Machine tools abstract
Methods of and apparatuses for mitigating chatter vibrations in
machine tools or components thereof. Chatter therein is suppressed
by periodically or continuously varying the stiffness of the cutting
tool (or some component of the cutting tool), and hence the resonant
frequency of the cutting tool (or some component thereof). The varying
of resonant frequency of the cutting tool can be accomplished by
modulating the stiffness of the cutting tool, the cutting tool holder,
or any other component of the support for the cutting tool. By periodically
altering the impedance of the cutting tool assembly, chatter is
mitigated. In one embodiment, a cyclic electric (or magnetic) field
is applied to the spindle quill which contains an electro-rheological
(or magneto-rheological) fluid. The variable yield stress in the
fluid affects the coupling of the spindle to the machine tool structure,
changing the natural frequency of oscillation. Altering the modal
characteristics in this fashion disrupts the modulation of current
tool vibrations with previous tool vibrations recorded on the workpiece
surface.
Machine tools claims
We claim:
1. A method for controlling chatter in a machine tool assembly
during a machining operation, the machine tool assembly having a
plurality of components, the plurality of components comprising
a variable compliant component made with a material having a variable
compliance, the plurality of components having a mechanical impedance,
the method comprising the step of modulating the mechanical impedance
of at least one of the plurality of components, wherein the machine
tool assembly further comprises a chuck, the chuck in mechanical
communication with the variable compliant component, wherein the
step of modulating the mechanical impedance comprises modulating
the resonant frequency of the plurality of components by changing
the variable compliance of the variable compliant component to mitigate
regenerative chatter vibrations.
2. The method of claim 1 wherein the at least one of the plurality
of components is a cutting tool.
3. The method of claim 2 wherein the at least one of the plurality
of components is a machine tool support adapted to receive the cutting
tool.
4. The method of claim 3 wherein the at least one of the plurality
of components is a workpiece to be operated on by the cutting tool.
5. The method of claim 3 wherein the step of modulating the mechanical
impedance further comprises the step of using piezo electric materials
and making intermittent contact with the machine tool support.
6. The method of claim 1 wherein the mechanical impedance is time
varying.
7. A method for controlling chatter in a machine tool assembly
during a machining operation, the machine tool assembly having a
plurality of components, the plurality of components having a mechanical
impedance, the method comprising the step of modulating the mechanical
impedance of at least one of the plurality of components to mitigate
regenerative chatter vibrations, wherein the step of modulating
the mechanical impedance further comprises the step of using a chuck
and means for modulating the resonant frequency of the plurality
of components, the means having a variable compliance.
8. The method of claim 7 wherein the means is an electro-rheological
fluid.
9. The method of claim 7 wherein the means is a magneto-rheological
fluid.
10. The method of claim 7 wherein the means is a metal containing
a shape memory alloy, the metal having an elastic modulus that changes
with phase.
11. The method of claim 10 wherein the step of modulating the
mechanical impedance further comprises the step of modulating the
elastic modulus of the metal.
12. The method of claim 7 wherein the chuck has an annulus adapted
to receive a material having a variable compliance.
13. The method of claim 7 wherein the at least one of the plurality
of components comprises a cutting tool, and wherein at least one
of the plurality of components is a machine tool support adapted
to receive the cutting tool, wherein the step of modulating the
mechanical impedance further comprises the step of providing a signal
to the cutting tool, the signal having a square wave form.
14. The method of claim 7 wherein the at least one of the plurality
of components comprises a cutting tool, and wherein at least one
of the plurality of components is a machine tool support adapted
to receive the cutting tool, wherein the step of modulating the
mechanical impedance further comprises the step of subjecting the
cutting tool to a field, the cutting tool having at least one blade,
the field being characterized by a sinusoidal form.
15. The method of claim 7 wherein the at least one of the plurality
of components comprises a cutting tool, and wherein the step of
modulating the mechanical impedance further comprises the step of
providing a random signal to the cutting tool.
16. A method for controlling chatter in a machine tool assembly
during a machining operation, the machine tool assembly having a
plurality of components, the plurality of components having a mechanical
impedance, the method comprising the step of modulating the mechanical
impedance of at least one of the plurality of components, wherein
the step of modulating the mechanical impedance further comprises
modulating the resonant frequency of the plurality of components
to mitigate regenerative chatter vibrations, wherein the at least
one of the plurality of components comprises a cutting tool, and
wherein at least one of the plurality of components is a machine
tool support adapted to receive the cutting tool, wherein the step
of modulating the mechanical impedance further comprises the step
of providing a signal to the cutting tool, the signal having a square
wave form.
17. A method for controlling chatter in a machine tool assembly
during a machining operation, the machine tool assembly having a
plurality of components, the plurality of components having a mechanical
impedance, the method comprising the step of modulating the mechanical
impedance of at least one of the plurality of components, wherein
the step of modulating the mechanical impedance further comprises
modulating the resonant frequency of the plurality of components
to mitigate regenerative chatter vibrations, wherein the at least
one of the plurality of components comprises a cutting tool, and
wherein at least one of the plurality of components is a machine
tool support adapted to receive the cutting tool, wherein the step
of modulating the mechanical impedance further comprises the step
of subjecting the cutting tool to a field, the cutting tool having
at least one blade, the field being characterized by a sinusoidal
form.
18. A method for controlling chatter in a machine tool assembly
during a machining operation, the machine tool assembly having a
plurality of components, the plurality of components having a mechanical
impedance, the method comprising the step of modulating the mechanical
impedance of at least one of the plurality of components, wherein
the step of modulating the mechanical impedance further comprises
modulating the resonant frequency of the plurality of components
to mitigate regenerative chatter vibrations, wherein the at least
one of the plurality of components comprises a cutting tool, wherein
the step of modulating the mechanical impedance further comprises
the step of providing a random signal to the cutting tool.
Machine tools description
BACKGROUND OF THE INVENTION
The present invention relates generally to the field of machine
tools. More specifically, the present invention relates generally
to methods of and apparatuses for suppressing or preventing chatter
in a machine tool assembly by modulating the mechanical impedance
(time-varying impedance) of a component of the machine tool assembly.
The literature discussing regenerative chatter associated with
metal cutting is extensive. Chatter is a regenerative instability
associated with the playback of irregularities on the workpiece
(machined part) from previous cuts to the machine (cutting) tool.
FIG. 1A is a representative depiction of the chatter associated
with machine tool assembly 400 and operation thereof, which, as
illustrated, can be an undesirable nuisance to one's ears. Encountered
in many types of metal removal processes, chatter is a dangerous
condition that results from the interaction of the cutting dynamics
with the modal characteristics of the machine-workpiece assembly.
Machine tool vibrations are recorded on the surface of the workpiece
during metal removal, imposing a waviness that alters the chip thickness
during subsequent cutting passes. Deviations from the nominal chip
thickness effect changes in the cutting force which, under certain
conditions, can further excite vibrations. The costs of chatter
include the variability of end product, waste of parts, tool breakage
or wear, and reduced production.
The mechanical process appears to be well understood. (See, e.g.,
H. E. Merritt , "Theory of Self-Excited Machine Tool Chatter",
Journal of Engineering for Industry , Vol. 87 pp. 447-453 November
1965; R. A. Thompson, "On the Doubly Regenerative Stability
of a Grinder: The Theory of Chatter Growth ", Journal of Engineering
for Industry , Vol. 108 pp. 75-82 May 1986 ; and J. Tlusty and
F. Ismail, "Special Aspects of chatter in Milling", Journal
of Vibration, Acoustics, Stress, and Reliability in Design Vol.
105 pp. 24-32 May 1986.) The physics of the problem can be understood
qualitatively through analogy with the old Edison wax phonographs
as depicted in FIG. 1; 85 the playing and recording of a wax phonograph
record 15 is analogous to the machine tool chatter process. Referring
to FIG. 1 as the track moves under the needle 10 undulations within
the groove 5 (dotted line) excite vibration in a speaker and sound
is conveyed into the air. Similarly, sounds in the air are conveyed
through the speaker and needle 10 into the wax of the record 15.
Any mechanical oscillations (resonances) in the mechanical system
will also be recorded on the record 15. If mechanical resonances
are inadvertently recorded, then playback of the record 15 could
further excite those resonances and the large signal will be re-recorded
into the wax. This regenerative process is referred to as chatter.
An illustration of this effect in milling is shown in FIG. 2. The
kinematics of the cutting process 200 illustrates the manner in
which surface texture due to tool oscillations on previous cuts
contributes to the forces on the cutting tool 410. Oscillation or
vibration of cutting tool 410 imposes a waviness (represented by
lines 230 and 235) on the surface of workpiece 210 during each pass
of the cutting tooth 220. (The ideal cut is represented by dotted
lines 260 and 265.) The waviness (lines 230 and 235) imposed then
excites tool vibration during the subsequent tool pass by altering
the nominal chip thickness and the resultant cutting forces. Surface
waviness can further excite vibrations by altering the instantaneous
chip thickness, h (where h=h.sub.nominal +z(t)-z(t-T) (see FIG.
2). The stability of cutting conditions depends on the interaction
of the current and past vibrations.
Additionally, chatter avoidance and suppression have been extensively
considered. (See, e.g., S. K. Choudhury and M. S. Sharath, "On
Line Control of Machine Tool Vibration During Turning Operation,"
Journal of Material Processing Technology , Vol. 47 pp. 251-259
1995; S. Smith and T. Delio, "Sensor Based Chatter Detection
and Avoidance by Spindle Speed Selection", Journal of Dynamic
Systems, Measurement, and Control, Vol. 114 pp. 486-492 1992;
and S. G. Tewani, B. L. Walcott, and K. E. Rouch, "Cutting
Process Stability of A Boring Bar with Active Dynamic Absorber",
Proceedings of the 13th Biennial Conference on Mechanical Vibration
and Noise, DE Vol. 37 pp. 205-213 1991.) The primary approaches
are suggested by the classical stability chart for turning processes,
which is defined by the dynamic stiffness of the cutting tool. A
nominal stability chart 300 is shown in FIG. 3 which displays a
cutting stability boundary (represented by curve 310) as a function
of the cutting-tool spindle speed (rpm) and axial depth-of-cut (mm).
In FIG. 3 all cutting processes that fall below curve 310 represent
stable cutting conditions. Conversely, all cutting processes that
fall above the curve 310 represent unstable cutting conditions (regenerative
chatter). Several techniques have been explored in the past for
maintaining cutting stability including:
(1) Small depth-of-cut. Taking such shallow cuts that the process
stays within the stability domain (see FIG. 3) regardless of cutting
speed;
(2) Speed selection. Adjusting the depth-of-cut for a particular
cutting speed to stay below envelope of stability curve 310;
(3) Low tool speed. Adjusting the speed to remain within stability
lobes for a given depth of cut (the CRACK method is a version of
this technique (See S. Smith and T. Delio, "Sensor Based Chatter
Detection and Avoidance by Spindle Speed Selection", Journal
of Dynamic Systems, Measurement, and Control, Vol. 114 pp. 486-492
1992.)); and
(4) High Stiffness/Damping. Maximizing the dynamic stiffness of
the cutting machine and the part support through design, which results
in raising the stability curve in the "depth-of-cut" direction
and increasing the number of stable process conditions.
Another method, not suggested by nominal stability chart 300 of
FIG. 3 is that of perturbing the cutting speed about the nominal
speed to disrupt the modulation of current and previous tool vibrations
that can lead to chatter. (See K. Jemelniak and A. Widota, "Suppression
of Self-Excited Vibration by Spindle Speed Variation Method",
International Journal of Machine Tool Design and Research",
Vol. 24 No.3 pp. 207-214 1984.) This process can be summarized
by a horizontal line segment 320 on nominal stability chart 300
with the cutting speed moving cyclically from side to side.
BRIEF SUMMARY OF THE INVENTION
The present invention comprises a method for controlling chatter
in a machine tool assembly during a machining operation, the machine
tool assembly having a plurality of components, the plurality of
components having a mechanical impedance, the method involving the
step of modulating the mechanical impedance of at least one of the
plurality of components. The plurality of components include a cutting
tool, a machine tool support, a workpiece, or any component of the
machine tool assembly. The present invention also comprises an apparatus
for controlling chatter in the machine tool assembly during a machining
operation. In the apparatus, the machine tool support is adapted
to receive a cutting tool, the machine tool support has a resonant
frequency; and the apparatus also includes means for modulating
the resonant frequency of the machine tool support and/or the cutting
tool. The machine tool support is specially adapted to contain a
means for modulating the resonant frequency of the machine tool
assembly. Chatter therein is suppressed by periodically or continuously
varying the stiffness of the machine tool (or some component of
the machine tool), and hence, the resonant frequency of the cutting
tool (or some component thereof). Results from a simulated milling
process reveal that significant reductions in vibration amplitude
can be achieved through proper selection of varying means and excitation
frequency.
The novel features of the present invention will become apparent
to those of skill in the art upon examination of the following detailed
description of the invention or can be learned by practice of the
present invention. It should be understood, however, that the detailed
description of the invention and the specific examples presented,
while indicating certain non-limiting embodiments of the present
invention, are provided for illustration purposes only because various
changes and modifications within the spirit and scope of the invention
will become apparent to those of skill in the art from the detailed
description of the invention and claims that follow.
BRIEF DESCRIPTION OF THE DRAWINGS
The accompanying figures, in which like reference numerals refer
to identical or functionally-similar elements throughout the separate
views and which are incorporated in and form part of the specification,
further illustrate the present invention and, together with the
detailed description of the invention, serve to explain the principles
of the present invention.
FIG. 1 is a perspective view of a Edison-type phonograph system
for use as an analogy in understanding the present invention.
FIG. 1A is a perspective view of an exemplary machine tool assembly.
FIG. 2 is a diagram of the kinematics of the cutting process illustrating
the manner in which surface texture due to tool oscillation on previous
cuts contributes to the forces on the machine (cutting) tool.
FIG. 3 is a graph of a nominal stability chart illustrating stable
and unstable cutting condition regions as a function of the cutting
tool spindle speed (rpm) and axial depth-of-cut.
FIG. 4 is a perspective view of an embodiment in accordance with
the present invention in which an electric field is applied.
FIG. 5 is an exploded view of a portion of the embodiment illustrated
in FIG. 4 in accordance with the present invention.
FIG. 5A is an exploded view of a portion of an alternate embodiment
(magneto-rheological fluid versus electro-rheological fluid as in
FIG. 5) in accordance with the present invention.
FIG. 6 is a graph of the Bingham fluid and the Newtonian fluid
with shear stress .tau. and shear strain rate .gamma..
FIG. 7 is a sectional view of a cutting tool and cutting tool support
(spindle) taken along view line 7--7 in FIG. 4.
FIG. 8 is a diagram of the shear flow and plug flow regions within
a slit.
FIG. 9 is a graph of dimensionless force versus dimensionless lateral
relative velocity of an inner cylinder with respect to an outer
cylinder.
FIG. 10 is a cutting tool vibration history graph of the normal
displacement (mm) of the tool as a function of time (secs) for the
case of no applied electric field, which exhibits large amplitude
chatter.
FIG. 11 is a cutting tool vibration history graph of the normal
displacement (mm) of the cutting tool as a function of time (secs)
for the case of a steadily-applied electric field, which exhibits
lower amplitude chatter.
FIG. 12 is a cutting tool vibration history graph of the normal
displacement (mm) of the cutting tool as a function of time (secs)
for the case of an applied electric field turned on for every-other
cutting tool passing, which exhibits even lower amplitude chatter.
FIG. 13 is a cutting tool vibration history graph of the normal
displacement (mm) of the cutting tool as a function of time (secs)
for the case of an oscillating applied electric field, which exhibits
very little amplitude chatter (almost entirely suppressed).
DETAILED DESCRIPTION OF THE INVENTION
Referring to FIG. 4 the present invention controls or suppresses
the occurrence of chatter in a machine tool assembly 400 by modulating
the mechanical impedance (time-varying impedance) of a component
of machine tool assembly 400 during the machining process. The component
can be either a cutting tool 410 any portion of a machine tool
support (spindle) 420 which holds cutting tool 410 workpiece 220
(machined part), a workpiece support 205 or any component of machine
tool assembly 400 that is mechanically connected to the assembly
400. For convenience in introducing the invention only, however,
the discussion of the present invention will utilize cutting tool
410 as the subject to be modulated. In the art, and as used herein,
spindle assembly is understood to include the spindle (machine tool
support 420) and the chuck 425 which can be provided as either
as one component or as two or more components. Additionally, although
the present invention is demonstrated herein using cutting-type
tools as the machine tools to introduce the invention, it should
be understood that "machine tools" can include cutting
tools, grinding tools, lathes, abrasion tools (any removal of a
substance through abrasion), planing tools, etc. The present invention
can be applied to any type of rotational device machine tool that
contacts or operates on a workpiece.
As discussed above, chatter is a regenerative instability associated
with the playback of irregularities on workpiece 220 from previous
cuts to cutting tool 410. Because this process requires the correlation
between the resonance of machine tool assembly 400 and the played
back irregularities, this process can be suppressed by continuously
varying the stiffness of cutting tool 410 (or some component of
or attached to machine tool assembly 400), and hence the resonant
frequency of the machine tool assembly 400. The varying of resonant
frequency of machine tool assembly 400 an be accomplished by modulating
the stiffness of cutting tool 410 machine tool support 420 workpiece
210 workpiece support 205 or any other component of or in contact
with the machine tool assembly 400. For convenience only, however,
the discussion of the present invention will utilize cutting tool
410 as the subject to be modulated. The method of the present invention
draws on the logic of the spindle speed variation approach described
above in the Background of the Invention. In the present invention,
instead of continuously changing the speed of the spindle assembly
(i.e., machine tool support 420 and chuck 425), however, the present
invention continuously changes the resonant frequency of machine
tool assembly 400. Stiffness modulation of machine tool assembly
400 e.g., cutting tool 410 (i.e., modulating the mechanical impedance
of cutting tool 410), can be accomplished by any means having variable
compliance, including but not limited to, the class generally known
as "smart" materials (ones whose flow or deformation compliance
can be changed very rapidly), such as electro-rheological fluids,
magneto-rheological fluids, shape memory alloys, metals or composites
containing shape memory alloys (the metal or composite having an
elastic modulus that changes with phase, which is influenced by
several factors including temperature), piezo electric crystals
to make or break mechanical contact depending on actuation, or actuators
that modulate the extension and contact of systems of springs or
other elastic structures.
One embodiment for modulated impedance (time-varying impedance)
of machine tool assembly 400 which exploits the modulated flow properties
of electro-rheological fluids, is shown in FIG. 4. The machine tool
support 420 is mechanically connected to a milling machine 490.
In FIG. 4 cutting tool and machine tool support 420 are coupled;
cutting tool 410 is received by machine tool support 420. A magnified
view (500 in FIG. 5) (interface region in dotted-line box) of an
annulus 430 formed between cutting tool 410 and machine tool chuck
425 is shown in FIG. 5. By way of example only, chuck 425 is fabricated
by extending a standard/conventional chuck 425 as is often done
to increase the stiffness of cutting tool 410 (See K. Z. Y. Yen
and W. C. Hsueh, "A Study of Regenerative Chatter Vibration
in Turning Workpiece," Second International Conference on Motion
and Vibration Control, Yokohama, pp. 259-264 1994.). However, a
difference in the present invention is that the extended domain
of chuck 425 is bored out to create annulus 430 between cutting
tool 410 and machine tool chuck 425. In the configuration shown
in FIGS. 4 and 5 annulus 430 is filled with electro-rheological
fluid 434 and a perforated frustrumal shell electrode 432. The electrodes
432 are charged through slip rings 422 on the outside of tool support
420 to create an electric field (electrical contact is effected
by contact 424 and slip rings 422). Altering the electrical field
applied to the fluid induces a change in the variable compliance
of the fluid, and thus the natural frequencies of the system. A
cyclic variation of the field causes a side to side shift of the
stability lobes as the process parameters are held constant.
To understand the significance of the present invention, one must
examine some of the properties of electro-rheological fluids. These
are fluids that are mildly shear thinning, viscous materials in
the absence of an electrical field, but behave similarly to a Bingham
fluid in the presence of electric fields. In the presence of an
electric field, these materials appear to have a yield stress before
flow and an elevated viscosity during flow. In the analysis performed
for the present invention, the electro-rheological fluid--27% zeolite
in silicon oil--is approximated as a Bingham fluid having a distinct
yield stress (.tau..sub.0) and a perfectly Newtonian viscosity (.eta.)
in the presence of an electric field as shown in FIG. 6. FIG. 6
is a graph of the Bingham fluid and the Newtonian fluid with shear
stress .tau. versus shear strain rate .gamma.. The difference between
the viscosities in the presence and the absence of electric field
is ignored. Magneto-rheological fluids behave in much the same manner
in response to applied magnetic fields and can be used in the present
invention. Referring to FIG. 5A, an alternate embodiment is shown
in which magneto-rheological fluid is used instead of electro-rheological
fluid as in FIG. 5 and configured accordingly (e.g., electrodes
are not necessary for the alternate embodiment). The general principles
regarding the chuck 425 design are applicable in this alternate
embodiment. A magnetic field 510 is created by charging wires 515
that circumferentially surround the cutting tool 410 to change the
compliance of the fluid, and thus the frequencies of the system.
Referring to FIGS. 4 and 5 charging slip rings 422 creates an
electric field through electro-rheological fluid 434 increases
the flow resistance through annulus 430 and enhances both mechanical
energy dissipation and mechanical coupling between machine tool
support 420 chuck 425 and cutting tool 410. That increased coupling
corresponds to an increased natural frequency to the system. The
strategy proposed by the present invention is to modulate the electric
field in annulus 430 between each cutting tool 410 passing. Thus,
the oscillations recorded in the part by one tooth 220 passing are
encountered by tool 410 with a different resonant frequency during
the next tooth 220 passing.
The resultant system is modeled assuming that the length of annulus
430 is large compared to the radius of annulus 430 and that the
radius of the annulus 430 is large compared to the gap dimension.
Under those assumptions, the flow within annulus 430 resulting from
the relative motion V between the cylinders can be assumed to be
entirely in the circumferential directions (flow is volume conserving)
and the lubrication approximation can be invoked. With the lubrication
approximation, the flow-rate to pressure relationship is approximated
by that of flow through a slit 710 (corresponding to gap width 2H
in FIG. 8) as shown in FIG. 7. The relative displacement of the
centers of the cylinders is .DELTA.y as shown in FIG. 7. The relative
velocity of the two cylinders V determines the flow rate Q per unit
depth through each cross section through conservation of volume
according to Eq. (1).
Flow of a Bingham fluid through a slit involves two shear flow
domains, 80 and 82 and a plug flow domain 85 as shown in FIG. 8.
The solution of the flow rate/pressure relationship for such problems
requires identification of the two shear flow 80 and 82 domains
and plug flow domain 85 within the slit, which is accomplished with
the aid of the following Eq. (2) (See R. B. Bird, R. C. Armstrong,
and O. Hassager, Dynamics of Polymer Liquids, VI, John Wiley and
Sons, p. 229 1987.):
where Q*=Q/Q.sub.0 ##EQU1## 2H is the slit thickness as shown
in FIG. 8 (2H=(R.sub.2 -R.sub.1)+.DELTA.y cos .theta.), and 2b is
the thickness of the plug flow region 85 as shown in FIG. 8. Once
the flow domains are determined, the pressure gradient .gradient.p
is found from: ##EQU2##
Having the pressure gradient around the annulus as a function of
relative velocity, V, one integrates to find the pressure and integrates
the vertical component of traction to obtain the net force, F, between
the two cylinders. One integrates non-dimensional pressures to achieve
non-dimensional force-velocity plots as shown in FIG. 9. This is
all performed numerically, summarized in FIG. 9 which shows a chart
of dimensionless force, ##EQU3## versus dimensionless relative velocity
##EQU4##
For the purpose of our analysis, we approximate the above curve
of FIG. 9 by Eq. (4):
where m=-75.36 and b=-11.9. For the Newtonian fluid we set b=0.
Determination of Fluid Forces on Cutting Tool 410
For simplicity, the machine tool support 420 is assumed rigid (single
mode of deformation) in the following calculations (employing Lagrange
analysis). We also consider a single cantilevered deformation mode
of the tool in each of the lateral directions. Letting the axial
spatial coordinate be z and the lateral spatial coordinates be x
and y, the velocity of cutting tool 410 is:
where i and j are unit vectors along the x and y axes as defined
in FIG. 2. We assume symmetry of cutting tool 410 so that:
The magnitude of the velocity along cutting tool 410 is given by
Eq. (7):
in which
Substituting these kinematics into the non-dimensional force given
in Eq. (4), we have an expression for the magnitude of force per
unit length of cutting tool 410: ##EQU5## where any flow along the
axial direction has been neglected. The vector force per unit length
acts opposite the instantaneous velocity with the x and y components
given by Eq. (10). ##EQU6##
In formulating the Lagrange equations for motion of cutting tool
410 we compute the generalized forces associated with the resisting
force of the Bingham fluid by assuming that the spindle assembly,
specifically chuck 425 envelopes the entire cutting tool 410 from
the point of cantilevered attachment to the end of annulus 430 (end
of chuck 425). Integrating over this effective tool length L yields
the generalized forces according to Eq. (11):
In our calculations, we use the first vibrational mode shape for
a circular cantilevered Euler-Bernouli beam, given by Eq. (12):
##EQU7## in which .rho. is the bar mass density and .gamma.=1.875
is the first solution to the characteristic Eq. (13):
Simulated Milling Enhancements
The particular values and configurations discussed in this non-limiting
example can be varied and are cited merely to illustrate an embodiment
of the present invention and are not intended to limit the scope
of the invention. In the following example of this embodiment, a
computer simulation of the present invention demonstrated the features
of the present invention.
Incorporating the generalized forces Q.sub.x and Q.sub.y as well
as the cutting and inertial forces into the governing equations
for the Euler Bernoulli beam, we solve for the generalized degrees
of freedom q.sub.x (t), and q.sub.y (t). The cutting forces are
assumed to be proportional to the instantaneous depth of cut, which
is influenced by current and previous vibrations. (See S. Smith
and J. Tlusty, "An Overview of Modeling and Simulation of the
Milling Process," Journal of Engineering for Industry, Vol.
113 pp. 169-175 May 1991). The corresponding metal removal is
recorded and exploited as a boundary condition in subsequent tool
passes.
The above formulation was implemented in a computer code and used
to simulate a slotting cut made with the paradigm system defined
in the following Table 1 (Cutting tool 410 System Parameters) (see
also FIG. 7):
TABLE 1 ______________________________________ Cutting tool 410
System Parameters ______________________________________ Tool Metal
Steel Length, L 12 in Inner Bar Radius, R.sub.1 1 in Gap Size, R.sub.2
-R.sub.1 1 mm Number of Flutes 4 Part Metal Aluminum Electro-Rheological
Fluid 27% Zeolite in Silicon Oil Viscosity, .eta. 0.1 Pa-s Yield
Stress at 0 V, .tau..sub.0 0 Pa Yield Stress at 2.5 kV, .tau..sub.0
500 Pa Revolution Speed 375 rpm Depth of Cut 4 mm Chip Loading 0.3
mm/tooth ______________________________________
(For a discussion of Zeolite properties as electro-rheological
Fluid, see A. Kollias and A. D. Dimarogonas, "Properties of
Zeolite- and Cornstarch-Based electro-rheological Fluids at High
Shear Strain Rates", Advances in Intelligent material Systems
and Structures, V2: Advances in Electro-rheological Fluids, edited
by Melvin A. Kohudic, Technomic Publishing Co., pp. 86-93 1994.)
The first case considered is that in which no electric field is
imposed. The compliance of cutting tool 410 is fairly large and
a characteristic chatter response (according to the parameters of
Table 1) evolves as illustrated in FIG. 10. The normal displacement
(in the feed direction) of the tool tip shows maximum displacements
on the order of 0.4-mm. The static component of cutting tool 410
deflection is due to a steady state normal force pushing cutting
tool 410 against workpiece 210.
The next case considered is that of the imposition of a steady
state electric field of 2.5-kV. The vibration amplitude decreases
as the mechanical impedance of the cutting system is increased as
shown in FIG. 11 (according to the parameters of Table 1). However,
this reduced level of vibration is still caused by regenerative
chatter.
The next case considered is that of a modulated electric field
and the resulting modulated impedance of cutting tool 410 which
is particularly interesting. In this case, the mechanical impedance
is changed as a step function for each tool passing by alternating
the electric field applied to fluid 434 between 0 and 2.5 kV. As
shown in FIG. 12 (according to the parameters of Table 1), the amplitude
of chatter is significantly reduced, though there does seem to be
a visible inception of oscillation each time the field is turned
off. It is important to note that the resultant tip displacement
is visibly less than that of the case of the continuously-stiff
structure.
The next case considered is that of an electric field applied sinusoidally
so that it achieves its maximum of 2.5-kV on every-other tool passing
and hits zero at the center of the remaining tool passings. Because
the tool frequency changes throughout each tooth 220 pass, the waviness
imposed on the workpiece 210 as a result of tool vibrations is more
random in nature. Consequently, the regeneration of waviness is
entirely suppressed and the chatter vibration for this case have
been eliminated as shown in FIG. 13 (according to the parameters
of Table 1).
In summary, the present invention presents a strategy for mitigating
regenerative chatter vibrations in machine tool assemblies. The
method of the present invention is based on altering the modal characteristics
of cutting tool 410 to disrupt the regeneration of waviness on the
surface of workpiece 210. An exemplary embodiment of the variable-impedance
machine tool was presented using an electro-rheological fluid to
influence the coupling of the tool to the sleeve. The viability
of the present invention was demonstrated using computer simulations
of a milling process with a variable-impedance machine tool. Results
indicate that significant reductions in chatter vibrations can be
achieved by varying the electric field applied to the fluid during
each pass of a tooth of cutting tool 410. Although a step variance
in the electric field proved moderately successful at reducing chatter
vibrations, best results were obtained through a sinusoidally-varying
field with a the half-period of oscillation equal to the time interval
between cutting passes. Success was demonstrated using only modest
changes in the mechanical impedance of tool 410 which can be achieved
using commercially-available technology.
Other variations and modifications of the present invention will
be apparent to those of skill in the art, and it is the intent of
the appended claims that such variations and modifications be covered.
For example, it is contemplated that hardware implementation of
the present invention could involve a magneto-rheological fluid
(as shown in FIG. 5A) instead of the electro-rheological fluid discussed
herein. The particular values and discussed above can be varied
and are cited merely to illustrate a particular embodiment of the
present invention and are not intended to limit the scope of the
invention. It is contemplated that the use of the present invention
can involve components having different characteristics as long
as the principle, the presentation of a method and apparatus for
suppressing vibration in machine tools by modulating impedance,
is followed. It is intended that the scope of the present invention
be defined by the claims appended hereto. |