Advertisement

Nonlinear springs with dynamic frictionless contact

  • Jeongho Ahn
  • Jay Mayfield
Article
  • 27 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

Primary 65L20 Secondary 74H20 74H15 74M20 

Notes

Acknowledgements

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

References

  1. 1.
    Ahn, J.: Damageable linear elastic buckling bodies (preprint)Google Scholar
  2. 2.
    Ahn, J., Calhoun, J.: Dynamic contact of viscoleastic bodies with two obstacle: mathematical and numerical approaches. Electron. J. Differ. Equ. 1–23, 2013 (2013)Google Scholar
  3. 3.
    Ahn, J., Shin, D., Park, E.: \({C}^{0}\) interior penalty finite element methods for damageable linear elastic plates (in preparation)Google Scholar
  4. 4.
    Ahn, J., Wolf, J.R.: Dynamic impact of a particle. Involve - a journal of mathematics 6(2), 147–167 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ballard, P.: The dynamics of discrete mechanical systems with perfect unilaeral constraints. Arch. Ration. Mech. Anal. 154, 199–274 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ballard, P.: Formulation and well-posedness of the dynamics of rigid-body systems with perfect unilateral constraints. Phil. Trans. R. Soc. Lond. A 359, 2327–2346 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New Delhi (1972)zbMATHGoogle Scholar
  8. 8.
    Dontchev, A., Lempio, F.: Difference methods for differential inclusions. A survey. SIAM Rev. 34, 263–294 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dumont, Y., Goeleven, D., Rochdi, M., Shillor, M.: Fricitonal contact of a nonlinear spring. Math. Comput. Model. 31, 83–97 (2000)CrossRefGoogle Scholar
  10. 10.
    Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. II of Springer Series in Operations Research. Springer, New York (2003)Google Scholar
  11. 11.
    Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Side. Kluwer Academic, Norwell (1988)CrossRefGoogle Scholar
  12. 12.
    Frémond, M.: Non-Smooth Thermo-Mechanics. Springer, Berlin (2002)CrossRefGoogle Scholar
  13. 13.
    Kachanov, K.L.: Introduction to the Theory of Damage. Martinus Nijhoff (New revised printing 1990), The Hague (1986)Google Scholar
  14. 14.
    Klarbring, A., Mikelić, A., Shillor, M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci 26(8), 811–832 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lempio, F., Veiov, V.: Discrete approximations of differential inclusions, vol. 42. Bayreuther Mathematische Shriften, (1998)Google Scholar
  16. 16.
    Martins, J.A.C., Odens, J.T.: Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface law. Nonlinear Anal. Theory Methods Appl. 11(3), 407–428 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Maury, B.: A time-stepping scheme for inelastic collisions. Numerische Mathematik 102(4), 649–679 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Migorski, S., Ochal, A., Shillor, M., Sofonea, M.: A model of a spring-mass-damper system with temperature-dependent friction. Eur. J. Appl. Math. 25, 45–64 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Moreau, J.J., Panagiotopoulos, P.D., Strang, G.: Topics in Nonsmooth Mechanics. Birkhäuser Verlag, Basel (1988)CrossRefGoogle Scholar
  20. 20.
    Rockafellar, R.T.: Integrals which are convex functionals. Pac. J. Math. 24, 525–539 (1968)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rockafellar, R.T.: Integrals which are convex functionals ii. Pac. J. Math. 39, 439–469 (1971)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Royden, H.L.: The Real Analysis, 3rd edn. Prentice Hall Inc, New Jersey (1988)zbMATHGoogle Scholar
  23. 23.
    Schatzman, M.: A class of nonlinear differential equations of second order in time. Nonlinear Anal. Theory Methods Appl. 2(3), 353–373 (1978)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schatzman, M.: Uniqueness and continuous dependence on data for one-dimensional impact problems. Math. Comput. Model. 28(4–8), 1–18 (1998)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Shillor, M., Han, W., Sofonea, M.: Aanlysis and Approximaions of Contact Problems with Adhesion or Damage. Pure and Applied Mathematics, vol. 276. Chapman & Hall/CRC, Boca Raton (2006)Google Scholar
  26. 26.
    Shillor, M., Sofonea, M., Telega, J. J.: Models and Analysis of Quasistatic Contact, volume Lecture Notes in Physics, p. 655. Springer, Berlin (2004)Google Scholar
  27. 27.
    Stewart, D.E.: Rigid-body dynamics with friction and impact. SIAM Rev. 42(1), 3–39 (2000)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Stewart, D.E.: Differentiating complementarity problems and fractional index convolution complementarity problems. Houst. J. Math. 33(1), 301–321 (2007)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Zeidler, Eberhard.: Applied Functional Analysis. Applied Mathematical Sciences, vol. 108. Springer, New York (1997)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

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

Personalised recommendations