PED-Vol. 64, MANUFACTURING SCIENCE AND ENGINEERING
ASME 1993
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A MECHANISTIC APPROACH TO THE PREDICTION OF
MATERIAL REMOVAL RATES IN ROTARY ULTRASONIC MACHINING
Z. J. Pei, D. Prabhakar, P. M. Ferreira
Department of Mechanical and Industrial Engineering
University of Illinois at Urbana-Champaign
M. Haselkorn
Technical Center
Caterpillar Inc.
Peoria, Illinois
ABSTRACT
ratio, high hardness, corrosion resistance, and oxidatio
An approach to modeling the material removal rateresistance)
(MRR)
result in superior performance which in tur
during rotary ultrasonic machining (RUM) of ceramics
translates
is
to significant cost savings.
proposed and applied to predicting the MRR for the case
The of
very properties of ceramics that make them attracti
magnesia stabilized zirconia. The model, a first attempt
from aatproduct performance standpoint are also responsible
predicting the MRR in RUM, is based on the assumption
difficulties
that
encountered in shaping/machining them to
brittle fracture is the primary mechanism of material precise
removal.
size and shape (often demanded by such application
To justify this assumption, a model parameter (whichPresently,
models the machining cost associated with certain ceram
the ratio of the fractured volume to the indented volume
components
of a
can be as high as 90 percent of the total co
single diamond particle) is shown to be invariant for
(Jahanmir
most
et al., 1992). Additionally, the machining o
machining conditions. The model is mechanistic in the
shaping
sense process is often responsible for strength degradat
that this parameter can be observed experimentally from
of the
a fewceramic components.
This can increase the
experiments for a particular material and then used
susceptibility
in
of a ceramic component to sudden failur
prediction of MRR over a wide range of process parameters.
making it necessary to use processes which involve very lo
This is demonstrated for magnesia stabilized zirconia,
pressures
where (e.g., grinding and lapping). Such processes tend
very good predications are obtained using an estimate
beof
extremely
this
slow and typically expensive.
single parameter. On the basis of this model, relations
The above-mentioned difficulties associated with the use
between the material removal rate and the controllable
advanced ceramics are addressed by two different areas
machining parameters are deduced. These relationships
ceramic
agreeresearch: processing and machining. In machining,
well with the trends observed by experimental observations
is apparent that there is a crucial need for the developmen
made by other investigators.
processes which are capable of relatively high materia
removal rates while maintaining the sub-surface damage to
ceramics at an acceptable level.
1. INTRODUCTION
Studies (Prabhakar, 1992; Stinton, 1988) of various mater
The demand for improved product performance has removal
led to theprocesses applicable to ceramics indicated that rot
emergence of advanced ceramics as an important class
ultrasonic
of
machining (RUM) has the potential for high
materials in the latter half of the twentieth century. Currently,
material removal rates while maintaining low cutting pressu
advanced ceramics are increasingly used for commercial
and resulting in little surface damage and consequently lit
applications in the aerospace, automotive, electronics,
strength
and reduction.
Further, the potential exists o
cutting tools industry. The superior properties of the implementing
advanced
RUM on conventional machines with some
ceramics (such as chemical inertness, high strength
modifications.
and
stiffness at elevated temperatures, high strength to weight
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Coolant In
MRR (mm3/sec)
6
Vibration
Rotation
Coolant Out
Core Drill
5
4
3
2
1
0
0
5
10 15 20 25 30 35 40
Amplitude (mm)
800 rpm
1500 rpm
Workpiece
3000 rpm
Magnesia stabilized zirconia
Machining pressure = 0.155 MPa
FIGURE 2 INFLUENCE OF VIBRATION AMPLITUDE ON
MRR
Constant Force
FIGURE 1 SCHEMATIC ILLUSTRATION OF ROTARY
ULTRASONIC MACHINING
influences of the ultrasonic vibration amplitude, the appli
static pressure, the rotating speed, the diamond type, grit s
and bond type, etc., on the material removal rate (MRR) ha
been investigated experimentally (Kubota et al., 1977; Legg
In rotary ultrasonic machining process, a rotating core
drill
1966;
Markov, 1966; Markov and Ustinov, 1972; Markov e
with metal bonded diamond abrasives is ultrasonicallyal.,
vibrated
1977; Petrukha et al, 1971; Prabhakar et al., 1992
by means of an ultrasonic transducer while the workpiece
Tyrrell, 1970). To the authors' best knowledge, no theoreti
being fed towards the core drill at a constant pressure.
models have been published to explain the material remov
Coolant pumped through the core of the drill washes mechanism
away the
and predict the MRR for RUM. Attempts to
swarf, prevents jamming of the drill and keeps it cool.develop
This is theoretical models to predict the MRR for RUM ar
illustrated in Figure 1. Light pressures and clean cuts produced
desirable since they would help in understanding th
by this process make it ideal for ceramic machining (Stinton,
mechanism of RUM and in the optimization of parameters
1988). Experimental results (Prabhakar, 1992) haveobtain
shown
required performance from the process.
that the machining rate obtained from rotary ultrasonic
The aim of this paper is to develop a mechanistic model
machining is nearly 6-10 times higher than that predict
from a the MRR in rotary ultrasonic machining. The paper
conventional grinding process under similar conditions,
see
organized
into five sections. Section 1 (this section) is a
Figure 2. Important parameters involved in the process
include
general introduction to rotary ultrasonic machining o
ultrasonic vibration amplitude, static pressure or static
force, Section 2 outlines the approach used for mod
ceramics.
rotating speed of the tool, grit size and grit number
of the
development,
while Section 3 develops the model. Section
abrasives, and the bond type. The frequency of vibrations
can the experiments performed and the verification of
discusses
also be considered a process parameter.
However,
hypotheses posed in Section 3. Conclusions are drawn up
experimental evidence indicates that at frequenciesSection
above5 along with a discussion of future directions of work
15kHz (which is typically the case for RUM), it has no
observable effects on the process.
RUM has been around for more than twenty 2.
years
APPROACH TO MODEL DEVELOPMENT
(Anonymous, 1964; Hards, 1966; Kubota et al., 1977; Legge,
RUM might be considered as a combination of the ultrason
1965; Legge, 1966; Markov, 1966) and many investigators
machining process and the diamond grinding process. Hen
have reported their studies on it. However, these studies
have
there
are two principal approaches to developing a model
been primarily experimental with little or no attempt
to
predicting
MRR for RUM: one considers the process as bein
develop a model governing material removal mechanism. The
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predominantly ultrasonic machining and superimposes the
effect of rotating motion of the tool, the other reverses the two
Load
primary processes. The first approach is used here.
To develop the model for RUM process, the following steps
are carried out:
Diamond
(1) Estimate δ, the depth to which the diamond particles
Particle
penetrates into the workpiece surface;
(2) Estimate F, the contact force between the diamond
d
particles and the workpiece;
(3) Estimate W, the volume removed by one diamond particle
in a single cycle of the ultrasonic vibration;
δ
(4) Aggregate the effect of all the diamond particles to obtain
the material removal rate of the process.
Workpiece
Similar steps have been followed by other investigators
(Kanith et al., 1979; Shaw, 1956), in developing theoretical
models to predict material removal rates for conventional
ultrasonic machining. However they used different methods to
FIGURE 3 SCHEMATIC ILLUSTRATION OF THE
calculate δ, F and W. One of the primary differences betweenABRASIVE-WORKPIECE INTERFACE
conventional ultrasonic machining and rotary ultrasonic
machining is the rotary motion of the tool which makes it
possible for diamond particles to hit the workpiece at equation
different can be used to relate the contact force with th
locations in consecutive cycles. Unlike the conventional
indentation depth. Although the Hertz equations are stric
ultrasonic machining, the rotary ultrasonic machining
does
applicable
only up to the point of initial surface fractur
not use an abrasive slurry. Therefore, in the absence Sheldon
of rotaryand Finnie (1966) showed that "the Hertz relatio
motion of the tool, the diamond particles would always
hit be
theused to predict the penetration of a particle into
may
workpiece in the same locations, making the material surface
removaleven after cracking occurs" within certain ranges.
rate very low. Our analysis will model the rotation of the
toolindentation depth, δ, is the maximum depth to whi
If the
and demonstrate how it affects the MRR. This effect diamond
will be particles penetrate the workpiece surface, as shown
discussed in greater detail later in this paper.
Figure 3, then, according to Timoshenko and Goodier (1970
δ can be calculated as follows (The Young's modulus of
diamond Ed is taken to be much larger than the Young
3. DEVELOPMENT OF THE MODEL
modulus of workpiece material E. So, E/Ed approaches zero.)
As Sheldon and Finnie (1966) pointed out, "Before analyzing
1/3
any material removal process it is convenient to idealize the
9 (F/n)21-ν22
material's behavior as either 'ductile' or 'brittle'. δ =This
(
(
) )
(1)
1 6 d/2 E
approach is clearly an oversimplification, since materials may
exhibit both ductile and brittle behavior, depending on testing
where,
conditions, but it enables solutions to be developed which are
F -- the maximum contact force between tool and workpiec
useful in many practical situations." In a ductile material,
N;
large plastic strains precede fracture and material is removed by
n -- the number of active abrasive particles across the to
the displacement action of the cutting tool or abrasive
face;
particles. In contrast, for an ideally brittle material, no plastic
d -- the diameter of the abrasive particles, mm;
deformation is present and material is removed by the
E -- Young's modulus of workpiece material, MPa;
propagation and intersection of cracks ahead of and around the
ν -- Poisson's ratio of workpiece material.
cutting tool or abrasive particles. The model developed below
In
the right side of the above equation, all the parameters
is based on several simplifications:
known
except F, the maximum contact force.
The workpiece material is an ideally brittle material;
The material is removed by Hertz fracture;
The diamond abrasive particles are assumed to be rigid
3.2 Estimation
of the Maximum Contact Force F
spheres of the same size, and all the particles are assumed
to
The
workpiece
is fed into the tool by a constant pressure
take part in cutting during each ultrasonic cycle.
static force Fs . The tool is not in continuous contact with t
workpiece due to its oscillatory motion. When an abrasiv
particle has penetrated the workpiece surface to maximu
3.1 Estimation of δ
In each ultrasonic cycle of the tool, each diamond depth,
particlethe force transmitted between tool and workpiec
through
will make contact with the workpiece. The classical Hertz a single abrasive particle will be F/n, where F
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30
0
t1 t2
t
Second)
y=Asin2πft
25
−6
δ
(10
Indentation Time
Dt
y
A
A- δ
20
Equation 4
15
10
Equation 3
5
0
0
Core drill
0.2
0.4
0.6
0.8
1
Indentation Depth/Amplitude
Mean
position
0
FIGURE 5 DIFFERENCES OF INDENTATION TIME
CALCULATED BY EQUATIONS (3) AND (4)
A
This has the advantage of simplicity and might be applicab
when the motion of abrasives relative to the workpiece is on
vertical (due to the oscillation of the tool). However , whe
the tool is also rotating with respect to the workpiece (as
the case with RUM), large inaccuracies in volume calculation
can result, especially when δ is very small relative to A.
The tip of the core drill oscillates with an amplitude A and
frequency f. The motion is considered to be sinusoidal and
position of the tool tip relative to its mean position may
described by the following equation:
A
y
Workpiece
δ
FIGURE 4 CALCULATION OF Dt
y = A sin (2πft)
maximum contact force between tool and workpiece and n the
number of active abrasive particles across the tool face. It will take the tool tip Dt/2 to move from y = A - δ to y = A
Assuming that the diamond particle is incompressible,
thecan be accurately calculated using the following
So, Dt
impulse for one cycle in terms of the maximum contactequation
force F (see Figure 4).
is:
∫ Fdt =~ F Dt
cycle
δ
1 π
Dt = 2 (t2 - t1) ={ - arcsin(1 )}
2
A
πf
(2)
(4)
For different values of δ/A, Dt is calculated by equations (3
where Dt is the period of time during which the particle has
and
(4) and shown in Figure 5. It can be seen that th
penetrated the workpiece surface. Some investigators (for
percentage
differences, specially at lower values of
example, Shaw, 1956) have used a very simple relation to
quite large. The MRR of rotary ultrasonic machining is mor
calculate Dt .
sensitive to the value of Dt than that of conventiona
ultrasonic machining, because the distance L moved by
δ 1
Dt = 2
(3)
diamond particle (due to the rotating motion of the tool) wh
Α 4f
penetrating the workpiece surface is also dependent on
might be noted that attempts to predict MRR using
where,
A -- the ultrasonic vibration amplitude, mm;
calculated by equation (3) exhibits a larger deviation from
f -- the ultrasonic vibration frequency, Hz;
experimental data than using Dt calculated by equation (4).)
δ -- the indentation depth, mm.
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L
Diamond grit
d
2
d
2
δ
Workpiece
d+L
2
FIGURE 6 CALCULATION OF CONTACT LENGTH L
FIGURE 7 EFFECTIVE SIZE OF DIAMOND GRIT DUE TO
THE ROTATING MOTION OF THE TOOL
Substituting (4) into (2), we get the impulse for one cycle in
terms of the maximum contact force F:
where,
D -- the tool diameter, mm;
cycle
S -- the rotating speed, rpm;
Dt -- the period of time during which the particle has inden
The impulse for one cycle in terms of the static force Fs is:
into the workpiece surface, second.
Substituting (4) into the above equation, we get:
1 Fs
Fs =
f f
δ
DS π
L=
{ - arcsin(1 )}
(7)
60f 2
A
By equating the two impulses, we get the following relation:
F π
δ
∫ Fdt =~πf {2 - arcsin(1A)}
(5)
Now, during the period of time Dt, the penetration of th
particle increases from 0 to δ and decreases to 0 while th
π
δ
particle moves through a distance L on the surface of th
( ּ-ּarcsin(1ּ-ּ
))
2
A
workpiece. As a result, the width of the "intersection crate
between the workpiece and the particle will also increase fro
δ and F are the only two unknown terms in equationszero
(1) to
and
some maximum value and decrease to zero. In short,
(6), and hence they can be solved for by using equations
shape (1)
of the intersection crater will be a part of an ellipso
and (6). Or
The volume of this part of the ellipsoid, or, the indentati
volume, can be calculated by integration (see Figure 7).
1/3
π
91 (F/n)21-ν22
F { - arcsin(1 ((
) )
)} = πFs
2
A 1 6 d/2 E
Ld δ
V = π (1 + ) ( - ) δ2
(8)
d 2 3
can be solved for F, the maximum contact force, by trial and
where,
error. δ can be obtained by substituting for F in equation
(1).
d -- the diameter of the spherical abrasive, mm;
δ −− the depth of maximum penetration, mm;
3.3 Estimation of the Indentation Volume V
L -- the distance moved by the particle during penetrati
Having developed an approach to estimating the maximum
into the workpiece due to the rotary motion of the tool, mm
contact force and consequently the maximum depth of
indentation, this sub-section deals with estimating the volume
of penetration due to a single diamond particle in 3.4
a single
Estimation of MRR
ultrasonic cycle.
Once the indentation volume per particle per cycle has be
Due to the rotating motion of the tool, the diamondobtained,
particle the MRR can be predicted if the relationship betwe
would move a distance L while in contact with the workpiece
the fractured volume and the indentation volume is know
(see Figure 6).
However, such relationship is not available in literature
perhaps due to its complexity and the number of facto
πDS
influencing it. We propose a simplified possibility which
L=
Dt
60
aggregates these influences into a single parameter. If W is t
F=
πFs
(6)
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Electric Motor Ultrasonic Spindle Kit
Spindle
Power SupplySpeed Controller
Vibrations
Core Drill
Workpiece
Rotary Motion
Constant Pressure Feed System
Feed Control
Panel
Milling Machine Table
Coolant System
Pump
Coolant Tank
FIGURE 8 SCHEMATIC ILLUSTRATION OF THE ROTARY ULTRASONIC MACHINING SETUP
volume of the fractured zone, it can be consideredvolume
to be is usually larger than indentation volume since t
proportional to V. Therefore,
cracks may initiate and propagate outside the indentati
volume. k1 and k 2 have the opposite effects (k1 decreasin
W = kV
and k2 increasing) on the value of k.
The MRR is given by the product of the volume of materia
where the constant of proportionality, k, could conceivably
removed
be by one particle, W, the frequency of vibration, f, a
a function of the material properties, process parameters,
the number of active diamond particles, n. Hence, the MRR
probability of causing fracture, etc.. However, for V given
to be by
a the following equation:
useful intermediate parameter in the estimation of the MRR in
RUM, k would have to stay relatively constant for a MRR
given= nfW = nfkV
material over a wide range of process parameters. At this point
we will assume that k is independent of process parameters,
Substituting (8) into this equation, we get:
proceed to develop the equation for MRR and then
experimentally verify that the assumption on the invariance is
Ld δ
MRR = knfπ( 1 )+ ( - ) δ2
(9)
indeed valid.
d 2 3
If the assumption made above is valid then k can be
considered to be a function of two primary effects, i.e.,
k =
where
δ can be obtained from equations (1) and (6). L can
f(k1 , k2 ), where k1 is the probability coefficient which
brings from equation (7). On the right hand of equation
obtained
into consideration the fact that it may take more than
only one
k has not been decided. As mentioned earlier, this val
indentation to remove the volume W. k 2 is the volume
will have to be experimentally determined.
coefficient which accommodates the fact that the fractured
776
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Actuating Circuit
Core Drill
Workpiece
Pressure Regulator
Pressure Gage
AC
Die-Set
Upper Plate
Lower Plate
Feed Direction
Hydraulic Cylinder
Valve
Milling Machine Table
Air/Oil Tanks
FIGURE 9 CONSTANT PRESSURE FEED SYSTEM
machine setup, in which the drill is fed down into th
workpiece, in the present setup the workpiece is raised u
towards the drill. Further, the feed pressure is controlle
4.1 Experimental Setup
instead of the feedrate. The support fixtures aid in securing
Figure 8 schematically illustrates the rotary ultrasonic
lower plate to the milling machine table. The workpiece
machining setup.
It consists of an ultrasonic spindle
clamped
kit, a firmly on the upper plate. The only moving parts
constant pressure feed system and a coolant system. the feed system are the upper die-set plate (work-table) and
The ultrasonic spindle kit comprises of an ultrasonicworkpiece.
spindle,
The feed (static) pressure is preset before the st
a power supply and a motor speed controller. The ultrasonic
of machining operation. When the circuit is activated th
spindle is mounted on a milling machine (replacing
piston
therises from the hydraulic cylinder raising the workpie
toolhead of the milling machine). The spindle contains
towards
an the drill.
ultrasonic transducer. The power supply converts The
50Hzexperimental setup employs the coolant system of t
electrical supply to high frequency (20kHz) AC output.milling
This ismachine.
fed to the piezoelectric transducer located in the spindle. The
transducer converts electrical input into mechanical
4.2 Design
of Parametric Set of Experiments
vibrations. By changing the setting of the output control
of
the power supply, the amplitude of the ultrasonic vibration
For can
the purpose of estimating k, data are used from
be adjusted. The rotational motion of the tool is supplied
parametric
by
study conducted on RUM. If k is independent
the motor attached atop the spindle and different speeds
machining
can be parameters, as assumed in the model developme
obtained by adjusting the motor speed controller.
then theoretically only one experiment is needed to get
Figure 9 shows a schematic illustration of the constant
value. However, to verify that it is indeed independent
pressure feed system. The basic purpose of the system
machining
is to
parameters, a number of different experiments
feed the workpiece towards the tool at a constant pressure.
various
Thecombination of machining parameters are neede
system consists of a four-pillar die-set, an actuating Further,
circuit, because of the large variations that usually occurs
and support fixtures. The die-set consists of two plates
any experiments
of
that involves ceramics, to get a bette
which the upper plate serves as the work-table. Thisestimate,
plate is a number of experiments will be needed. A
moved vertically by means of a double acting hydraulic
systematic way of doing this would require a proper
cylinder. The cylinder is driven by an actuating circuit
experimental
which
design. The design of experiments involved fi
consists of an electronic pressure regulator, pressure gage,
control
and
variables (machining parameters) shown in Table
a three port-two way valve. Unlike a conventional drilling
Therefore, for a two-level full-factorial design, at least 3
4. VERIFICATION OF THE MODEL
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TABLE 1 CONTROL VARIABLES AND THEIR LEVELS IN
EXPERIMENTS
10
Experimental MRR (mm
3
Control Variable
Low level High Level
Static Force (N)
111
356
Vibration Amplitude (mm)
0.023
0.033
Rotating Speed (rpm)
1000
3000
Abrasive Bond Type Brass-copperIron-nickel
Abrasive Grit Size (mm) 0.05
0.22
8
6
4
experiments need to be performed. Each test is replicated once,
bringing the total number of tests to 64. The levels shown in
Table 1 represent the typical high and low settings for the
process parameters.
2
The following variables were held constant during all test
runs.
Workpiece Material: Magnesia Stabilized Zirconia (Young's
0
modulus E = 205000MPa, Poisson's ratio ν=0.31);
0
2
4
6
8
10
12
Coolant: A water based semi-synthesis emulsifier;
2
3
nfπ(1 + L/d)(d/2 - δ/3)δ (mm /s)
Abrasive: Diamond particles.
In each test the time taken to drill a hole of 12.7mm in
FIGURE 10 CALCULATION OF K
diameter into the 6.35mm thick workpiece was recorded. The
material removal rate for any machining operation is given
by:
the process parameters. However, we suspect that Hert
fracture, as modeled above, is the dominating influence. As
VolumeּofּMaterialּRemoved
first approximation, we are assuming that it is independent
MRR =
ּTime
machining parameters. The purpose of this section is t
validate this claim for one material and to estimate the value
Hence, for rotary ultrasonic machining, i.e., drilling operation
k for the material using the data obtained from the experim
using core drills, MRR is
described in the previous section.
For the purpose of evaluating the assumption that k sta
2
D h 2 ּ-ּD d πH
relatively constant over the parameter ranges, the data from
MRR =
two-level, full-factorial experiment (described above) is use
4t
so that all process parameters are varied over their ranges. D
from all the experiments will be used to estimate k as the slo
where,
of the least-squares straight line (Neter et al., 1990) passi
Dh -- the hole diameter, mm;
through the origin and relating the observed MRR for eac
Dd -- the machined core diameter, mm;
d δ 2L
H -- the thickness of the workpiece, mm;
experiment with the corresponding )nfπ(
( - 1) +
δ value
d 2 3
t -- the time consumed during machining, second.
forofthe
For further details of the experiments, the calibration
theexperiment (see Figure 10). The value of k for th
overall
setup, measurement of variables and results the reader isdata is 0.618. Next, to test if the value is in fac
constant,
referred to Prabhakar (1992) and Prabhakar et al. (1992).
In for each parameter, the data will be divided into t
those studies, the experimentally observed effects ofgroups,
each of corresponding to the high and low level of th
parameter. The parameter k will be estimated for each group
the parameters on the MRR were studied. Further, an empirical
a manner similar to that described above. If our assumpti
models was given. In this paper, the results of the experiment
that
k is
are used for estimating k and testing the hypothesis
that
it constant is true, the estimated values for each of
groups will not differ significantly from the value determin
does stay constant for a particular material.
for the overall data.
From Figure 10, it can be seen that there exists some
deviation of the predicted MRR values from the experiment
4.3 Analysis of the Experimental Results
The scattering of the data may be the results of th
The model developed relies on the assumption that kdata.
remains
following.
invariant across a broad range of process parameters
for a
particular material. Strictly speaking, k might be a function of
(
)
778
0.943
3
-2
0.575
0.587
0.597
0.639
0.703
0.523
0.6
Rotating Speed -
-1
0.6
0
Amplitude +
0.618
0.8
0.644
1
0.781
1
2
k
Estimated Cumulative Probability
Converted to Standard Normal Variable
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0.4
-3
-3
-2
-1
0
1
2
3
Standard Value of the Differences
Between the Predicted and Experimental MRR 0.2
Grit Size +
Grit Size -
Force +
Force -
Bond Type +
Bond Type -
Rotating Speed +
Amplitude -
0
Total Data
FIGURE 11 Q-Q PLOT
(1) Large dispersion of the data may result from unbiased
measurement errors and the probabilistic nature of the flaw
distribution in ceramics.
(2) All abrasives are assumed to take part in machining, with
all the particles having the same shape and size. In reality, it
is more likely that only a fraction of particles take part in
machining.
(3) Other assumptions previously described.
Parameter Level
First, a check needs to be made to test whether there exist any
particular trends in the residuals, i.e., differences between the
FIGURE 12 INFLUENCE OF PARAMETER LEVELS ON K
MRR values predicted by the straight line and the observed
values. For this a Q-Q plot (DeVor et al., 1992) is made and
shown in Figure 11. Clearly, the residuals are random in nature
the material removal mechanism at very low contact force
and no underlying trends are detected. Next, for each
Our initial analysis suggests that this could be plastic
parameter, the data is separated into two groups corresponding
deformation. As the contact force is reduced, the depth
to the high level and low level of that parameter. Figure 12
penetration of the abrasives into the work material keep
shows the values of k estimated for each of these groups. At
reducing. At some contact forces, this could be close to th
the 0.001 level of significance, the hypothesis that the value
ductile-brittle transition for the material. (For the mater
of k is not significantly different from 0.618 has not been
used, this is roughly 0.1 to 0.4 microns). This suggests that
rejected for all groups except the group that corresponds to the
lower forces a different model might be required.
low level of static force. Based on this analysis, one can state
that the assumption of k being constant for a particular
material is reasonable and can be applied as a first
4.4 The Influence of Different Parameters on
approximation to evaluating the material removal rate for a
MRR
given material and set of process parameters.
In the previous sections, we have developed a simplifie
If k were to be thought of as the efficiency of the rotary
analytical model for MRR in rotary ultrasonic machining und
ultrasonic machining for a particular material, then at low
the assumption that a particular model parameter, k, rema
forces, even though the MRR is low, a statistically significant
constant for a given material. In this section, we will use th
higher efficiency is obtained. This significant change in the
model to study how individual machining parameters influe
"efficiency" of the process might be attributable to a
the MRR and compare the trends predicted by the model w
mechanism other than Hertz fracture, that begins to dominate
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F/n (N)
5
4
30
20
10
3
0
8
2
6
δ (mm)
MRR (mm3/s)
6
1
2
0
0.01
0.02 0.03 0.04
Amplitude (mm)
Fs=600N
Fs=400N
0.05
16
L (mm)
0
0
4
Fs=200N
Fs=100N
12
8
4
0
S=1000rpm, d=0.05mm, n=100
3
MRR (mm3/s) V (10-6
mm )
5
4
3
FIGURE 13 RELATION BETWEEN AMPLITUDE AND MRR
2
1
those observed by other researchers. It must be noted here that
0
only trends can be compared, because the precise experimental
conditions for the experiments (reported by other6
investigators) are not known. The model is applied to predict
the relations between the MRR and the different parameters 4
for
rotary ultrasonic machining of magnesia stabilized zirconia.
2
The value of k is taken as 0.618.
The predicted relation between MRR and the amplitude of the
0
ultrasonic vibration has been plotted in Figure 13. Figure 14
10 20
30
40
50
shows the variation of different important components of0
equation (9) (the MRR equation) with variation in amplitude.
Amplitude (mm)
Specifically, the indentation force/indenter, the depth of
S=1000rpm, Fs=600N,
indentation and the distance moved by an indenter when in
d=0.05mm, n=100
contact with the workpiece (length of contact) due to the
rotational motion of the tool are shown in this figure. Finally
the volume of the indentation and the MRR are also shown.
FIGURE 14 INFLUENCE OF AMPLITUDE
Two important effects are visible. First, the indentation force
and depth increase at a decreasing rate with amplitude. Second,
the length of contact decreases with increasing amplitude.
This is certainly true, however, our model (which does n
These two effects cause the MRR to increase at a decreasing
consider wear) suggests that in addition, the proces
rate, suggesting that at some amplitude the curve will
flatten
mechanics,
explained above, is also responsible for such
and possibly begin to drop. The experimental data reported
by
behavior.
Markov and Ustinov (1972) shows that further increases
The of
relation between MRR and the static force is shown i
ultrasonic amplitude above a certain value will result
in a15. It can be seen that the predicted MRR will alway
Figure
reduction in the MRR. The reason for this has been explained
increase with the static force. The experimental data repor
as "due to an excessive increase in alternate loadingby
onMarkov
the
and Ustinov (1972) also show that MRR first
diamond grits and a weakening of the bond" and increases
further with the static force until reaching a certain valu
increase of ultrasonic amplitude "may result in complete
andfailure
then decreases with the static force. No explanation h
of the diamond core bits as a result of the high cycle stresses".
been given for this phenomenon by them. According to o
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F/n (N)
3
1
10
5
0
100
200
300
400
Static Force (N)
A=0.035mm
A=.025mm
4
2
0
8
500
L (mm)
0
15
0
6
2
δ (mm)
MRR (mm3/s)
4
A=0.015mm
A=0.005mm
6
4
2
0
MRR (mm3/s) V (10-6mm3
S=1000rpm, d=0.05mm, n=100
FIGURE 15 RELATION BETWEEN STATIC FORCE AND
MRR
4
3
2
1
analysis, it may also be due to the rapid wear of the core drill.
0
Over the region of comparable forces, the trends of the
3
proposed model and the experimental trends are similar. The
difference at higher values is due to the fact that our model does
2
not account for such factors as tool wear. Figure 16 shows the
variation of different components of the MRR equation with
1
static force. The almost linear trend is due to the fact that both
the indentation depth and the contact length increase with
static force causing the indentation volume to increase0
0
100 200 300 400 500
steadily.
The predicted relation between MRR and the rotating speed
Static Force (N)
has been plotted in Figure 17. It is seen that MRR increases as
S=1000rpm, A=0.025mm,
the rotating speed is increased. Markov and Ustinov (1972)
d=0.05mm, n=100
state that "material removal rate increases as the peripheral
speed of the drill is increased". They do not report on the
characteristics. Our model agrees with this qualitativeFIGURE 16 INFLUENCE OF STATIC FORCE
statement. Figure 18 shows the different components of the
MRR equation. From this figure, one can observe that the
However,
indentation force and depth are unaffected by the rotating
speedthe number of indenters increases to offset th
rendering MRR almost independent of
which only causes a change in the length of contact.reduction
As the
concentration. It must, however, be pointed out that th
rotating speed increases, the indentation volume changes
independence is observed for the particular constants (mate
proportionally and the MRR also increases.
properties)
chosen. A harder or softer material may result i
The predicted relation between MRR and the diamond
grit
decrease
concentration (through the grit number) has been plotted
inor increase of MRR with diamond concentration.
Figureּ19. Figureּ20 shows the components of the The
MRRpredicted relation between MRR and the diamond gr
diameter
equation. It can be observed that as the grit number increases, has been plotted in Figure 21. The components
the MRR
the indentation force per indenter and the indentation
depthequation are shown in Figure 22. It can be seen th
force increases as the diameter of the diam
decrease, which in turn reduces the contact length. the
All indentation
this
particles increases. However, the indentation depth reduces
causes a reduction in the indentation volume per indenter.
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6
F/n (N)
6
4
2
1
2
Fs=600N
Fs=400N
4
2
0
16
1000
2000
3000
4000
Rotating Speed (rpm)
L (mm)
0
0
4
0
6
3
δ (mm)
MRR (mm3/s)
5
Fs=200N
Fs=100N
12
8
4
0
MRR (mm3/s) V (10-6mm3
A=0.025mm, d=0.05mm, n=100
FIGURE 17 RELATION BETWEEN ROTATING SPEED
AND MRR
0.6
0.4
0.2
causes the contact length to do the same. The particle diameter,
however, plays a dominant role in the calculation of the 0
6
indentation volume causing it to increase which increases the
MRR. No experimental data on these relations are available
4
in literature.
Figures 13, 15, 17, 19 and 21 collectively suggest that
2
second order effects exist between the process variables which
is in agreement with the results obtained from a full factorial
0
experiment reported by Prabhakar et al. (1992). For example, 0
the positive effect of machining pressure (or static force) on
the effectiveness of increased rotating speed in improving
MRR is clearly evident in Figure 17.
3
4
1
2
Rotating Speed (1000 rpm)
Fs=600N, A=0.025mm,
d=0.05mm, n=100
5. CONCLUSION
A theoretical model to predict MRR for RUM has beenFIGURE 18 INFLUENCE OF ROTATING SPEED
developed. The model is based on the assumption that the
brittle fracture mechanism is the dominant mode of material
case, no statistically significant difference was observed in t
removal. It might be only one of the mechanisms actually
estimate of the model parameter. This leads us to believe th
effecting material removal during rotary ultrasonic machining
spite of the assumptions that workpiece material is idea
of ceramics. The model requires the estimation of ainsingle
brittle
and the diamond particles are rigid spheres of the sa
parameter (which is dependent on the properties of the
material
size being oversimplifications of the actual situation, th
being machined).
modelthe
is, at least, a good first approximation of the mater
An experimental investigation was conducted to estimate
removal
model parameter and verify that it was, in fact, constant
over mechanism. The model parameter which can b
estimated for different materials can be thought of as th
the entire range of machining parameters for magnesia
efficiency of the RUM process for that material.
stabilized zirconia. Statistical analysis of the experimental
Thethe
model has been used to study the influence of differe
data indicated that it was reasonable to assume that
machining
parameter is constant. For all cases except the low static
force parameters on the MRR. The trends predicted by
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F/n (N)
5
4
δ (mm)
2
1
4
2
0
300
600
900
Grit Number
Fs=400N
Fs=300N
2
1
0
6
1200
L (mm)
0
6
0
3
3
Fs=200N
Fs=100N
S=1000rpm, A=0.025mm, d=0.05mm
FIGURE 19 RELATION BETWEEN GRIT NUMBER AND
MRR
MRR (mm3/s) V (10-6mm3
MRR (mm3/s)
6
4
2
0
0.8
0.4
model are consistent with those reported in experimental
0
investigations. Mechanistic explanations for these idealized
1
trends are given in our discussion.
This investigation, being the first attempt to theoretically
predict the MRR in rotary ultrasonic machining, leaves a
0.5
number of avenues for follow-up work. First, models which
account for plastic flow (especially at very low static forces
and consequently very small indentation depths) are required.
Next, a study of different materials for the estimating the 0
0
efficiency of RUM in machining them is required. MRR is
often not the only consideration in process design. Tool wear
and workpiece surface damage are important factors which need
to be studied along with MRR.
2
4
6
8 10 12
Grit Number (100)
S=1000rpm, Fs=100N,
A=0.025mm, d=0.05mm
ACKNOWLEDGMENT
FIGURE 20 INFLUENCE OF GRIT NUMBER
Financial assistance was provided by Caterpillar Inc.,
Hards, K. W., 1966, "Ultrasonic Speed Diamond Machining"
through the Manufacturing Research Center of the University
of Illinois, and by the National Science Foundation PYI Ceramic
Award Age, Vol. 82, No. 12, pp. 34-36.
Jahanmir, S., Ives, L. K., Ruff, A. W. and Peterson, M. B.,
(DDM - 9157191).
1992, "Ceramic Machining: Assessment of Current Practice
and Research Needs in the United States", NIST Special
Publication 834.
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Statistical Quality Design and Control: Contemporary
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783
PED-Vol. 64, MANUFACTURING SCIENCE AND ENGINEERING
ASME 1993
And to appear at The ASME Transactions, Journal of Engineering for Industry
F/n (N)
8
6
4
δ (mm)
MRR(mm3/s)
10
2
0.2
0.3
0.4
Grit Diameter (mm)
Fs=400N
Fs=300N
0.5
L (mm)
0.1
2
1
0
6
0
0
10
8
6
4
2
0
3
Fs=200N
Fs=100N
4
2
0
MRR (mm3/s) V (10-6mm3
S=1000rpm, A=0.025mm, n=100
FIGURE 21 RELATION BETWEEN GRIT DIAMETER AND
MRR
2
1
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0
Legge, P., 1965, "Ultrasonic Drilling of Ceramics",
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0
Ultrasonic Diamond Drilling of Non-Metallic Materials",
0
0.1 0.2 0.3 0.4 0.5
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