Assessment of Structural Monitoring by Analyzing Some Modal Parameters: An Extended Inventory of Methods and Developments

The SHM (structural health monitoring) evaluation consists to determining the modes (resonances) of vibration characteristic of the structure and each of them is represented by its modal parameters which can be obtained experimentally and can be analyzed by different procedures. In the present paper (except subsections 2.1.1, 3.1.1) the coefficients (including the coefficient of displacement) are constant; in this regard, it is made an inventory of some methods of classical and non-classical mathematics with the specific computing scheme. All methods of classical mathematics that were considered, i.e., second-order linear nonhomogeneous differential equations, Laplace operational method, analytical conditional form, approximations with error evaluation (according to contraction principle) are inventoried and then developed-and the first two methods by comparison. As methods of the non-classical mathematics the dyadic wavelet method, the approximation (transform) fuzzy method and the grammatical evolution method are inventoried and then the first is developed. In addition, there are illustrated the calculus techniques by few examples and also computing wavelet coefficients for healthy and damaged structures. Notice that in the subsection 2.1.1 in which the coefficients (including the coefficient of displacement) are considered variable there are some transformations of (non)homogeneous equations in the scaled forms. In addition in the subsection 3.1.1 exact solutions, respectively approximations with error evaluation (according to contraction principle) relative to remarkable equations in literature (including the corresponding standard form) are obtained by using the results of 2.1.1.