Nonlinear dynamics in a camfollower impacting system
Ricardo Alzate Ph.D. Student
University of Naples FEDERICO II (SINCRO GROUP)
NLD in a cam-follower impacting system
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Outline
• Background
• Results
• Conjectures and open problems
R. Alzate - Barcelona, 2007
NLD in a cam-follower impacting system
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Background
R. Alzate - Barcelona, 2007
NLD in a cam-follower impacting system
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:
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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:
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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
R. Alzate - Barcelona, 2007
NLD in a cam-follower impacting system
The experimental rig
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NLD in a cam-follower impacting system
Results
(circular shape case)
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Dynamics overview
Low velocity region - contact
condition
Detachment
regime
and
periodic
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
Newton’s 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.
R. Alzate - Barcelona, 2007
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Equations
unconstrained
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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
R. Alzate - Barcelona, 2007
numerical
NLD in a cam-follower impacting system
Dynamics: simulation
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NLD in a cam-follower impacting system
Conjectures
and open problems
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Conjectures: bif. cascade 1
ω [150,155] rpm
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Is particular transition to chaos on scenario 1 consequence of
interrupted chattering sequences?
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Is such interruption of chattering due to grazing?
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Is scenario 1 the particular transition observed in bifurcation map
obtained experimentally?
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Conjectures: bif. cascade 1
ω [150,155] rpm (2)
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Fingered patterns
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Conjectures bif. cascade 2
ω [135,160] rpm
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How to predict analytically conditions for existence of the period
doubling cascade? [Note that vector field under study is highly nonlinear]
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Is scenario 2 detectable experimentally?
R. Alzate - Barcelona, 2007
NLD in a cam-follower impacting system
Open problems
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Development of a more accurate simulation algorithm and of an
appropriated continuation algorithm
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Open problems (2)
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Continuation of stable and unstable solutions and detection of
smooth and DIB
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Analysis for coexistence of attractors and computation for related
basins of attraction
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Experimental evidence of solutions
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Analysis of discontinuous-cam case
R. Alzate - Barcelona, 2007
NLD in a cam-follower impacting system
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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 !!!
[email protected]
R. Alzate - Barcelona, 2007