Automated Upgraded Generalized Full-Discretization Method …
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4.2 Stability Diagrams
A 200 by 200 computational grid is created on the axes of spindle speed and depth
of cut on which the spectral radius at each grid point is computed. Contour curves
(which are the stability curves) are then placed on the plane to distinguish the spectral
radii into those higher than unity in magnitude (the unstable ones) from those less
than unity in magnitude (the stable ones). Since the best computational accuracy is
expected in the range defined by p c = 1 to 10 and p d = 1 to 10, the square unification
cases p = 1, 2, . . . , 10 are considered first where square unification implies p c =
p d . The full scale results for the unidirectional and bidirectional models of workpiece
at the beginning of tool pass are shown in Fig. 5a, b. Each of the stability curves is
identified with the discretization integer k = 40. The red reference stability curves
are determined with the monodromy matrix (3, 3) and the discretization integer
k = 100. The identical results for the unidirectional and bidirectional models of the
elastic flexible workpiece confirm the expectation that chatter in the feed direction
is negligible because of the very high ratio of length to thickness typical of elastic
flexible workpiece. Figure 6, which is an enlargement of the initial low speed domain
of 1000–1800 rpm in Fig. 5a or Fig. 5b, shows that accuracy improved up to the third
order and decreased beyond the third order. Figure 7 shows that all the methods within
the stable range of interpolation order converge at the intermediate speeds but the
methods beyond the ninth order fail to converge at the high speed range.
There is still a question of whether the sensitivity of stability lobes precision is
affected the same way or differently by independent variations in p c and p d under
rectangular unification where p c = p d . Figures 3 and 4 suggest a differing sensitivity
when either p c or p d approaches zero. Error surface reflected higher magnitude
when p c approaches 0 than when p d approaches 0. To investigate further, error
surfaces are plotted for a low speed and an intermediate speed process parameter
coordinates in Fig. 8 to clearly reveal a difference of sensitivities of stability lobe
precision to variations in p c and p d . It should be recalled that the axes for spectral
radii computation error in Fig. 8 are also given on logarithmic scale to base ten.
Keeping p c constant at 3 and varying p d from 0 to 10, the stability boundary curves
in Figs. 9 and 10 were generated. As presented in Figs. 11 and 12, the corresponding
results were generated while keeping p d constant at 3 and varying p c from 0 to
10. Comparison of Figs. 9 and 11 confirms the suggestion of the error surfaces that
stability lobes precision is more sensitive to variation in p d than variation in p c ,
especially at the low speed domain. Also, the comparison agrees with the error
surfaces that p c = 0 is numerically unstable while p d = 0 is not only numerically
stable but can be acceptable. The specific deductions from the comparison include
the following:
• The precision of stability lobes is relatively resistant to change in p c with the best
accuracy expected for p c in the range of 1 to 9.
• The precision of stability lobes is strongly sensitive to change in p d , especially at
the low speed range.
95
4.2 Stability Diagrams
A 200 by 200 computational grid is created on the axes of spindle speed and depth
of cut on which the spectral radius at each grid point is computed. Contour curves
(which are the stability curves) are then placed on the plane to distinguish the spectral
radii into those higher than unity in magnitude (the unstable ones) from those less
than unity in magnitude (the stable ones). Since the best computational accuracy is
expected in the range defined by p c = 1 to 10 and p d = 1 to 10, the square unification
cases p = 1, 2, . . . , 10 are considered first where square unification implies p c =
p d . The full scale results for the unidirectional and bidirectional models of workpiece
at the beginning of tool pass are shown in Fig. 5a, b. Each of the stability curves is
identified with the discretization integer k = 40. The red reference stability curves
are determined with the monodromy matrix (3, 3) and the discretization integer
k = 100. The identical results for the unidirectional and bidirectional models of the
elastic flexible workpiece confirm the expectation that chatter in the feed direction
is negligible because of the very high ratio of length to thickness typical of elastic
flexible workpiece. Figure 6, which is an enlargement of the initial low speed domain
of 1000–1800 rpm in Fig. 5a or Fig. 5b, shows that accuracy improved up to the third
order and decreased beyond the third order. Figure 7 shows that all the methods within
the stable range of interpolation order converge at the intermediate speeds but the
methods beyond the ninth order fail to converge at the high speed range.
There is still a question of whether the sensitivity of stability lobes precision is
affected the same way or differently by independent variations in p c and p d under
rectangular unification where p c = p d . Figures 3 and 4 suggest a differing sensitivity
when either p c or p d approaches zero. Error surface reflected higher magnitude
when p c approaches 0 than when p d approaches 0. To investigate further, error
surfaces are plotted for a low speed and an intermediate speed process parameter
coordinates in Fig. 8 to clearly reveal a difference of sensitivities of stability lobe
precision to variations in p c and p d . It should be recalled that the axes for spectral
radii computation error in Fig. 8 are also given on logarithmic scale to base ten.
Keeping p c constant at 3 and varying p d from 0 to 10, the stability boundary curves
in Figs. 9 and 10 were generated. As presented in Figs. 11 and 12, the corresponding
results were generated while keeping p d constant at 3 and varying p c from 0 to
10. Comparison of Figs. 9 and 11 confirms the suggestion of the error surfaces that
stability lobes precision is more sensitive to variation in p d than variation in p c ,
especially at the low speed domain. Also, the comparison agrees with the error
surfaces that p c = 0 is numerically unstable while p d = 0 is not only numerically
stable but can be acceptable. The specific deductions from the comparison include
the following:
• The precision of stability lobes is relatively resistant to change in p c with the best
accuracy expected for p c in the range of 1 to 9.
• The precision of stability lobes is strongly sensitive to change in p d , especially at
the low speed range.