100
C. Ozoegwu and P. Eberhard
Fig. 15 Identified stability
curves at the start of tool pass
of the bidirectional model for
the orders p = p c = p d
0.5
1
1.5
2
2.5
spindle speed [rpm]
10
4
0
5
10
15
20
25
30
depth of cut [mm]
ref
p=1
p=2
p=3
p=4
p=5
p=6
p=7
p=8
p=9
p=10
CTRS
Fig. 16 Time-domain
simulation of nodal
responses at [13630 rpm, 20
mm] indicated by star in
Fig. 15
0
0.01 0.02
time [s]
-600
-400
-200
0
200
400
600
regenerative displacement [ m]
0
0.01 0.02
time [s]
-1.5
-1
-0.5
0
0.5
1
1.5
regenerative velocity [m/s]
instability is seen. The time-domain simulation validates the reference curve. As seen
in Fig. 18, time-domain simulation also validates the reference stability diagram at
the two-thirds of the tool pass.
Though the generalized computerization allows arbitrary choices of p c and p d
for any given case study, such arbitrary choices show that there is a threshold interpolation order at which numerical instability becomes overbearing. Stability curves
are generated for p c = p d = p > 10 and compared against the reference in Figs. 19
and 20. It is seen that p c = p d = p = 9 is the threshold interpolation order beyond
which the FDM fails for the studied system. The failure starts from the high speed
domain and extends deeper into the low speed domain for higher orders p, and the
magnitude of the failure is higher for higher orders p at every spindle speed.