Colombeau algebra as a mathematical tool for investigating step load and step deformation of systems of nonlinear springs and dashpots

TL;DR

This paper demonstrates how Colombeau algebra can be used as a mathematical tool to analyze the step response of nonlinear spring-dashpot systems, extending classical distribution methods to nonlinear contexts.

Contribution

It applies Colombeau algebra to study nonlinear mechanical systems' responses to step inputs, providing a new approach beyond classical distribution theory.

Findings

Successfully models nonlinear system response to step loads

Extends mathematical analysis tools for nonlinear systems

Provides a framework for future nonlinear system analysis

Abstract

The response of mechanical systems composed of springs and dashpots to a step input is of eminent interest in the applications. If the system is formed by linear elements, then its response is governed by a system of linear ordinary differential equations, and the mathematical method of choice for the analysis of the response of such systems is the classical theory of distributions. However, if the system contains nonlinear elements, then the classical theory of distributions is of no use, since it is strictly limited to the linear setting. Consequently, a question arises whether it is even possible or reasonable to study the response of nonlinear systems to step inputs. The answer is positive. A mathematical theory that can handle the challenge is the so-called Colombeau algebra. Building on the abstract result by (Pr\r{u}\v{s}a & Rajagopal 2016, Int. J. Non-Linear Mech) we show how to use the theory in the analysis of response of a simple nonlinear mass--spring--dashpot system.