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

  • Vít Průša
  • Martin Řehoř
  • Karel Tůma


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. In the linear case, the mathematical method of choice for the analysis of the response 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 Průša and Rajagopal (Int J Non-Linear Mech 81:207–221, 2016), we show how to use the theory in the analysis of response of nonlinear spring–dashpot and spring–dashpot–mass systems.


Mechanical systems Nonlinear ordinary differential equations Jump discontinuities Colombeau algebra 

Mathematics Subject Classification

46F30 34A36 34A37 70G70 


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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPraha 8 – KarlínCzech Republic
  2. 2.Institute of Fundamental Technological Research, Polish Academy of SciencesWarszawaPoland

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