Biomedical Engineering Reference
In-Depth Information
Bayly and his colleagues started research on the torsional axial model in
2001, [3] by developing a mathematical model of the torsional-axial coupling
that causes chatter in drilling. In this work, researchers have taken into
consideration three translations and one rotation, leaving the bending moments
out of the equation.
The equation of motion can be written as:
..
.
M x
C x
Kx
F
where
:
T
x
u
,
u
,..,
u
v
,
v
,..,
v
,
w
,
w
,..,
w
,
,..,
1
2
N
,
1
2
N
1
2
N
1,
2
N
T
F
F
,
F
,..,
F
,
F
,
F
,..,
F
,
F
,
F
,..,
F
,
M
,
M
,..,
M
x
1
x
2
xN
y
1
y
2
yN
z
1
z
2
zN
1
2
N
u, v, w - deflections in x, y, z , θ- twist about the axis.
The time varying component of the modal equation can be written as:
where: m, k, c - modal mass, stiffness and damping, ʱ = C
2
/C
1
+ θ
Np
R
av
, θ
Np
-
torsional axial coupling parameter, R
av
- average radius of the cutting force,
C
1,
C
2
-tangential and thrust forces per unit area of uncut chip, b- radial depth of
cut, η
p
- modal coefficient of the p
th
mode, η - time delay between passage of
cutting edges of the drill.
The difference between this equation and classical chatter equations is in
the constant ʱ which includes the effects of tangential and axial forces as well
as torsional axial coupling. Another difference lays in the fact that -ʱC
1
which
presents the effective cutting pressure is not always positive as it is in milling
and turning operations.
Predictions of stability regions have been done by frequency domain
analysis and confirmed by both experimental work and computer simulations.
The drill dynamics were described by the Frequency Response Function
(FRF), which is defined as displacement at the given location in response to a
unit force, as a function of excitation frequency.
Radial depth of cut is expressed as:
