Nonlinear dynamics in a camfollower impacting system

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Nonlinear dynamics in a cam-follower impacting
system

• Ricardo Alzate Ph.D. Student
• University of Naples FEDERICO II (SINCRO GROUP)

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Outline

• Background
• Results
• Conjectures and open problems

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• Background

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Application problem

• Valve floating phenomena in cam-shaft based
  engines (performance reduction, piece-wearing and
  damage)

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Theoretical context

• Cam-follower systems are particular cases of
• Hybrid automata (constrained and unconstrained
  modes reset)
• Impact oscillator with surface forcing (nonsmooth
  dynamics and complex behaviour)
• Broad application (mechanical tools)

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The system features

• Experimental rig to validate theoretical
  approaches
• Flat-faced sliding follower driven by a rotating,
  shape selectable, eccentric cam (circular and
  discontinuous) with spring preload
• High performance sensors (encoders and gyroscope)
  and actuator
• Flexible storage and programming tools

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The experimental rig
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• Results
• (circular shape case)

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Dynamics overview
Low velocity region - contact condition
Which phenomenon is behind this abrupt
transition?
Detachment and periodic regime
A-periodic behaviour and chaos
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Modeling

• Mathematical representation of main events (see
  1 for details)
• Development of expression for unconstrained state
  vector evolution using Lagrangian approach
• Transition rule between modes impact law based
  on a single Newtons restitution model
• Numeric evaluation with event-driven algorithm
  (G. Osorio)
• 1. R. Alzate, M. Di Bernardo, U. Montanaro and
  S. Santini. Experimental and numerical
  verification of bifurcations
• and chaos in a cam-follower impacting
  system, Nonlinear dynamics, Springer. 2007.

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Equations
unconstrained
constrained
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Identification

• Assignment of quantitative values for model
  parameters searching agreement with experimental
  data
• Geometric parameters
• Physical parameters
• Restitution coefficient

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Validation
110 rpm
150 rpm
experimental
numerical
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Dynamics simulation
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• Conjectures
• and open problems

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Conjectures bif. cascade 1 ? ?150,155 rpm

• Is particular transition to chaos on scenario 1
  consequence of interrupted chattering sequences?
• Is such interruption of chattering due to
  grazing?
• Is scenario 1 the particular transition observed
  in bifurcation map obtained experimentally?

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Conjectures bif. cascade 1 ? ?150,155 rpm (2)

• Fingered patterns

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Conjectures bif. cascade 2 ? ?135,160 rpm

• How to predict analytically conditions for
  existence of the period doubling cascade? Note
  that vector field under study is highly
  nonlinear
• Is scenario 2 detectable experimentally?

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Open problems

• Development of a more accurate simulation
  algorithm and of an appropriated continuation
  algorithm

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Open problems (2)

• Continuation of stable and unstable solutions and
  detection of smooth and DIB
• Analysis for coexistence of attractors and
  computation for related basins of attraction
• Experimental evidence of solutions
• Analysis of discontinuous-cam case

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Conclusion

• We have studied a cam follower system with
  continuous cam profile, involving highly
  nonlinearities in associated model
• We have detected two main bifurcation scenarios
• Scenario 1 involves sudden transition to chaos
  with chattering solutions
• Scenario 2 involves a period doubling route to
  chaos
• The open problem regards numerical and
  experimental study of coexistence between such
  dynamics

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• Thanks !!!
• r.alzate_at_unina.it