An
approach to modeling ductile mode
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The material
removal in rotary ultrasonic machining of ceramic materials was
exclusively attributed to brittle fracture in the published literature
prior to the 1980's (Kubota et al., 1977; Markov and Ustinov, 1972; Markov
et al., 1977; Petrukha et al., 1970). Recently, it has been shown (Pei et
al., 1995b) that ductile flow can contribute to the material removal
process in rotary ultrasonic machining of ceramic materials.
A model
accounting for the ductile mode of material removal has been developed
based on these assumptions:
1.
Workpiece material is rigid-plastic;
2.
Diamond abrasives are rigid spheres with the same diameter and all the
active abrasive particles take part in cutting in each ultrasonic
vibration cycle;
3.
The volume of the material removed by one abrasive particle in one
vibration cycle is equal to the intersection volume of the
abrasive-particle-swept-envelope (APSE) and the workpiece.
As shown in
Figure 15, APSE can be formed by a circle in u-v-w coordinate system which
moves along a sine wave curve in x-y-z coordinate system:
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Figure 15
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(5)
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where, r is
radius of the spherical abrasive.
The equations
describing the sine curve are as follows:
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(6) |
where, D is the
tool diameter; S rotational speed of the tool; A amplitude of ultrasonic
vibration; f frequency of ultrasonic vibration; t time.
The
transformation of any point from the u-v-w coordinate system into the
x-y-z coordinate system is:
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(7) |
or
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(8) |
where
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(9) |
Since the
circle w2+ v2=
r2can also be expressed as:
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(10) |
we can express
APSE as follows:
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(11) |
Figure 16 shows
the three-dimensional graph of APSE.
As shown in
Figure 17, the intersection volume between APSE and workpiece, W, is the
volume above the lower portion of APSE and below the workpiece. This
volume can be calculated numerically.
The MRR can be
calculated by:
MRR = n f W
(12)
From this
model, the relations between the MRR and the process parameters can be
obtained. One of these relations has been verified against a set of pilot
experiments. The experimental data and the calculated MRR from the model
are plotted in Figure 18. It can be seen that the calculated MRR agrees
fairly well with the experimental MRR in the low force range (when the
static force is less than 200 N). As the static force increases, say,
greater than 300 N, the prediction errors increase dramatically. In other
words, the model works reasonably well in the low static force range (when
the ductile mode dominates the material removal), but fails to predict the
MRR correctly in the high static force range (when the brittle fracture is
the predominating mode of material removal).
The observation
of the machined surfaces under the Scanning Electron Microscope (SEM) has
revealed the following. For the pilot experiments, the ductile mode
dominated the material removal in the range that the static force was less
than 200 N. As the static force increased, more and more portion of the
material was removed by the brittle fracture mode.
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Figure 16 |
Figure 17 |
Figure 18 |