Nonlinear Feedback Control and Artificial Intelligence Computational Methods applied to a Dissipative Dynamic Contact Problem

Daniela Danciu, Andaluzia Cristina Matei, Sorin Daniel Micu, Ionel Rovenţa

2014

Abstract

In this paper we consider a vibrational percussion system described by a one-dimensional hyperbolic partial differential equation with boundary dissipation at one extremity and a normal compliance contact condition at the other extremity. Firstly, we obtain the mathematical model using the Calculus of variations and we prove the existence of weak solutions. Secondly, we focus on the numerical approximation of solutions by using a neuromathematics approach – a well-structured numerical technique which combines a specific approach of Method of Lines with the paradigm of Cellular Neural Networks. Our technique ensures from the beginning the requirements for convergence and stability preservation of the initial problem and, exploiting the local connectivity of the approximating system, leads to a low computational effort. A comprehensive set of numerical simulations, considering both contact and non-contact cases, ends the contribution.

References

  1. Andersson, L.-E. (1991). A quasistatic frictional problem with normal compliance. Nonlinear Analysis TMA, (16):347-370.
  2. Andersson, L.-E. (1995). A global existence result for a quasistatic contact problem with friction. Advanced in Mathematical Sciences and Applications, (5):249- 286.
  3. Chua, L. and Roska, T. (1993). The cnn paradigm. IEEE Trans. Circuits Syst. I, 40(3):147-156.
  4. Danciu, D. (2013a). A CNN Based Approach for Solving a Hyperbolic PDE Arising from a System of Conservation Laws - The Case of the Overhead Crane, Advances in Computational Intelligence, volume 7903, pages 365-374. Springer.
  5. Danciu, D. (2013b). Numerics for hyperbolic partial differential equations (pde) via cellular neural networks (cnn). In 2nd IEEE Int. Conf. on Systems and Computer Science ICSCS'2013, pages 183-188, Villeneuve d'Ascq, France.
  6. Danciu, D. and Ra?svan, V. (2014). Delays and Propagation: Control Liapunov Functionals and Computational Issues, chapter 10. Advances in Delays and Dynamics. Springer.
  7. Galushkin, A. I. (2010). Neural Network Theory. Springer.
  8. Gilli, M., Roska, T., Chua, L. O., and Civalleri, P. P. (2002). Cnn dynamics represents a broader class than pdes. I. J. Bifurcation and Chaos, (10):2051-2068.
  9. Halanay, A. and Ra?svan, V. (1981). Approximations of delays by ordinary differential equations. Recent advances in differential equations, pages 155-197.
  10. Han, W. and Sofonea, M. (2002). Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics 30, American Mathematical Society, Providence, RI-International Press, Somerville, MA.
  11. Hyman, J. (1979). A method of lines approach to the numerical solution of conservation laws. Advances in Delays and Dynamics. IMACS Publ. House.
  12. Kikuchi, N. and Oden, J. (1988). Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia.
  13. Kim, J. U. (1989). A boundary thin obstacle problem for a wave equation. Commun. P. D. E., (14):1011-1026.
  14. Klarbring, A. Mikelic, A. and Shillor, M. (1988). Frictional contact problems with normal compliance. Int. J. Engng. Sci., (26):811-832.
  15. Klarbring, A. Mikelic, A. and Shillor, M. (1991). A global existence result for the quasistatic frictional contact problem with normal compliance. G. del Piero and F. Maceri, eds., Unilateral Problems in Structural Analysis Vol. 4, Birkhäuser.
  16. Lions, J.-L. (1969). Quelques methodes de resolution des problmes aux limites non lineaires. Dunod et Gauthier-Villars, Paris.
  17. Martins, J. A. and Oden, J. T. (1987). Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Analysis TMA, (11):407-428.
  18. Rochdi, M. Shillor, M. and Sofonea, M. (1998). Quasistatic viscoelastic contact with normal compliance and friction. Journal of Elasticity, (51):105-126.
  19. Szolgay, P., Voros, G., and Eross, G. (1993). On the applications of the cellular neural network paradigm in mechanical vibrating systems. IEEE Trans. Circuits Syst. I, 40(3):222-227.
  20. Timoshenko, S. (1937). Vibration problems in engineering. D.Van Nostrand Company.
Download


Paper Citation


in Harvard Style

Danciu D., Matei A., Micu S. and Rovenţa I. (2014). Nonlinear Feedback Control and Artificial Intelligence Computational Methods applied to a Dissipative Dynamic Contact Problem . In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO, ISBN 978-989-758-039-0, pages 528-539. DOI: 10.5220/0005055005280539


in Bibtex Style

@conference{icinco14,
author={Daniela Danciu and Andaluzia Cristina Matei and Sorin Daniel Micu and Ionel Rovenţa},
title={Nonlinear Feedback Control and Artificial Intelligence Computational Methods applied to a Dissipative Dynamic Contact Problem},
booktitle={Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},
year={2014},
pages={528-539},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005055005280539},
isbn={978-989-758-039-0},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,
TI - Nonlinear Feedback Control and Artificial Intelligence Computational Methods applied to a Dissipative Dynamic Contact Problem
SN - 978-989-758-039-0
AU - Danciu D.
AU - Matei A.
AU - Micu S.
AU - Rovenţa I.
PY - 2014
SP - 528
EP - 539
DO - 10.5220/0005055005280539