Quasiperiodicity and chaos in course of grinding: analysis of a model based on nonlinearly coupled Van der Pol-like oscillators

Abstract

We consider a known dynamical model of grinding process that takes
into account an interaction between a tool and a workpiece as sliding
friction. Within this approach an effective work of the tool expended
for cutting roughnesses is negligibly small. Both the grinding tool
and the workpiece are described by vertical (normal) and horizontal
(tangential) coordinates and velocities, so the phase space is eight
dimensional An interaction between the grinding tool and the workpiece
occurs via a Coulomb friction and a lifting force. The Coulomb
friction is exerted horizontally and depends on vertical distance
between the tool and the workpiece. The lifting force in turn is
vertical and depends on their relative velocity. Both of the forces
nonlinearly depend on coordinates and when it is taken into account up
to quadratic terms the resulting model consists of four second order
ODE's whose structure reminds Van der Pol equation. The interaction
between partial equation occurs via nonlinear terms. This system has a
single fixed point at the origin and its stability properties is
studied well. We analyze its other regimes for a special parameter
area where the system is split into two independent subsystems each of
two equations. We demonstrate quasiperiodic and chaotic regimes of the
subsystems, transition to chaos via destruction of torus and also
reveal parameter areas where quadratic approximation of nonlinear
forces become inappropriate. Author acknowledges support from Russian
Science Foundation, grant No 20-19-00299.

Speaker

Pavel Kuptsov
Mechanical Engineering Research Institute of the Russian Academy of Sciences (IMASH RAN) Moscow, Russia
Russia

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