Transcript Chattering: a novel route to chaos in cam-follower impacting systems Ricardo Alzate Ph.D.

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
5th
8th World Congress on Computational Mechanics WCCM8
European Congress on Computational Methods in Applied Sciences and Engineering
ECCOMMAS 2008
Minisymposium on Computational Methods in Nonlinear Dynamics
Venice, June 2008
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Outline
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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 2008
Impact oscillators
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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, 1983.
[2] J. Thompson and H. Stewart, Nonlinear Dynamics and Chaos, John Wiley, New York,
1986.
[3] A. B. Nordmark, “Non-periodic motion caused by grazing incidence in impact
oscillators”, Journal of sound and vibration, vol. 2, pp. 279–297, 1991.
[4] G. Whiston, “Singularities in vibro-impact dynamics”, Journal of sound and vibration, vol.
152, pp. 427–460, 1992.
[5] F. Peterka, “Transition to chaotic motion in mechanical systems with impacts”, Journal
of sound and vibration, vol. 154, pp. 95–115, 1992.
[6] Chris Budd and F. Dux, The dynamics of impact oscillators, Ph.D. thesis, University of
Bristol, 1992.
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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 2008
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Application case: valve floating
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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 2008
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Application case: valve floating
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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 2008
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The model
Reset law
Forcing shape
Equation of motion
[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 2008
Simulation environment
Features:
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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. [email protected]
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Bifurcation behaviour
Main zones on stroboscopic bifurcation
diagram:
A.
B.
C.
coexistence of attractors
period-doubling cascade to chaos
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. [email protected]
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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. [email protected]
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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 1994.
[12] A. Nordmark and R. Kisitu, “On chattering bifurcations in 1 dof impact oscillator
models”, Royal Institute of Technology, Sweden, 2003.
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Conclusions
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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
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Ongoing and future work
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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.
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To extend the results on the chattering map to the case of
discontinuous-periodic forcing; e.g. a corner-chattering bifurcation.
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...?
Arrivederci e grazie !!!
http://wpage.unina.it/r.alzate
Chattering: a novel route to chaos in cam-follower impacting systems