructuralDamage Detection of Space Truss Structures Us-ing Best Achievable Eigenvectors", AIAA Journal,Vol. 32, No. 5, May 1994, pp. 1049 - 1057.

[9] T.W. Lim, "Structural Damage Detection Using aConstrained Eigenstructure Assignment," Journalof Guidance, Control, and Dynamics, Vol. 18(3),1995, pp. 411-418.

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[17] E.J. Williams and J.S. Green, "A Spatial Curve-fitting Technique for Estimating Rotational De-grees of Freedom", Proceedings of the 8th AnnualIMAC, 1990, pp.376-381.

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Documents

[American Institute of Aeronautics and Astronautics 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit - Long Beach,CA,U.S.A. (20 April