Philip V. Bayly Associate Professor, Mem. ASME, Mechanical Engineering, Box 1185, Washington University, 1 Brookings Drive, St. Louis, MO 63130 e-mail: pvb@me.wustl.edu Keith A. Young Technical Specialist, The Boeing Company St. Louis, MO 63130 Sean G. Calvert Research Assistant, Washington University, St. Louis, MO 63130 Jeremiah E. Halley Associate Technical Fellow, The Boeing Company St. Louis, MO 63130 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, re- spectively, to forward or backward whirl of the tool at N cycles/rev. Other whirl harmon- ics  2N cycles/rev, e.g.) are occasionally seen as well. The quasi-static model is moti- vated 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. DOI: 10.1115/1.1383551 Introduction The desire to improve the quality and extend the lifetime of aerospace products has elevated the importance of precision in machining. Reaming is performed to increase the precision and roundness of drilled holes. Hundreds of thousands of holes may be drilled and reamed in the manufacture of a single aircraft. These holes must often be made to tight tolerances in size and roundness in order to ensure correct ﬁts during assembly and ad- equate performance of the ﬁnished product. Previous studies 1,2 have suggested that hole quality affects the fatigue life of joints with fasteners. The cost of repairing or re-making parts with poor quality holes also contributes signiﬁcantly to the total cost of the product. In experimental investigations of reaming, Kiyota and Sakuma 3 and Sakuma and Kiyota 4,5 observed that elliptical motion of the tool at N cycles/revolution N/rev, where N is the number of evenly-spaced cutting ﬂutes, leads to lobed holes with N -1 or N +1 lobes Fig. 1. Other harmonics (2 N +1,2N -1) typically are also present in proﬁles. The use of reaming tools with uneven spacing between teeth has been somewhat effective in reducing roundness error. The choice of tooth spacing is often based on trial and error, although algorithms have been proposed, on the basis of qualitative physical arguments 6,7, to maximize the spectral content of the tooth pattern. Dynamic modeling has been used widely to investigate the ‘‘chatter’’ stability of related metal cutting processes. Chatter is self-excited regenerative vibration of the cutting tool at frequen- cies near and above its natural frequency. Analyses of simpliﬁed models in the frequency domain 8–14, as well as detailed time- domain simulations 10,15, have been used to ﬁnd stability lobes and uncover regions where large depths of cut can be achieved at high spindle speeds. Stability analyses include studies of boring 13,14, a process similar to reaming. Though chatter is an impor- tant phenomenon, lobed hole proﬁles exist even in the absence of chatter, and at very low cutting speeds. Furthermore, the number of lobes corresponds to the tooth-passing frequency, not to the natural frequency of the tool, as it would in chatter. Studies of reaming via simulation include an early paper by Friedmann and Wu 16 which examines the ‘‘rounding’’ mecha- nism of reaming. Many papers on the related problem of drilling have also been published in recent years: these include kinematic analyses and simulation studies. Lee and co-workers 17 and Ze- lentsov 18 assumed elliptical motion of a drill bit with frequency equal to the tooth passing frequency and obtained results that were consistent with test. These kinematic analyses do not explain the forces that produce the assumed motion. Reinhall and Storti 19 and Basile 20 model a drill as an impacting rotating rod, and show that polygonal proﬁles may result. However neither Basile 20 nor Reinhall and Storti 19 include the regenerative nature of the cutting forces in their models. Fujii and colleagues 21–23 performed simulations of drilling in which regenerative effects were included; they observed tool oscillation and lobed hole pro- ﬁles in the output of their simulation. An analytical treatment that explains the predominance of N -1 and N +1-lobed patterns in reamed holes, the existence of related harmonics 2 N -1,2N +1, e.g., and the mechanisms re- sponsible for them has remained lacking. In the present paper, we introduce a method to ﬁnd the fundamental solutions of the equa- tions of motion for the reamer at low cutting speeds. A quasi- static model of the cutting process leads to a matrix eigenvalue problem in which the characteristic equation for the eigenvalues is transcendental. The eigenvalues and eigenvectors are found nu- merically, by reﬁnement of approximate solutions. The eigen- solutions thus obtained correspond closely to the modes of behav- ior observed in practice and in simulation. Contributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received July 1998; revised November 2000. Associate Editor: K. Ehmann. Fig. 1 Examples of lobed holes made with a 6-ﬂute reamer Journal of Manufacturing Science and Engineering AUGUST 2001, Vol. 123 Õ 387 Copyright © 2001 by ASME