D. N. Dilley
1
Mechanical Engineering,
Box 1185, Washington University,
1 Brookings Drive,
St. Louis, MO 63130
e-mail: dave@d3vibrations.com
D. A. Stephenson
General Motors Powertrain,
Pontiac, MI 48340
P. V. Bayly
Associate Professor
Mechanical Engineering,
Box 1185, Washington University,
1 Brookings Drive,
St. Louis, MO 63130
A. J. Schaut
2
Advanced Manufacturing R&D,
The Boeing Company,
St. Louis, MO
Frequency Shift in Drilling due to
Margin Engagement
Drill chatter degrades hole roundness, hole size, and tool life. This wastes time and
money in tools, scrap, and hole rework. Chatter prediction in milling and turning has
shown significant benefit to industry; however, researchers have been unable to accurately
predict chatter in drilling applications. In the past, the drill, including the chisel edge,
was modeled as either a fixed-fixed or fixed-pinned beam (Tekinalp, O., and Ulsoy, A. G.,
1989, ‘‘Modeling and Finite Element Analysis of Drill Bit Vibrations,’’ ASME J. Eng.
Indust. 111, pp. 148 – 154), but more recent research (Dilley, D. N., Bayly, P. V., and
Schaut, A. J., 2005, ‘‘Effects of the Chisel Edge on the Chatter Frequency in Drilling,’’J.
Sound Vib., 281, pp. 423 – 428) has shown that a fixed-embedded model using springs
improves frequency matching. The effects of the drill margins on dynamics have not been
studied. The fixed-fixed or fixed-pinned model will be shown to be inappropriate for
modeling the effects of margin engagement, while the spring-end boundary condition can
better approximate the frequency increase observed experimentally as the drill margins
engage deeper into the hole. In addition, the shifted frequency is well below the frequency
found from an analytical fixed-fixed or fixed-pinned beam. Evidence that the margins
cause the frequency shift is seen in three-dimensional waterfall plots that show this shift
for pilot hole drilling (in which the margins are engaged), but not for tube drilling (in
which margins are not engaged). DOI: 10.1115/1.1863255
1 Introduction
Dynamic drill models can provide engineers and machinists
with useful insight to help design drilling processes. Past cutting
force mechanics research 1,2 has provided information to im-
prove machine component design, such as spindles, cutting tools,
holders, and work holding fixtures. High-speed machining theory
uses a dynamics-based model that predicts stable cutting speeds
for increased metal removal rates and more accurate parts 3–7.
Stability charts are commonly used industrial design tools for
milling and turning. The ability to match predicted stability re-
gions in drilling with experiment has had less success; however,
analytical methods for bending chatter have been presented 8.
Complete structural models combined with empirical cutting
force models have been used in time domain simulations for pre-
dicting milling motion 4–6, but generally, simulations have
been unsuccessful in accurately predicting drill motion and chat-
ter. Modeling difficulties include effects from axial feed and the
complicated chisel edge and margin geometry.
Whirling motion was experimentally observed to create lobed
holes 9. Kinematic models showed that lobed holes are created
by low-frequency elliptical drill motion 10, but these models
have little predictive capability as they are based only on tool tip
geometry and tool motion that must be known a priori. Quasi-
static drilling and reaming models, in which motion depends on
tool stiffness and cutting forces, can qualitatively predict low-
frequency lobed holes 11,12; however, accurate chatter predic-
tion near tool natural frequencies has not been achieved.
This paper uses an existing dynamic model that includes accel-
eration and velocity terms 8,13–15, and then adds an end sup-
port of increased spring stiffness to model the margin and hole
interaction as hole depth increases. The spring is assumed to
model the linear force caused by the elastic deflection of the tool
and hole material; nonlinearity due to plastic deflection is not
considered in this initial model.
2 Tool Dynamics
The equation of motion for a single-mode rotating two-degree-
of-freedom DOF drill, Fig. 1a, in the tool fixed frame, -, at
the cutting edge has been developed 8,13–15. The values for
mass, damping, and stiffness can be approximated using the first
bending natural frequency from experimental modal analysis, col-
loquially referred to as tap testing. The simplest model for a drill
in a rotating reference frame at a constant speed, accounting for a
single mode in each direction, without structural coupling, has the
following form:
m
0
0 m
¨
¨
+
c
-2 m
2 m
c
˙
˙
+
k
-m
2
-c
c
k
-m
2
=F
ext
(1)
The stiffening effect from the margins will be modeled by add-
ing a spring constraint near its tip, as shown in Fig. 1b. The
margin model will be dependent on the depth into the hole, where
z is the displacement m. The stiffness factor, 1/m, will be
determined from cutting experiments. The equation of motion
with stiffness from margin interaction for a constant-rpm drilling
process is
M
r
¨
¨
+C
r
˙
˙
+K
rm
=F
ext
(2)
where
M
r
=
m
0
0 m
C
r
=
c
-2 m
2 m
c
1
Corresponding author. Present address: D3 Vibrations Inc., Royal Oak,
MI 48067, dave@d3vibrations.com
2
Present address: Alcoa, 100 Technical Drive, Alcoa, PA 15069
Contributed by the Manufacturing Engineering Division for publication in the
JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received
February 12, 2003; revised May 14, 2004. Associate Editor: D.-W. Cho.
Journal of Manufacturing Science and Engineering MAY 2005, Vol. 127 Õ 271
Copyright © 2005 by ASME