Philip V. Bayly 1 e-mail: pvb@mecf.wustl.edu Michael T. Lamar 2 Sean G. Calvert Mechanical Engineering, Box 1185, Washington University, 1 Brookings Drive, St. Louis, MO 63130 Low-Frequency Regenerative Vibration and the Formation of Lobed Holes in Drilling Large-amplitude vibrations in drilling often occur at frequencies near multiples of the rotation frequency, even when these are much lower than the system’s ﬁrst natural fre- quency. These vibrations are responsible for out-of-round, ‘‘lobed’’ holes. A simpliﬁed model of the mechanics of this phenomenon is presented in this paper. The model includes cutting and ‘‘rubbing’’forces on the drill, but inertia and damping of the tool are ne- glected at low speeds. This quasi-static model remains dynamic because of the regenera- tive nature of cutting; the force on each cutting element depends on both the tool’s current position and its position at the time of the previous tooth passage. Characteristic solu- tions, including unstable retrograde ‘‘whirling’’modes, are found in terms of eigenvalues and eigenvectors of a discrete state-transition matrix. These unstable modes correspond closely to behavior observed in drilling tests. DOI: 10.1115/1.1459087 Introduction Drilling is an extremely important metal cutting operation. Hundreds of thousands of holes are drilled in the process of manu- facturing a single passenger jet airframe. Tool vibration during drilling can cause errors in hole size and form that require holes to be repaired or remade. Demand for increased productivity and high quality has stimulated interest in automated drilling equip- ment for high-speed and high-precision operations. However, progress in precision drilling is hampered by incomplete under- standing of the mechanisms responsible for tool vibration and poor hole quality. Two different types of vibration can be distinguished in drilling: 1 low-frequency vibrations associated with lobed holes, and 2 chatter. One of the most common roundness problems in drilled holes is the existence of regularly spaced ‘‘lobes’’ Fig. 1. Al- though the diameter of such holes, measured between any two opposite points, may be within tolerance, the hole will not accept a round pin of the corresponding size. The phenomenon of lobed hole formation is different from clas- sical chatter in metal cutting. Chatter is self-excited vibration of the cutting tool at frequencies near and above its natural fre- quency; it is due to the ‘‘regeneration of waviness’’ 1 as the vibrating tool encounters the surface left on its previous pass. Frequency-domain analysis has been used to ﬁnd stability regimes for high-speed milling: regions where large depths of cut can be achieved at high spindle speeds 1,2,3,4. Stability predictions have been conﬁrmed and reﬁned by detailed time-domain simula- tions 1, e.g.. Several recent papers 5,6,7 describe experimen- tal observations and strategies for coping with chatter in drilling. Although chatter is important under some conditions, lobed hole proﬁles exist even in the absence of chatter, and at very low cutting speeds. Also, the number of lobes corresponds to the tooth-passing frequency, and not to the natural frequency of the tool, as it would in chatter. The absence of dynamic models that explain the formation of lobed holes is particularly important. Oxford, Jr. 8 and Shaw and Oxford, Jr. 9 performed seminal studies of the steady-state mechanics of drilling, including torque and thrust predictions. Their work was extended by Armarego et al. 10, Makhecha et al. 11, Watson 12, and Stephenson and Agapiou 13, among others. These recent authors describe steady force predictions based on division of the drill’s cutting edges into a ﬁnite number of oblique cutting edges with constant properties, using empirical expressions to relate cutting force on each ele- ment to chip load, and summing forces on each element to obtain torque and thrust. Studies focusing on vibration in drilling include experimental papers 14,15 that describe observations of oscillations at 2 cyc/ rev associated with the commonly-observed 3-lobed hole. In reaming, a process use to enlarge drilled holes, Sakuma and Kiyota 16,17 observed in experiments that whirling motion of the reamer at N cyc/rev, where N is the number of cutting ﬂutes, leads to lobed holes with N -1 or N +1 lobes. This motion ap- pears at N 1 cycles/rev in the rotating tool frame. Purely kine- matic models of lobing oscillations have been proposed 18,19 which show that, if the proper backward retrograde whirling tool motion is speciﬁed, out-of-round holes of the correct shape result. However they do not explain the forces that produce the assumed motion. Reinhall and Storti 20 and Basile 21 model the drill as an impacting rotating rod, and show that polygonal proﬁles may result. Tekinalp and Ulsoy 22,23 and Rincon and Ulsoy 24,25 developed dynamic ﬁnite element models of the drill and have explored the effects of process parameters on drill vibrations. These models do not include the regenerative effects of the cutting process. The current paper builds on the experimental and numerical work of Fujii et al. 26,27 in drilling, and on a recent model and analysis of reaming 28. These papers consider regenerative cut- ting forces as well as forces due to contact and friction ‘‘rub- bing’’ on the clearance faces. Fujii and colleagues 27 performed 1 Corresponding author 2 Currently with Applied Physics Laboratory, Baltimore, MD Contributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received December 2000; Revised July 2001. Associate Editor: Y. Altintas. Fig. 1 Photograph and measured proﬁle of a three-lobed hole Journal of Manufacturing Science and Engineering MAY 2002, Vol. 124 Õ 275 Copyright © 2002 by ASME