Petroleum Institute, Abu Dhabi, United Arab Emirates
Joshua Gaerke†, Phillip Cooley† and Joseph C. Slater‡
Wright State University, Dayton, OH 45435
The goal of this research is to further develop a Structural Health Monitoring (SHM)system for detection of fatigue cracks in blades and bladed disks using harmonic responsescaused by crack nonlinearities. Driving weakly nonlinear systems, such as a cracked struc-
ture, results in the generation of harmonics. When these harmonics coincide with a res-onance, the resulting information can be used to identify the location of the crack. Thispaper presents analytical results obtained for a simple beam model illustrating the phe-nomenon. The results suggest that superharmonic resonances caused by weak nonlinearityare a suitable crack detection feature.
I. Introduction
A means of an automated Structural Health Monitoring (SHM) system for turbomachinery componentswould greatly decrease the cost of maintenance inspections and increase the likeliness of structural failureprevention. Degradation in structures generally involves corrosion, cracking, delamination of compositelayers, and loosened fasteners. In the case of turbine engine disks, blades, and other components, a SHMsystem would be greatly advantageous for the purposes of fatigue crack detection and evaluation. Fatiguecracks are one of the most common causes of failures in turbomachinery (and in many other mechanicalsystems) and yet one of the hardest types of damage to detect. Due to the large amount of kinetic energystored in the moving parts of turbomachinery, fatigue cracks can cause uncontained failures, which may havecatastrophic consequences. Debris flying out of the engines may damage vital aircraft systems leading toloss of control and a consequent crash,1 or fatalities on board the aircraft after penetrating the fuselage.2
Therefore, development of reliable techniques for detection of damage is highly critical.New sensor and actuator capabilities have helped in the development of damage detection. Damage
detection can be performed optically, or using x-ray, ultrasound, and vibration methods. Vibration charac-teristics of structures, however, has struggled in its development due to focus on behaviors that are sensitiveto effects other than damage, such as thermal effects. Currently, vibration techniques for damage detectioninclude using time, frequency, and modal concepts,3–6 most of which do not exploit the inherent nonlinearbehavior of fatigue cracks which would help distinguish damage from other benign phenomenon. In many
instances linear vibration testing for damage detection is not sensitive enough to the very localized behav-ior that comes from hairline fatigue cracks. Mode shapes and especially natural frequencies can be veryinsensitive to fatigue cracks until the damage is too far along.
Unlike many types of damage, fatigue cracks are typically hard to find by inspection due to little if anymaterial being lost. They do become observable, however, when looking at the nonlinear elastic behaviorthat results from the opening and closing of the crack. This nonlinear behavior is very localized and actslike a bilinear local modulus when strained. This nonlinearity is well known to be true of beams with cracks
∗Assistant Professor, Petroleum Institute, AIAA Member†Research Assistant, Department of Mechanical and Materials Engineering.‡Professor, Department of Mechanical and Materials Engineering, [email protected]. AIAA Associate Fellow
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also.7, 8 Shiryayev and Slater have exploited this phenomena using changes in statistics of the Randomdecsignatures cause by the onset of nonlinearity due to a crack. This method was developed and then verifiedexperimentally.9–11 The SHM community has also investigated other nonlinear response features (see forexample12–16).
Previous work by Meier, Shiryayev and Slater17,18 identified candidate features that can be used todetect fatigue cracks in bladed fan or turbine disks from vibration data. In this document we describe theextension of the crack detection approach by considering a spectral fingerprint of the structure obtainedby continuously changing the excitation frequency. We describe further investigations of these frequency
response features utilizing a model of a beam. We consider a simple structure instead of a full FE modelof a hypothetical compressor disk18 due to its computational efficiency. This is a necessary step towardsdeveloping effective automated monitoring techniques.
II. Nonlinear Response Features Resulting From Harmonic Excitations
The bilinear type weak nonlinearity that occurs due to the presence of an opening and closing crack canbe roughly approximated by a system that has a quadratic stiffness term in addition to the linear one asshown in Equation (1). Higher polynomial terms can also be included to improve the approximation. Onlyterms that are even functions will have non-zero coefficients.
y + 2ζω0 y + ω20y + αy2 = F (t) (1)
In this equation ω0 is the natural frequency of the linear system, and α is small relative to ω20. One of the
differences between responses of linear and nonlinear systems due to harmonic excitations is that responsesof nonlinear systems may contain harmonics other than those at the driving frequencies. Nayfeh and Mook19
provide a detailed analysis of systems with quadratic and cubic nonlinearities under harmonic excitations.If the system with quadratic nonlinearity is excited by a two term harmonic excitation such as in Equation(2),
F (t) = F 1(t) + F 2(t) = A1 cos(Ω1t + Θ1) + A2 cos(Ω2t + Θ2) (2)
then besides the primary resonance (Ω1 or Ω2 close to ω0) several interesting phenomena can be observedin the response depending on the relationships between driving frequencies Ω1 and Ω2, as well as the linearnatural frequency of the system ω0.
• Superharmonic resonance can be excited if Ω1 or Ω2 ≈ 1/2ω0.
• Subharmonic resonance can be excited when Ω1 or Ω2 ≈ 2ω0.
• Combination resonance occurs when Ω1 + Ω2 ≈ ω0, or Ω1 − Ω2 ≈ ω0.
An alternative approach to account for the nonlinearity introduced by a crack is to consider the forcesresulting from the crack motion directly. When a structure is excited, it experiences alternating states of tension and compression. Since a crack cannot support a tensile load, it opens as the tension grows in thatsection of the structure and closes when tension reverts to compression. As the crack closes, a graduallyincreasing compressive force is applied to the structure. When subject to sinusoidal excitation, his non-linearbehavior is best approximated by the half-wave rectified sinusoid shown in Equation 3. The peaks in the firstplot of Figure 1 correspond to compression and the flat regions correspond to tension. The Fourier seriescoefficients are shown in the stem plot in Figure 1 and indicate that the presence of a crack will induce asignificant excitation at twice the driving frequency. The zeroth term and fourth term will also amplify theresponse but may not be significant enough to be useful.
F c(t) = A sin(Ω1t) + abs (A sin(Ω1t))
2 =
A
π +
A
2 sin(Ω1t) −
∞n=2
2A
π(n2 − 1) cos(nΩ1t) (3)
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This phenomenon will cause a resonant response to occur when a cracked beam is excited at one half a natural frequency and accounts for the additional peak for the cracked beam in Figure 2. Contrary toanticipated, but not observed, behavior in previous work by Meier et al., it should be noted in Figure 1that there is no term corresponding to half of the driving frequency. This implies that there will be nosubharmonic resonance for a breathing crack despite the prediction of such by the quadratic stiffness model.Further, there is no n = 3 harmonic, but there is a small n = 4 harmonic that is unlikely to be useful.
0 50 100 150 200 250−80
−60
−40
−20
0
Frequency, Hz
A B S f f t , d B
Linear Beam
0 50 100 150 200 250−80
−60
−40
−20
0
Frequency, Hz
A B S f f t , d B
Cracked Beam
Figure 2. Combination resonance excited by two driving functions.
The amplitude of the superharmonic excitation introduced by the crack depends on several factors. Thefirst is the location of the excitation. Away from resonances, operational deflection shapes (responses toharmonic excitations) are not close to mode shapes. Instead they are linear combinations of mode shapes.Further, the shapes are highly dependent on location of excitation, as illustrated by the sensitivity of anti-resonances to excitation location. As a result, moving the excitation has a strong direct impact on the strainamplitude in the vicinity of the crack. As the strain amplitude in the vicinity of the crack is directly related to
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i Y i(x)q i(t) is the sum of modal components representingdisplacement with respect to the spring hinge. Substituting Eq. (5) into Eq. (4), dividing by Aρ andrearranging gives
i
Y i(x)q i(t) + Dc
Aρ
i
Y i(x)q i(t) + E I
Aρ
i
d4Y i(x)
dx4 q i(t) = −w(t) −
Dc
Aρ w(t) (6)
In this derivation we assume the mode shapes are similar to the undamped case and are written as
Y i(x) = a1i cos β ix + a2i sin β ix + a3ie−βix + a4ie
βix (7)
One can verify that d4Y i(x)dx4 = β 4i Y i(x), then:
i
Y i(x)q i(t) + Dc
Aρ
i
Y i(x)q i(t) + E I
Aρ
i
β 4i Y i(x)q i(t) = −w(t) − Dc
Aρ w(t) (8)
Figure 4. Spring hinge beam configuration.
The eigenvalue problem for the clamped-free beam can be formulated by considering the boundaryconditions (see Fig. 4).
at x = 0 : Y i(0) = 0, EI d2Y i(0)
dx2 = K c
dY i(0)
dx (9)
at x = L : d2
Y i(0)dx2
= 0, d3
Y i(0)dx3
= 0 (10)
where K c is the torsional stiffness of the spring hinge. A set of four equations can be obtained from conditionsin Eqs. (9) and (10). Characteristic equation is obtained by setting the determinant of the resulting matrixto zero. Solution of the characteristic equation was calculated using Matlab.25 The resulting mode shapesare expressed as:
Y i(x) = ci
− (1 + i)cos β ix +
2β iEI
K c+ 1 +
2β iEI
K c− 1
i
sin β ix
+ e−βix + ieβix
(11)
where coefficients i are defined as
i =
2βiEI K c
(sin β iL + cos β iL) + 2sin β iL
2βiEI K c
(− sin β iL − cos β iL) + 2 cos β iL + 2eβiL
(12)
Coefficients β i are related to the natural frequencies as ωi = β 2i
EI /(Aρ). Constant ci is chosen to normalizethe mode shapes L
0
AρY 2i (x)dx = 1 (13)
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Utilizing orthogonality of the modes, we multiply Eq. (6) by AρY i(x) and integrate over the length of thebeam, which results in the set of modal equations shown below.
q i(t) + Dc
Aρ q i(t) +
E I
Aρβ 4i q i(t) =
L0
AρY i(x)dx
−w(t) −
Dc
Aρ w(t)
(14)
In a more common form Eq. (14) can be rewritten as
q i(t) + 2ζ iωi q i(t) + ω
2
i q i(t) = L
0 AρY i(x)dx·
(−
w(t)−
2ζ iωi w(t)) (15)
where ζ i and ωi are the corresponding modal damping ratio and natural frequency. Note that the right-hand-side of Eq. (15) represents modal forcing, which is dependent on the motion of the clamp.
III.B. Open crack case
When the crack is open (see Fig. 5), the beam is divided into two continuous segments connected by atorsional spring as shown in Fig. 6. According to Sundermeyer and Weaver,26 the stiffness of the torsionalspring K T is calculated based on the concepts of linear fracture mechanics and Castigliano’s theorem as
K T = Ebh2
72πF 1(ac/h) (16)
where F 1
(ac
/h) is the shape factor that depends on geometry and loading. The shape factor is calculated as
F 1(ac/h) = 19.6(ac/h)10 − 40.69(ac/h)9 + 47.04(ac/h)8 − 32.99(ac/h)7
+ 20.29(ac/h)6 − 9.975(ac/h)5 + 4.602(ac/h)4 (17)
− 1.047(ac/h)3 + 0.6294(ac/h)2
Figure 5. Geometry of a transverse edge crack.
Figure 6. Representation of the beam with an open crack.
The mode shapes for the two beam segments in Fig. 6 are written as
Y 1i(x1) = a1i sin β ix1 + a2i cos β ix1 + a3ie−βix1 + a4ie
βix1 (18)
Y 2i(x2) = b1i sin β ix2 + b2i cos β ix2 + b3ie−βix2 + b4ie
βix2 (19)
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The eigenvalue problem is formed in order to calculate the mode shapes and natural frequencies byconsidering boundary conditions and continuity conditions at the location of the crack. At the clamped end(x1 = 0) and the free end (x2 = 0):
at x1 = 0 : Y 1(0) = 0, dY 1(0)
dx1= 0 (20)
at x2 = 0 : d2Y 2(0)
dx22
= 0, d3Y 2(0)
dx32
= 0 (21)
Continuity of displacement, shear force and bending moment is enforced at the crack location (x1 = X c,x2 = L − X c):
Y 1(X c) = Y 2(L − X c) (22)
d3Y 1(X c)
dx31
= −d3Y 2(L − X c)
dx32
(23)
d2Y 1(X c)
dx21
= −K T EI
dY 1(X c)
dx1+
d Y 2(L − X c)
dx2
(24)
d2Y 2(L − X c)
dx22
= −K T EI
dY 1(X c)
dx1+
d Y 2(L − X c)
dx2
(25)
Eqs. (20-25) are written in matrix form and the characteristic equation is formed by equating the determinantto zero. Solution of characteristic equation is calculated numerically in Matlab. The resulting mode shapesare expressed as
Y 1i = δ i
−Ai cos β ix +
2 −Ai
1 − 2βiEI
K c
sin β ix + e−βix +
(Ai − 1) eβix
, 0 < x < X c (26)
Y 2i = δ iαi
−B i cos β i(L − x) − (B i + 2) sin β i(L − x) + e−βi(L−x) −
(B i + 1)eβi(L−x), X c x < L (27)
where coefficients αi are expressed as
αi =−Ai cos β iX c +
2 −Ai
1 − 2βiEI
K c
sin β iX c + e−βiXc + (Ai − 1) eβiXc
−B i cos β i(L − X c) − (B i + 2)sin β i(L − X c) + e−βi(L−Xc) − (B i + 1)eβi(L−Xc) (28)
Coefficients Ai and B i are expressed as shown in the appendix, and δ i are normalization constants such thatEq. (29) holds. Xc
0
AρY 21i(x)dx +
LXc
AρY 22i(x)dx = 1 (29)
The parameters for the beam were chosen to replicate the experimental setup that was used in previouswork.17 The beam was made of 2024 aluminum (E = 70 GPa, ρ = 2700 kg/m3), with a free length of 0.66 m,
and a rectangular cross-section of 0.025×0.0125 m. The distance from the clamp to the crack was X c = 0.257m. The stiffness of the torsional spring K t that represents flexibility of the shaker head assembly has beenvaried to achieve natural frequencies that are close to those observed in the past series of experiments.17 Themodels use the modal damping ratios obtained from the experimental data. In Table 1, a large difference isobserved between the model and experimental value at the 1st natural frequency. However, higher modeshave frequency values that match experimental data to within 2%. In this work we are mostly concernedwith higher modes because they are the ones that are most affected by the presence of cracks.
Figure 7 illustrates the mode shapes obtained for the the open and closed crack submodels. As expectedthe first several modes are mostly unaffected by the presence of the crack. However, modes 4 and 5 havemore noticeable differences between the baseline and the open crack states.
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Mode Experimental value Baseline beam Cracked beam Damping ratio
(cracked beam) model model, a/h=0.2
1 15.20 22.56 22.52 0.0401
2 139.0 142.2 141.8 0.0083
3 399.9 399.9 399.2 0.0018
4 770.6 786.8 786.0 0.0012
Table 1. Natural frequencies (Hz) and damping ratios of experimental setup and beam models.
The numerical algorithm for the simulation of the cracked beam model utilizes two submodels withinwhich the beam behaves linearly. It is the switching between two different linear submodels that is responsbilefor the bilinear behavior of the system. The state of the crack is checked at every time step based on thecurvature of the beam near the crack. The curvature of the beam near the crack is calculated based on thedisplacements of two points on each side of the crack and the point at the crack location. A coordinatetransformation is performed to maintain continuity of displacement and velocity fields at the intstants whenthe state of the crack changes.
IV. Results
In this work we compare the data obtained from the model of the baseline healthy beam and the modelof the cracked beam. We consider exciting superharmonic resonance near the third mode of the structure atapproximately 400 Hz, hence the driving frequency is varied from 195 Hz to 205 Hz with an increment of 0.5Hz. Excitation to the structure is provided in terms of acceleration of the shaker head with an amplitude of 40 m/s2. For each excitation frequency, a 5 seconds long time history displacement data has been obtainedfor the point 5 mm away from the free end of the beam. The response reaches the steady state withinthe first 3 seconds, hence for further analysis we considered the data in the range of [4, 5] seconds. Thedisplacement response data was windowed using a Hann window and the FFT was calculated to obtain thefrequency content of the response.
Figure 8 demonstrates the spectral contents of the responses obtained from the models of healthy beamand cracked beams with the crack depths of 10% and 20% of the cross-section height respectively. The dataobtained from the model of the healthy beam represents typical spectral content of a linear system excitatedaway from resonances. There is a signle well-defined peak at the excitation frequency, which could be easily
observed on both the waterfall and contour plots.The waterfall and contour plots in Figures 8(c)-8(f) suggest that response data obtained from cracked
beam models contains additional harmonics besides the one at the excitation frequency. There are clearpeaks at 2× and 4× the excitation frequency. There is also a spectral line at 3× the excitation frequency,but it is not nearly as well-defined compared to the ones at even multiples of excitation frequency.
From Figure 8 it is difficult to observe if the superharmonic resonance has been excited when the driving
frequency passed through the region near 0.5ω(3)n at approximately 200 Hz. In order to observe excitation of
superharmonic resonances, magnitudes of the harmonics at 2× and 4× the excitation frequency are plottedin Figure 10 versus the excitation frequency. As the driving frequency passes through the region near 200Hz, the magnitude of the 2× harmonic is about 10 dB larger than compared to its values when the drivingfrequency is 195 Hz or 205 Hz. A similar observation can be made for the magnitude of the 4× harmonic.To consider this further, another case was evaluated from 95 Hz to 105 Hz to observe the impact on the4× harmonic when it coincides with a natural frequency. As seen in Figure 9, the 4× harmonic is moreprominent than the 2× under these circumstances and reaches a maximum as the driving frequency passesthrough one quarter the natural frequency. These observations suggest that the superharmonic resonancescan be observed in the system with bilinear stiffness characteristic. Also, as the crack becomes larger themagnitudes of both harmonics increase substantially. The magnitudes of 2× and 4× harmonics for the beamwith a crack 20% deep are about 15 dB higher than those from the beam with a crack that is 10% deep.
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In this work we considered a spectral fingerprint of the structure for detection of fatigue cracks. Thisfingerprint can be obtained by exciting the structure away from any resonances. The fingerprint of a linearundamaged structure contains a single well-defined harmonic due to excitation. The fingerprints of a crackedbeam contain additional easily observed harmonics at even multiples of excitation frequency. The magnitudeof these harmonics increases by almost an order of magnitude when excitation is near half of a specific naturalfrequency, targeting superhamonic resonance condition. In practical setting, such spectral fingerprints can
be easily obtained for the structure of interest.Future work will focus on experimental implementation of this method. Initially, we will focus on val-
idating analytical results obtained from the beam models using a setup similar to the model described inthis paper: the beam mounted in a clamp on a shaker. Ideally, a non-contact sensor will be used in theexperiment.
VI. Appendix
Coefficients Ai and B i in Eq. (26, 27, 28, 29) are expressed as
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