Chattering: a novel route to chaos in camfollower impacting systems

1
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
• 8th World Congress on Computational Mechanics
  WCCM8
• 5th 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

• Background
• The cam-follower model
• Simulation and bifurcation diagram
• Chattering and local mapping
• Open problems
• Conclusions

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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. 129155,
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. 279297, 1991. 4 G. Whiston,
Singularities in vibro-impact dynamics, Journal
of sound and vibration, vol. 152, pp. 427460,
1992. 5 F. Peterka, Transition to chaotic
motion in mechanical systems with impacts,
Journal of sound and vibration, vol. 154, pp.
95115, 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

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Application case valve floating

• Performance of internal combustion engines
• Preloaded forces, wearing
• Can be modeled as an impact oscillator

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Application case valve floating

• Performance of internal combustion engines
• Preloaded forces, wearing
• Can be modeled as an impact oscillator

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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. 409429, November 2007.
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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.
petri.piiroinen_at_nuigalway.ie
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Bifurcation behaviour

• Main zones on stroboscopic bifurcation
• diagram
• 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. stsantin_at_unina.it
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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. r.alzate_at_unina.it
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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. 365389, 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

• 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.

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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.

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