Nonlinear vibrations due to time delay effects in metal cutting system such as turning, milling, and drilling operations are undesirable as it might lead to a poor finish and low precision on the workpiece. In this paper, the authors study a very simplified model for turning processes. In the model, the exact geometry of the workpiece surface are taking into consideration, while other nonlinear factors such as distributed time delay, exponential cutting force, and state-dependent delay are neglected. Thus, the time delay system at hand simply suffers nonlinearities from the loss of contact effects. To deal with this system, a novel concept are proposed based on which the system can be described by a combination of a partial differential equation (PDE) and a ordinary differential equation (ODE). Comparing with literature, the PDE-ODE model can be concise to include the time-delay, loss of contact, and multiple-regenerative effects. A high dimensional map system from the PDE-ODE model is obtained via semi-discretization method. By iterating of the map, the high dimensional behavior of the cutting system are studied in the time domain. And the simulations show the route from stable cutting to bifurcation, chaos, and hyperchaos. The Poincare maps and the bifurcation diagrams from the high dimensional attractors also illustrate the rich nonlinear dynamical behavior of this very simplified system.
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