# Differential Evolution: An Inverse Approach for Crack Detection

Authors: Prabir Kumar Jena; Dhirendra Nath Thatoi; Dayal R. Parhi
Journal: Advances in Acoustics and Vibration
 (2013)

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## Abstract

This paper presents a damage detection technique combining analytical and experimental investigations on a cantilever aluminium alloy beam with a transverse surface crack. Firstly, the first three natural frequencies were determined using analytical methods based on strain energy release rate. Secondly, an experimental method was adopted to validate the theoretical findings. The damage location and severity assessment is the third stage and is formulated as a constrained optimisation problem and solved using the proposed differential evolution (DE) algorithm based on the measured and calculated first three natural frequencies as inputs. Numerical simulation studies indicate that the proposed method is robust and can be used effectively in structural health monitoring (SHM) applications.


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## Body

## 1. Introduction

A crack is a potential source of catastrophic failure in structures. Extensive investigations by researchers have been done to develop structural integrity monitoring techniques. Vibration measurement and analysis being an effective and convenient way to detect cracks in structures is mostly being used for development of various such techniques.Several nondestructive techniques (NDT) are available for local damage detection [1] using experimental methods like radiography, the magnetic field method, the acoustic method, and so forth. However, for health monitoring of critical and complex structures, in such experimental methods which require prior knowledge of the damage vicinity, accurate predictions may not be suitable. This has led to the development of quantitative global damage detection methods which are based on modal analysis [2, 3]. Researchers [4] also argue that in view of prohibitive costs and efforts involved in predicting damage to a high level accuracy a better idea is to roughly locate damage in the structure and then use standard NDT methods for closer analysis of the damaged area.Recently a lot of work has been done using modal analysis to detect, locate, and predict crack severity to a greater accuracy level. Dimarogonas [2] in 1996 provided a comprehensive review of vibration based mode shape analysis followed by some recent reviews [4–7]. The damage detection problem can be defined as a nonlinear inverse problem [3]. In conventional model-based detection methods, the minimization of an objective function is defined in terms of the differences between the vibration data obtained by modal testing and those computed from the analytical model. These conventional optimization methods are gradient based and usually lead to a local minimum only. For a more accurate and reliable solution, global optimization techniques are developed such as genetic algorithms (GAs) [8, 9], artificial neural networks (ANNs) [10, 11], fuzzy logic [12, 13], particle swarm optimization (PSO) [14–19], and some hybrid methods like genetic fuzzy systems [20]. All these approaches have their strengths and weaknesses. However, there is a growing preference for the soft computing approaches based on dfferential evolution (DE) [21, 22] due to faster and accurate predictions of global optimization to the objective function. A differential quadrature method combined with an evolutionary optimization algorithm has been proposed for crack detection in cylindrical shell structures. The circumferential crack, which is assumed to be open, is modeled by the extended rotational spring. For acceptability of this method an experimental analysis is also carried out [23]. A comparison has been made between the coupled local minimizers (CLM) method and the differential evolution (DE) algorithm to perform FE model updating for the damage detection in a cracked beam. CLM method is a gradient-based method with multiple local optimization runs whereas the DE algorithm is a direct search approach which uses a population of solution vectors collecting the design parameters [24].In the current work, a systematic procedure has been developed to calculate the natural frequencies and mode shapes of a cracked cantilever beam with a transverse crack. The process endeavours to study the influence of a crack on natural frequencies and mode shapes. For different relative crack depths and locations, natural frequencies and the mode shapes of the cracked cantilever beam specimen have been found out using a theoretical method.The rest of this paper is organized as follows. Section2 introduces the theoretical analysis of a cracked beam structure. The experimental results are presented in Section 3, followed by Section 4 which includes the proposed DE algorithm. The results and discussions as well as conclusions are discussed in Sections 5 and 6, respectively.


## 2. Mathematical Formulation

### 2.1. Computation of Flexibility Matrix of a Damaged Beam Subjected to Complex Loading

A beam with cracks has smaller stiffness than that of a normal beam. This decreased local stiffness can be formulated as a matrix. The dimension of the matrix would depend on the degrees of freedom in the problem. Figure1 shows a cantilever beam of width W and height T, having a transverse surface crack of depth b1. The beam experiences combined longitudinal and transverse motion due to the axial force P1 and bending moment P2. Here we consider two degrees of freedom, leading to a 2 × 2 local stiffness matrix.Figure 1
Beam model.The relationship between strain energy release rateJ(b) and stress intensity factors (GIi, i=1 to 2) at the crack section is given by Tada et al. [25] as
(1)J(b)=1E′(GI1+GI2)2,
where E′=E/(1-ν2), for plane strain condition, E′=E, for plane stress condition,GI1 = stress intensity factor for opening mode I due to load P1,GI2 = stress intensity factor for opening mode I due to load P2.From earlier studies [25], the values of stress intensity factors are
(2)GI1=P1WTπb(F1(bt)),GI2=6P2WT2πb(F2(bt)),000000000000000000000000000000000≤b≤b1,where the experimentally determined functionsF1 and F2 are expressed as follows:
(3)F1(bT)=(2Tπbtan(πb2T))0.5×{0.752+2.02(b/T)+0.37(1-sin(πb/2T))3cos(πb/2T)}(4)F2(bT)=(2Tπbtan(πb2T))0.5×{0.923+0.199(1-sin(πb/2T))4cos(πb/2T)}.The strain energy release rate (also called strain energy density function) at the crack location is defined asJ(b)=∂Ut/∂(b×W), where (b×W) is the newly created surface area of the crack.⇒
d
U
t
=
J
(
b
)
d
(
b
×
W
)
=
W
J
(
b
)
d
b, since the width of the cross-section of the beam is constant.So the strain energy release (Ut) due to the crack of depth b1 is calculated as
(5)Ut=W∫0b1J(b)db.
Then from Castigliano’s theorem, the additional displacement along the force Pi is
(6)Si=∂Ut∂Pi.
From (1) and (3), thus we have
(7)Si=∂∂Pi[W∫0b1J(b)db]=W∂∂Pi[∫0b1J(b)db].The flexibility influence coefficientCij will be by definition
(8)Cij=∂Si∂Pj=∂∂Pj[W∂∂Pi{∫0b1J(b)db}]=W∂2∂Pi∂Pj∫0b1J(b)db.
Substituting (1) in (6), we have
(9)Cij=WE′∂2∂Pi∂Pj∫0b1(Gl1+Gl2)2db.
Using δ=(b/T), dδ=db/T, the following expressions are found out db=Tdδ, and, when b=0, δ=0; b=b1, δ=b1/T=δ1.From the above condition, (9) can be written as
(10)Cij=WTE′∂2∂Pi∂Pj∫0δ1(Gl1+Gl2)2dδ,
where Cij = flexibility influence coefficient in i direction (x-direction or y-direction) due to the load in j direction (P1 or P2).CalculatingC11, C12(=C21), and C22 is as follows:
(11)C11=WTE′∫0δ1πbW2T22F12(δ)dδ=2πWE′∫0δ1δF12(δ)dδ,C12=C21=12πE′TW∫0δ1δF1(δ)F2(δ)dδ,C22=72πE′WT2∫0δ1δF22(δ)dδ.The local stiffness matrix can be obtained using the inverse of compliance matrix(12)K=[K11K12K21K22]=[C11C12C21C22]-1.Converting the influence coefficient into dimensionless form we get(13)C11¯=C11WE′2π,C12¯=C12E′TW12π=C21¯,C22¯=C22E′WT272π.


### 2.2. Governing Equations for Vibration Mode of the Cracked Beam

The cantilever beam as mentioned in Section2.1 is being considered for free vibration analysis. A cantilever beam of length “L,” width “W,” and depth “T” with a crack of depth “b1” at a distance “Lc” from the fixed end is considered as shown in Figure 1. Considering S1(x,t) and S2(x,t) as the amplitudes of longitudinal vibration for the sections before and after the crack position and V1(x,t) and V2(x,t) as the amplitudes of bending vibration for the same section as shown in Figure 2, the free vibration of an Euler-Bernoulli beam of a constant rectangular cross-section is given by the following differential equations:longitudinal vibration(14)∂2S∂t2=(Eρ)∂2S∂x2,lateral vibration(15)EI∂4V∂x4-ρω2V=0.Figure 2
Beam model with deflections.The normal functions for the cracked beam in nondimensional form for both the longitudinal and bending vibrations in steady state can be defined as(16a)S¯1(x-)=B1cos(H¯sx-)+B2sin(H¯sx-),(16b)S¯2(x-)=B3cos(Hs¯x-)+B4sin(Hs¯x-),(16c)V¯1(x-)=B5cosh(Hv¯x-)+B6sinh(Hv¯x-)+B7cos(Hv¯x-)+B8sin(Hv¯x-),(16d)V2¯(x-)=B9cosh(Hv¯x-)+B10sinh(Hv¯x-)+B11cos(Hv¯x-)+B12sin(Hv¯x-),where(17)x-=xL,S-=SL,V-=VL,α=LcL,Hs¯=ωLDs,Ds=(Eρ)1/2,HV¯=(ωL2Dv)1/2,Dv=(EIμ)1/2,μ=Aρ.B
i
(
i
=
1
,
12
) are the constants to be determined, from boundary conditions. The boundary conditions of the cantilever beam considered in the present study are(18a)S1¯(0)=0,V1¯(0)=0,V1¯′(0)=0,(18b)S¯′2(1)=0,V2¯′′(1)=0,V2¯′′′(1)=0.At the cracked section(19)S¯1(α)=S¯2(α),V¯1(α)=V¯2(α),V1¯′′(α)=V2¯′′(α),V1¯′′′(α)=V2¯′′′(α).
Also at the cracked section, we have
(20)AEdS1(Lc)dx=K11(S2(Lc)-S1(Lc))+K12(dV2(Lc)dx-dV1(Lc)dx).Multiplying both sides of the above equation byAE/LK11K12 we get
(21)N1N2S1¯′(α)=N2(S2¯(α)-S1¯(α))+N1(V2¯′(α)-V1¯′(α)).
Similarly,
(22)EId2V1(Lc)dx2=K21(S2(Lc)-S1(Lc))+K22(dV2(Lc)dx-dV1(Lc)dx).
Multiplying both sides of the aforementaioned equation by EI/L2K22K21 we get
(23)N3N4V1¯′′(α)=N3(S2¯(α)-S1¯(α))+N4(V2¯′(α)-V1¯′(α)),
where
(24)N1=AELK11,N2=AEK12,N3=EILK22,N4=EIL2K21.The normal functions ((16a)–(16d)) along with the boundary conditions as mentioned above ((18a) to (23)) yield the characteristic equation of the system as
(25)|Q|=0,
where Q is a 12 × 12 matrix as given below, whose determinant is a function of natural circular frequency (ω), the relative location of the crack (α), and the local stiffness matrix (K), which in turn is a function of the relative crack depth δ1=(b1/T):(26)
Q
=
[
1
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
G
3
G
4
-
G
7
-
G
8
0
0
0
0
0
0
0
0
G
4
G
3
G
8
-
G
7
0
0
0
0
G
1
G
2
-
G
5
-
G
6
-
G
1
-
G
2
G
5
G
6
0
0
0
0
G
2
G
1
G
6
-
G
5
-
G
2
-
G
1
-
G
6
G
5
0
0
0
0
G
1
G
2
G
5
G
6
-
G
1
-
G
2
-
G
5
-
G
6
0
0
0
0
S
1
S
2
S
3
S
4
-
G
2
-
G
1
G
6
-
G
5
S
5
S
6
S
7
S
8
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
-
T
8
T
7
0
0
0
0
0
0
0
0
-
T
6
T
5
T
6
-
T
5
S
9
S
10
S
11
S
12
S
13
S
14
S
15
S
16
S
17
S
18
-
T
5
-
T
6
]
,where(27)S1=G2+N3H-VG1,S2=G1+N3H-VG2,S3=-G6-N3H-VG5,S4=G5-N3H-VG6,S5=N34H-VT5,S6=N34H-VT6,S7=-N34H-VT5,S8=-N34H-VT6,S9=N12H-VG2,S10=N12H-VG1,S11=-N12H-VG6,S12=N12H-VG5,S13=-N12H-VG2,S14=-N12H-VG1,S15=N12H-VG6,S16=-N12H-VG5,S17=T5-N1H-ST6,S18=T6+N1H-ST5,G1=Cosh(H-Vα),G2=Sinh(H-Vα),G3=Cosh(H-V),G4=Sinh(H-V),G5=Cos(H-Vα),G6=Sin(H-Vα),G7=Cosh(H-V),G8=Sin(H-V),T5=Cos(H-Sα),T6=Sin(H-Sα),T7=Cos(H-S),T8=Sin(H-S),N12=N1N2,N34=N3N4.


## 2.1. Computation of Flexibility Matrix of a Damaged Beam Subjected to Complex Loading

A beam with cracks has smaller stiffness than that of a normal beam. This decreased local stiffness can be formulated as a matrix. The dimension of the matrix would depend on the degrees of freedom in the problem. Figure1 shows a cantilever beam of width W and height T, having a transverse surface crack of depth b1. The beam experiences combined longitudinal and transverse motion due to the axial force P1 and bending moment P2. Here we consider two degrees of freedom, leading to a 2 × 2 local stiffness matrix.Figure 1
Beam model.The relationship between strain energy release rateJ(b) and stress intensity factors (GIi, i=1 to 2) at the crack section is given by Tada et al. [25] as
(1)J(b)=1E′(GI1+GI2)2,
where E′=E/(1-ν2), for plane strain condition, E′=E, for plane stress condition,GI1 = stress intensity factor for opening mode I due to load P1,GI2 = stress intensity factor for opening mode I due to load P2.From earlier studies [25], the values of stress intensity factors are
(2)GI1=P1WTπb(F1(bt)),GI2=6P2WT2πb(F2(bt)),000000000000000000000000000000000≤b≤b1,where the experimentally determined functionsF1 and F2 are expressed as follows:
(3)F1(bT)=(2Tπbtan(πb2T))0.5×{0.752+2.02(b/T)+0.37(1-sin(πb/2T))3cos(πb/2T)}(4)F2(bT)=(2Tπbtan(πb2T))0.5×{0.923+0.199(1-sin(πb/2T))4cos(πb/2T)}.The strain energy release rate (also called strain energy density function) at the crack location is defined asJ(b)=∂Ut/∂(b×W), where (b×W) is the newly created surface area of the crack.⇒
d
U
t
=
J
(
b
)
d
(
b
×
W
)
=
W
J
(
b
)
d
b, since the width of the cross-section of the beam is constant.So the strain energy release (Ut) due to the crack of depth b1 is calculated as
(5)Ut=W∫0b1J(b)db.
Then from Castigliano’s theorem, the additional displacement along the force Pi is
(6)Si=∂Ut∂Pi.
From (1) and (3), thus we have
(7)Si=∂∂Pi[W∫0b1J(b)db]=W∂∂Pi[∫0b1J(b)db].The flexibility influence coefficientCij will be by definition
(8)Cij=∂Si∂Pj=∂∂Pj[W∂∂Pi{∫0b1J(b)db}]=W∂2∂Pi∂Pj∫0b1J(b)db.
Substituting (1) in (6), we have
(9)Cij=WE′∂2∂Pi∂Pj∫0b1(Gl1+Gl2)2db.
Using δ=(b/T), dδ=db/T, the following expressions are found out db=Tdδ, and, when b=0, δ=0; b=b1, δ=b1/T=δ1.From the above condition, (9) can be written as
(10)Cij=WTE′∂2∂Pi∂Pj∫0δ1(Gl1+Gl2)2dδ,
where Cij = flexibility influence coefficient in i direction (x-direction or y-direction) due to the load in j direction (P1 or P2).CalculatingC11, C12(=C21), and C22 is as follows:
(11)C11=WTE′∫0δ1πbW2T22F12(δ)dδ=2πWE′∫0δ1δF12(δ)dδ,C12=C21=12πE′TW∫0δ1δF1(δ)F2(δ)dδ,C22=72πE′WT2∫0δ1δF22(δ)dδ.The local stiffness matrix can be obtained using the inverse of compliance matrix(12)K=[K11K12K21K22]=[C11C12C21C22]-1.Converting the influence coefficient into dimensionless form we get(13)C11¯=C11WE′2π,C12¯=C12E′TW12π=C21¯,C22¯=C22E′WT272π.


## 2.2. Governing Equations for Vibration Mode of the Cracked Beam

The cantilever beam as mentioned in Section2.1 is being considered for free vibration analysis. A cantilever beam of length “L,” width “W,” and depth “T” with a crack of depth “b1” at a distance “Lc” from the fixed end is considered as shown in Figure 1. Considering S1(x,t) and S2(x,t) as the amplitudes of longitudinal vibration for the sections before and after the crack position and V1(x,t) and V2(x,t) as the amplitudes of bending vibration for the same section as shown in Figure 2, the free vibration of an Euler-Bernoulli beam of a constant rectangular cross-section is given by the following differential equations:longitudinal vibration(14)∂2S∂t2=(Eρ)∂2S∂x2,lateral vibration(15)EI∂4V∂x4-ρω2V=0.Figure 2
Beam model with deflections.The normal functions for the cracked beam in nondimensional form for both the longitudinal and bending vibrations in steady state can be defined as(16a)S¯1(x-)=B1cos(H¯sx-)+B2sin(H¯sx-),(16b)S¯2(x-)=B3cos(Hs¯x-)+B4sin(Hs¯x-),(16c)V¯1(x-)=B5cosh(Hv¯x-)+B6sinh(Hv¯x-)+B7cos(Hv¯x-)+B8sin(Hv¯x-),(16d)V2¯(x-)=B9cosh(Hv¯x-)+B10sinh(Hv¯x-)+B11cos(Hv¯x-)+B12sin(Hv¯x-),where(17)x-=xL,S-=SL,V-=VL,α=LcL,Hs¯=ωLDs,Ds=(Eρ)1/2,HV¯=(ωL2Dv)1/2,Dv=(EIμ)1/2,μ=Aρ.B
i
(
i
=
1
,
12
) are the constants to be determined, from boundary conditions. The boundary conditions of the cantilever beam considered in the present study are(18a)S1¯(0)=0,V1¯(0)=0,V1¯′(0)=0,(18b)S¯′2(1)=0,V2¯′′(1)=0,V2¯′′′(1)=0.At the cracked section(19)S¯1(α)=S¯2(α),V¯1(α)=V¯2(α),V1¯′′(α)=V2¯′′(α),V1¯′′′(α)=V2¯′′′(α).
Also at the cracked section, we have
(20)AEdS1(Lc)dx=K11(S2(Lc)-S1(Lc))+K12(dV2(Lc)dx-dV1(Lc)dx).Multiplying both sides of the above equation byAE/LK11K12 we get
(21)N1N2S1¯′(α)=N2(S2¯(α)-S1¯(α))+N1(V2¯′(α)-V1¯′(α)).
Similarly,
(22)EId2V1(Lc)dx2=K21(S2(Lc)-S1(Lc))+K22(dV2(Lc)dx-dV1(Lc)dx).
Multiplying both sides of the aforementaioned equation by EI/L2K22K21 we get
(23)N3N4V1¯′′(α)=N3(S2¯(α)-S1¯(α))+N4(V2¯′(α)-V1¯′(α)),
where
(24)N1=AELK11,N2=AEK12,N3=EILK22,N4=EIL2K21.The normal functions ((16a)–(16d)) along with the boundary conditions as mentioned above ((18a) to (23)) yield the characteristic equation of the system as
(25)|Q|=0,
where Q is a 12 × 12 matrix as given below, whose determinant is a function of natural circular frequency (ω), the relative location of the crack (α), and the local stiffness matrix (K), which in turn is a function of the relative crack depth δ1=(b1/T):(26)
Q
=
[
1
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
G
3
G
4
-
G
7
-
G
8
0
0
0
0
0
0
0
0
G
4
G
3
G
8
-
G
7
0
0
0
0
G
1
G
2
-
G
5
-
G
6
-
G
1
-
G
2
G
5
G
6
0
0
0
0
G
2
G
1
G
6
-
G
5
-
G
2
-
G
1
-
G
6
G
5
0
0
0
0
G
1
G
2
G
5
G
6
-
G
1
-
G
2
-
G
5
-
G
6
0
0
0
0
S
1
S
2
S
3
S
4
-
G
2
-
G
1
G
6
-
G
5
S
5
S
6
S
7
S
8
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
-
T
8
T
7
0
0
0
0
0
0
0
0
-
T
6
T
5
T
6
-
T
5
S
9
S
10
S
11
S
12
S
13
S
14
S
15
S
16
S
17
S
18
-
T
5
-
T
6
]
,where(27)S1=G2+N3H-VG1,S2=G1+N3H-VG2,S3=-G6-N3H-VG5,S4=G5-N3H-VG6,S5=N34H-VT5,S6=N34H-VT6,S7=-N34H-VT5,S8=-N34H-VT6,S9=N12H-VG2,S10=N12H-VG1,S11=-N12H-VG6,S12=N12H-VG5,S13=-N12H-VG2,S14=-N12H-VG1,S15=N12H-VG6,S16=-N12H-VG5,S17=T5-N1H-ST6,S18=T6+N1H-ST5,G1=Cosh(H-Vα),G2=Sinh(H-Vα),G3=Cosh(H-V),G4=Sinh(H-V),G5=Cos(H-Vα),G6=Sin(H-Vα),G7=Cosh(H-V),G8=Sin(H-V),T5=Cos(H-Sα),T6=Sin(H-Sα),T7=Cos(H-S),T8=Sin(H-S),N12=N1N2,N34=N3N4.


## 3. Experimental Results

To verify the integrity of the proposed crack detection method and to find out the errors associated with the modeling and measurements, several experiments also have been conducted in the laboratory. Figure3 shows a schematic diagram of the experimental setup and its description. A cracked cantilever beam has been rigidly clamped to the concrete foundation base. The geometry and material properties of the beam are presented in Table 1. The free end of the beam is excited with a vibration exciter. The vibration exciter is excited by the signal from the function generator. The signal is amplified by a power amplifier before being fed to the vibration exciter. The amplitude of vibration of the uncracked and cracked cantilever beam is taken by the accelerometer and is fed to the vibration indicator for vibration analysis. The vibration signatures are analyzed graphically by vibration indicators.Table 1
Material properties of Aluminium-Alloy, 2014-T4.
Young’s modulus,E
Density,ρ
Poisson’s ratio,μ
Length,L
Width,W
Thickness,T
72.4 GPa
2.8 gm/cc
0.33
800 mm
50 mm
6 mmFigure 3
Schematic block diagram of experimental setup for the cantilever beam.Several tests are conducted using the experimental setup on aluminum beam specimens (800 × 50 × 6 mm) with a transverse crack for determining the natural frequencies and mode shapes for different crack locations (i.e., 200 mm, 400 mm, and 600 mm from the clamped end) and crack depths varying from 1 mm to 5 mm by a step of 1 mm. The cracks were prepared by fine saw cuts perpendicular to the longitudinal axis. This ensures that the crack remains open during the vibrations. At each step the first three bending natural frequencies of the cracked beams were measured. Table1 gives the corresponding bending natural frequencies of the intact and cracked beams. These specimens are set to vibrate under the 1st, 2nd, and 3rd modes of vibrations and the corresponding amplitudes are recorded in the vibration indicator. Experimental results of frequencies of transverse vibration at various locations along the length of the beam are recorded by positioning the vibration pickup and tuning the vibration generator at the corresponding resonant frequencies.DE was employed to detect cracks utilizing the results from the experimental study. A comparison was made between the experimentally measured natural frequencies of the damaged beam and the ones obtained through the cracked beam model using the objective function as discussed in Section4.4.The theoretical results are better than the experimental ones, because of measurement errors. The proposed methods have been applied to nine damage cases obtained by combining three different crack positions and three different crack depths.Then theoretical and experimental results are compared by using the differential evolution method.


## 4. Differential Evolution

The differential evolution (DE) algorithm is a population based evolutionary algorithm developed by nondifferentiable continuous space functions [26]. Like the genetic algorithm, the optimization process in differential evolution is subject to three basic operations: mutation, crossover, and selection. DE, at the start, randomly initializes a population of size NP solution vectors, which is evolved over G generations to reach an optimal solution. The population size NP is kept constant during the minimization process. Each individual Xi of the population is a vector that contains a set of D optimization parameters as the problem decision variables and can be expressed as
(28)Xi,G={xi,G1,…,xi,GD}.In the crack detection problem the search space is bounded by the parameters such as crack location and depth in the cantilever beam. The minimum and maximum parameter bounds areXmin={xmin1,…,xminD} and Xmax={xmax1,…,xmaxD}, respectively. The initial value of the jth parameter in the ith individual at the iteration G=0 can be expressed as
(29)xi,0j=xminj+rand(xmaxj-xminj),
where rand represents a uniformly distributed random variable within the range [0,1].


### 4.1. Mutation Operation

For each target vectorXi,G,i=1,2,…,NP, a mutant vector is generated according to the following mutation strategy that is used in the present work:
(30)Vi,G+1=Xi,G+F·(Xbest,G-Xi,G)+F·(Xr1,G-Xr2,G),
Where r1,r2∈{1,2,…,NP} are mutually different integer numbers and are also different from the index i.The indicesr1i, r2i, r3i, r4i, and r5i are mutually exclusive integers randomly generated within the range [1, NP], which are also different from the index i. These indices are randomly generated once for each mutant vector. The scaling factor F is a positive control parameter for scaling the difference vector.


### 4.2. Crossover Operation

The crossover operation is introduced in the DE algorithm, in order to increase the diversity of the vectors. The crossover operation is carried out by randomly exchanging between the original vectors of the populationXi,G and those of the mutant population Vi,G+1 to obtain the trial vectors Zi,G+1=(zi,G+11,…,zi,G+1D). The trial vector is determined by a parameter called crossover probability (CR∈[0,1]) as follows:
(31)zi,Gj={vi,G,jif(rand≤CR)orj=jrand,xi,G,jotherwise,
where jrand is a randomly chosen integer in the range [1,D].


### 4.3. Selection

In order to decide if a vectorZi may be the element of the new population of generation G+1, each vector zi,G+1 is compared to the previous corresponding target vector xi,G. If vector zi,G+1 yields a smaller objective function value than xi,G, then xi,G+1 is set to zi,G+1; otherwise, the old value xi,G is retained. The selection operation can be expressed as
(32)Xi,G+1={Zi,G,iff(Zi,G)≤f(Xi,G)Xi,G,otherwise.
with i=1,2,…,NP.The aforementioned steps are repeated generation after generation until some specific termination criteria are satisfied. The algorithmic description of DE is summarized in Table1.


### 4.4. Objective Function Based on Vibration Data

Damage in a structure makes changes in vibrational parameters such as natural frequencies and mode shapes. In this current study natural frequencies are taken as the damage indicator as they are easier to measure than mode shapes. The objective function chosen for damage estimation is a minimization optimization problem. The objective function based on natural frequencies can be expressed as(33)F=∑i=1n1wi|(fic-fim)|,
where fim and fic are the measured natural frequencies and natural frequencies obtained from a theoretical spring model based on the assumed crack location and depth, respectively. The n is the number of natural frequencies used to evaluate the objective function and wi is a weighting factor, whose values are considered to be i. In the present work, the first three natural frequencies are considered to determine the objective function.Stopping Rule. The iterative procedure is terminated when a predefined number of generations or a computational error of 10-4 is reached.Pseudo code for DE
(1)
Generate a population of solution vectors.
(2)
Evaluate the best member of the populationXbest,i.
(3)
Carry out the mutation operationVi,G+1.
(4)
Perform the crossover strategyZi,G+1.
(5)
Check the bound constraint.
(6)
Do the selection.
(7)
Evaluate the best member of the population after selectionXbest,i.
(8)
If the stopping criterion is reached, then print the output results and stop; otherwise repeat Steps  2–7.
(9)
Then go to Step  3.


## 4.1. Mutation Operation

For each target vectorXi,G,i=1,2,…,NP, a mutant vector is generated according to the following mutation strategy that is used in the present work:
(30)Vi,G+1=Xi,G+F·(Xbest,G-Xi,G)+F·(Xr1,G-Xr2,G),
Where r1,r2∈{1,2,…,NP} are mutually different integer numbers and are also different from the index i.The indicesr1i, r2i, r3i, r4i, and r5i are mutually exclusive integers randomly generated within the range [1, NP], which are also different from the index i. These indices are randomly generated once for each mutant vector. The scaling factor F is a positive control parameter for scaling the difference vector.


## 4.2. Crossover Operation

The crossover operation is introduced in the DE algorithm, in order to increase the diversity of the vectors. The crossover operation is carried out by randomly exchanging between the original vectors of the populationXi,G and those of the mutant population Vi,G+1 to obtain the trial vectors Zi,G+1=(zi,G+11,…,zi,G+1D). The trial vector is determined by a parameter called crossover probability (CR∈[0,1]) as follows:
(31)zi,Gj={vi,G,jif(rand≤CR)orj=jrand,xi,G,jotherwise,
where jrand is a randomly chosen integer in the range [1,D].


## 4.3. Selection

In order to decide if a vectorZi may be the element of the new population of generation G+1, each vector zi,G+1 is compared to the previous corresponding target vector xi,G. If vector zi,G+1 yields a smaller objective function value than xi,G, then xi,G+1 is set to zi,G+1; otherwise, the old value xi,G is retained. The selection operation can be expressed as
(32)Xi,G+1={Zi,G,iff(Zi,G)≤f(Xi,G)Xi,G,otherwise.
with i=1,2,…,NP.The aforementioned steps are repeated generation after generation until some specific termination criteria are satisfied. The algorithmic description of DE is summarized in Table1.


## 4.4. Objective Function Based on Vibration Data

Damage in a structure makes changes in vibrational parameters such as natural frequencies and mode shapes. In this current study natural frequencies are taken as the damage indicator as they are easier to measure than mode shapes. The objective function chosen for damage estimation is a minimization optimization problem. The objective function based on natural frequencies can be expressed as(33)F=∑i=1n1wi|(fic-fim)|,
where fim and fic are the measured natural frequencies and natural frequencies obtained from a theoretical spring model based on the assumed crack location and depth, respectively. The n is the number of natural frequencies used to evaluate the objective function and wi is a weighting factor, whose values are considered to be i. In the present work, the first three natural frequencies are considered to determine the objective function.Stopping Rule. The iterative procedure is terminated when a predefined number of generations or a computational error of 10-4 is reached.Pseudo code for DE
(1)
Generate a population of solution vectors.
(2)
Evaluate the best member of the populationXbest,i.
(3)
Carry out the mutation operationVi,G+1.
(4)
Perform the crossover strategyZi,G+1.
(5)
Check the bound constraint.
(6)
Do the selection.
(7)
Evaluate the best member of the population after selectionXbest,i.
(8)
If the stopping criterion is reached, then print the output results and stop; otherwise repeat Steps  2–7.
(9)
Then go to Step  3.


## 5. Results and Discussions

The theoretical analysis and proposed DE algorithm were implemented using MATLAB 7.0 [27]. Based on the results obtained from the numerical, experimental, and proposed DE model, the following discussions can be made. A crack in a beam structure causes a change in the stiffness of the beam, which is in turn a function of crack location and crack depth. As discussed in Section 2, the dimensionless compliance matrix is used to evaluate the stiffness matrix. The dimensionless compliances (C-11,C-12=C-21,C-22) increasing with the increase in relative crack depth are as shown in Figure 4.Figure 4
Dimensionless compliance (ln(C11¯) versus relative crack depth (δ1).For crack location (e.g., 100 mm) and relative crack depths(δ1=b1/T) (e.g., 0.1 ), the first three mode shapes are presented graphically in Figure 5,  where it is observed that there are reasonable changes in mode shapes due to the presence of a crack in the beam.(a) Relative amplitude versus distance from the fixed end (1st mode of vibration),Lc=100mm, b1/T=0.1. (b) Magnified view of Figure 4(a) at the location. (c) Relative amplitude versus distance from the fixed end (2nd mode of vibration), Lc=100mm, b1/T=0.1. (d) Relative amplitude versus distance from the fixed end (3rd mode of vibration), Lc=100mm, b1/T=0.1.
(a)
(b)
(c)
(d)The numerical results for the relative amplitude of transverse vibration at different locations of aluminium alloy 2014-T4 cracked specimens for the first three modes are obtained using the theoretical model as per (16a) to (16d) with the help of computer programming. These results for cracked and uncracked specimens are presented in Figure 5 for comparison. It can be seen that the deviations are more prominent with higher modes of vibrations. Table 1 presents the first three natural frequencies obtained from the theoretical model, which is subsequently used in the DE algorithm for crack identification.


### 5.1. Computational Results/Simulation Results

Different crack conditions have been taken to evaluate the performance of the proposed DE. Simulation results of 4 test points have been presented in this paper.Figure6 shows the convergence trend of the fitness value of the objective function with the number of iterations. Figures 6(a)–6(c) show the convergence trend for the theoretical model and Figure 6(d) is for the experimental study. As it can be seen from Figures 6(a)–6(c), the number of iterations for convergence increases with the increase in crack location for the same crack depth. Figure 6(d) shows that the fitness value does not converge for the predefined number of iterations.(a) Loc = 200, depth = 2.4. (b) Loc = 600, depth = 2.4. (c) Loc = 400, depth = 2.4. (d) Loc = 300, depth = 2.4.
(a)
(b)
(c)
(d)


### 5.2. Simulation Results

Different crack conditions have been taken to evaluate the performance of DE. Simulation results of 6 test points have been presented in this paper. The error is calculated using the following formula:(34)%error=|predictedvalue-actualvalueactualvalue|×100.Table4 represents the validation of the proposed DE model with the results obtained as per the theoretical model. The percentage deviations in the results obtained by the experimental study and DE model are presented in Table 5.Table2 shows the first three natural frequencies of the cracked cantilever beam at different crack locations and depths. The theoretical natural frequencies of the test specimens are presented in Table 3. It is found that the estimation error is reduced by increasing the crack depths for the same locations (Table 5). Also by increasing the distance of the crack from the fixed end of the beam, the estimation error has been found to be decreasing for similar crack depth. Tables 4 and 5 represent the validation of the proposed DE model with the results obtained through the theoretical and experimental models, respectively.Table 2
Measured natural frequencies for single crack cantilever beam.
Sl. No.
Crack
Bending frequencies (Hz)
Location (mm)
Depth (mm)
f
1
f
2
f
3
1
No crack
0.0
7.0683
44.2951
124.0278
2
300
1.2
7.0589
44.2267
123.8194
3
300
1.8
7.0466
44.1380
123.5518
4
300
2.4
7.0265
43.9916
123.1154
5
450
1.2
7.0654
44.2564
123.9431
6
450
1.8
7.0617
44.0162
123.8339
7
450
2.4
7.0556
43.7567
123.6551
8
600
1.2
7.0678
44.2532
123.6751
9
600
1.8
7.0673
44.1985
123.2185
10
600
2.4
7.0665
44.1068
122.4656Table 3
Calculated frequencies using theoretical analysis.
Sl. No.
Crack
Bending frequencies (Hz)
Location (mm)
Depth (mm)
f
1
f
2
f
3
1
No crack
0.0
7.6829
47.1839
132.1165
2
200
1.8
7.4123
46.6920
130.2206
3
400
1.8
7.3648
46.6467
130.7678
4
600
1.8
7.489 8
46.6006
130.5848
5
200
2.4
7.3745
46.6821
130.3828
6
400
2.4
7.3547
46.1335
130.7672
7
600
2.4
7.2969
46.5039
129.1213Table 4
Crack detection results of the theoretical study by applying DE.
Measured crack
Predicted results (mm) using (DE)
Location (mm)
Depth (mm)
Location (mm)
% err
Depth (mm)
% err
200
1.8
200.012
0.006
1.8002
0.011
200
2.4
200.103
0.034
2.4006
0.025
400
1.8
399.987
0.003
1.8000
0.000
400
2.4
400.054
0.014
2.3964
0.151
600
1.8
600.106
0.018
1.8006
0.034
600
2.4
599.898
0.017
2.4005
0.021Table 5
Crack detection results of the experimental study by applying DE.
Measured crack
Predicted results (mm) using (DE)
Location (mm)
Depth (mm)
Location (mm)
% err
Depth (mm)
% err
300
1.8
265.14
11.62
1.961
8.94
300
2.4
332.04
10.68
2.604
8.50
450
1.8
399.96
11.12
1.633
8.72
450
2.4
496.58
10.35
2.588
7.83
600
1.8
537.66
10.39
1.940
7.78
600
2.4
540.72
9.88
2.228
7.16


## 5.1. Computational Results/Simulation Results

Different crack conditions have been taken to evaluate the performance of the proposed DE. Simulation results of 4 test points have been presented in this paper.Figure6 shows the convergence trend of the fitness value of the objective function with the number of iterations. Figures 6(a)–6(c) show the convergence trend for the theoretical model and Figure 6(d) is for the experimental study. As it can be seen from Figures 6(a)–6(c), the number of iterations for convergence increases with the increase in crack location for the same crack depth. Figure 6(d) shows that the fitness value does not converge for the predefined number of iterations.(a) Loc = 200, depth = 2.4. (b) Loc = 600, depth = 2.4. (c) Loc = 400, depth = 2.4. (d) Loc = 300, depth = 2.4.
(a)
(b)
(c)
(d)


## 5.2. Simulation Results

Different crack conditions have been taken to evaluate the performance of DE. Simulation results of 6 test points have been presented in this paper. The error is calculated using the following formula:(34)%error=|predictedvalue-actualvalueactualvalue|×100.Table4 represents the validation of the proposed DE model with the results obtained as per the theoretical model. The percentage deviations in the results obtained by the experimental study and DE model are presented in Table 5.Table2 shows the first three natural frequencies of the cracked cantilever beam at different crack locations and depths. The theoretical natural frequencies of the test specimens are presented in Table 3. It is found that the estimation error is reduced by increasing the crack depths for the same locations (Table 5). Also by increasing the distance of the crack from the fixed end of the beam, the estimation error has been found to be decreasing for similar crack depth. Tables 4 and 5 represent the validation of the proposed DE model with the results obtained through the theoretical and experimental models, respectively.Table 2
Measured natural frequencies for single crack cantilever beam.
Sl. No.
Crack
Bending frequencies (Hz)
Location (mm)
Depth (mm)
f
1
f
2
f
3
1
No crack
0.0
7.0683
44.2951
124.0278
2
300
1.2
7.0589
44.2267
123.8194
3
300
1.8
7.0466
44.1380
123.5518
4
300
2.4
7.0265
43.9916
123.1154
5
450
1.2
7.0654
44.2564
123.9431
6
450
1.8
7.0617
44.0162
123.8339
7
450
2.4
7.0556
43.7567
123.6551
8
600
1.2
7.0678
44.2532
123.6751
9
600
1.8
7.0673
44.1985
123.2185
10
600
2.4
7.0665
44.1068
122.4656Table 3
Calculated frequencies using theoretical analysis.
Sl. No.
Crack
Bending frequencies (Hz)
Location (mm)
Depth (mm)
f
1
f
2
f
3
1
No crack
0.0
7.6829
47.1839
132.1165
2
200
1.8
7.4123
46.6920
130.2206
3
400
1.8
7.3648
46.6467
130.7678
4
600
1.8
7.489 8
46.6006
130.5848
5
200
2.4
7.3745
46.6821
130.3828
6
400
2.4
7.3547
46.1335
130.7672
7
600
2.4
7.2969
46.5039
129.1213Table 4
Crack detection results of the theoretical study by applying DE.
Measured crack
Predicted results (mm) using (DE)
Location (mm)
Depth (mm)
Location (mm)
% err
Depth (mm)
% err
200
1.8
200.012
0.006
1.8002
0.011
200
2.4
200.103
0.034
2.4006
0.025
400
1.8
399.987
0.003
1.8000
0.000
400
2.4
400.054
0.014
2.3964
0.151
600
1.8
600.106
0.018
1.8006
0.034
600
2.4
599.898
0.017
2.4005
0.021Table 5
Crack detection results of the experimental study by applying DE.
Measured crack
Predicted results (mm) using (DE)
Location (mm)
Depth (mm)
Location (mm)
% err
Depth (mm)
% err
300
1.8
265.14
11.62
1.961
8.94
300
2.4
332.04
10.68
2.604
8.50
450
1.8
399.96
11.12
1.633
8.72
450
2.4
496.58
10.35
2.588
7.83
600
1.8
537.66
10.39
1.940
7.78
600
2.4
540.72
9.88
2.228
7.16


## 6. Conclusions

Based on the results and discussions thereof, the following conclusions can be made. The mode shapes and bending frequencies of the cracked elastic structures are strongly influenced by the crack location and its intensity. Though significant changes in mode shapes are observed in the vicinity of crack location, these deviations in mode shapes cannot be used as a measuring tool in the prediction of crack location and its intensity. In the present work, the proposed DE algorithm is found to be an efficient method for damage quantification in terms of crack location and crack depth by minimizing the error between measured and predicted frequencies.  It was also observed that the error associated with the prediction is less in the theoretical model as compared to the experimental model. The present work can be implemented for damage assessment in different structures.



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*Source: 321931-2013-11-13.xml*