Differential Evolution: An Inverse Approach for Crack Detection

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.

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. Section 2 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. Figure 1 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 rate 𝐽(𝑏) and stress intensity factors (𝐺_𝐼𝑖, 𝑖=1 to 2) at the crack section is given by Tada et al. [25] as 𝐽(𝑏)=1𝐸^′𝐺_𝐼1+𝐺_𝐼2², (1) where 𝐸^′=𝐸/(1−𝜈²), for plane strain condition, 𝐸^′=𝐸, for plane stress condition, 𝐺_𝐼1 = stress intensity factor for opening mode I due to load 𝑃₁, 𝐺_𝐼2 = stress intensity factor for opening mode I due to load 𝑃₂.

From earlier studies [25], the values of stress intensity factors are 𝐺_𝐼1=𝑃₁√𝑊𝑇𝜋𝑏𝐹₁𝑏𝑡,𝐺_𝐼2=6𝑃₂𝑊𝑇²√𝜋𝑏𝐹₂𝑏𝑡,0≤𝑏≤𝑏₁, (2)

where the experimentally determined functions 𝐹₁ and 𝐹₂ are expressed as follows: 𝐹₁𝑏𝑇=2𝑇𝜋𝑏tan𝜋𝑏2𝑇^0.5×0.752+2.02(𝑏/𝑇)+0.37(1−sin(𝜋𝑏/2𝑇))³()𝐹cos𝜋𝑏/2𝑇₂𝑏𝑇=2𝑇𝜋𝑏tan𝜋𝑏2𝑇^0.5×0.923+0.199(1−sin(𝜋𝑏/2𝑇))⁴().cos𝜋𝑏/2𝑇 (3)(4)

The strain energy release rate (also called strain energy density function) at the crack location is defined as 𝐽(𝑏)=𝜕𝑈_𝑡/𝜕(𝑏×𝑊), where (𝑏×𝑊) is the newly created surface area of the crack.

⇒𝑑𝑈_𝑡=𝐽(𝑏)𝑑(𝑏×𝑊)=𝑊𝐽(𝑏)𝑑𝑏, since the width of the cross-section of the beam is constant.

So the strain energy release (𝑈_𝑡) due to the crack of depth 𝑏₁ is calculated as 𝑈_𝑡=𝑊^𝑏1₀𝐽(𝑏)𝑑𝑏. (5) Then from Castigliano’s theorem, the additional displacement along the force 𝑃_𝑖 is 𝑆_𝑖=𝜕𝑈_𝑡𝜕𝑃_𝑖. (6) From (1) and (3), thus we have 𝑆_𝑖=𝜕𝜕𝑃_𝑖𝑊^𝑏1₀𝐽(𝑏)𝜕𝑑𝑏=𝑊𝜕𝑃_𝑖^𝑏1₀𝐽(𝑏)𝑑𝑏. (7)

The flexibility influence coefficient 𝐶_𝑖𝑗 will be by definition 𝐶_𝑖𝑗=𝜕𝑆_𝑖𝜕𝑃_𝑗=𝜕𝜕𝑃_𝑗𝑊𝜕𝜕𝑃_𝑖^𝑏1₀𝐽(𝑏)𝜕𝑑𝑏=𝑊²𝜕𝑃_𝑖𝜕𝑃_𝑗^𝑏1₀𝐽(𝑏)𝑑𝑏. (8) Substituting (1) in (6), we have 𝐶_𝑖𝑗=𝑊𝐸^′𝜕²𝜕𝑃_𝑖𝜕𝑃_𝑗^𝑏1₀𝐺_𝑙1+𝐺_𝑙2²𝑑𝑏. (9) 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 𝐶_𝑖𝑗=𝑊𝑇𝐸^′𝜕²𝜕𝑃_𝑖𝜕𝑃_𝑗^𝛿1₀𝐺_𝑙1+𝐺_𝑙2²𝑑𝛿, (10) where C_𝑖𝑗 = flexibility influence coefficient in 𝑖 direction (𝑥-direction or 𝑦-direction) due to the load in 𝑗 direction (𝑃₁ or 𝑃₂).

Calculating 𝐶₁₁, 𝐶₁₂(=𝐶₂₁), and 𝐶₂₂ is as follows: 𝐶₁₁=𝑊𝑇𝐸^′𝛿10𝜋𝑏𝑊²𝑇²2𝐹²₁(𝛿)=𝑑𝛿2𝜋𝑊𝐸^′𝛿10𝛿𝐹²₁(𝛿)𝐶𝑑𝛿,₁₂=𝐶₂₁=12𝜋𝐸^′𝑇𝑊^𝛿10𝛿𝐹₁(𝛿)𝐹₂(𝛿)𝐶𝑑𝛿,₂₂=72𝜋𝐸^′𝑊𝑇^2𝛿10𝛿𝐹²₂(𝛿)𝑑𝛿. (11)

The local stiffness matrix can be obtained using the inverse of compliance matrix 𝐾𝐾=¹¹𝐾₁₂𝐾₂₁𝐾₂₂=𝐶₁₁𝐶₁₂𝐶₂₁𝐶₂₂⁻¹. (12)

Converting the influence coefficient into dimensionless form we get 𝐶₁₁=𝐶₁₁𝑊𝐸^′,2𝜋𝐶₁₂=𝐶₁₂𝐸^′𝑇𝑊=12𝜋𝐶₂₁,𝐶₂₂=𝐶₂₂𝐸^′𝑊𝑇².72𝜋 (13)

2.2. Governing Equations for Vibration Mode of the Cracked Beam

The cantilever beam as mentioned in Section 2.1 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 “𝐿_𝑐” 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 𝜕²𝑆𝜕𝑡²=𝐸𝜌𝜕²𝑆𝜕𝑥², (14) lateral vibration 𝜕𝐸𝐼⁴𝑉𝜕𝑥⁴−𝜌𝜔²𝑉=0. (15)

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 as𝑆₁(𝑥)=𝐵₁cos𝐻_𝑠𝑥+𝐵₂sin𝐻_𝑠𝑥,𝑆₂(𝑥)=𝐵₃cos𝐻_𝑠𝑥+𝐵₄sin𝐻_𝑠𝑥,𝑉₁(𝑥)=𝐵₅cosh𝐻_𝑣𝑥+𝐵₆sinh𝐻_𝑣𝑥+𝐵₇cos𝐻_𝑣𝑥+𝐵₈sin𝐻_𝑣𝑥,𝑉₂(𝑥)=𝐵₉cosh𝐻_𝑣𝑥+𝐵₁₀sinh𝐻_𝑣𝑥+𝐵₁₁cos𝐻_𝑣𝑥+𝐵₁₂sin𝐻_𝑣𝑥, (16a)(16b)(16c)(16d)

where 𝑥𝑥=𝐿,𝑆𝑆=𝐿,𝑉𝑉=𝐿𝐿,𝛼=^𝑐𝐿,𝐻_𝑠=𝜔𝐿𝐷_𝑠,𝐷_𝑠=𝐸𝜌^1/2,𝐻_𝑉=𝜔𝐿²𝐷_𝑣^1/2,𝐷_𝑣=𝐸𝐼𝜇^1/2,𝜇=𝐴𝜌. (17)

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 are𝑆₁(0)=0,𝑉₁(0)=0,𝑉₁^′(0)=0,𝑆^′₂(1)=0,𝑉₂^′′(1)=0,𝑉₂^′′′(1)=0. (18a)(18b)

At the cracked section 𝑆₁(𝛼)=𝑆₂(𝛼),𝑉₁(𝛼)=𝑉₂(𝛼),𝑉₁^′′(𝛼)=𝑉₂^′′(𝛼),𝑉₁^′′′(𝛼)=𝑉₂^′′′(𝛼). (19) Also at the cracked section, we have 𝐴𝐸𝑑𝑆₁𝐿_𝑐𝑑𝑥=𝐾₁₁𝑆₂𝐿_𝑐−𝑆₁𝐿_𝑐+𝐾₁₂𝑑𝑉₂𝐿_𝑐−𝑑𝑥𝑑𝑉₁𝐿_𝑐.𝑑𝑥 (20)

Multiplying both sides of the above equation by 𝐴𝐸/𝐿𝐾₁₁𝐾₁₂ we get 𝑁₁𝑁₂𝑆₁^′(𝛼)=𝑁₂𝑆₂(𝛼)−𝑆₁(𝛼)+𝑁₁𝑉₂^′(𝛼)−𝑉₁^′(𝛼). (21) Similarly, 𝑑𝐸𝐼²𝑉₁𝐿_𝑐𝑑𝑥²=𝐾₂₁𝑆₂𝐿_𝑐−𝑆₁𝐿_𝑐+𝐾₂₂𝑑𝑉₂𝐿_𝑐−𝑑𝑥𝑑𝑉₁𝐿_𝑐.𝑑𝑥 (22) Multiplying both sides of the aforementaioned equation by 𝐸𝐼/𝐿²𝐾₂₂𝐾₂₁ we get 𝑁₃𝑁₄𝑉₁^′′(𝛼)=𝑁₃𝑆₂(𝛼)−𝑆₁(𝛼)+𝑁₄𝑉₂^′(𝛼)−𝑉₁^′(𝛼), (23) where 𝑁₁=𝐴𝐸𝐿𝐾₁₁,𝑁₂=𝐴𝐸𝐾₁₂,𝑁₃=𝐸𝐼𝐿𝐾₂₂,𝑁₄=𝐸𝐼𝐿²𝐾₂₁. (24)

The normal functions ((16a)–(16d)) along with the boundary conditions as mentioned above ((18a) to (23)) yield the characteristic equation of the system as |𝑄|=0, (25) 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):

𝑄=1010000000000101000000000000𝐺₃𝐺₄−𝐺₇−𝐺₈00000000𝐺₄𝐺₃𝐺₈−𝐺₇𝐺0000₁𝐺₂−𝐺₅−𝐺₆−𝐺₁−𝐺₂𝐺₅𝐺₆𝐺0000₂𝐺₁𝐺₆−𝐺₅−𝐺₂−𝐺₁−𝐺₆𝐺₅𝐺0000₁𝐺₂𝐺₅𝐺₆−𝐺₁−𝐺₂−𝐺₅−𝐺₆𝑆0000₁𝑆₂𝑆₃𝑆₄−𝐺₂−𝐺₁𝐺₆−𝐺₅𝑆₅𝑆₆𝑆₇𝑆₈0000000010000000000000−𝑇₈𝑇₇00000000−𝑇₆𝑇₅𝑇₆−𝑇₅𝑆₉𝑆₁₀𝑆₁₁𝑆₁₂𝑆₁₃𝑆₁₄𝑆₁₅𝑆₁₆𝑆₁₇𝑆₁₈−𝑇₅−𝑇₆, (26)

where 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=Cosh𝐻_𝑉𝛼,𝐺₂=Sinh𝐻_𝑉𝛼,𝐺₃=Cosh𝐻_𝑉,𝐺₄=Sinh𝐻_𝑉,𝐺₅=Cos𝐻_𝑉𝛼,𝐺₆=Sin𝐻_𝑉𝛼,𝐺₇=Cosh𝐻_𝑉,𝐺₈=Sin𝐻_𝑉,𝑇₅=Cos𝐻_𝑆𝛼,𝑇₆=Sin𝐻_𝑆𝛼,𝑇₇=Cos𝐻_𝑆,𝑇₈=Sin𝐻_𝑆,𝑁₁₂=𝑁₁𝑁₂,𝑁₃₄=𝑁₃𝑁₄. (27)

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. Figure 3 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.

Figure 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. Table 1 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 Section 4.4.

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 𝑋_𝑖 of the population is a vector that contains a set of 𝐷 optimization parameters as the problem decision variables and can be expressed as 𝑋_𝑖,𝐺=𝑥¹_𝑖,𝐺,…,𝑥^𝐷_𝑖,𝐺. (28)

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 are 𝑋_min={𝑥¹_min,…,𝑥^𝐷_min} and 𝑋_max={𝑥¹_max,…,𝑥^𝐷_max}, respectively. The initial value of the 𝑗th parameter in the 𝑖th individual at the iteration 𝐺=0 can be expressed as 𝑥^𝑗_𝑖,0=𝑥^𝑗_min+rand𝑥^𝑗_max−𝑥^𝑗_min, (29) where rand represents a uniformly distributed random variable within the range [0,1].

4.1. Mutation Operation

For each target vector 𝑋_𝑖,𝐺,𝑖=1,2,…,NP, a mutant vector is generated according to the following mutation strategy that is used in the present work: 𝑉_𝑖,𝐺+1=𝑋_𝑖,𝐺+𝐹⋅𝑋_best,𝐺−𝑋_𝑖,𝐺+𝐹⋅𝑋_𝑟1,𝐺−𝑋_𝑟2,𝐺, (30) Where 𝑟₁,𝑟₂∈{1,2,…,𝑁𝑃} are mutually different integer numbers and are also different from the index 𝑖.

The indices 𝑟^𝑖₁, 𝑟^𝑖₂, 𝑟^𝑖₃, 𝑟^𝑖₄, and 𝑟^𝑖₅ 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 population 𝑋_𝑖,𝐺 and those of the mutant population 𝑉_𝑖,𝐺+1 to obtain the trial vectors 𝑍_𝑖,𝐺+1=(𝑧¹_𝑖,𝐺+1,…,𝑧^𝐷_𝑖,𝐺+1). The trial vector is determined by a parameter called crossover probability (CR∈[0,1]) as follows: 𝑧^𝑗_𝑖,𝐺=𝑣^𝑗_𝑖,𝐺,if(rand≤CR)or𝑗=𝑗_rand,𝑥^𝑗_𝑖,𝐺,otherwise, (31) where 𝑗_rand is a randomly chosen integer in the range [1,𝐷].

4.3. Selection

In order to decide if a vector 𝑍_𝑖 may be the element of the new population of generation 𝐺+1, each vector 𝑧_𝑖,𝐺+1 is compared to the previous corresponding target vector 𝑥_𝑖,𝐺. If vector 𝑧_𝑖,𝐺+1 yields a smaller objective function value than 𝑥_𝑖,𝐺, then 𝑥_𝑖,𝐺+1 is set to 𝑧_𝑖,𝐺+1; otherwise, the old value 𝑥_𝑖,𝐺 is retained. The selection operation can be expressed as 𝑋_𝑖,𝐺+1=𝑍_𝑖,𝐺,if𝑓𝑍_𝑖,𝐺≤𝑓𝑋_𝑖,𝐺𝑋_𝑖,𝐺,otherwise. (32) 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 Table 1.

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 𝐹=^𝑛_𝑖=11𝑤_𝑖𝑓^𝑐_𝑖−𝑓^𝑚_𝑖, (33) where 𝑓^𝑚_𝑖 and 𝑓^𝑐_𝑖 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 𝑤_𝑖 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 population 𝑋_best,𝑖.(3)Carry out the mutation operation 𝑉_𝑖,𝐺+1.(4)Perform the crossover strategy 𝑍_𝑖,𝐺+1.(5)Check the bound constraint.(6)Do the selection.(7)Evaluate the best member of the population after selection 𝑋_best,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 (𝐶₁₁,𝐶₁₂=𝐶₂₁,𝐶₂₂) 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)

(b)

(c)

(d)

Figure 5 

(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.

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.

Figure 6 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)

(b)

(c)

(d)

Figure 6 

(a) Loc = 200, depth = 2.4. (b) Loc = 600, depth = 2.4. (c) Loc = 400, depth = 2.4. (d) Loc = 300, depth = 2.4.

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: %error=predictedvalue−actualvalueactualvalue×100. (34)

Table 4 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.

Table 2 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.

Table 3 

Calculated frequencies using theoretical analysis.

Table 4 

Crack detection results of the theoretical study by applying DE.

Table 5 

Crack detection results of the experimental study by applying DE.

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.