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

摘要

The response of mechanical systems composed of springs and dashpots to a stepinput is of eminent interest in the applications. If the system is formed bylinear elements, then its response is governed by a system of linear ordinarydifferential equations, and the mathematical method of choice for the analysisof the response of such systems is the classical theory of distributions.However, if the system contains nonlinear elements, then the classical theoryof distributions is of no use, since it is strictly limited to the linearsetting. Consequently, a question arises whether it is even possible orreasonable to study the response of nonlinear systems to step inputs. Theanswer is positive. A mathematical theory that can handle the challenge is theso-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 theanalysis of response of a simple nonlinear mass--spring--dashpot system.