Abstract

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.

摘要

通过由全局分段平滑函数构成的广义基来分析自由振动模式下包含内部复杂性的机械部件。2002 年引入的此类基础（如 GPSF）在 2015 年进行了自适应，以减少计算工作量（A-GPSF），而 2018 年的一项最新研究使用补充分析条件和 A-GPSF 来研究薄壁梁的自由振动和盘子。在本文中，A-GPSF 的广义基以自包含的方式介绍（i），id est，不需要任何补充分析条件，甚至 (ii) 扩展了对包含内部跳跃的机械组件进行建模的可能性。展示了如何使用这种广义的 A-GPSF 来模拟具有不同平滑度的连续函数；即使是不连续的函数（即包含跳跃）也可以建模，并且总是不涉及显着的近似或导致任何吉布斯现象。因此，机械部件的自由振动分析（包括工程感兴趣的内部复杂性）成为基于全局近似的简单统一程序的一部分，其中未知特征值和特征函数只能通过 A-GPSF 的广义基来挑选；然后将先前公式的结果作为特定情况获得。