Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

Nonlinear Dynamics

Volumn 51, Issue 4, 2008, Pages 551-583

  Leine, R I ^a   Van De Wouw, N ^b  

^a ETH ZURICH   (Switzerland)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Attractivity; Lyapunov stability; Measure differential inclusion; Unilateral constraint

Indexed keywords

CONSTRAINT THEORY; DEGREES OF FREEDOM (MECHANICS); DIFFERENTIAL EQUATIONS; FRICTION; LYAPUNOV FUNCTIONS;

ATTRACTIVITY; LYAPUNOV STABILITY; MEASURE DIFFERENTIAL INCLUSION; UNILATERAL CONSTRAINT;

NONLINEAR SYSTEMS;

EID: 38349134760     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-007-9244-z     Document Type: Article

References (48)

1. Adly, S., Goeleven, D.: A stability theory for second-order nonsmooth dynamical systems with application to friction problems. J. Mathematiques Pures et Appliquées 83, 17-51 (2004)
2. Alvarez, J., Orlov, I., Acho, L.: An invariance principle for discontinuous dynamic systems with application to a Coulomb friction oscillator. J. Dyn. Syst. Meas. Control 122, 687-690 (2000)
3. Anosov, D.V.: Stability of the equilibrium positions in relay systems. Autom. Remote Control 20, 130-143 (1959)
4. Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Pure and Applied Mathematics. Wiley, Berlin (1984)
5. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis, vol. 2, Systems and Control: Foundations and Applications. Birkhäauser, Boston (1990)
6. Bacciotti, A., Ceragioli, F.: Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESIAM: Control Optim. Calculus Variations 4, 361-376 (1999)
7. Bainov, D.D., Simeonov, P.S.: Systems with Impulse Effect: Stability, Theory and Applications. Wiley, New York (1989)
8. Brogliato, B.: Nonsmooth Mechanics, 2nd edn. Springer, London (1999)
9. Brogliato, B.: Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings. Syst. Control Lett. 51, 343-353 (2004)
10. Brogliato, B., Goeleven, D.: The Krakovskii-LaSalle invariance principle for a class of unilateral dynamical systems. Math. Control Signals Syst. 17(1), 57-76 (2005)
11. Brogliato, B., Niculescu, S.-I., Orhant, P.: On the control of finite-dimensional mechanical systems with unilateral constraints. IEEE Trans. Autom. Control 42(2), 200-215 (1997)
12. Camlibel, M., Heemels, W., van der Schaft, A., Schumacher, J.: Switched networks and complementarity. IEEE Trans. Circuits Syst. 50(8), 1036-1046 (2003)
13. Chareyron, S., Wieber, P.B.: A LaSalle's invariance theorem for nonsmooth Lagrangian dynamical systems. In: Proceedings of the ENOC-2005 Conference, 7-12 August, Eindhoven, The Netherlands (2005)
14. Chareyron, S., Wieber, P.B.: Stability and regulation of nonsmooth dynamical systems. IEEE Trans. Autom. Control (2007) submitted
15. Davrazos, G.N., Koussoulas, N.T.: A review of stability results for switched and hybrid systems. In: 9th Mediterranean Conference on Control and Automation, 27-29 June, Dubrovnik, Croatia (2001)
16. Filippov, A.F.: Differential Equations with Discontinuous Right-hand Sides. Kluwer Academic, Dordrecht, The Netherlands (1988)
17. Glocker, Christoph.: On frictionless impact models in rigid-body systems. Phil. Trans. R. Soc. Lond. A 359, 2385-2404 (2001)
18. Glocker, Christoph.: Set-Valued Force Laws, Dynamics of Non-Smooth Systems, vol. 1, Lecture Notes in Applied Mechanics. Springer-Verlag, Berlin (2001)
19. Glocker, Christoph.: An introduction to impacts. In: Haslinger, J., Stavroulakis, G.E. (eds.), Nonsmooth Mechanics of Solids, vol. 485, CISM Courses and Lectures. Springer, Wien, New York, pp. 45-101 (2006)
20. Goeleven, D., Brogliato, B.: Stability and instability matrices for linear evolution variational inequalities. IEEE Trans. Autom. Control 49, 521-534 (2004)
21. Goeleven, D., Brogliato, B.: Necessary conditions of asymptotic stability for unilateral dynamical systems. Nonlinear Anal. 61(6), 961-1004 (2005)
22. Haslinger, J., Stavroulakis, G.E. (eds.): Nonsmooth Mechanics of Solids, vol. 485, CISM Courses and Lectures. Springer, Wien, New York (2006)
23. Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Autom. Control 43(4), 555-559 (1998)
24. Kane, T.R., Levinson, D.A.: Dynamics, Theory and Applications. McGraw-Hill, New York (1985)
25. Khalil, H.K.: Nonlinear Systems. Prentice-Hall, Englewood Cliffs, NJ (1996)
26. Kunze, M.: Non-Smooth Dynamical Systems. Springer, Berlin (2000)
27. Leine, R.I.: Bifurcations in discontinuous mechanical systems of Filippov-Type. Ph.D. Thesis, Eindhoven University of Technology, The Netherlands (2000)
28. Leine, R.I.: On the stability of motion in non-smooth mechanical systems. Habilitation Thesis, ETH Zäurich, Switzerland (2007)
29. Leine, R.I., Brogliato, B., Nijmeijer, H.: Periodic motion and bifurcations induced by the Painleváe paradox. Eur. J. Mech. A Solids 21(5), 869-896 (2002)
30. Leine, R.I., Glocker, Christoph.: A set-valued force law for spatial Coulomb-Contensou friction. Eur. J. Mech. A Solids 22, 193-216 (2003)
31. Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems, vol. 18, Lecture Notes in Applied and Computational Mechanics. Springer-Verlag, Berlin (2004)
32. Moreau, J.J.: Liaisons unilatáerales sans frottement et chocs inelastiques. Comptes Rendu des Sáeances de l'Academie des Sciences 296, 1473-1476 (1983)
33. Moreau, J.J.: Bounded variation in time. In: Moreau, J.J., Panagiotopoulos, P.D., Strang, G. (eds.), Topics in Nonsmooth Mechanics, pp. 1-74. Birkhäuser Verlag, Basel, Boston, Berlin (1988)
34. Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J.J., Panagiotopoulos, P.D. (eds.) Non-Smooth Mechanics and Applications, vol. 302, CISM Courses and Lectures, pp. 1-82. Springer, Wien, New York (1988)
35. Moreau, J.J.: Some numerical methods in multibody dynamics: Application to granular materials. Eur. J. Mech. A Solids 13(4, suppl.), 93-114 (1994)
36. Papastavridis, J. G.: Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems: For Engineers, Physicists and Mathematicians. Oxford University Press, New York (2002)
37. Payr, M., Glocker, C.: Oblique frictional impact of a bar: Analysis and comparison of different impact laws. Nonlinear Dyn. 41, 361-383 (2005)
38. Pettersson, S., Lennartson, B.: Hybrid system stability and robustness verification using linear matrix inequalities. Int. J. Control 75(16-17), 1335-1355 (2002)
39. Pfeiffer, F., Glocker, Christoph.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996)
40. Quittner, P.: An instability criterion for variational inequalities. Nonlinear Anal. 15(12), 1167-1180 (1990)
41. Rockafellar, R.T.: The Theory of Subgradients and its Applications to Problems of Optimization. Convex and Nonconvex Functions, vol. 1, Research and Education in Mathematics. Heldermann Verlag, Berlin (1970)
42. Rockafellar, R.T., Wets, R.-B.: Variational Analysis. Springer, Berlin (1998)
43. Shevitz, D., Paden, B.: Lyapunov stability of nonsmooth systems. IEEE Trans. Autom. Control 39(9), 1910-1914 (1994)
44. Tornambè, A.: Modeling and control of impact in mechanical systems: theory and experimental results. IEEE Trans. Autom. Control 44(2), 294-309 (1999)
45. Van de Wouw, N., Leine, R.I.: Attractivity of equilibrium sets of systems with dry friction. Int. J. Nonlinear Dyn. Chaos Eng. Syst. 35(1), 19-39 (2004)
46. van de Wouw, N., van den Heuvel, M., van Rooij, J., Nijmeijer, H.: Performance of an automatic ball balancer with dry friction. Int. J. Bif. Chaos 15(1), 65-82 (2005)
47. Yakubovich, V.A., Leonov, G.A., Gelig, A.K.: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities, vol. 14, Series on Stability, Vibration and Control of Systems, Series A. World Scientific, Singapore (2004)
48. Ye, H., Michel, A., Hou, L.: Stability theory for hybrid dynamical systems. IEEE Trans. Autom. Control 43(4), 461-474 (1998)