Chattering: a novel route to chaos in cam-follower impacting systems Ricardo Alzate Ph.D. Student University of Naples FEDERICO II, ITALY Prof. Mario di Bernardo University of Naples FEDERICO II, ITALY / University of Bristol, U.K. Dr. Petri T. Piiroinen National University of Ireland – Galway, ROI 8 th World Congress on Computational Mechanics WCCM8 5 th European Congress on Computational Methods in Applied Sciences and Engineering ECCOMMAS 2008 Minisymposium on Computational Methods in Nonlinear Dynamics Venice, June 2008

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS 20082/14 Outline Background The cam-follower model Simulation and bifurcation diagram Chattering and local mapping Open problems Conclusions

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS 20083/14 Impact oscillators Constrained harmonic oscillators Interaction between rigid bodies Instantaneous collisions: impacts Reset law: Newton Periodic and continuous forcing Single degree of freedom models Valid reduced representation [1] S. W. Shaw and P. J. Holmes, “A periodically forced piecewise linear oscillator”, Journal of sound and vibration, vol. 90, pp. 129–155, [2] J. Thompson and H. Stewart, Nonlinear Dynamics and Chaos, John Wiley, New York, [3] A. B. Nordmark, “Non-periodic motion caused by grazing incidence in impact oscillators”, Journal of sound and vibration, vol. 2, pp. 279–297, [4] G. Whiston, “Singularities in vibro-impact dynamics”, Journal of sound and vibration, vol. 152, pp. 427–460, [5] F. Peterka, “Transition to chaotic motion in mechanical systems with impacts”, Journal of sound and vibration, vol. 154, pp. 95–115, [6] Chris Budd and F. Dux, The dynamics of impact oscillators, Ph.D. thesis, University of Bristol, 1992.

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS 20084/14 Dynamics Parameter dependence: - Clearance, - Forcing frequency and amplitude, - Restitution coefficient Traditional bifurcation scenarios: - PD, SN Discontinuity induced phenomena - Grazing and Chattering Novel routes to chaos: - Period-adding cascades

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS 20085/14 Application case: valve floating Performance of internal combustion engines Preloaded forces, wearing Can be modeled as an impact oscillator

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS 20086/14 Application case: valve floating Performance of internal combustion engines Preloaded forces, wearing Can be modeled as an impact oscillator

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS 20087/14 The model Forcing shape Equation of motion Reset law [7] R. Alzate, M. di Bernardo, U. Montanaro and S. Santini, “Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems”, Nonlinear Dynamics - Springer. The Netherlands, vol. 50, No 3, pp. 409–429, November 2007.

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS 20088/14 Simulation environment Features: Event-driven based simulation (Matlab® ODE45). Complete-chattering mapping: event. Conditions applied on Lie derivatives of the interaction between system flow and the discontinuity surface, defining motion states and transitions. Extended variables for overcoming singularities on Jacobian of system, close to zero velocity impacts. [8] A. B. Nordmark and P. T. Piiroinen, “Simulation and stability analysis of impacting systems with complete chattering”, Submitted.

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS 20089/14 Bifurcation behaviour Main zones on stroboscopic bifurcation diagram: A.coexistence of attractors B.period-doubling cascade to chaos C.transition from complete to incomplete chattering. [9] R. Alzate, M. di Bernardo, G. Giordano, G. Rea and S. Santini, “Experimental and Numerical Investigation of coexistence, novel bifurcations and chaos in a cam-follower system. Submitted to SIAM.

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS /14 Chattering bifurcation Continuation of the multi-impacting branch, employing the sticking time as test function. Perturbation on a single direction, flowing forward one forcing period: single-return one-dimensional approach. Repetitive pattern, with a fundamental component translated and scaled. [10] R. Alzate, P. T. Piiroinen and M. di Bernardo, “Transition from complete to incomplete chattering in impacting systems: the case of a representative cam-follower device”, In preparation.

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS /14 Local mapping: analytical The structure predicted numerically can be explained theoretically in terms of variational equations, by expanding in series near the releasing point. Reduction of dimensionality is included by working on an impact based mapping. Such local analysis can be generalized to any periodically-forced impact oscillator. [11] C. Budd and F. Dux, “Chattering and related behaviour in impact oscillators”, Philosophical transactions: physical sciences and engineering., vol. 347, No 1683, pp. 365–389, May [12] A. Nordmark and R. Kisitu, “On chattering bifurcations in 1 dof impact oscillator models”, Royal Institute of Technology, Sweden, 2003.

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS /14 Conclusions A combination of numerical and analytical tools, have been employed to uncover the dynamics of a practical impact oscillator: the cam-follower system. A sudden transition to chaos has been detected both numerically and experimentally. Such a transition, has been demonstrated to be consequence of interruption of complete chattering sequences, giving rise to a discontinuity-induced bifurcation characterized by a chain of grazing events.

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS /14 Ongoing and future work To perform accurate calculations on the linear equivalent of the map for the remaining part of the trajectory (global behaviour), in order to derive the composed equivalent global Poincaré map describing the overall dynamics. To extend the results on the chattering map to the case of discontinuous-periodic forcing; e.g. a corner-chattering bifurcation.

Chattering: a novel route to chaos in cam-follower impacting systems Venice, WCCM8/ECCOMAS /14 Arrivederci e grazie !!!  ?