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.
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 [
Recently a lot of work has been done using modal analysis to detect, locate, and predict crack severity to a greater accuracy level. Dimarogonas [
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
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
Beam model.
The relationship between strain energy release rate
From earlier studies [
where the experimentally determined functions
The strain energy release rate (also called strain energy density function) at the crack location is defined as
So the strain energy release (
The flexibility influence coefficient
From the above condition, (
Calculating
The local stiffness matrix can be obtained using the inverse of compliance matrix
Converting the influence coefficient into dimensionless form we get
The cantilever beam as mentioned in Section
longitudinal vibration
lateral vibration
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
where
At the cracked section
Multiplying both sides of the above equation by
The normal functions ((
where
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
Material properties of AluminiumAlloy, 2014T_{4}.
Young’s modulus, 
Density, 
Poisson’s ratio, 
Length, 
Width, 
Thickness, 

72.4 GPa  2.8 gm/cc  0.33  800 mm  50 mm  6 mm 
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
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
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.
The differential evolution (DE) algorithm is a population based evolutionary algorithm developed by nondifferentiable continuous space functions [
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
For each target vector
The indices
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
In order to decide if a vector
The aforementioned steps are repeated generation after generation until some specific termination criteria are satisfied. The algorithmic description of DE is summarized in Table
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
Generate a population of solution vectors.
Evaluate the best member of the population
Carry out the mutation operation
Perform the crossover strategy
Check the bound constraint.
Do the selection.
Evaluate the best member of the population after selection
If the stopping criterion is reached, then print the output results and stop; otherwise repeat Steps 2–7.
Then go to Step 3.
The theoretical analysis and proposed DE algorithm were implemented using MATLAB 7.0 [
Dimensionless compliance (
For crack location (e.g., 100 mm) and relative crack depths
(a) Relative amplitude versus distance from the fixed end (1st mode of vibration),
The numerical results for the relative amplitude of transverse vibration at different locations of aluminium alloy 2014
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
(a) Loc = 200, depth = 2.4. (b) Loc = 600, depth = 2.4. (c) Loc = 400, depth = 2.4. (d) Loc = 300, depth = 2.4.
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:
Table
Table
Measured natural frequencies for single crack cantilever beam.
Sl. No.  Crack  Bending frequencies (Hz)  

Location (mm)  Depth (mm) 




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  122.4656 
Calculated frequencies using theoretical analysis.
Sl. No.  Crack  Bending frequencies (Hz)  

Location (mm)  Depth (mm) 




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  129.1213 
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  0.021 
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 
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.