Frequency measurement based damage detection

methods applied to different cracks configurations

Laxmikant Kannappan∗ and Krishnakumar Shankar†

University of New South Wales, Australian Defence Force Academy, Canberra, ACT, 2600, Australia

With advancements in sensor technology and structural health monitoring systems, theapplication of modal response measurements for online damage detection and assessmentappears more promising than ever. This paper examines the use of natural frequencymeasurements of the structure for detection of cracks in beams. Two different crack con-figurations are considered, partial thickness through width edge cracks and partial widththrough thickness centre cracks. Here, crack is modelled as a torsional spring whose stiff-ness is proportional to the size of the crack. Changes in modal frequencies depend on thelocation and the size of crack. With three or more measured natural frequencies, the loca-tion as well as the spring stiffness and thus the size of crack is determined. Two differentapproaches, equilibrium and energy, are used in determination of location of cracks andthe results are compared. Previous applications of the energy approach rely on analyticalmodelling of the structure to determine the mode shapes and modal strain energies. Thishas the disadvantage of being applicable only to simple structures with simple boundaryconditions whose mode shapes can be determined analytically. In the present work it isdemonstrated that discrete values of deflection mode shapes, which may be obtained frommeasurements on the structure, can be employed to determine the location and assess thesize of the damage. This makes the proposed technique suitable for possible extension tocomplex structures which are not amenable to analytical modelling. A method to assess thecrack size in elastic beams with through thickness-crack is also formulated. ExperimentalModal Analysis (EMA) was performed on beams with simulated edge and centre cracksand the measured frequency changes are used to determine the damage location and size.Good agreement is obtained between predicted and actual locations and sizes of cracks forboth crack configurations.

I. Introduction

With both Civilian and military operators seeking more cost effective and efficient means of aircraftmaintenance, the new approach to airframe structural maintenance involves implementation of on-line healthmonitoring systems, where the structure is monitored for damage continuously in real time. This reducesthe aircraft’s down time and offers further cost savings in terms of time, skilled labour and equipment re-quired for conventional non-destructive evaluation. Though techniques like use of fiber optic sensors andacoustic emission are being explored, the most promising technique for on-line health monitoring appearsto be the measurement of vibration parameters, due to their high reliability and sensitivity to initiationand progression of damage in the structure. It is well known that a crack or any damage in a structurechanges its dynamic characteristics, viz. natural frequencies, mode shapes and damping. The changes inthese dynamic properties depend on the location and size of damage. Hence, by monitoring the change inany or all of these parameters, damage can be characterised. Vibration techniques currently being developedfor structural health monitoring are based on the measurement of changes caused by the damage in eitherthe natural frequencies or the resonant mode shapes of the structure. Mode shape methods require arduousand meticulous measurement of displacements or accelerations of the undamaged and damaged structures,or of the structure before and after it is damaged. Usually the presence of a local defect produces only an

∗Graduate student, School of Aerospace, Civil and Mechanical Engineering, Northcott Drive, Australia.†Senior Lecturer, School of Aerospace, Civil and Mechanical Engineering, Northcott Drive, Australia.

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48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br> 15th23 - 26 April 2007, Honolulu, Hawaii

AIAA 2007-2058

Copyright © 2007 by Laxmikant Kannappan. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

insignificant variation in the overall mode shape, but a sharp discrepancy in the second derivative (curva-ture) due to its direct relation to the stiffness, which experiences a local discontinuity at the location ofthe defect. Hence damage detection using comparison of second derivatives is usually more efficient thanexamining differences in deflection values. A major disadvantage of damage assessment using mode shapeis that the measurements are prone to noise. This makes the accurate determination of second derivativesby numerical differentiation difficult. Secondly, differences in the values of mode shape deflections of thedamaged and undamaged structures is often of the same order of magnitude or lower than the noise level ineach measurement, making it difficult to identify small defects reliably. Further, variations in mode shapevalues caused by other factors such as changes in boundary conditions and environmental conditions, may bemore prominent than changes due to local damage, once again making the identification of damage difficult.On the other hand, frequency based damage detection methods are easier to implement in practice since theyonly require measurement of resonant frequencies which can be performed quickly using a single sensor1 ata single location in contrast to measurement of displacements or accelerations over the entire structure formode shape. Further, changes in modal frequencies can be recorded with greater accuracy and more reliablyas the effect of external noise is minimal in comparison to mode shapes. As a forward problem, Vandiver2

was the first to examine the change in the frequencies due to the presence of damage. The presence ofdamage in an offshore light station tower was identified using the first two bending modes and first torsionalmode. Cawley and Adams3 were the early researchers to solve the inverse problem of characterising thedamage from dynamic responses of the damaged structure.

There are two approaches in using frequency as a candidate for damage detection: numerical and analyti-cal. Numerical approaches include Finite Element Model Updating, where Finite Element Analysis (FEA) isperformed iteratively on a numerical model, with the stiffness of the elements in the model changed system-atically until the natural frequencies of the numerical model matches those measured on the real structure4

and FEA based sensitivity analysis where the sensitivity of the modal frequencies to induced damage arecomputed from the elements of the stiffness matrix of the model5 . In both cases the disadvantage is thatan accurate numerical model of the structure is needed, which is not always easy to generate. In the ana-lytical approach, the damage is modelled as a spring when formulating the vibration theory, assuming thatit produces a discontinuity in stiffness, i.e., a finite increment in the deflection gradient at the location ofthe damage. Rizos et al.6 modelled through width edge crack as a rotational spring, whose stiffness wasproportional to the extent of damage. Including the effect of the spring in the Euler-Bernoulli beam vibrationtheory, a characteristic equation was derived. They solved the inverse problem of finding the location ofcrack and spring stiffness using Newton-Raphson’s method. Liang et al.7 adopted the same methodologybut solved the inverse problem by separating the matrix of the characteristic equation, avoiding any non-linear solvers. Ostachowicz and Krawczuk8 derived the characteristic equation for beams containing multiplecracks. But the number of cracks present must be known a priori.

Energy based formulation was proposed by Liang and Hu9 for damage detection in beams containingmultiple edge cracks. They derived the relationship between changes in natural frequencies of a structure andchanges in strain energy stored in the structure, due to the occurrence of damage. The change in energy dueto a crack was obtained in terms of the energy stored in the rotational spring due to the finite increment inrotation at the location of crack. The energy stored in the undamaged structure was obtained by integrationof the curvature distribution on the structure. The curvatures were obtained from mode shape deflectionsof the undamaged structure, which were obtained by solving the governing equations with the appropriateboundary conditions. Though this methodology of crack detection is promising, its application to complexstructures is restricted since it requires the analytical solution of mode shapes for that structure.

To overcome this disadvantage, in the present work, numerical data of the undamaged structure’s modeshapes is employed to calculate the strain energies. Combining the use of mode shapes of undamagedstructure and the measured changes in frequencies obtained from experiments, this new hybrid techniqueretains the reliability and accuracy of the frequency based methods, while eliminating the need for analyticalor numerical modelling to determine the mode shapes. It is to be noted that unlike traditional mode shapetechniques which rely on differences between the measured mode shapes of the undamaged and damagedstructures, in the present method we use only the mode shapes from the undamaged (or the damaged)structure to determine the strain energy of vibration. As the differences in mode shapes caused by thedamage are no longer required, the mode shapes can be smoothed without fear that the damage signaturewill be lost in the process.

The classic problem of identifying the size of edge cracks from the spring stiffness has been addressed by

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several researchers7–10 . Kannappan et al.11,12 extended the rotational spring model with force equilibriumequations to through thickness centre cracks and using numerical simulation and experimental testing showedthat even with small recorded changes in natural frequencies, cracks can be located as well as sized. Thenatural frequencies of the undamaged and damaged beam are obtained from experiments and numericalmode shapes are obtained from FEM. In this paper, damage detection by both equilibrium and the hybrid

energy method is carried out to find the location and size of cracks in cantilever beams containing edge andcentre cracks. Good results are obtained in both cases.

II. Damage location determination

Cracks can be modelled as discontinuity in stiffness which again can be represented by a massless,rotational spring (Fig. [1]) with stiffness, kr, which is proportional to the size of the crack. Where everthere is a crack, the beam is segmented, but connected by the spring. The deflection, curvature, bendingmoment and shear force distributions remain continuous across the damage location, but the the presenceof the spring introduces a finite increment in rotation and thus the gradient of the deflection of the beam.This is represented as

Slope,∂X2

∂β=∂X1

∂β+

1

K

∂2X2

∂2β(1)

where, X1 and X2 are the deflection functions before and after the crack location, xc is the location of thecrack, L is the length of the beam, β=x/L is the normalised co-ordinate along the location of the beam andK is the non-dimensionalised spring stiffness derived as K = krL/EI. The same methodology is adopted forbeams containing either edge or centre cracks.

Figure 1. Crack represented as torsional spring

A. Equilibrium method

The concept of representing a crack as a rotational spring is applied to the Euler-Bernoulli beam vibrationtheory. Here, in this study, the theory is applied to beam with fixed-free boundary condition. At the fixedend, the deflection and slope are zero and at the free end, the bending moment and shear force are zero13 .According to Euler-Bernoulli vibration theory, the governing equation for transverse beam vibration is

EIδ4y

δx4= −ρAδ

2y

δt2(2)

Since the beam is segmented along the crack location, the deflection of the two segments to the left andright of the crack satisfying the governing differential equation can be assumed to be of the form

X1(x) = a1sinλx+ a2cosλx+ a3sinhλx+ a4coshλx (3)

X2(x) = b1sinλx+ b2cosλx+ b3sinhλx+ b4coshλx (4)

Here, λ is the normalised natural frequency of the damaged beam defined as λ4 = ω2

c2 , ω is the natural

frequency (Hz) of the damaged beam and c =√

EIρA . The coefficients a1 to a4 and b1 to b4 are to be

determined by substituting the eight boundary conditions, the conditions of zero deflection and slope at thefixed end, vanishing shear force and bending moment at the free end, continuity in deflection, shear force

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and bending moments across the crack and discontinuity in gradient deflection across the crack given byEqn. [1]. A characteristic equation is obtained substituting these conditions in Eqns. [3] and [4] which canbe written as

λ∆1 +K∆2 = 0 (5)

∆1 and ∆2 are shown in Eqns. [A-2] and [A-3] in the appendix.If the damage location and size are known, the frequency of transverse vibration of the damaged beam

corresponding to any mode can be determined from Eqn. [5]. In case of damage assessment, the inverseproblem of characterising the damage from the measured changes in natural frequencies has to be solved.The methodology for this is explained in Section. C.

B. Energy method

Since equilibrium method cannot be directly extended for damage detection in plates or complex structuresor for multiple crack detection, an alternate formulation based on energy distribution of the structures isused for damage detection. Using perturbation theory, Gudmundson14 derived the relation between theeigen frequency changes and the strain energy in uncracked and cracked structure as

ω′2n

ω2n

= 1 − ∆Un

U0n(6)

where,ωn is nth mode natural frequency of the uncracked structure,ω′

n is nth mode natural frequency of the cracked structure,∆Un is the increase in nth mode strain energy due to the finite increase in rotation at the crack, equal

to the strain energy stored in the spring, and,U0n is the strain energy of the undamaged structure in nth mode.A first order approximation of Eqn. [6] yields

∆ωn

ωn=

1

2

∆Un

U0n(7)

where, ∆ωn = ωn − ω′

n.The strain energy of the uncracked structure, U 0n, is given by

U0 = L

∫ 1

0

ψ(β)dβ (8)

The increase in strain energy because of the presence of crack can be obtained by Castigliano’s theoremas

∆U =EIψ

kr(9)

Here kr is the spring stiffness (moment per unit rotation), E is the Young’s modulus, I is the secondmoment of inertia and ψ is the strain energy per unit length specific to each mode ’n’. For beam structuresψ can be computed as

ψn(β) =1

2EI

[

φ′′

n(β)]2

(10)

where, φ′′

n is the second derivative of the nth mode deflection mode shape, φn.Substituting Eqn. [8] and Eqn. [9] in Eqn. [7]

∆ωn

ωn=

1

2K

ψn(β)∫ 1

0ψn(β)dβ

(11)

The strain energies of the undamaged beam and that stored in the rotational spring are calculated,from Eqns. [8] and [9], using the undamaged beam’s mode shapes. Previous studies based on analyticalformulation obtained the mode shapes by solving the governing differential equation of the beam with theappropriate boundary conditions using symbolic computation9 or manually15 . Here, it is proposed thatthe strain energies can be calculated from the measured mode shapes of either the damaged or undamagedstructure avoiding complex analytical derivations. In this paper this hypothesis is validated by employingmode shapes obtained from experiments simulated using finite element modelling (FEA).

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C. Solution to the inverse problem

0 0.2 0.4 0.6 0.8 1

0

200

400

600

800

1000

1200

β (x/L)K

Mode1Mode2Mode3

β=0.417b/w=0.2

Figure 2. Typical K vs β plot

To solve the inverse problem, the naturalfrequencies are extracted from the undam-aged and damaged structure. In equilibriummethod, the measured natural frequenciesof the cracked structure are scaled by the ra-tio of the natural frequencies obtained fromtheory to those measured experimentally onthe uncracked structure. The scaled fre-quency values are employed in Eqn. [5] andthe non-dimensional spring stiffness values,K, are calculated as a function of the nor-malised beam coordinate for each mode. Inthe energy method, the normalised (∆ω/ω)change in frequency values are employed inEqn. [11] to determine the spring stiffnessK as a function of for each mode. Since themagnitude of the spring stiffness K (whichis governed by the severity of the crack) andthe actual crack location are independent ofthe vibration mode, their values can be ob-tained by triangulation. Plots of K versusβ for different modes will all pass thorough a common point if there is a damage. The abscissa and ordinateof the point of intersection of the curves, respectively, give the location and the spring stiffness, which isindicative of the size of the damage. A typical K vs β curve is shown in Fig. [2].

III. Damage size determination

A. Edge cracks

Stiffness of the spring, which was used to represent the crack, was analytically modelled such that it wasproportional to the extent of damage. To quantitatively represent the extent of damage, a relation betweenthe spring stiffness and crack length for beams with part thickness through width cracks (edge cracks),vibrating transversely as shown in Fig. [3], has been derived by a few researchers6,8, 16 .

Figure 3. Transverse vibration of beam with edge crack

Ostcahowicz and Krawczuk8 derived the relation as

K =L

6πhf(a/h)(12)

where, f(a/h) = 0.64(a/h)2 − 1.035(a/h)3 + 3.72(a/h)4 − 5.18(a/h)5 + 7.55(a/h)6 − 7.33(a/h)7 + 2.5(a/h)8

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B. Centre cracks

As opposed to edge crack detection, not much attention has been given to the determination of crack lengthfor structures with through thickness, part width, center cracks Fig. [4]. There is no current equationrelating spring stiffness and crack sizes for this crack configuration which, in this paper, is accomplished.

If KI is the Stress Intensity Factor (SIF), E the Young’s modulus and A the area of the crack, then, thechange in elastic deformation energy due to the crack, as given by17 , is

∆U =1

E

∫ A

K2I dA (13)

The change in strain energy can also be defined as

∆U =1

2Mθ =

1

2

M2

kr(14)

where, θ is the finite increase in rotation due to the applied bending moment, M.

Figure 4. Transverse vibration of beam with centre crack

Boduroglu and Erdogan18 have derived the SIF equations as

KI =3M

wh2f(γ)

√b (15)

where, w and h are semi-width and thickness of the beam respectively, γ=b/w, b being semi-crack lengthand f(γ) is the finite width correction factor which is a function of the semi-width to thickness ratio (w/h).The SIF equation including finite width correction factor, f(γ), for different ratios of w/h are derived inBoduroglu and Erdogan18 and for thin plates, f(γ) can be obtained from19,20 . After incorporating Eqn.[15] in Eqn. [13]

∆U =9M2

Ew2h4

∫ A

0

f(γ)2bdA =9M2b2

Ew2h3g(γ) (16)

Comparing Eqns. [16] and [14]

Spring constant, K =1

3

L

w

1

γ2g(γ)(17)

and

g(γ)w/h=7.8 = 110.7γ10 + 63.8γ9 − 329.1γ8 + 110.2γ7 + 282.4γ6 −343.3γ5 + 200γ4 − 82.5γ3 + 25.8γ2 − 5.1γ + 1

(18)

Note from Eqn. [17] that the non-dimensionalised spring stiffness K is independent of the Young’sModulus of the beam, and is a function of the length to width ratio (L/w), width to thickness ratio (w/h)as well as the crack-length to width ratio (b/w).

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IV. Experimental results

Experimental Modal Analysis (EMA) was conducted on beams with different crack configurations andfrom the experimentally extracted frequencies, damage was detected using both equilibrium and energyformulations.

A. Edge cracks

For damage detection in edge cracks, EMA was conducted on steel beams. The dimensions of the beam were500mm X 10mm X 24.9mm and density and Young’s modulus of the beam were measured experimentallyas 7808Kg/m3 and 192.29GPa respectively. Simulated part thickness through width, edge cracks wereintroduced in the beams using wire-cut electrical discharge machining (EDM) process. Ten different damagecases were studied for possible damage detection, varying the locations and sizes slots. Slot widths as smallas 0.2 mm were introduced in each of the beams. A rigid fixture was designed to simulate the beam in fixed-free boundary condition. Experiments were done using B&K 4374 miniature accelerometers to measure theacceleration signal and StarModal software was used to extract the first three natural frequencies of thebeam [Table. 1]. The obtained natural frequencies are used to predict the location and size of crack usingboth equilibrium and energy method.

Once the location and sizes of the cracks are deduced, error is calculated as the difference between thepredicted and actual value expressed as a percentage of the beam length or beam width when calculatingcrack location error or crack size error respectively. This helps reduce the multiplicity when comparing withvery small actual values12 .

Table 1. Measured natural frequencies of undamaged and damaged beam with edge cracks

Location of crack Crack length Mode1 Mode2 Mode3

βc (a/h)

Undamaged - - 77.68 482.59 1338.71

Case1 0.05 0.08 76.75 479.79 1331.38

Case2 0.05 0.16 75.34 477.46 1326.41

Case3 0.05 0.24 73.77 475.06 1319.91

Case4 0.05 0.32 70.86 470.12 1314.72

Case5 0.25 0.16 76.38 481.51 1322.93

Case6 0.25 0.24 75.19 480.84 1310.02

Case7 0.25 0.32 74.55 480.09 1294.59

Case8 0.5 0.16 77.10 477.30 1335.30

Case9 0.5 0.24 76.71 471.98 1332.78

Case10 0.5 0.32 75.70 461.55 1324.68

1. Equilibrium Method

The extracted natural frequencies are input into Eqn. [5]. The spring stiffness, K, values are calculated foreach mode along the length of the beam depending on the how many segments the beam is divided into.Here the beam is divided into 1000 segments and the time to calculate K for three modes and 1000 segmentsis very less (0.25 seconds). Once the K values are calculated, they are plotted along the length of the beamfor all modes. Since the spring will have the same stiffness irrespective of the mode of vibration, the pointof intersection of the curves gives the location and spring stiffness. If the curves do not meet at a point, thecentroid of the triangle formed by the intersection of the curves is assumed to be the intersection point21

. The same procedure is repeated for all damage cases and the location of damage and spring stiffness iscalculated. From spring stiffness, using Eqn. [12], the crack length can be determined. The results from theequilibrium method are shown in Table. [2].

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Table 2. Comparison of actual and predicted crack locations and crack sizes (Equilibrium method - Edgecracks)

CaseNo.

Location of crack, βc Crack length, a/h

Actual Predicted % Error Actual Predicted % Error K

1 0.05 0.044 0.6 0.08 0.099 1.9 195.98

2 0.05 0.060 1.0 0.16 0.152 0.8 83.07

3 0.05 0.061 1.1 0.24 0.190 5.0 54.84

4 0.05 0.073 2.3 0.32 0.280 4.0 25.29

5 0.25 0.270 2.0 0.16 0.186 2.6 55.87

6 0.25 0.265 1.5 0.24 0.257 1.7 28.95

7 0.25 0.271 2.1 0.32 0.310 1.1 19.75

8 0.5 0.430 7.0 0.16 0.183 2.2 58.11

9 0.5 0.434 6.6 0.24 0.247 0.7 31.43

10 0.5 0.430 6.9 0.32 0.347 2.7 15.04

2. Energy Method

In energy based formulation for damage detection, the spring stiffness K is obtained using Eqn. [11]. Thestrain energies for the beam are calculated using Eqn. [10]. Due to the disadvantages in the method adoptedby previous studies by Liang and Hu9 , and Patil and Maiti15,22 , where it is time consuming or analyticallyintensive, the strain energies are calculated numerically using the mode shapes. In this paper simulated modeshapes obtained from finite element analysis is used. This method can thus be adopted for plate structuresor complex structures without having to analytically model the structure to obtain its mode shape equation.Also the time needed to calculate the strain energies is reduced because the large time taken to symbolicallycompute them is totally avoided. In this study, the mode shapes of the first three modes of the cantileverbeam are extracted from commercial FEA software, ANSYS 10. Just with deflections in each mode for 20points along the length of the beam and measured natural frequencies from experiments, the strain energiesare calculated in each mode.

Thus the spring stiffness, K, is also calculated for each point on the beam for the first three modes.Hereagain, triangulation of K Vs β curves is performed to extract the damage parameters. From the K Vsβ curves adopting the same procedure as in equilibrium method, the locations and spring stiffnesses arededuced for all damage cases. Using Eqn. [12], the crack lengths are calculated from the spring stiffnesses.The results are shown in Table. [3].

Table 3. Comparison of actual and predicted crack locations and crack sizes (Energy method - Edge cracks)

CaseNo.

Location of crack, βc Crack length, a/h

Actual Predicted % Error Actual Predicted % Error K

1 0.05 0.065 1.5 0.08 0.110 3.0 154.912

2 0.05 0.076 2.6 0.16 0.167 0.7 69.181

3 0.05 0.078 2.8 0.24 0.210 3.0 43.991

4 0.05 0.086 3.6 0.32 0.286 3.4 23.136

5 0.25 0.273 2.3 0.16 0.197 3.7 50.236

6 0.25 0.266 1.6 0.24 0.266 2.6 26.886

7 0.25 0.280 3.0 0.32 0.316 0.5 18.641

8 0.50 0.423 7.7 0.16 0.182 2.2 58.874

9 0.50 0.430 7.0 0.24 0.242 0.2 32.882

10 0.50 0.423 7.7 0.32 0.334 1.4 16.383

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B. Centre cracks

EMA is conducted on a Aluminium beam (Al-7075) whose dimensions were 600mm X 50.15mm X 3.2mm.The density and Young’s modulus of the material were determined experimentally as 2772.53Kg/m3 and69.6GPa respectively. Here, through thickness part width center cracks are simulated by introducing slots aswide as 0.08mm using die-sink EDM. The same procedure as in the previous section was followed to extractthe first three natural frequencies of the beam and are shown in Table. [4].

Table 4. Measured natural frequencies of undamaged and damaged beam with centre cracks

Location of crack Crack length Mode1 Mode2 Mode3

βc γ = b/w

Undamaged beam - - 7.151 45.074 126.468

Case 1 0.167 0.1 7.1362 45.0420 126.3540

Case 2 0.167 0.2 7.1169 45.0255 126.3501

Case 3 0.167 0.4 7.0608 44.9749 126.2384

Case 4 0.250 0.4 7.0811 45.0092 125.3861

Case 5 0.333 0.4 7.0924 44.8872 125.2820

Case 6 0.417 0.2 7.1370 44.9256 126.1980

Case 7 0.417 0.4 7.1078 44.7371 126.1283

1. Equilibrium Method

Using the natural frequencies of the undamaged and damaged beam obtained from experiments and substi-tuting in Eqn. [5], K vs β curves were obtained and thus the location and stiffness of the spring representingthe crack size. Here, crack length is determined from the spring stiffness using Eqn. [17]. The results areshown in Table. [5].

Table 5. Comparison of actual and predicted crack locations and crack sizes (Equilibrium method - Centrecracks)

CaseNo.

Location of crack, βc Crack length, γ

Actual Predicted % Error Actual Predicted % Error K

1 0.167 0.163 0.4 0.1 0.132 3.2 715.79

2 0.167 0.153 1.4 0.2 0.250 4.9 227.00

3 0.167 0.161 0.6 0.4 0.431 3.1 78.04

4 0.250 0.274 2.4 0.4 0.412 1.2 85.37

5 0.333 0.348 1.4 0.4 0.447 4.7 72.57

6 0.417 0.400 1.7 0.2 0.256 5.6 215.91

7 0.417 0.416 0.1 0.4 0.429 2.9 78.89

2. Energy Method

Adopting the procedure as explained for edge crack detection using energy method, K is calculated. Usingthe experimentally measured frequencies of the beam and mode shapes from numerical modelling from whichstrain energies are calculated in each mode, the variation of spring stiffness, K, along the length of the beamis calculated. Thus from the obtained K vs β plots, the location and spring stiffnesses are calculated. FromEqn. [17], the crack length is deduced from the spring stiffnesses. The results are shown in Table. [6].

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Table 6. Comparison of actual and predicted crack locations and crack sizes (Energy method - Centre cracks)

CaseNo.

Location of crack, βc Crack length, γ

Actual Predicted % Error Actual Predicted % Error K

1 0.167 0.163 0.3 0.1 0.174 7.4 436.41

2 0.167 0.160 0.7 0.2 0.265 6.5 201.27

3 0.167 0.161 0.6 0.4 0.431 3.1 78.04

4 0.250 0.274 2.4 0.4 0.412 1.2 85.38

5 0.333 0.317 1.7 0.4 0.419 1.9 82.71

6 0.417 0.400 1.7 0.2 0.256 5.6 215.83

7 0.417 0.416 0.1 0.4 0.428 2.8 79.08

V. Conclusion

In this paper, both equilibrium and energy formulation based methods were applied to the measuredfrequency data obtained from experiments on beams containing different crack configurations. ∆1 and∆2 derived using equilibrium method is specific for that boundary conditions and similarly mode shapeequations used to calculate strain energies are also specific to the boundary conditions. A method to avoidthe dependency of the algorithm on the boundary conditions is proposed in this paper. The strain energiescan be calculated by measuring the mode shapes of the structure. Since this method does not depend onthe difference between the mode shapes of the structure before and after damage, the mode shapes of eitherundamaged or damaged structure can be smoothed without having to fear the loss of the damage signature.This technique also gives the advantage of extending the strain energy based damage detection method toplates and more complex structures.

From experiments, the order of error between the predicted and actual location and sizes were found tobe similar in both the detection methods. The maximum error in predicting the locations was 7.7% in caseof edge cracks and 2.4% in case of centre cracks. Both these damage detection methods were applied tobeams containing through width and part thickness, edge cracks and part width through thickness, centrecracks. A new methodology was proposed to calculate the crack length in case of beams containing centrecracks. The maximum error in predicting the sizes was 3.7% in case of edge cracks and 7.4% in case of centrecracks.

Appendix

At fixed end,

Deflection, X1 = 0

Slope,δX1

δx= 0

At free end,

Bending moment, EIδ2X2

δx2= 0

Shear force,δ

δx

[

EIδ2X2

δx2

]

= 0 (A-1)

Along the crack location,

Deflection, X1 = X2

Bending moment, EIδ2X1

δx2= EI

δ2X2

δx2

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Shear force,δ

δx

[

EIδ2X1

δx2

]

=δ

δx

[

EIδ2X2

δx2

]

Slope,∂X2

∂x=

∂X1

∂x+EI

kr

∂2X2

∂2xor

Slope,∂X2

∂β=

∂X1

∂β+

1

K

∂2X2

∂2β

∆1 and ∆2 are obtained by substituting the boundary conditions (shown in [A-1]) in Eqns. [3] and [4].These ∆1 and ∆2 are specific to fixed-free boundary conditions.

|∆1| =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

1 1 0 0 0 0 0 0

0 0 1 1 0 0 0 0

0 0 0 0 −cosλ coshλ −sinλ sinhλ

0 0 0 0 sinλ sinhλ −cosλ coshλ

cosα coshα sinα sinhα −cosα −coshα −sinα −sinhα−cosα coshα −sinα sinhα cosα −coshα sinα −sinhαsinα sinhα −cosα coshα −sinα −sinhα cosα −coshα−sinα sinhα cosα coshα sinα −sinhα −cosα −coshα

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(A-2)

|∆2| =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

1 1 0 0 0 0 0 0

0 0 1 1 0 0 0 0

0 0 0 0 −cosλ coshλ −sinλ sinhλ

0 0 0 0 sinλ sinhλ −cosλ coshλ

cosα coshα sinα sinhα −cosα −coshα −sinα −sinhα−cosα coshα −sinα sinhα cosα −coshα sinα −sinhαsinα sinhα −cosα coshα −sinα −sinhα cosα −coshα−cosα coshα −sinα sinhα 0 0 0 0

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(A-3)

where, α = λβ

References

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