Contribution to new numerical integration in dynamics

Katarína Tvrdá

doi:10.21595/vp.2019.20669

Abstract

The problem of time integration in the dynamics of building structures is primarily a mathematical problem of the accuracy of the numerical integration of the acceleration time course. All existing methods are trying more or less successfully to master this problem. The best known and most widely used is the Newmark method, which by its very nature is only a trapezoidal rule applied to both acceleration and speed. The most effective method of numerical integration of repeated integrals was introduced mathematically [1], namely the modification of numerical integration methods of the Gaussian type for use on repeated integrals. In this work is introduced their simple use in problem solution of mechanics and dynamics of structures and consequently comparison of the results with the traditional Newmark Direct Time Integration method is presented.

Highlights

The transformation of repeated integrals into simple ones is analytically derived in published work [1]
There is listed a modification of some existing numerical methods for their efficient solution
The present paper shows the possibilities of a new method of numerical integration of repeated integrals [1] and their practical use in nonlinear calculations in statics and dynamics of structures

Summary

Introduction

The transformation of repeated integrals into simple ones is analytically derived in published work [1]. The detailed explanation of the modified Gaussian type numerical methods for repeated integrals the reader can find in [1]. Seeing that it is a Gaussian quadrature that has been splitted and transformed by Cauchy’s theorem [2] to solving repeated integrals [1], the error term R is according to [4]: R. From a practical point of view it is important to note that the standard Gaussian quadrature with the m number of points accurately integrates all polynomial functions of the order of equal and less than (2m 1). The Eq (1), as shown below, allows us to solve numerically the technical tasks whose relationships are defined by the higher order integrals ( the problem of forced vibration, too) by applying simple algebraic operations

Illustrative use of modified Gaussian integration

Direct integration of the d’Alembert’s differential equation

Dynamic analysis of the reinforced concrete cantilever

Conclusions