Biomedical Engineering Reference
In-Depth Information
2.3. Combination of the Bending and Torsional-Axial Models
As the continuation of the previously presented work, reference [24]
presents a combination of the effects of torsion and bending on chatter in
drilling using ―short‖ and ―long‖ drills. The analytical work regarding the
torsional limits of stability in drilling, have been developed using Bayly's
work as the basis [3].
Bending equations with two degrees of freedom have been combined with
the equation of motion in the axial direction (z):
..
.
.
m x t
( )
c x t
( )
(
k
m
) ( )
x t
2
m
y t
( )
c
y t
( )
F
( )
t
x
..
.
.
m y t
( )
c y t
( )
(
k
m
)
y t
( )
2
m
x t
( )
c
x t
( )
F
( )
t
y
..
.
m
z t
( )
c
z t
( )
k z t
( )
F
( )
t
a
a
a
z
where: , -variations of cutting forces, x(t), y(t), z(t) - drill
deflections in the X, Y and Z directions, x(t), y(t) - deflections of the middle of
the drills, z(t) - axial elongation of the tip, m, c and k - dynamic properties of
the bending modes, m
a
, c
a
, k
a
- dynamic properties of the axial mode, Ω - drill
rotational speed.
A time domain simulation of chatter drilling has shown strong agreement
between the experiments and analytical derivations for the ―long‖ drills, but
not for the ―short‖ drills due to inaccurate mathematical modeling. It has been
experimentally determined that there were two sets of chatter marks for the
―long‖ drill, corresponding to the bending mode and the torsional mode. For
the ―short‖ drill all the chatter marks correspond to the torsional mode. This
was explained by the fact that the support in the case of the ―short‖ drill acts
more like a spring rather than a rigid pin and therefore making the bending
mode less important. There are several parameters that haven't been included
in the study: process damping due the interaction between the flank faces of
the drill and the just cut surface (the ‗‗rubbing'' mechanism tends to dampen
the vibration and it plays an important role in the stability of the tool), and the
chisel edge in cases where there is no previous pilot hole as well as edge
forces.
Dynamics, kinematics, and stability in drilling operations have been
covered in a two-part paper by Roukema and Altintas. The first part of the
paper [6] deals with the problem of combined lateral drill deflection, lateral
'
()
'
()
x
Ft
z
Ft
()
y
Ft
