Biomedical Engineering Reference
In-Depth Information
where: θ - vibration angle, θ - shear plane angle, ʱ - rake angle and
du
cos(
)
0
.
cos(
)
d
du
0
V
cos
d
cos
2
(
)
0
The critical depth of cut is expressed as:
k
(
2)
crit
2
p
cos
0
where : + is used for p
o
cosθ>0 and - for p
o
cosθ<0, p
o
- constant coefficient of
G, G - force penetration function, Γ - dimensionless damping coefficient, η -
shear stress of the material. As the rake angle is increased from ʱ<ʱ
crit
to
ʱ>ʱ
crit
the response frequency becomes less than the natural frequency and a
shift in the stability threshold occurs. This mathematical phenomenon is seen
on the shop floor as the tool starts digging into the material as opposed to
being forced away from it.
Stone and Campbell continued Stone and Askari's work in [16] by
studying torsional-axial chatter. This work confirmed Bayly's finding that for
the drilling case, the onset of chatter oscillation will happen at a frequency less
than the natural frequency of the vibration mode, which is in contrast to the
traditional case where the onset chatter frequency will be greater than the
natural frequency. Authors pointed out that the previous research had an
approximation that βp
1
<γ (β - bifurcation parameter, p
1 -
linear coefficient of
G,
γ - dimensionless damping coefficient) which causes the difference
between the cases, resulting in the presence of closed loops in the traditional
case.
The criticality of the Hopf bifurcation has been analyzed considering the
sign of the quantity:
2
2
2
2
a
(0)(3
f
f
)
(0)(
f
3
f
(
(0)
(0) )
f
(
f
f
)
2
(0)
(0)(
f
f
)
12
111
122
22
112
222
12
22
12
11
22
12
22
22
11
where: The f
jk
and f
ijk
are functions of the parameters β, T, γ, p
0
, p
1
, p
2
(constant, linear and quadratic coefficients of G), θ(vibration angle), the Hopf
frequency ω, and the centre manifold coefficients. By analyzing this equation
