Apparatus and method for adaptive suppression of vibrations in mechanical systems

A method and apparatus are provided for stabilizing at least one element of a mechanical system against echo-like responses to mechanical disturbances. One or more error signals are provided by sensing the motion of the system at one or more points. At least one adaptive filter is operated in response to the error signal or signals, and in response to at least one non-advanced reference signal that is directly related to some motion of the system. The adaptive filter produces a corrective signal for driving a mechanical actuator, thereby to apply to the element a stabilizing generalized force. By non-advanced is meant that there is a zero or negative time-delay between the presence of a given signal at the reference-sensing location and the arrival of the same, or a similar, signal at the error-sensing location. In particular embodiments of the invention, the reference-sensing location is the same as the error-sensing location.

FIELD OF THE INVENTION 
This invention relates to active control techniques for suppressing 
vibrations in mechanical systems. Specific embodiments of the invention 
relate to the suppression of toolpiece vibrations during the machining of 
rotating workpieces. 
ART BACKGROUND 
Unwanted mechanical vibrations have for many years plagued designers of 
mechanical systems that include moving parts, or that are, in use, liable 
to be mechanically coupled to sources of vibrational noise. Such systems 
include, notably, machines for cutting rotating metal workpieces. Such 
systems further include other machines for the subtractive shaping of 
workpieces, as well as optical instruments and their support frames, 
lithographic and other manufacturing tools and their support frames, 
imaging systems of various kinds and their support frames, and 
self-propelled vehicles. 
In metal-cutting operations, for example, the quality of the surface finish 
that can be achieved on a rotatable workpiece is often limited by the 
propensity of the cutting tool to exhibit chatter, or some other 
vibrational instability. This problem is especially severe in boring 
operations, which require the cutting tool to be mounted at the end of a 
relatively long, cantilever-supported bar. Because structures of this kind 
are rich in troublesome mechanical resonances, chatter has proven to be an 
important limitation for the surface finishes achievable within machined 
articles having cylindrical bores such as engines and projectile 
launchers. 
Real-time signal processing has been applied to the problem of unwanted 
vibrations in mechanical systems. Typically, motion sensors are used to 
generate signals that contain information about the unwanted vibrations. 
These signals are transmitted to digital signal processors, which use the 
transmitted information to generate corrective signals for driving 
electromechanical actuators. These actuators, in turn, produce responses 
in the mechanical system that tend to oppose the unwanted vibrations. 
Modem Control Theory is one well-known technique that is applied in the 
course of digital signal processing in order to generate corrective 
signals for active vibration control. Very briefly, Modern Control Theory 
(MCT) involves generating corrective actuator drive signals from linear 
combinations of the sensor signals, scaled in magnitude by fixed 
real-valued coefficients. Thus, the corrective drive signals are nearly 
instantaneous representations of the state of the error-sensor output. 
This leads to a wideband feedback-control system. Stated briefly, MCT is a 
multi-dimensional extension of single-sensor, single-actuator feedback 
control. 
For example, MCT is applied in an active control device for machine-tool 
elements described in U.S. Pat. No. 5,170,103, which issued on Dec. 8, 
1992 to K. E. Rouch et al. (hereinafter, the "Rouch patent"). This device 
includes a sensor for producing, respectively, boring-bar displacement and 
velocity signals, a reaction mass mounted near the free end of the boring 
bar, a sensor for producing, respectively, reaction-mass displacement and 
velocity signals, and an actuator for displacing the reaction mass in such 
a manner as to counteract the undesirable vibrations of the boring bar. In 
a signal processor, the two velocity signals and the two displacement 
signals are scaled and combined according to methods of MCI to generate a 
corrective signal. 
Through various applications of Modern Control Theory, practitioners in the 
art have achieved significant advances in the suppression of unwanted 
vibrations. However, there remain certain sources of vibration in, e.g., 
machining operations that have hitherto not been entirely suppressed by 
these methods. 
SUMMARY OF THE INVENTION 
In a broad aspect, the invention involves a method and apparatus for 
stabilizing at least one element of a mechanical system against echo-like 
responses to mechanical disturbances. One or more error signals are 
provided by sensing the motion of the system at one or more points. At 
least one adaptive filter is operated in response to the error signal or 
signals, and in response to at least one non-advanced reference signal 
that is directly related to some motion of the system. The adaptive filter 
produces a corrective signal for driving a mechanical actuator, thereby to 
apply to the element a stabilizing generalized force. By non-advanced is 
meant that there is a zero or negative time-delay between the presence of 
a given signal at the reference-sensing location and the arrival of the 
same, or a similar, signal at the error-sensing location. In particular 
embodiments of the invention, the reference-sensing location is the same 
as the error-sensing location.

DETAILED DESCRIPTION 
A. Glossary of Terms 
As used herein, each of the following terms has the meaning described 
below: 
An adaptive filter is a time-varying, self-adjusting, digital signal 
processing device for controlling the performance of a system. This device 
acts upon an input signal (sometimes referred to as a reference signal) 
and produces an output signal. The system performance depends, at least in 
part, on this output signal. The filter automatically optimizes its 
processing of the input signal (i.e., it adapts) in order to minimize the 
difference between the actual and desired system performance. 
A specific type of adaptive filter, referred to as a transversal filter, 
processes the input signal by linearly combining sequential time-samples 
of the input signal at various fixed delays, with respective variable 
weights. 
An echo-like response to a mechanical disturbance of a system means a 
response that exhibits a detectable self-correlation at one or more time 
delays, where the self-correlation is independent of the waveform of the 
original mechanical disturbance, and is instead a consequence of temporal 
correlation in the impulse response of the system itself. 
A generalized force is a force, pseudoforce, torque, or bending moment 
generated by any means, including a reaction mass or an intrastructural 
mechanical actuator. 
Advanced reference signal refers to a reference signal for an adaptive 
filter in an adaptive regulator loop. This is best understood with 
reference to FIGS. 1 and 2. 
FIG. 1 illustrates, in a broad sense, the use of controller 1.1 to produce 
a corrective signal which combines in plant 1.2 with disturbance noise n 
in such a way as to reduce the resulting error signal e. As shown, plant 
1.2 comprises disturbance path 1.3 and actuator path 1.4. Absent a signal 
input to plant 1.2, the output displacement is the error signal e (for a 
specific spatial location on the physical plant). This error signal 
represents the difference between the noise-only displacement response d 
and the actuator-only displacement response y. That is, e=d-y. In the case 
of a plant that is linear, responses d and y combine at the physical 
measurement location represented by the summing point 1.5. 
One form of controller 1.1 is an adaptive transversal filter implementing 
the filtered-x least mean square (FXLMS) algorithm, which is well-known to 
those skilled in the art. This form of controller is illustrated in FIG. 
2. (Certain essential features of implementations of the FXLMS algorithm 
have been omitted from the figure to simplify it.) 
It will be appreciated that there are common elements, denoted by similar 
reference numerals, in FIGS. 1 and 2. However, adaptive filter 2.1 has 
been substituted in FIG. 2 for controller 1.1. Moreover, line 2.3 has been 
added, bringing reference signal x from tap point 2.2 to the reference 
input part of the adaptive filter. 
The configuration shown in FIG. 2 is one conventional in the art. The 
reference signal x is advanced in the sense that each segment (in time) of 
signal x is received at filter 2.1 before the corresponding segment of 
error signal e is received at filter 2.1. Signal e arrives after a delay 
due to the latency inherent in the disturbance path 1.3. In conventional 
adaptive regulator configurations, it is considered desirable for signal x 
to be advanced in order to compensate for the combined latency inherent in 
the adaptation process of filter 2.1 and the actuator path 1.4. This makes 
it possible for filter 2.1 to remove broad-band noise from the error 
signal by cancelling noise components that correlate with signal x. 
In practical implementations, tap point 2.2 is advantageously situated at a 
point on a mechanical structure that lies as close as physically possible 
to the entry point of the disturbing force on the structure. For example, 
an error sensor and an actuator may be situated on the roof of a high-rise 
building for stabilizing the sway of the building against earthquake 
loading. In such a case, a useful location for tap point 2.2 would be at 
ground level, where a suitable transducer, such as a seismic 
accelerometer, would provide an electrical reference signal. 
Thus, when a reference signal is said to be an advanced reference signal, 
what is meant is that there is a positive time delay between the presence 
of a given signal at the reference location, and the later arrival of the 
same, or a similar, signal at the error location. Stated another way, if 
an impulsive force were applied to the structure at the entry point of 
disturbance forces, then the reference sensor would respond before the 
error sensor produced an indication of a structural response. 
Non-advanced reference signal is best understood with reference to FIG. 3. 
It will be appreciated that line 2.3 and tap point 2.2 are absent from 
FIG. 3, and instead, the error signal e also functions as the reference 
signal x. This represents a departure from adaptive control methods of the 
prior art, in that the reference signal does not arrive at filter 2.1 in 
advance of the error signal. This is one instance of a non-advanced 
reference signal. 
In practice, a non-advanced reference signal may be taken not only directly 
from the error signal, but also from, as but one example, a tachometer 
which generates a narrowband (typically, sinusoidal) signal or signals 
directly related to shaft rotation frequency in a rotating machine that is 
to be stabilized 
Another example of a non-advanced reference signal is the output of a 
roof-level sensor on a high-rise building that generates a broadband 
signal related to building sway, and acts in concert with an error sensor 
and an actuator situated within the building or near ground level. 
In a general sense, to say that a reference signal is non-advanced means 
that there is a zero or negative time-delay between the presence of a 
given signal at the reference location and the arrival of the same, or a 
similar, signal at the error location. Thus, the reference sensor would 
respond to an impulsive force applied at the disturbance-force entry point 
simultaneously with or after the error sensor responded to the same 
impulsive force. 
Regenerative feedback is best understood with reference to FIG. 4, in which 
disturbance path 1.3 has been expanded to include H(s) (Box 4.4), the 
structural response function assuming infinite-impedance (i.e., 
reflection-free) boundary conditions; A(s)e.sup.sT(s), which is indicated 
in Box 4.1 and represents a structural boundary condition response that 
produces an echo-like response at a frequency-dependent time delay T(s), 
leading to structural resonant dynamics; and .mu.e.sup.-sT, which is 
indicated in Box 4.2 and represents an echo effect that has a fixed time 
delay T which is independent of the structural resonances. (Such a fixed 
time delay may, for example, be the period of a rotating toolpiece in a 
machining system, as discussed below.) It will be understood that .mu. is 
a frequency independent amplitude, A(s) is frequency-dependent amplitude, 
and s is the Laplace-transform frequency variable. In the specific context 
of machining operations, .mu. may be the fractional overlap 
(0.ltoreq..mu..ltoreq.1) between successive cuts. 
As shown in FIG. 4, both Box 4.1 and Box 4.2 are included in respective 
feedback loops that return a portion of the structural noise response to 
summing point 4.3. Both of these loops lead to echo-like responses to the 
disturbance n. However, the resonant feedback represented by Box 4.1 is 
not regenerative feedback according to our meaning of this term. On the 
other hand, the fixed-time-delay feedback represented by Box 4.2 is 
regenerative feedback if a portion of the response of the system that is 
real-valued and of a fixed magnitude adds to the disturbing noise in a 
purely time-delayed (periodic or quasiperiodic) manner. 
Resonant feedback represented by Box 4.1 adds to the disturbing noise via 
the filtering mechanism A(s)e.sup.-sT(s), where A(s) is a complex-valued 
function which, in conjunction with the frequency-dependent time-delay 
term T(s), gives rise to stable resonant dynamics. 
The regenerative feedback loop, on the other hand, can produce an unstable 
output response d. This will occur if the relevant loop gain exceeds unity 
and the phase angle between disturbance noise n and displacement response 
d exceeds 180 degrees. 
Significantly, positive regenerative feedback can readily destabilize a 
resonant system, because the regenerative loop gain tends to be high at 
the resonant frequency. For this reason, the concurrence of a regenerative 
feedback loop 4.2 with a resonant feedback loop 4.1 can produce an 
unstable output response d. (An unstable output response is characterized 
by a continuously-growing magnitude of the output over some significant 
length of time, such as, for example, a time interval that is long 
relative to a resonant period. 
B. Adaptive Control of Echo-Like Mechanical Vibration Phenomena 
The controller configuration depicted in FIG. 3 involves operation of 
adaptive filter 2.1 with a reference signal x that is tapped directly from 
error signal e, and therefore is neither advanced nor delayed relative to 
the error signal. Theoretically, the bandwidth of vibrational frequencies 
over which this configuration is effective will depend upon the degree of 
self-correlation in the disturbance signal d, because the adaptive filter 
is operating effectively only insofar as it is removing self-correlated 
(or resonant) components that it finds in the error signal e. 
This theoretical limitation does not generally apply to the conventional 
use of an adaptive filter with an advanced reference signal. However, it 
should be noted in this regard that adaptive filters structured as in FIG. 
3, with the reference signal tapped directly from the error signal, have 
not hitherto been used for controlling vibrations in mechanical 
structures. One obstacle to such an application is the incorrect 
assumption that an advanced reference signal is required in order to make 
any adaptive FXLMS controller perform effectively. 
By contrast, we have demonstrated that control of structural resonant 
response, as well as control of regenerative feedback effects, can be 
achieved using a single error-sensing location which also serves as the 
reference input to the filter. Thus, we have shown that it is feasible to 
use the control structure of FIG. 3 to solve certain vibration-control 
problems. 
It should be noted in this regard that the control structure of FIG. 3 is 
particularly useful when a time-advanced reference signal is not 
physically attainable, for example in machining operations in which the 
error sensor should be situated as close as practicable to the cutting 
tip. This control structure is also particularly useful where resonant 
dynamics are to be controlled, and a cost savings is provided by the 
single-sensor approach. 
The controller configuration depicted in FIG. 3, and more generally, 
adaptive controller configurations in which a non-advanced reference 
signal is applied to the adaptive filter, are advantageously applied to 
solve the broad class of vibration problems represented in FIG. 4. These 
are problems in which the presence of one or more structural resonances 
(Box 4.1), or the presence of regenerative feedback (Box 4.2), produces 
vibrational instabilities at or near the structural resonances. 
Removal by the adaptive filter of self-correlated components from the error 
signal (where there is one error/reference sensor) or of components 
cross-correlated with the reference signal (where the reference signal is 
non-advanced and from a different sensor) may be understood as removal of 
those signal components that are associated with the time-delayed feedback 
paths 4.1 and 4.2. When, for example, adaptive filter 2.1 (see FIG. 3) is 
well adapted, error signal e will behave approximately as the output of 
the reflection-free structural response function H(s) (Box 4.4 of FIG. 4), 
driven by noise source n, with the echo-path effects (i.e., those due to 
Boxes 4.1 and 4.2) removed or significantly reduced within the controller 
bandwidth BW.sub.con. 
The controller bandwidth can be estimated from the following 
considerations: 
(i) For a natural resonant response of a system to be controllable, the 
total actuator path delay T.sub.DEL should be less than one-half the 
resonant period T.sub.RES ; i.e., T.sub.DEL &lt;1/2T.sub.RES. The delay 
T.sub.DEL includes contributions from computer-sampling delay, 
signal-conditioning delay such as filter delay, and delay in the 
actuators. 
(ii) For a regenerative vibration to be controllable, the actuator path 
delay should be less than a period T.sub.REV of the machine rotation or 
other periodic input of energy that is driving the instability. That is, 
T.sub.DEL &lt;T.sub.REV. 
In view of these considerations, it is evident that in operation, the 
non-advanced reference signal is effectively advanced in time relative to 
a pertinent echo period (characterized by T.sub.REV or T.sub.RES, or, in 
some cases, a multiple thereof). 
Thus, the effect of the vibration controller may be understood as removing 
the natural resonant behavior of the plant when it is excited by a 
finite-bandwidth noise source. Conditionally, the controller may be 
further understood as removing the periodic influence of regenerative 
feedback. This is so on condition that the length of the adaptive filter 
(i.e., the total length of time spanned by the taps of the filter plus any 
intervening circular buffers or other programmed delays in lieu of unused 
taps) is great enough to encompass at least one period T. 
We believe that our vibration controller is useful for reducing vibration 
in a broad range of mechanical structures including, without limitation, 
machinery for cutting, grinding, milling, and drilling metal workpieces, 
optical and electromagnetic projections systems, space frames, bridges, 
other truss or beam structures, rotating propulsive engines, and 
spacecraft antennas. (In reference to the last-named item, we believe that 
our vibration controller will be useful for reducing the well-known 
phenomenon of jitter in spacecraft antennas.) 
A general approach for such applications is illustrated in FIG. 5. Each of 
L actuators 5.1 is driven by a respective adaptive filter 5.2. Each of M 
error sensors 5.3 sends a respective error signal to each of the L 
adaptive filters. For each of the adaptive filters, a respective one of 
the M error sensors provides the reference input for that filter. The 
convergence step of each adaptive filter includes a contribution from each 
of the M error signals. The size of this contribution is related to an 
estimate of the transfer function between the relevant error sensor and 
the relevant actuator. This is explained in greater detail below. 
Various kinds of mechanical motion may be sensed by the error sensors, 
including bending modes (typically of two orthogonal types referred to, 
respectively, as parallel and tangential), torsional modes, axial modes 
(at least in structural members that are significantly compressible in the 
axial direction), and shell modes. Respective ones of the multiple error 
sensors may detect different kinds of motion at the same location, the 
same kind of motion at different locations, or different kinds of motion 
at different locations, or there may be some combination of these various 
schemes. Similarly, the L actuators may be adapted to drive different 
kinds of motion at the same location, the same kind of motion at different 
locations, different kinds of motion at different locations, or some 
combination thereof. 
C. Regenerative Chatter in Machining Operations 
One source of unwanted vibrations in machining operations is regenerative 
feedback that occurs when a past feature of the motion of the toolpiece 
makes a reinforcing contribution to the toolpiece motion at a later time. 
Such a time-delayed, positive feedback mechanism can arise, for example, 
during metal-turning operations in which the current width-of-cut overlaps 
a portion of the cut made during the preceding revolution of the 
workpiece. 
The resulting toolpiece vibrations, which are referred to as "chatter," 
tend to limit the quality of finish that can be obtained on the tooled 
surface of the workpiece. 
An early, theoretical description of this phenomenon was proposed in H. E. 
Merritt, "Theory of Self-Excited Machine-Tool Chatter," Journal of 
Engineering for Industry, (November 1965). In this work, Merritt 
introduced a regenerative feedback coefficient .mu..sub.M based on the 
fractional overlap of cutting width from one workpiece revolution to the 
next. 
The Merritt model is illustrated schematically in FIG. 6. As shown, the 
primary feedback path 10 relates the tool displacement Y.sub.d (s) to the 
instantaneous cutting depth u(s). (It will be understood that the variable 
s is the frequency variable well-known from Laplace-transform techniques.) 
The regenerative feedback path 20 is characterized by the coefficient 
.mu.M and the delay factor e.sup.-sT, which represents a delay by one 
rotational period T. The variable F.sub.c (s), indicated in the figure, 
represents the frequency-domain cutting forces, which are related to the 
instantaneous cutting depth via the cutting stiffness K.sub.c. The tool 
motion is the response to these forces. The cutting-path dynamics G(s) 
relate the tool response to the applied cutting force. These dynamics 
typically will represent tool dynamic properties during the machining of a 
relatively stiff or thick-walled workpiece. 
We have discovered that there are at least two kinds of chatter that are 
driven by regenerative feedback. We refer to these kinds of chatter, 
respectively, as "broadband regenerative chatter" and "narrowband 
regenerative chatter." Significantly, both of these kinds of chatter 
exhibit substantial self-correlations at time delays that are multiples of 
a rotational period. In this sense, they both are echo-like responses to 
mechanical disturbance. Some understanding of regenerative chatter can be 
gained from the power spectra of, for example, tool displacement during 
the machining of a rotating workpiece. In such spectra, both broadband and 
narrowband chatter exhibit fine structure with spectral lines that are 
regularly spaced at increments equal to the rotational frequency. 
Narrowband chatter is typically observed during the machining of relatively 
hard materials such as nickel alloys and titanium, at lower rotational 
speeds. By contrast, broadband chatter is typically observed during the 
machining of relatively soft metals such as aluminum and steel at 
relatively high rotational speeds. However, there is no distinct division 
between a regime of hardness and speed that pertains to narrowband 
chatter, and such a regime that pertains to broadband chatter. 
One distinction between broadband and narrowband chatter is apparent from 
the power spectra mentioned above. A spectrum of broadband chatter will 
exhibit a main peak centered upon a frequency that lies, typically, 
10%-30% above a natural frequency of the boring bar. Such a peak is 
evident at a fundamental frequency of 318.5 Hz in FIG. 7, together with a 
peak near the first harmonic. (It will be understood that each of these 
peaks is a composite of multiple spectral lines as discussed above.) By 
contrast, a spectrum of narrowband chatter will typically exhibit narrower 
peaks centered at one or more resonant frequencies of the tool or 
workpiece. Such a spectrum is provided in FIG. 8. 
The Merritt model has achieved some success in elucidating the mechanisms 
responsible for broadband regenerative chatter. However, no application of 
the techniques of active vibration control has hitherto been able to 
reduce either broadband or narrowband chatter sufficiently to provide the 
quality of surface finish demanded by customers of advanced machining 
operations. 
We have discovered that when relatively hard metals are cut (under 
conditions leading to narrowband chatter), the regenerative loop 20 (see 
FIG. 6) in the disturbance path tends to create an instability in the 
plant at one of the structural resonances (at any given time). We have 
found that the technique of FIG. 3 (exemplarily using the error signal as 
a non-advanced reference signal) is effective for reducing the 
regenerative feedback effect while also reducing the structural resonant 
energy. This is illustrated by the various features of the idealized power 
spectrum of FIG. 9, in which a resonant peak is subdivided into a portion 
600 attributable to the natural resonant response to cutting noise, and a 
portion 610 attributable to regenerative feedback. The controlled 
bandwidth is indicated in the figure as range 620, and the controlled 
response of the mechanical structure is indicated by curve portion 630 and 
amplitude 640. 
We have further discovered that when softer metals are cut at relatively 
high rotational velocities (under conditions leading to broadband 
chatter), loop 20 (see FIG. 1) creates an instability in the plant at a 
collection of frequencies that lie above a free-bar resonant frequency. In 
this instance, we have found that the technique of FIG. 3 will counteract 
the regenerative loop only if the adaptive filter is long enough to span 
at least one rotational period of the workpiece. 
This is illustrated by the various features of the power spectrum of FIG. 
10, in which curve 700 represents the idealized free-bar impact response, 
curve portion 710 represents an unstable cutting operation, and curve 
portion 720 represents a corresponding, controlled cutting operation. 
Range 730 represents the controlled bandwidth. 
D. Illustrative Embodiments 
Our technique differs from the technique of the Rouch patent in that, inter 
alia, we do not apply Modern Control Theory to generate an actuator 
control signal. Instead, as noted above, we use an adaptive transversal 
filter to automatically update the coefficients that characterize a 
corrective signal to be applied to the actuator. We believe that our own 
technique is effective for suppressing echo-like responses to mechanical 
disturbances in many kinds of mechanical systems. In the specific context 
of machining operations, our invention is effective for suppressing both 
broadband and narrowband regenerative chatter. 
By applying well-known computational methods such as the FXLMS algorithm, 
the adaptive filter operates upon an appropriate reference signal to 
generate the corrective signal. Each coefficient specifies the fractional 
contribution, or weight, of a component of the corrective signal that is 
generated by delaying the reference signal by a respective increment. 
(These increments are typically designed or programmed into the filter. By 
analogy to an analog delay line, each increment is often said to relate to 
a respective "tap" of the filter.) The weights are periodically updated in 
such a manner as to drive downward the magnitude of an error signal. 
It is a significant feature of our invention that the adaptive filter 
receives a non-advanced reference signal. In fact, in certain embodiments 
the reference signal and the error signal both correspond substantially to 
the same time-varying descriptor of toolpiece motion, and can, in fact, be 
provided by the same tool-motion sensor. This descriptor is typically 
either the displacement function or the acceleration function of the 
toolpiece. (The acceleration function is the second derivative of the 
displacement function.) 
Embodiments of the invention that use the same sensor to provide both the 
error and reference signals are particularly useful for suppressing 
broadband chatter. In such an application, there is a known correlation 
between current toolpiece deflections caused by regenerative feedback and 
those deflections that will occur one rotational period later. The filter 
tap whose corresponding delay most closely matches the rotational period 
of the workpiece will typically make a substantial contribution to the 
corrective signal. (Taps lying near submultiples of the rotational period, 
i.e., near multiples of the rotational frequency, will also contribute 
significantly to the corrective signal, although their contribution will 
generally be smaller.) In fact, in at least some cases the convergence of 
the filter coefficients (i.e., during adaptation) can be improved by 
augmenting the filter with an optional delay line adjusted to delay the 
reference signal by one rotational period (and thus, in effect, to add one 
rotational period to each of the filter taps). 
We now describe an advantageous embodiment of our invention for the purpose 
of suppressing chatter in machining operations in which a stationary 
toolpiece cuts a rotating metal workpiece. It should be noted that this 
description is illustrative and not limiting. In fact, we believe that our 
invention is advantageously applied for suppressing vibrations in other 
kinds of machining operations, including those in which the workpiece is 
stationary and the toolpiece rotates, as in various milling, drilling, and 
grinding operations. More generally, we believe that our invention is 
advantageously applied for suppressing echo-like responses to mechanical 
disturbances in mechanical systems of many kinds, as noted previously. 
As depicted in FIG. 11, a typical metal-turning installation includes a 
boring bar 30 mounted at one end 35. Mounted at the opposite end of the 
boring bar is a cutting bit 40. The support 45 for the boring bar is 
mounted on a movable carriage 50. By movement of the carriage, the cutting 
bit can be brought into contact with a workpiece 55. Means (not shown) are 
provided for rotating the workpiece with a rotational period T seconds and 
a rotational velocity F Hz, wherein F=1/T. 
Also shown in the figure is an electromechanical actuator 60 for displacing 
the cutting bit in accordance with corrective signals issued from signal 
processor 65 and amplified by amplifier 70. At least one sensor is 
required for sensing the motion of the tool bit or boring bar. 
Two illustrative sensors are shown in the figure. One of these is normal 
accelerometer 75, which senses acceleration of the boring bar, at a point 
near the tool bit, in the direction normal to the workpiece surface (at 
the point of application of the cutting tool). The other of these is 
tangential accelerometer 80, which senses acceleration of the boring bar 
in the direction tangential to the workpiece surface (and normal to the 
long axis of the boring bar). The acceleration signal is readily used 
directly as the descriptor of tool-bit motion. Alternatively, a related 
signal, such as a velocity or displacement signal, can be used as the 
descriptor. We currently prefer to use a displacement signal X(t), which 
is proportional to the displacement of the cutting bit, because this 
signal is directly related to the resulting surface finish. 
If the motion sensor is an accelerometer, it is necessary to twice 
integrate the accelerometer output in order to provide a displacement 
signal X(t). This operation is advantageously performed by signal 
processor 65, as described in greater detail below. 
It will be appreciated that various other mechanical motions of the cutting 
bit and boring bar may be of interest in the application of the methods 
described herein. Such other motions may include, for example, torsion of 
the boring bar, and flexion of the boring bar in the directions normal and 
tangential to the workpiece surface. In addition, it may be advantageous 
to measure any of these motions at locations on the boring bar that are 
removed from the position of the toolpiece. It will be further appreciated 
that although the use of accelerometers is currently preferred, other 
kinds of motion sensors are available, and their use in this context will 
be readily apparent to the skilled practitioner. Such other sensors may 
include, for example, optical sensors and piezoelectric strain gauges. 
Significantly, we have found that normal displacement signals are generally 
more effective for controlling broadband chatter, whereas tangential 
displacement signals are generally more effective for controlling 
narrowband chatter. 
As noted, the output of at least one sensor is provided as input to the 
signal processor. A tachometer 90 is also advantageously provided, and its 
output signal also advantageously provided to the signal processor. The 
purpose of the tachometer is to provide a current reading of the 
rotational velocity F. 
Actuator 60 is exemplarily an electrodynamic shaker. (In such a device, the 
current through a magnetic winding is directly proportional to the force 
imparted to a coil and to a piston attached to the coil. This piston is 
sometimes referred to as a "stinger.") It will be appreciated that other 
kinds of actuator are useful in this context, as will be readily apparent 
to the skilled practitioner. Other such actuators include, for example, 
piezoelectric stacks used as force drivers for inertial actuator masses, 
or for articulated clamps which direct the actuation force through the 
base of the boring bar. 
Significantly, we have found that for controlling broadband chatter, it is 
generally most effective to arrange the actuator such as to produce 
toolpiece displacements normal to the surface of the workpiece. On the 
other hand, for controlling narrowband chatter, we have found that 
tangential displacements of the toolpiece are generally more effective. 
Turning now to FIG. 12, a simple way to provide a corrective signal F.sub.a 
(s) is to feed back the tool displacement signal. For correcting broadband 
chatter (but not, in general, for correcting narrowband chatter), this 
signal is fed back after applying a delay .DELTA. approximately equal to 
the rotational period T. This delay is produced in signal processing 
element 100, which may be an analog delay line, but is preferably a 
digital signal processor having analog-to-digital (A/D) conversion on its 
input end, and digital-to-analog (D/A) conversion on its output end. 
The corrective signal (after being delayed, if appropriate) is amplified in 
inverting amplifier 110 and applied to the actuator (modeled in the figure 
as block 120) to produce a corrective displacement y.sub.a (s). This 
corrective displacement is summed at the cutting bit with the other 
displacements inherent in the cutting system to produce the total 
displacement y.sub.c (s). A sensor, such as accelerometer 75 or 80 of FIG. 
11 (together with an appropriate signal integrator, if required) provides 
displacement signal X(t) which is proportional to the cutting-bit 
displacement of interest. 
The delay .DELTA. and the amplifier gain K are adjusted (manually or 
automatically) to minimize observed chatter of the toolpiece. As noted, 
the optimum value of .DELTA. for this purpose will be equal to the 
rotational period T. 
Although the corrective system of FIG. 12 can afford significant noise 
reduction, still further improvements are achieved with the system of FIG. 
13, which is currently preferred. In this system, the acceleration signal 
X(t) (i.e., the second derivative of the displacement signal), (after A/D 
conversion in box 200), is fed to digital adaptive filter 205 as both 
error signal 210 and reference signal 215. 
As noted above, reference signal 215 is optionally subjected to a time 
delay .DELTA. before it is input to the adaptive filter. This delay is 
exemplarily provided by circular buffer 220. Updated estimates of the 
rotational period T (to which .DELTA. is to be set) are provided to the 
circular buffer by a tachometer after A/D conversion (if required) as 
shown in box 225. (As noted, time-delay element 220 will not generally 
used in a corrective system for narrowband chatter.) 
As discussed above, adaptive filter 205 generates a corrective signal 230, 
which is applied to the actuator after D/A conversion, as shown in box 
235. The "plant," denoted by the symbol "Y" in box 240 of the figure, is 
the transfer function that relates the actual motion of the toolpiece to 
the electrical input to the actuator. Plant estimate Y, which is a 
mathematical model of the plant Y, is advantageously provided, as shown in 
box 245, as a component of the corrective system. The reference signal is 
filtered in box 245 to produce filtered reference signal 250. Signal 250 
and error signal 210 are provided as input for updating the weights of the 
adaptive filter, as represented by box 255. The weights are updated 
according to an algorithm to be described below. 
As shown in the figure, adaptive filter 205, weight-updating unit 255, 
plant estimate 245, optional circular buffer 220, A/D converters 200 and 
225, and D/A converter 235 are included within a functionality 260, which 
is referred to herein as a "digital controller." Although these various 
functions, either individually or in subcombinations, may be provided by 
separate components, it is currently preferred to have these functions 
performed by one or more digital signal processors. Such a processor or 
group of processors is to be identified with digital controller 260. 
As is well known in signal sampling arts, anti-aliasing filters 265 and 270 
are advantageously included to remove artifacts of the sampling process 
from the error signal and tachometer signal, respectively. Reconstruction 
filter 275 is advantageously included to smooth the corrective signal 230 
and to remove digital artifacts introduced during the digital processing 
stage. 
We currently prefer to use the well-known Filtered-X Least Mean Square 
(FXLMS) algorithm for updating the weight coefficients of the adaptive 
filter. This algorithm is described, for example, in B. Widrow and S. D. 
Stearns, Adaptive Signal Processing, Prentice-Hall (1985). Other, more 
computationally intensive algorithms could be used, for example to provide 
faster convergence to optimal weight vectors. However, such algorithms 
would tend to make greater demands on the computational power of the 
digital processor. It is significant in this regard that the number of 
calculations required to operate the adaptive filter tends to increase as 
the square of the number of filter taps. 
According to the FXLMS algorithm, the equation governing the updating of 
the weight coefficients is: 
EQU w.sub.k+1.sup.(i) =.alpha.w.sub.k.sup.(i) +2.mu..sub.filt e.sub.k 
x.sub.k.sup.(i) ; 
wherein w.sub.k+1.sup.(i) is the updated weight vector for the adaptive 
filter, w.sub.k.sup.(i) is the weight vector from the previous sample 
period, .mu..sub.filt is the convergence step size of the adaptive filter, 
e.sub.k is the current sample-period error, and x.sub.k.sup.(i) is the 
reference signal vector after filtering through plant estimate 245. The 
symbol .alpha. represents a so-called leak factor having a positive value 
less than or equal to 1. A typical value of .alpha. used in our 
investigations is 0.9. 
More specifically, the vector X.sub.k (i) is related to the error e.sub.k 
and the plant estimate Y according to: 
EQU x.sub.k.sup.(1) =e.sub.k * Y;x.sub.k.sup.(i) =x.sub.k-1.sup.(i-1). 
The * symbol represents the convolution operation. Conventionally, the 
signal that is convolved with Y is the reference signal, from a distinct 
reference sensor. Instead, we have indicated, here, that Y is to be 
convolved with the signal e.sub.k from the error sensor. 
The index (i) runs from 1 to N, where N is the number of taps of the 
adaptive filter. An exemplary value for N is 1024. We have found that this 
value is effective for achieving wideband frequency rejection in the 
operation of the adaptive filter for controlling regenerative feedback in 
applications where broadband chatter is predominant. 
More generally, N should be large enough to encompass at least one 
rotational period of the workpiece, and preferably encompasses two or more 
rotational periods. 
In the case of multiple filters and multiple actuators, the above-described 
equations are generalized to the following: 
##EQU1## 
Here, L is the number of actuators, M is the number of sensors, the index 
.lambda. ranges from 1 to L, and the index m ranges from 1 to M. The 
quantity Y.sub..lambda.j is the transfer-function estimate between 
actuator .lambda. and sensor j. For each adaptive filter, one error sensor 
serves to provide the reference input. That is the sensor whose output 
(e.sub.k).sub.m is convolved with the transfer-function estimate. 
Generally, some residual error will be present in the tool-displacement 
signal (or, equivalently, in the accelerometer signal) even after the 
filter has adapted and the values of the weight coefficients have 
stabilized. This error represents the noise that is uncorrelated between 
successive workpiece rotations. It is explainable as the uncorrelated 
portion of the response of the cutting system to the cutting of fresh 
material. 
The corrective system described above is optionally augmented by a linear 
regulator feedback loop around adaptive filter 205. Because such a 
feedback loop can compensate natural dynamics of the boring bar, it may 
further improve the surface finish by suppressing linear response noise 
that remains in the error signal. 
It should be noted in this regard that each of the respective feedback 
loops (i.e., the FXLMS loop and the linear regulator loop) will affect the 
plant transfer function of the other. Thus, one or more iterative cycles 
may be required in order to determine stable plant estimates for the 
respective loops. In an exemplary such loop, the adaptive filter is first 
allowed to converge, then a plant estimate is determined for the linear 
regulator loop, and then a new plant estimate is determined for the FXLMS 
loop. 
EXAMPLE 
We performed experimental tests of our controller using the arrangement 
depicted in FIG. 14. Boring bar 300 was secured in clamp 305, which was 
attached, in turn, to a lathe carriage (not shown) driven at a constant 
feedrate by a lathe motor. A ring clamp (not shown) fastened normal shaker 
310 and tangential shaker 315 to the boring bar. At the end of the boring 
bar, as shown, we attached accelerometer 320 for measuring tangential bar 
motion, and accelerometer 325 for measuring normal bar motion. In this 
context, the normal direction is the direction normal to the surface of 
rotating workpiece 327 at the point of application of cutting tip 330, and 
the tangential direction is the direction tangential to the workpiece 
surface and parallel to the workpiece motion at the point of application 
of the cutting tip. It is apparent from FIG. 14 that a third direction, 
the axial direction (i.e., parallel to the longitudinal axis of the boring 
bar) may also be parallel to the workpiece surface. We did not make any 
effort to control deflections of the cutting tip in this axial direction, 
because any chatter that might be attributable to such deflections was far 
outweighed by normal chatter, or tangential chatter, or both. Axial 
control could readily be implemented in structures having a boring bar (or 
other important structural element) exhibiting significant axial 
compressibility. 
Narrowband Chatter Test 
The workpiece was made of Inconel 718. We have found that when cutting this 
or other nickel alloys (using a boring bar of symmetrical cross-section), 
narrowband chatter first emerges as a tangential deflection concentrated 
near the fundamental frequency of the boring bar and harmonics thereof, 
superimposed on the background cutting noise. 
However, as the chatter grows, normal deflections (also concentrated at bar 
resonances) appear. Significantly, it is the normal chatter that more 
directly relates to the quality of the surface finish that is achievable. 
We found that controlling the tangential deflections can be effective for 
reducing first-mode chatter in the normal direction, thereby improving the 
resulting surface finish. 
Our controller implemented the standard, reference-power normalized version 
of the FXLMS algorithm, updating the weights of the adaptive filter once 
each sample period. The reference signal was tapped from the output of the 
error sensor. (In this instance, the error sensor was the tangential 
accelerometer.) 
FIG. 15 is a frequency spectrum of normal chatter magnitude during the 
machining of 718 Inconel (Rockwell Hardness of 38) with the controller off 
and with the controller on. The workpiece rotates at 0.47 Hz, the depth of 
cut is 0.51 mm, and the feedrate is 0.25 mm per revolution. The boring bar 
is steel, with an overhang ratio of 10. Tangential acceleration is used as 
the error signal (without integration which would otherwise convert 
acceleration to, e.g., displacement). The adaptive filter length was 256 
taps, representing a total time of 32 ms at a sample rate of 8 kHz. The 
fundamental chatter frequency, evident near 100 Hz, is the first mode 
frequency of the boring bar. 
FIG. 16 is a frequency spectrum of the corresponding tangential chatter 
magnitude. 
We found that as rotational velocity was increased still further, there 
emerged higher-order chatter, at higher resonant modes of the boring bar. 
We found it desirable, in suppressing chatter at higher than the 
fundamental mode, to control both normal and tangential deflections. We 
found it effective to use independent normal and tangential control loops, 
without cross-coupling between them. FIG. 5 helpfully illustrates our use 
of dual control loops, if, for example, error sensor e.sub.1 is taken as 
the normal error sensor, error sensor e.sub.2 is taken as the tangential 
error sensor, e.sub.1 is connected only to Adaptive Filter 1, e.sub.2 is 
connected only to Adaptive Filter 2, Actuator 1 is a normal actuator, and 
Actuator 2 is a tangential actuator. 
Broadband Chatter Test 
We found that in our tests, normal control was more effective than 
tangential control for reducing broadband chatter. 
FIG. 17 is a frequency spectrum of normal chatter magnitude with the 
controller on and off during the cutting of 4140 steel. The workpiece 
rotates at 5.75 Hz, the depth of cut is 1 mm, and the feedrate is 0.125 mm 
per revolution. The adaptive filter length was 1024 taps, representing 256 
ms at a sample rate of 4 kHz. 
Inertial Actuator 
We achieved qualitatively similar results when an inertial actuator, 
contained within the boring bar, was used in place of the shaker (which is 
mounted external to the boring bar, as shown, e.g., in FIG. 14). The 
positioning of inertial actuator 400 within boring bar 410 is depicted in 
FIG. 18.