Fault Diagnosis of Cracked Cantilever Composite Beam by Vibration Measurement and RBFNN

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Ÿ Length of the Beam (L) = 800 mm; Width of the beam (W) = 50 mm; Thickness of the Beam (H) = 6 mm;

Ÿ Relative crack depth (ψ₁= a₁/H) = from 0.0833 to 0.5;

Ÿ Relative crack depth (ψ₂= a₂/H) = from 0.0.833 to 0.5;

Ÿ Relative crack location (β₁= L₁/L) = from 0.625 to 0.875;

Ÿ Relative crack location (β₂= L₂/L) = from 0.125 to 0.9375;

Ÿ Properties of Graphite fiber reinforced polyimide composite material in analysis:

Ÿ Young’s modulus of fiber = E_f = 275.6 GPa;

Ÿ Young’s modulus of matrix = E_m = 2.756 GPa;

Ÿ Modulus of rigidity of fiber = G_f= 114.8 GPa;

Ÿ Modulus of rigidity of matrix = G_m = 1.036 GPa;

Ÿ Poisson’s ratio = ν_f = 0.2; Poisson’s ratio = ν_m= 0.33;

Ÿ Density of fiber = ρ_f = 1.9 gr/cm³; Density of matrix = ρ_m = 1.6 gr/cm³;

The numerical analysis is accepted by using the finite element software ANSYS in the frequency domain and obtain natural frequencies, and mode shapes.

A higher order 3-D, 8 node element having three degrees of freedom at each node: translations in the nodal x, y, and z directions (Specified as SOLSH 190 in ANSYS) was selected and used throughout the analysis. Each node has three degrees of freedom, making a total twenty four degrees of freedom per element. The layers stacking in ANSYS shown in Figure 2. The results of the numerical analysis for the first three mode shapes for un-cracked and cracked beam (ψ₁= 0.166, ψ₂= 0.333 and β₁= 0.25, β₂= 0.5) are shown in the Figure 3.

Figure 2. Layers Stacking in ANSYS

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Figure 3 a. Relative Amplitude vs. Relative crack location from fixed end (1^st mode of vibration). b. Relative Amplitude vs. Relative crack location from fixed end (2^nd mode of vibration). c. Relative Amplitude vs. Relative crack location from fixed end (3^rd mode of vibration)

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Figure 4. RBFNN Model

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3. Radial Basis Function Neural Network (RBFNN) Analysis

The radial basis function neural network, as a type of feed-forward neural network has recently attracted extensive research interest because of its simple architecture, high approximation and regularization capability, and good local specialization and global generalization ability. The parameter of RBFNN involves the numbers of neurons in input layer, hidden layer and output layer, RBF centers and width of neuron in hidden layer and linear weight connected to the hidden layer and output layer. Each neuron in hidden layer of RBFNN produces a radically symmetric response around a node parameter victor called a center. As well known the performance of RBFNN critically relies on the selection of RBF centers. RBF center located through clustering techniques by which RBF centers are selected randomly from input data and adjusted constantly by clustering algorithm until they no longer change. The RBFNN controller has been developed for detection of the relative crack locations and relative crack depth having six input parameters and two output parameters as shown in Figure 4. The linguistic term used for the inputs are as follows; Relative first natural frequency = “rfnf”; Relative second natural frequency = “rsnf”; Relative third natural frequency = “rtnf”; Relative first mode shape difference = “rfmd”; Relative second mode shape difference = “rsmd”; Relative third mode shape difference = “rtmd. The linguistic term used for the outputs are as follows; Relative first crack location = “rcl1” Relative second crack location = “rcl2” Relative first crack depth = “rcd1” Relative second crack depth = “rcd2”. The input layer has six neurons, three for first three relative natural frequencies and other three for first three average relative mode shape difference. The output layer has four neurons, which represents relative crack locations and relative crack depths.

The relative natural frequency and relative mode shape difference used in the above analysis can be defined as follows.

Figure 5. Schematic block diagram of experimental set-up

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Figure 6 a. Relative Amplitude vs. Relative crack location from fixed end (1^st mode of vibration). b. Relative Amplitude vs. Relative crack location from fixed end (2^nd mode of vibration). c. Relative Amplitude vs. Relative crack location from fixed end (3^rd mode of vibration)

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4. Experimental Investigation

To validate the numerical analysis result, an experiment on composite beam has been performed shown in Figure 5. A composite beam was clamped at a vibrating table. During the experiment the cracked and undamaged beams have been vibrated at their 1^st, 2^nd and 3^rd mode of vibration by using an exciter and a function generator. The vibrations characteristics such as natural frequencies and mode shape of the beams correspond to 1^st, 2^nd and 3^rd mode of vibration have been recorded by placing the accelerometer along the length of the beams and displayed on the vibration indicator. The surface specimens were cut with EDM wire cut machine to ensure high accuracy of cracked model. The experimental results are in close justification with neural analysis results. These results for first three modes are plotted in Figure 6. Corresponding numerical results for the cracked and un-cracked beam are also presented in the same graph for comparison. The comparison of results between RBFNN controller, numerical analysis and experimental analysis shown in Table 1.

Table 1. Comparison of results between RBFNN model, Numerical analysis and Experiment analysis

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