Summary
Two mathematical formulations of a model that describes the self-exited oscillations of milling tools are compared: (i) a delay differential equation (DDE) and (ii) a partial differential equation-ordinary differential equation (PDE-ODE) formulation. Although their mathematical expressions appear to be different, the two models should be consistent. Here, we show that the two models are equivalent under specific assumptions. Utilizing the boundary conditions implied by these assumptions, the PDE of the PDE-ODE system is solved analytically and the PDE-ODE formulation is accordingly transformed to a DDE. From this perspective, DDE models can be viewed as a simplifying approximation of PDE-ODE models. As a result, the DDE model is more efficient than the PDE-ODE model in most scenarios but can be less accurate in extreme cases.
