Each cylinder is measured three times under different boundary conditions by changing the orientation of a rectangular sponge inserted inside the cylinder. The number of sets of measurements taken for undamaged population is 60 (20 cylinders X 3 for different boundary conditions). 

The impulse and response data are processed using the Fast Fourier Transform (FFT) 
to convert the time domain impulse history and response data into the frequency domain. The data in the frequency domain are used to calculate the FRFs. From the FRFs, the modal properties are extracted using modal analysis and the pseudo modal energies are calculated using the integrals under the peaks for a given frequency bandwidth using the trapezoidal technique. The frequency spacing of the FRFs is 1.22Hz. When the pseudo modal energies are calculated, frequency ranges spanning over 6% of the natural frequencies are chosen. These bandwidths are as follows in Hz: 393-418, 418-443, 536-570, 1110-1180, 1183-1254, 1355-1440, 1450-1538, 2146-2280, 2300-2440, 2250-2401, 2500-2656, 3140-3340, 3350-3565, 3800-4039, and 4200-4458. The guidelines outlined in [2] are taken into consideration when choosing these frequency ranges. These guidelines state that the frequency bandwidth must be: (1) sufficiently narrow to capture the resonance behavior, (2) sufficiently wide to capture the smoothing out of zero-mean noise, and (3) must not include the regions of the anti-resonance, which are generally noisy. The pseudo modal energies are used to train the pseudo-modal-energy TS neuro-fuzzy model. The numbers of pseudo-modal-energies identified are 646 (corresponding to 17 natural frequencies X 19 measured mode-shape-co-ordinates X 2 for real and imaginary parts of the pseudo modal energy). The Statistical Overlap Factor (SOF) and the Principal Component Analysis (PCA) are used to reduce the dimension of the input data from 646X167 pseudo-modal-energies to 10X167 for both these data types. 

![0_image_0.png](0_image_0.png)

## Neuro-Fuzzy Model Optimisation

For the optimization of the neuro-fuzzy model a 10-fold cross validation method is used for model selection. Once the appropriate model has been selected, the training data is then used to train a neuro-fuzzy model with the correct model parameters. A first order TS neuro-fuzzy model with a Gaussian membership function has been implemented. The 