Mathematical models play an increasing role in understanding and predicting machining processes, in particular milling. However, despite the considerable efforts that have been dedicated to this problem, a majority of milling models still rely on simplifying assumptions to calculate the chip thickness. In this paper, the chip thickness is determined without these simplifications, based on a surface function that describes the milled surface and on information about the workpiece boundary. By combining the partial differential equation (PDE) governing the evolution of this surface function with the ordinary differential equations (ODE) governing the tool/machine dynamics, a mixed PDE–ODE formulation is proposed to describe the dynamics of the milling process. The coupled system of differential equations is solved using an algorithm that combines finite difference (ODE) and finite volume (PDE) methods. A case study is presented to compare the proposed approach with the classical delay differential equations (DDE) model formulation for milling processes based on a simplified chip thickness model. The PDE–ODE formulation represents an explicit mathematical model for milling process dynamics; it yields a theoretically exact chip thickness and offers a means to assess the validity of models based on DDE formulation. Moreover, the proposed formulation is capable of simulating transient tool behaviors when the tool is milling the outer region of the workpiece, which is in general neglected by the DDE-based models.
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- Chip thickness
- Modeling and numerical simulation
- PDE–ODE formulation
- Tool/machine dynamics