146 P. Seventekidis et al. where g ij θm|M is the numerical time-history of the introduced FE model and ˆyij is the respective experimental signal. Subscripts i correspond to the sensor (accelerometer) location and measurement direction, and j corresponds to the timestep instant. n is the total number of measured sensor locations and directions, whereas mis the total number of measured time-steps (number of observations). 16.3 Machine Learning with Neural Networks Artificial Neural Networks (ANN) comprise a relatively old idea [23] that mimics the biological function of neurons output and input. Mathematically a NN in its simplest form, called perceptron, computes an output based on a weighted sum of previous input as follows: Yi =f (Xi) and Xi =bi +Wiyi−1 (16.8) Where Yi is the output of neuron i computed with the transfer function f and Xi is weighted sum of its inputs yi −1 which are weighted over with an array Wi and bias b. The goal in a trivial machine learning problem like the above is to correctly choose the weight matrixWand this process is called NN training, which nowadays is usually performed through stochastic gradient optimization methods using the Back-Propagation (BP) algorithm [24]. The simple perceptron however would not be suitable or adequate in a SHM problem where raw signal is intended to be fed in the training process due to structured way it expects data and also due to the inability to extract training features, requiring high user pre-processing. The above difficulties however are handled conveniently by a class of ANN called Convolutional Neural Networks (CNN) that by applying learned filters to the input data, they can extract training features in an unstructured manner. The convolution is filtering process that this time the neurons produce as output and an acceleration signal a may be filtered at a convolutional layer as follows: X=b+conv1D(k,a) (16.9) Wherek would be the learned network filter, bthe neuron bias andconv1Drepresents the filtering process of the initial signal. The learning of the filters adds in as an extra task for network BP training, however filtered signals now contain enhanced information about the status of structure. Moreover, additional layers isolate and extract the important signal characteristics, which end up in a simple perceptron classifier. This time however, the necessary feature extraction and structuring has already been performed by the convolutional and the intermediate layers, accounting for the wide success and use of this type of networks [25]. 16.4 Experimental Application In order to examine the complexity and orthotropic material mechanical behavior of the used CFRP, dynamically induced excitation tests were conducted at a cantilever CFRP tubular beam as presented in Fig. 16.1. Specifically, Fig. 16.1 presents the cantilever CFRP small-radius tube along with two (2) tri-axial accelerometers, a strain gauge sensor and a load cell at the free end of the cantilever beam, where an electromagnetic shaker device is mounted. Both arrangements were introduced in order to acquire knowledge of the mechanical behavior of the CFRP material and thus characterize its orthotropic behavior. The CFRP is consisted of a stack of nine (9) plies with equal thickness and orientation angles apart from one ply. Specifically, plies 1 to 6 and 8 to 9 have a thickness of t = 0.175mm at θ = 55◦ and θ = − 55◦ orientation angles consecutively. Ply 7 has a thickness of t =0.16mmat θ =86◦. The nominal material parameters of the 2D orthotropic material used to model the CFRP was E1 = 146, 45GPa and E2 = 7.73GPa for the modulus of elasticity in X and Y direction respectively, vxy = vyx = 0.12 is the Poisson’s ration for in-plane bi-axial loading, and G12 = 3.54GPa, Gxz =3.95GPaandGyz =2.80GPa are the in-plane, transverse for shear in XZ plane and transverse for shear in YZ plane shear moduli and ρ =1600kgr/m3 is the density.
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