Biomedical Engineering Reference
In-Depth Information
it has been suggested that the bifurcation in the traditional case is subcritical
for high values of T and supercritical for intermediate and small values of T.
In the axial-torsional case it has been determined that the bifurcation is always
subcritical.
An analysis for high-speed stability limits has been performed stating very
interesting results, especially if one is interested in micro drilling, which
would be in the field of very high speeds. In high speed limits, T becomes zero
(large cutting speeds) and the equations do not provide further delay, but can
be analyzed by conventional techniques.
For the T→0 limit, the equation of motion is expressed as:
''
'
'
'2
(
p
p
p
)
0
1
2
if calculated, a=0 pointing out degeneracy of the system, suggesting that it
would be necessary to determine higher order terms in order to define stability.
These bifurcations re not well understood yet and are suggested as a good
topic for future research. Due to all afore presented, macro and micro drilling
cannot be analyzed the same way.
Stone and Askari's work was followed by research [17] which illustrates
the effect of varying thickness and speed with linear stability boundaries and
considers the change with varying cutting width. This paper is based on the
models developed in the previously mentioned work and defines the stability
regions with respect to depth of cut. The researchers have defined the
traditional case with a high cutting speed having a supercritical bifurcation,
while the case with low cutting speed is followed by a subcritical bifurcation.
In the axial-torsional vibration case, for all physically reasonable situations,
the bifurcation is shown to be subcritical. The implications of switching
between supercritical and subcritical Hopf bifurcation, influence the stability
of the cutting process. When approaching the supercritical Hopf bifurcation,
the linear stability boundary determines when the cutting will become
unstable. With the subcritical Hopf bifurcation, small perturbations can cause
the instability.
The equation of motion used has been derived in [15] and rescaled in
order to use depth of cut t
1
as the bifurcation parameter instead of width of cut
w and accounting for chip thickness variation:
w
''
'
'
' 2
t
( )
t
(
t
T
) cos(
) (
p
p
p
)
k
