Nonlinear springs with dynamic frictionless contact

  • Jeongho AhnEmail author
  • Jay Mayfield


This paper focuses on mathematical and numerical approaches to dynamic frictionless contact of nonlinear viscoelastic springs. This contact model is formulated by a nonlinear ordinary differential equation system and a pair of complementarity conditions. We propose three different numerical schemes in which each of them consists of several numerical methods. As a result, three groups of time-discrete numerical formulations are established. We use the coefficient of restitution to prove convergence of numerical trajectories, passing to the limit in the time step size. The Banach-fixed point theorem is applied to show the existence of global solutions satisfying all conditions. A new form of energy balance is derived, which is verified theoretically and numerically. All of the three schemes are implemented and their numerical results are compared with each other.


Nonlinear springs Signorini contact conditions Complementarity conditions Euler methods Runge–Kutta method 

Mathematics Subject Classification

Primary 65L20 Secondary 74H20 74H15 74M20 



The authors would like to acknowledge the valuable comments of an anonymous referee which have improved the presentation of the paper.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsArkansas State UniversityState UniversityUSA
  2. 2.Department of mathematicsIowa State UniversityAmesUSA

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