Reproducing kernel mesh-free collocation analysis of structural vibrations. Issue 3 (23rd January 2019) - Qi, Dongliang; Wang, Dongdong; Deng, Like; Xu, Xiaolan; Wu, Cheng-Tang
Abstract : Purpose: Although high-order smooth reproducing kernel mesh-free approximation enables the analysis of structural vibrations in an efficient collocation formulation, there is still a lack of systematic theoretical accuracy assessment for such approach. The purpose of this paper is to present a detailed accuracy analysis for the reproducing kernel mesh-free collocation method regarding structural vibrations. Design/methodology/approach: Both second-order problems such as one-dimensional (1D) rod and two-dimensional (2D) membrane and fourth-order problems such as Euler–Bernoulli beam and Kirchhoff plate are considered. Staring from a generic equation of motion deduced from the reproducing kernel mesh-free collocation method, a frequency error measure is rationally attained through properly introducing the consistency conditions of reproducing kernel mesh-free shape functions. Findings: This paper reveals that for the second-order structural vibration problems, the frequency accuracy orders are p and ( p − 1) for even and odd degree basis functions; for the fourth-order structural vibration problems, the frequency accuracy orders are ( p − 2) and ( p − 3) for even and odd degree basis functions, respectively, where p denotes the degree of the basis function used in mesh-free approximation. Originality/value: A frequency accuracy estimation is achieved for the reproducing kernel mesh-free collocation analysis of structural vibrations, which can effectively underpin the practical applications of this method.