Generalized A-GPSFS and modeling of vibrating elements with internal complexities

Mechanical components containing internal complexities in free vibration mode are analyzed through generalized bases made of global piecewise smooth functions. Such bases introduced in 2002, as GPSFs, were made adaptive in 2015 to reduce the computational efforts (A-GPSFs) while, a recent study in 2018 used supplementary analytical conditions along with A-GPSFs to study free vibrations of thin-walled beams and plates. In this paper, generalized bases of A-GPSFs are introduced (i) in a self-contained manner, id est, without needing any supplementary analytical conditions and even (ii) extending the possibility of modeling mechanical components containing internal jumps. It is shown how such generalized A-GPSFs can be used to model continuous functions which have different degrees of smoothness; even discontinuous functions (i.e. containing jumps) can be modeled and always without involving significant approximations or letting rise to any Gibbs phenomenon. Thus, a free vibration analysis of mechanical components including internal complexities of engineering interest becomes part of a simple unified procedure based on a global approximation where unknown eigenvalues and eigenfunctions can be singled out only through generalized bases of A-GPSFs; the results of previous formulations are then obtained as particular cases. The efficiency and ability of the proposed models result from the comparison between calculated eigen-parameters and those of other models presented in the open literature.