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Abstract

A large number of engineering systems include non-linear or even non-smooth behaviour. During development and optimisation of mechanical systems, a comprehensive dynamical analysis may be essential. Set-oriented Numerical Methods offer a new and efficient approach to global dynamical analysis without the requirement for a detailed a priori knowledge of the system dynamics. This paper shows the basic concepts behind this method and presents its application to a simple model of a vibro-impact device for ultrasonic/sonic drilling. In this drilling device an impacting mass transmits vibrational energy from a piezoelectric actuator as shock waves into a drill stem. Relative global attractors will be derived which show statistical distributions for the occurrence of all possible states. Periodic solutions can be identified parallel to the evaluation of chaotic behaviour. Thus a global understanding of the system performance can be obtained, offering new ways for model verification and system design.

Keywords

Impact non-linear non-smooth modeling

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References

Babitsky, V.I. , 1998, Theory of Vibro-Impact Systems and Applications, Springer-Verlag, Berlin.

Bar-Cohen, Y. , Sherrit, S. , Dolgin, B. , Bridges, N. , Bao, X. , Chang, Z. , Yen, A. , and Saunders, R. , 2001, “Ultrasonic/sonic driller/corer (USDC) as a sampler for planetary exploration,” Jet Propulsion Laboratory / California Institute of Technology, Pasadena, CA.

Brogliato, B. , 1999, Nonsmooth Mechanics, Springer-Verlag , London.

de Kraker, B. , van der Spek, J. and van Campen, D.H. , 2000, “Extensions of cell mapping for discontinuous systems,” in Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, M. Wiercigroch and B. de Kraker (eds.), World Scientific, London, pp. 61—102.

Dellnitz, M. and Hohmann, A. , 1997, “A subdivision algorithm for the computation of unstable manifolds and global attractors,” Numerische Mathematik 75, 293—317.

Dellnitz, M. and Junge, O. , 1999, “On the approximation of complicated dynamical behavior ,” SIAM Journal of Numerical Analysis 36, 491—515.

Dellnitz, M. , Froyland, G. , and Junge, O. , 2001, “The algorithms behind GAIO — Set oriented numerical methods for dynamical systems,” in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler (ed.), Springer-Verlag, Berlin, pp. 145—174.

Dellnitz, M. and Junge, O. , 2002, “Set oriented numerical methods for dynamical systems ,” in Handbook of Dynamical Systems III: Towards Applications , B. Fiedler , G. Iooss and N. Kopell (eds), World Scientific, London.

Hsu, C.S. , 1981, “A generalized theory of cell-to-cell mapping for nonlinear dynamical systems,” Journal of Applied Mechanics 48, 634—642.

10.

Leine, R.I. and Nijmeijer, H. , 2004, Dynamics and Bifurcations in Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, Vol. 18, Springer-Verlag, Berlin, Heidelberg, New York.

11.

Neumann, N. and Sattel, T. , 2005, “On the modelling and model analysis of a piezoelectric impact drilling device,” in Proceedings of 5th EUROMECH Nonlinear Dynamics Conference (ENOC), Eindhoven .

12.

Norris, J.R. , 1997, Markov Chains, Cambridge University Press, Cambridge.

On Set-oriented Numerical Methods for Global Analysis of Non-smooth Mechanical Systems

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