Analysis of Tool Oscillation and Hole Roundness Error in a Quasi-Static Model of Reaming

A quasi-static model of reaming is developed to explain oscillation of the tool during cutting and the resulting roundness errors in reamed holes. A tool with N evenly-spaced teeth often produces holes with $N+1$ or $N−1$ “lobes.” These profiles correspond, respectively, to forward or backward whirl of the tool at N cycles/rev. Other whirl harmonics $(2N$ cycles/rev, e.g.) are occasionally seen as well. The quasi-static model is motivated by the observations that relatively large oscillations occur at frequencies well below the natural frequency of the tool, and that in this regime the wavelength of the hole profile is largely independent of both cutting speed and tool natural frequency. In the quasi-static approach, inertial and viscous damping forces are neglected, but the system remains dynamic because regenerative (time-delayed) cutting and rubbing forces are included. The model leads to an eigenvalue problem with forward and backward whirl solutions that closely resemble the tool behavior seen in practice.