PSI - Issue 37

412 Deniss Mironovs et al. / Procedia Structural Integrity 37 (2022) 410–416 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 3 where ( ) represent singular vectors, while ( ) is a diagonal matrix of singular values (Chauhan (2015)). SVD is analogous to Principal Component Analysis (PCA) which give principal components (eigenvectors) of presented data (Sen et al (2019)) with corresponding energy factors (eigenvalues). In modal analysis eigenvectors represent mode shapes and eigenvalues represent corresponding frequencies. The above mentioned approach in operational modal analysis is called Frequency Domain Decomposition, first introduced by Brincker et al (2000) and widely used today, especially its enhanced version EFDD. Alternative modal parameter estimation approach is based in time domain and is called Stochastic Subspace Identification (SSI), explained by Brincker and Andersen (2006), where in case of covariance driven SSI, singular value decomposition is applied to a Hankel matrix, formed from vibrational data covariance functions matrices in time domain. In data driven SSI, Hankel matrix is formed from raw time data. When eigenvectors and eigenvalues are extracted from raw data, it is a matter of choosing the right peak of power spectrum, which is meant to represent the mode. Unfortunately, pick peaking is unreliable, there is a strong human factor involved, it depends on signal-to-noise ratio, also harmonics can be mistaken for modes. Above mentioned approaches develop mathematical polynomial models that best fit vibrational data (Hu et al (2012)), so modal parameters are estimated from models instead. 2.2. Anomaly detection Anomaly detection, also known as Novelty detection, as shown by Yan et al (2005) and Hu (2011) is a statistical method for signaling whether a new data set is similar to established reference data set. This is a type of unsupervised machine learning algorithm. The analysis first step is to create a representation of a normal condition of a structure. The second step is to observe subsequent data to see if they are significantly different from the normal condition. For this study anomaly detection data is formed as a matrix of modal frequencies and mode shape vectors. Other potential factors to add to this matrix are load, temperature, humidity, wind speed, etc. The matrix is subjected to calculation of mean values for each row and standard uncertainty for the presented data set of samples. One important assumption is that the studied structures modal characteristics and operational conditions influence are linear and thus have Gaussian distribution. A probability of each measured sample is calculated using multivariate Gaussian distribution equation (Do (2008)): ( , , Σ) = (2 ) 1 2 |Σ| 1 2 − 1 2 ( − ) Σ −1 ( − ) (4) where , , Σ are given sample vector, mean value vector and covariance matrix respectively, is the number of features. To satisfy multivariate condition for number of samples to be significantly larger than number of features, a 12 th degree polynomial approximation can be used for mode shape vectors. 3. Simulation The study is performed in the following way. A finite element model of a cantilever carbon fiber reinforced polymer composite beam with the spatial dimensions of 1×0.1 m a total thickness of 2 mm is constructed in Ansys software. There are 20 layers in the beam, which are layered orthogonally relative to each other. The laminate lay-up of the beam is (0/90) 5s with a ply thickness of 0.1 mm. The FE model of the beam is developed using 100×10 eight-node shear-deformable shell elements and the clamped boundary condition is applied at one edge of the beam. The material properties of the model are displayed in table 1. Modal analysis is performed for this structure and 5 modes are identified with representative modal parameters – frequency and mode shape. Mode shape is formed of strain data for each node, which forms strain matrix × 2 for the whole beam, where is the number of nodes equal to 1111 and number 2 represents X and Y strain directions. Strain matrix is normalized and approximated with 12 th degree polynomial using 202 elements.

Table 1. Material properties of the FE model.

direction x / xy plane

direction y / yz plane

direction z / xz plane