Numerical approach of free and forced elastic vibrations using high-regularity Hermitian partition of unities
- PDF / 3,876,704 Bytes
- 19 Pages / 595.276 x 790.866 pts Page_size
- 43 Downloads / 213 Views
(2020) 42:278
TECHNICAL PAPER
Numerical approach of free and forced elastic vibrations using high‑regularity Hermitian partition of unities Rudimar Mazzochi1 · Oscar Alfredo Garcia de Suarez1 · Rodrigo Rossi1 Received: 16 October 2019 / Accepted: 20 April 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract In this paper, shape functions with regularity up to C2 were developed for the four-node quadrilateral finite element considering the partition of unity property. This high-regularity approximation space was applied to approach the natural frequencies of free vibration of some in-plane elastic problems as well as the elastic wave response of forced vibration caused by the application of impulsive loading. The obtained results show that it was possible to numerically approximate a greater number of accurate natural frequencies with the devised procedure when a comparison is established with the C0 Lagrangian and serendipity elements of 4, 8 (serendipity), 16, and 25 nodes. Also, the numerical predictions for the elastic wave propagation problem using the derived approximation space presented small oscillations improving the representation of the displacement field. Keywords High-regularity Hermitian · FEM · Partition of unity · Eigenvalues · Natural frequencies · Elastic waves
1 Introduction The numerical determination of eigenvalues inside of the finite element method is a challenge. The approximation of eigenvalues, and consequently of natural frequencies of free vibration, through the finite element method is addressed in the literature, for example, by [22]. In this book, an a priori error estimator is presented which correlates the upper approximation limit of an eigenvalue with a parameter related to discretization (mesh size) and the properties of the approximation space (order of Hilbert space and polynomial degree of interpolation functions), see theorem 1 on page 7. This error estimator is also presented in Hughes [15] while Technical Editor: João Marciano Laredo dos Reis. Grant: CNPq 306058/2018-9 & FAPERGS 19/2551-0001954-8-1. * Rodrigo Rossi [email protected] Rudimar Mazzochi [email protected] Oscar Alfredo Garcia de Suarez [email protected] 1
Departamento de Engenharia Mecânica, Universidade Federal do Rio Grande do Sul, Rua Sarmento Leite, 425, Porto Alegre, RS 90046‑902, Brazil
describing the behavior of numerical results obtained for natural frequencies of free vibration for bars and beams. The results include graphs of normalized spectrum curves which are constructed by plotting the ratio between the approximated natural frequency and the exact one in the vertical axis and the ratio between the nth approximated mode and the total number of modes approximated in the horizontal axis. Such graphs show the progressive degeneration of approximate solution with the increasing vibration mode. The problem of approximation of natural frequencies through the finite element method was also addressed in Cottrell et al. [7], Cottrell et al. [8], Hughes et al.
Data Loading...
