The weak form of the SDOF and MDOF equation of motion, part I: Theory

TL;DR

This paper develops a theoretical framework for the weak form of single and multi-degree-of-freedom equations of motion, transforming initial conditions into boundary value problems and outlining a general numerical approach.

Contribution

It introduces a novel theoretical approach to formulating the weak form of equations of motion and proposes a general method for numerical solutions using arbitrary basis functions.

Findings

Transforming initial conditions into boundary value problems

Outlining a general numerical method with arbitrary basis functions

Foundation for subsequent numerical method development

Abstract

The weak form of the SDOF and MDOF equations of motion are obtained. The original initial conditions problem is transformed into a boundary value problem. The boundary value problem is then solved and transformed back to the initial conditions one. Subsequently, a general method for obtaining numerical methods using an arbitrary number of linearly independent approximating functions is outlined. This is part one of a series of three papers, in the second of which a numerical method is obtained, using Bernstein polynomials of arbitrarily high order. The numerical evidence for the convergence of the method will be presented in the third part paper.