Nonlinear dynamics in a cam- follower impacting system Ricardo Alzate Ph.D. Student University of Naples FEDERICO II (SINCRO GROUP)

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system2/22 Outline Background Results Conjectures and open problems

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system3/22 Background

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system4/22 Application problem Valve floating phenomena in cam-shaft based engines (performance reduction, piece-wearing and damage)

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system5/22 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)

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system6/22 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

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system7/22 The experimental rig

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system8/22 Results (circular shape case)

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system9/22 Dynamics overview Low velocity region - contact condition Which phenomenon is behind this abrupt transition? Detachment and periodic regime A-periodic behaviour and chaos

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system10/22 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

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system11/22 Equations unconstrainedconstrained

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system12/22 Identification Assignment of quantitative values for model parameters searching agreement with experimental data: Geometric parameters Physical parameters Restitution coefficient

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system13/22 Validation experimentalnumerical 110 rpm 150 rpm

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system14/22 Dynamics: simulation

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system15/22 Conjectures and open problems

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system16/22 Conjectures: bif. cascade 1 ω  [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?

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system17/22 Conjectures: bif. cascade 1 ω  [150,155] rpm (2) Fingered patterns

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system18/22 Conjectures bif. cascade 2 ω  [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?

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system19/22 Open problems Development of a more accurate simulation algorithm and of an appropriated continuation algorithm

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system20/22 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

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system21/22 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

R. Alzate - Barcelona, 2007 NLD in a cam-follower impacting system22/22 Thanks !!!