Detection of multiple cracks in cantilever beams using frequency
measurements
Kannappan, L., Shankar, K., Sreenatha, A.G.
School of Aerospace, Civil and Mechanical Engineering,
University of New South Wales, Australian Defence Force Academy,
Northcott drive, Canberra, ACT- 2600, Australia.
Phone: +61 2 6268 8258 Fax: +61 2 6268 8276 email: [email protected]
ABSTRACT
This paper examines the use of natural frequency measurements for detection and assessment of multiple cracks in beams
using the energy approach. The damage is modelled as a spring whose stiffness is inversely proportional to the damage size.
The previous applications of the energy approach in this regard have relied on analytical modelling of the structure to
determine the vibration mode shapes. This has the disadvantage of being applicable only to simple structures with simple
boundary conditions. It is proposed in the present work that discrete values of structure deflections, such as those measured
by experimental modal analysis or obtained by finite element modelling, can be employed to determine the location and
assess the size of the damage. The applicability of this approach is demonstrated through a numerical study, employing mode
shapes obtained by finite element analysis to determine locations and sizes of multiple through thickness centre cracks in
cantilever beams with greater than 95 percent accuracy. This paves the way for implementation of the technique to more
complex structures without the need to develop or employ complex analytical models.
Introduction
With both Civilian and military operators seeking more cost effective and efficient means of aircraft maintenance, the new
approach to airframe structural maintenance involves implementation of on-line health monitoring systems, where the structure
is monitored for damage continuously in real time. This reduces the aircraft’s down time assuring an uninterrupted operation
and further reduces the need for skilled people involved in conventional non-destructive evaluation. Though techniques like
use of fiber optic sensors and acoustic emission are being explored, the most promising technique for on-line health
monitoring is the measurement of vibration parameters, because they are highly sensitive to the initiation and progression of
damage in the structure. It is quite well known that a crack or any damage in a structure changes its dynamic characteristics,
viz. natural frequencies, mode shapes and damping. The changes in these dynamic properties depend on the location and
size of damage. Hence, by monitoring the change in any or all of these parameters, damage can be characterised. Frequency
can be measured with least uncertainity and high repeatability, using just one sensor [1], compared to mode shapes, which is
highly dependent on the number of sensors used and distribution of sensors.
One of the earliest applications of the vibration method was reported by [2], where damage was detected in an off-shore light
structure. In applying frequency measurements to crack detection, changes in measured frequencies due to the damage in the
structure have to be compared to predicted frequency changes, which may be obtained either by numerical simulation or
mathematical modelling. In the latter method, cracks are modelled as torsional springs whose stiffness kr is inversely
proportional to the extent of damage, and account for the discontinuity in slope at the location of the damage. This concept has
been studied by [3, 4] employing solution of the governing equation of equilibrium to determine the modal shape function for
the beam, and by [5, 6, 7], who use the energy approach which has the advantage of being applicable to detection of multiple
cracks.
In the energy approach, the key parameter representing the ratio of the energy of the spring to the energy of the beam, g( ), is
calculated from the mode shapes of the undamaged beam, which are obtained from Euler-Bernoulli beam vibration theory with
the appropriate boundary conditions, either using symbolic math [5, 6] or manual solution [7]. In either case analytical
modelling of the beam and its boundary conditions is necessary to obtain the mode shapes, which is not easily applicable to
real complex structures.
In the present paper, it is proposed that mode shapes measured by experimental modal analysis of the undamaged structure
can be employed to determine the strain energy parameter. This has the advantage that mathematical modelling of the
structure is not necessary, making it viable for application to real complex structures. At the same time, since the method does
not rely on measured differences in mode shapes or its derivatives as in traditional damage detection techniques using mode
shapes, but only on the mode shape of the undamaged structure, it is not susceptible to uncertainties due to noise or changes
in boundary or environmental conditions between measurements taken on the undamaged and damaged structures. The
method is in essence a hybrid of the frequency measurement and mode shape techniques, since it uses measured frequency
changes and measured mode shapes of the undamaged beam, but does not require numerical or mathematical modelling of
the structure and its boundary conditions. The applicability of this hybrid approach is verified by identifying the locations and
assessing the lengths of the two centre cracks in a cantilever beam. For validation purposes, the frequencies of the damaged
and undamaged beams and the mode shapes of the latter are obtained by numerical simulation using commercial Finite
Element Analysis (FEA) software, ANSYS 10.
THEORETICAL FORMULATION USING SPRING MODEL FOR CRACK
The crack is modelled as a discontinuity in stiffness represented by a massless, rotational spring with stiffness, kr, which is
inversely proportional to the size of crack [4]. In the present case, the crack is assumed to be full depth (through thickness)
extending over part of the beam width located at the centre (Figure. [1]). It is assumed that, wherever there is a crack, the
beam is segmented, but connected by the spring.
Cantilever beam with multiple surface cracks
kr1
kr2
kr3
Figure 1: Crack represented as a rotational spring
From Castigliano’s theorem, the finite change in bending rotation, , due to the presence of crack is given by
θ =
∂u
∂M
(1)
where, u is the strain energy because of the presence of crack and M is the applied bending moment.
The flexibility, C, because of the crack is given as
C=
1 ∂θ
∂ 2u
=
=
k r ∂M ∂M 2
(2)
Integrating Equation. [2],
u=
M2
2k r
(3)
The relation between the strain energy/unit length, , and bending moment can also be expressed as
ψ =
M2
2EI
M 2 = 2EIψ
(4)
Substituting for M in Equation. [3]
Strain energy, u=
EIψ
kr
(5)
Using perturbation theory, Gudmundson [8] derived the relation between the eigen values representing frequencies and the
strain energies of the uncracked and cracked structure as
ωn'2
u
= 1− n
2
ωn
u0 n
where,
´n
(6)
= nth mode natural frequency of the cracked structure,
n = nth mode natural frequency of the uncracked structure ,
un = increase nth mode strain energy due to the finite bending at the crack, equal to the strain
energy stored in the spring, given by Equation.[ 5], and,
u0n = is the strain energy of the undamaged structure in nth mode.
A first order approximation of Equation. [6] yields
∆ω n
=
ωn
un
2u0 n
(7)
The strain energy of the uncracked structure, u0n, in case of beam structures is given by
1
u0 n = L ψ ( β )d β
(8)
0
Substituting Equations. [5] and [8] in Equation. [7]
∆ω n
ωn
=
gn (β )
2K
where,
gn (β ) =
1
(9)
ψ n (β )
(10)
ψ n ( β )d β
0
and,
If
n
K=
kr L
EI
th
is the mode shape of the beam in its n mode, the equation for
1
2
(11)
n
can be obtained as
ψ n ( β ) = EIφn''2 ( β )
(12)
For a beam with multiple cracks, according to linear superposition principle, Equation. [9] can be written as
∆ω n
ωn
{
nX 1
= H }nXm {S}mX 1
(13)
where, n is number of modal frequencies used, m is number of beam segments, {S} is the damage parameter proportional to
the flexibility of each segment, Sm and
H=
n
i =1
gn ( βi )
Here, [6, 7] used the mode shape equations obtained from basic vibration theory, to compute the values of specific strain
energy
n( ) in Equation. [12], where the former obtains them using symbolic computation and the latter derives them
manually. In the present approach, the curvature terms on the right hand side of Equation. [10] are obtained by numerical
differentiation of ”measured” mode shapes, whose values, in this case, are provided by the FEA. The damage parameter
vector [S] is calculated from Equation. [13] using pseudo-inverse technique and K Vs is plotted for every mode. This process
is repeated for all segments with non-zero S value. Since the location of the crack and the value of the spring stiffness, which
is a function of crack size, have to be unique, all the K vs curves should pass through a common point on the K, space.
Thus the location of the crack and spring stiffness K are determined by the point of intersection of the K vs curves of the
different modes.
DETERMINATION OF CRACK LENGTH
The relation between the bending spring constant, kr and extent of crack in case of beams containing through width partial
thickness cracks (a/h) has been derived earlier [3, 9]. Here a similar methodology is applied to determine the crack size from
the spring stiffness by equating the spring energy to the energy stored in the crack, which is related to the Stress Intensity
Factor (SIF) at the crack tip.
If KI is the SIF, E the Young’s modulus and A the area of the crack, then, the change in elastic deformation energy due to the
crack, as given by [10], is
∆U =
A
E
(14)
K I dA
The SIF equations including the finite width correction factor, f( ), for different width to thickness ratios derived by Boduroglu
and Erdogan [11] is expressed as
KI =
Mo
M
f (γ ) b =
f (γ ) b
t
wt
(15)
where, Mo is the bending moment/unit width, t is the thickness of the beam and f( )w/t=7.8 =25.77 5+6.81
3.83 +1.01.
4
-32.83
3
+18.3
2
-
After incorporating Equation. 15 in Equation. 14
∆U =
A
9M
f (γ )2 b 2dA
Ew 2t 4 0
(16)
The change in strain energy can also be defined as
∆U =
Mθ =
M
kr
(17)
Comparing Equations. 16 and 17
Spring constant,kr =
Et
γ g (γ )
g(γ )w / t = . =110.7γ 10 +63.8γ 9 -329.1γ ^8 +110.2γ 7 +282.4γ 6 -343.3γ 5 +200γ 4 -82.5γ 3 +
(18)
25.8γ 2 -5.1γ +1
In Equations. (15,16,17,18), =b/w, where ’b’ is the semi-crack length and ’w’ is the half width of the beam. Equation. 18 can be
solved to obtain b/w from known values of kr.
NUMERICAL STUDY
The above explained theory is applied to frequency values of cantilever beams containing two cracks. The frequency values
were obtained from commercial Finite Element Analysis (FEA) software, ANSYS 10. For this numerical study, a beam 600mm
long, 50mm wide and 3.2mm thick is considered. It is assumed to be made of Aluminium, Al-7076, with Young’s modulus
69.6GPa and density 2.77X10−6kg/mm3. Cracks of different sizes are modelled in two different locations on the beam. The
first 6 natural frequency values, shown in Table. 1, are input to the damage detection algorithm. All computations including
calculation of g( ) are carried out using MATLAB.
In this damage detection technique, the beam is divided into m segments. Referring to Equation. 13, using the percentage
changes in each frequency values due to the presence of crack, the damage parameter vector {S} is calculated. Now, the
percentage change in frequency is recalculated based on each of the obtained damage parameter, Sm. In our case, since the
beam contains two cracks, only two of the m segments contain a non-zero damage parameter and thus two sets of percentage
change in frequency values are be obtained. Using this new percentage change in frequency values and Equation. [9], K Vs
curves are obtained for each of the segments identified as containing damage.
Table 1: Natural frequencies (in Hz.) of undamaged and damaged beam
Mode1
Mode2
Mode3
Mode4
Mode5
Mode6
Undamaged
beam
7.2453
45.3956
127.1966
249.6108
413.4333
619.0174
Case1: 1=0.1; 1=0.5;
2=0.5; 2=0.5 ;
7.0794
44.4391
126.9938
246.3295
412.5512
607.568
Case2: 1=0.1; 1=0.2;
2=0.5; 2=0.5;
7.2031
44.733
127.1689
246.3453
413.2966
610.5985
Case3: 1=0.1; 1=0.5;
2=0.5; 2=0.2;
7.0983
44.9736
126.9958
249.1464
412.5699
614.2713
Case4: 1=0.05; 1=0.5;
2=0.6; 2=0.3;
7.0769
44.5277
125.6498
247.997
410.5875
618.3205
Case5: 1=0.05; 1=0.2;
2=0.65; 2=0.4;
7.2192
45.0171
126.061
249.3991
410.8582
615.4631
Case6: 1=0.05; 1=0.3;
2=0.45; 2=0.4;
7.1729
44.8379
126.6495
247.685
411.6215
616.8088
Case7: 1=0.15; 1=0.4;
2=0.55; 2=0.5;
7.155
44.7015
126.8996
246.5063
408.6098
611.5779
Case8: 1=0.15; 1=0.6;
2=0.6; 2=0.4;
7.0513
44.8794
126.5848
247.3496
403.9866
608.919
Case9: 1=0.2; 1=0.3;
2=0.5; 2=0.6;
7.1768
44.4517
127.0313
243.8333
412.0974
606.5559
Case10: 1=0.2; 1=0.5;
2=0.5; 2=0.3;
7.1348
45.1883
126.6941
245.8816
409.561
614.7293
In this study, the beam is divided into 10 segments and the above mentioned procedure is followed to obtain K Vs curves.
Irrespective of the mode of vibration used, the spring which was modelled to represent the crack must be of the same stiffness.
Hence, the location and spring stiffness is deduced from the point of intersection of K Vs curves of all modes. From spring
stiffness, the crack length is the calculated using Equation. [18]. A comparison of the actual and predicted location as well as
b/w ratio is shown in Table 2. Here, error is calculated as the difference between the predicted and actual value expressed as
a percentage of the beam length and beam width when calculating crack location error and crack size error respectively. This
helps reduce the multiplicity when comparing with very small actual values [12]. Typical K Vs curves are also shown in
Figure. 2.
Table 2: Comparison of actual and predicted crack locations and crack sizes
Test
case
1
1
0.1
0.5
2
0.1
3
0.1
4
0.05
0.5
0.6
0.3
0.045
0.545
0.615
0.302
0.5
4.5
1.5
0.2
5
0.05
0.2
0.65
0.4
0.045
0.195
0.615
0.346
0.5
0.5
3.5
5.4
6
0.05
0.3
0.45
0.4
0.045
0.330
0.505
0.369
0.5
3.0
5.5
3.1
7
0.15
0.4
0.55
0.5
0.114
0.373
0.516
0.502
3.6
2.7
3.5
0.2
8
0.15
0.6
0.6
0.4
0.165
0.620
0.615
0.444
1.5
1.9
1.5
4.4
9
0.2
0.3
0.5
0.6
0.205
0.310
0.506
0.623
0.5
1.0
0.6
2.3
10
0.2
0.5
0.5
0.3
0.204
0.550
0.505
0.333
0.4
5.0
0.5
3.3
Actual data
1
2
Predicted data
1
2
2
1
0.5
0.5
0.095
0.551
0.2
0.5
0.5
0.114
0.5
0.5
0.2
0.114
% Error
1
2
2
1
2
0.506
0.551
0.5
5.1
0.6
5.1
0.229
0.514
0.552
1.4
2.9
1.4
5.2
0.546
0.514
0.250
1.4
4.6
1.4
5.0
K Vs β
400
350
250
250
200
200
150
150
100
100
50
0
Case 1
β2=0.5
γ2=0.5
300
K
K
350
Case 1
β1=0.1
γ1=0.5
300
K Vs β
400
50
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
β
0
0.45
0.46
0.47
0.48
0.49
0.5
0.51
0.52
0.53
0.54
0.55
β
Mode 1
Figure 2: K Vs
Mode 2
Mode 3
curve for case1
Conclusion
A method to predict the location and size of cracks in cantilever beam using energy formulation has been presented. This
method employs measured natural frequencies of the structure before and after damage and the mode shapes of the
undamaged structure which can be obtained from measurements or FEA. Thus, this method can be easily adopted for
damage detection in real complex structures. The accuracy of the method in characterising the damage is demonstrated with
results obtained from numerical simulation using FEA. The maximum error in prediction of location and crack length is 5%. The
method is currently being extended to detection of damage in plate structures.
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