Nonlinear dynamics in a cam-follower impacting system

Nonlinear dynamics in a cam-follower impacting systemRicardo Alzate Ph.D. StudentUniversity of Naples FEDERICO II (SINCRO GROUP)

NLD in a cam-follower impacting system

Outline

BackgroundResultsConjectures and open problems


Background


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


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)


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 preloadHigh performance sensors (encoders and gyroscope) and actuatorFlexible storage and programming tools


The experimental rig


Results (circular shape case)


Dynamics overviewLow velocity region - contact conditionWhich phenomenon isbehind this abrupt transition?Detachment and periodic regimeA-periodic behaviour and chaos


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.


Equationsunconstrainedconstrained


Identification

Assignment of quantitative values for model parameters searching agreement with experimental data:

Geometric parameters Physical parameters

Restitution coefficient


Validationexperimentalnumerical110 rpm150 rpm


Dynamics: simulation


Conjecturesand open problems


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?


Conjectures: bif. cascade 1 [150,155] rpm (2)

Fingered patterns


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?


Open problems

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


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


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


Thanks !!!

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Nonlinear dynamics in a cam-follower impacting system

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