Multiple regenerative effects in cutting process and nonlinear oscillations

A turning process paradigm is considered to study multiple regenerative effects in cutting operations. The workpiece is considered to be a spatially continuous element, while the cutter is modeled as a discrete-parameter element. The resulting system is described by a combined partial differential equation–ordinary differential equation (ODE) model with a surface function that is used for updating the workpiece. The time delay in this model is allowed to be any integer multiple of the tooth-pass period. Analysis of this system reveals that the loss of contact between the workpiece and the cutter results in two principal features, namely, a non-smooth cutting force and multiple regenerative effects. The model of the spatially continuous work piece is cast into a system of ODEs through the semi-discretization method. Subsequent analysis results in a high-dimensional, non-smooth discrete-time map. Iterations of this mapping show that the time delay can vary in a wide range, and due to the multiple regenerative effect, this delay can be as high as ten times the constant delay value. Through parametric studies, it is learned that the system can exhibit stable cutting behavior, as well as periodic, quasi-periodic, chaotic and hyperchaotic behavior. With the choice of the non-dimensional cutting coefficient as a control parameter, bifurcations of the system responses are examined. The system is observed to possess rich dynamics, including multiple solutions. Supported by computations of correlation dimension, Kaplan–Yorke dimension, and Lyapunov spectrum, limit cycle, torus, chaotic, and hyperchaotic attractors are observed to be present in the considered parameter range. The findings can help further our understanding of multiple regenerative effects in cutting and drilling operations.