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Патент USA US3051399

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Aug. 28, 1962
13 Sheets-Sheet 1
Filed Nov. 7, 1955
Georges G. Fuyo rd
Aug. 28, 1962
Filed Nov. 7, 1955
D1 .
13 Sheets-Sheet 2
Georges G. Fuyurd
M MIMIBW (1,174?»
Aug. 28, 1962
Filed Nov. 7, 1955
13 Sheets-Sheet 3
Georges G. Fuyord _
//M M Magnum/1%
Aug. 28, 1962
13 Sheets-Sheet 4
Filed Nov. 7, 1955
Aug. 28, 1962
Filed Nov. 7, 1955
13 Sheets-Sheet 5
Georges G. Fuyurd
Aug. 28, 1962
Filed Nov. 7, 1955
Georges G. Fayord
Aug. 28, 1962
13 Sheets-Sheet 8
Filed Nov. '7, 1955
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Gseerges G. Foyurd
Aug- 28, 1962
Filed Nov. 7, 1955
13 Sheets-Sheet 9
270 500 550 as lfindcgrees
Georges G. Fcyurd
Aug. 28, 1962
Filed Nov. 7, 1955
13 Sheets-Sheet l0
Georges G. Fuyord
1% MMIBM F7275,’
Aug- 28, 1962
Filed Nov. 7, 1955
13 Sheets-Sheet 11
FIG -18
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Georges G. Foyurd
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Aug. 28, 1962
Filed Nov. 7, 1955
13 Sheets-Sheet 12
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Unite St .tes atnt
Patented Aug. 28, 1962
by a suitable contour and to operate the command mech
anism for the machine tool provided by the invention only
between particular values of the parameter pertaining
Georges G. Fayard, Paris, France, assignor to O?ice
National d’Etudes et de Recherches Aéronautiqnes,
Chatillon-sous-Fagneux, France, a French body cor
Filed Nov. 7, 1955, Ser. No. 545,397
Claims priority, application France Nov. 6, 1954
Claims. (Cl. 235—180)
The present invention relates to automatically con
trolled machines, for example machines for the shaping
to the pro?le desired to be achieved while disabling it for
other values of the parametre. Similarly the invention is
applicable to the machining in part or whole of dies in
tended for stamping of metal sheets such as those used
in automobile bodies. In such applications the inven
tion obviates the use of the elaborate plaster models
hitherto employed with a feeler for machining of dies in
the automobile body industry.
The desired developments of the pro?le coordinates into
limited Fourier series can be obtained by computation.
of turbine blades and the like.
According to the invention however the coe?‘icients of
The invention provides a machine of this type which
dispenses with memory devices for the shapes of patterns 15 these developments are preferably obtained automatical
ly by the apparatus of the invention itself. For this pur
such as cams to be followed by a feeler or magnetic or
pose the pro?le data comprise not the coef?cients of the
perforated tapes which record a function of space for
limited Fourier developments but the parametric co
reproduction by the machine. Instead the machine re
ordinates of a certain number of points on the pro?le
quires only that the curve to be followed by the cutting
curve and the parametric values which correspond there
tool be known in the form of a parametric equation, or
to. The coef?cients of the Fourier developments being
alternatively that the coordinates of a ?nite number of
thus determined, the curve represented in parametric co
points on the curve be known.
In general terms a turbine blade or a compressor or
propeller blade can be considered as being made up of
a plurality of right cylinders having altitudes of any de
sired degree of smallness stacked together. The shape
and the dimensions of the right sections of these cylin
ders, hereinafter to be called “pro?les,” generally varies
ordinates by these developments passes exactly through
all of the given points but cannot coincide between such
with the theoretical curve. A better degree of
coincidence is obtained of course with a larger number
being simply displaced angularly with respect to each
of data points to begin with but particularly by a proper
choice of the data points along the parallel curve. The
pro?le of a blade to be reproduced always includes por
tions of unequal hydrodynamic or aerodynamic impor
tance. For example it is usually desirable that the shape
in the vicinity of the leading edge conform to the pattern
by control units generating signals representative of the
coefficients in the Fourier developments. This analyzer
from one cylinder or cylindrical element of the blade to
the next. Of course it is also possible for the shape and
dimensions of the successive pro?les to remain constant,
intended therefor more closely than is necessary at the
trailing edge or at certain parts of the extrados or in
The path followed by the cutter in milling a pro?le is a
trados. The number of data points being constant, it is
closed curve parallel to the pro?le, and it will hereinafter
nonetheless possible to increase the precision along par
be referred to as the “parallel curve.” The pro?le it
ticular portions of the desired pro?le without modifying
self may for example be of convex cylindrical type of two
in any way the system of parametric coordinates initial
chosen. It is sui?cient to space the points unequally,
This parallel curve can usually be de?ned in Cartesian
or polar parametric coordinates vwhich can be developed 4:0 increasing the number employed for de?nition of that
portion of the pro?le which must be reproduced the most
in the form of a Fourier series limited to a certain num
ber of terms. A complete description of the curve is
The apparatus of the invention comprises three prin
obtained by causing the parameter to vary between zero
cipal elements:
and 211-. If the curve is de?ned by p points, the number
(1) A harmonic analyzer operating by interpolation
of terms 2n of the limited series is equal to the number p 45
which receives as input data the numerical values of the
of points which are given for representation of the parallel
coordinates of a number of points along the parallel curve
curve in question.
the desired pro?le and which supplies the values of the
Prior attempts have been made to control machine tools
abscissa and the ordinate of a plane pro?le to be machined 50 effects summation of products of the form:
or of its parallel curve through Fourier analysis and syn
EYi 008mm)
thesis. The parameter chosen has been, so far as I am
aware, the polar angle of a point on the pro?le. This
has the advantage of making it possible to drive by the
same movement the pattern and workpiece support and 55
the analyzer and synthesizer of the control unit. But
practice has shown that, for hydrodynamic and aero
dynamic pro?les, the convergence of the Fourier series
in terms of such a parameter is rather poor.
in anion)
(2) A harmonic synthesizer which receives as input
data the numerical values of the Fourier coe?icients and
I have found that, by a proper choice of the angular 60 which supplies in the form of voltages the instantaneous
parameter in terms of which the Cartesian or polar co
coordinates of the parallel curve. This synthesizer ef
ordinates of the parallel curve are expressed, it is possi
fects summation of the products:
ble for a given limit of accuracy to obtain developments
having good convergence and a small number of terms.
2a,- cos ya-I-Eb; sin jOt
The invention is based on this fact. For a given curve
there are in general an in?nite number of possible develop
(3) Servomechanisms adjusting the machine tool to
ments in Cartesian or polar coordinates.
position the cutter with respect to the workpiece in ac
The invention is equally applicable to the production
of workpieces whose pro?les are not closed such as the
shaping of grooves or special cams. It is su?icient in
such cases to join the ends of the desired pro?le curve
cordance with these voltages.
Harmonic analyzers and synthesizers are well known
in the art. See for example “Harmonic Analyzer and
Synthesizer” by Jules Lehmann, Electronics, November
1949, pages 106-110 and “A One-Dimensional Fourier
of the machine were governed by command signals in
the form of voltages proportional to the coordinates
thus de?ned. In general however for various reasons, in
particular the convergence of the series, the coordinates
Analog Computer” by Leonid V. Azaroif, The Review of
Scienti?c Instruments, May 1954, pages 471-477. These
devices comprise essentially sinusoidal multiplying poten
tiometers or synchro transformers. The parameter
values in the terms to be added are arranged in arithmetic
progression and can be simultaneously materialized by
means of shafts coupled through gearing systems having
are developed in the form of limited Fourier series of a
and the ?rst one, which varies linearly with time. The
variation of the second angular parameter as a function
of time is then determined by a coupling which takes
into account the relation between the second parameter
shaft (basic shaft) whose rotation represents the variation
and the ?rst one, which varies linearily with time. The
of the parameter itself. The sum of the products can
servomechanism or other apparatus which then’ operates
be instantaneously obtained by adding in a common
to position the cutting tool in response to voltages pro
resistor the output voltages of the various potentiometers
portional to coordinates expressed as functions of the
or synchros. In the case of the synthesizer the param
second angular parameter thereupon operates so that the
eter varies continuously from zero to 211' and the basic 15 motion of the cutting tool along the parallel curve is
shaft turns continuously. In the case of the analyzer
the parameter takes on a series of discrete values which
It should however be noted that since in general the
are multiples of a quantity 1x1, and the basic shaft is
curvature of the pro?le itself is not constant, constant
successively turned to positions inclined to a zero orienta
speed of advance of the cutting tool along its trajectory
tion by one of these discrete values.
20 does not correspond to a constant speed of advance of
ratios arranged in arithmetic progression to a common
Sinusoidal potentiometers are well known and are avail
able in various types. It is possible for example to use
those described in US. Patent No. 2,434,057 which com
prises essentially a coil mounted ?at on a rectangular in
the tool along the pro?le at the cutting point. Nonethe
less the latter speed of advance does not differ greatly
from the rate of advance of the cutting tool along its
trajectory (e.g. of the axis of a rotating cutter) except
sulating plate and a rotating wiper whose aXis passes 25 perhaps in the vicinity of the leading and trailing edges
through the intersection of the diagonals of the plate.
of a blade. Since in the ?nal analysis it is more im
In synthesizers of the usual direct current and alternat
ing current types, both coef?cients and sines and cosines
‘of the argument and of its multiples of a Fourier series
are represented by current or voltage amplitudes, and
the voltage representative of a term of the series is the
product of the voltage representative of the coefficient
portant to obtain a constant rate of cut in the sense of a
constant rate of ?ow of chips than to have a constant
rate of advance of the tool along the pro?le, the outline
of the raw workpiece as it comes from the foundry is so
established that the thickness of material to be removed
is greater in the vicinity of the leading and trailing edges
and by voltage representative of the sine or cosine of the
of the blade than elsewhere, i.e. greater at those places
argument. In case alternating current is used, all the
where the speed of advance of the point of contact be
voltages representing the terms of the series are co 35 tween the cutting tool and the workpiece is substantially
phasal. The invention provides a further type of syn
lower than the motion of the cutting tool center along its
thesizer in which the coefficients of the Fourier series
trajectory. The invention further makes it possible to
are represented by amplitudes of alternating currents and
determine, on the curve parallel to the desired pro?le,
the argument and its multiples are represented by the
zones within each of which the speed of motion of the
phase and the multiples thereof. Consequently the fre 40 cutting tool (e.g. motion of the axis of a rotating cutter
quencies of the voltages representative of the successive
with respect to the workpiece) is constant, this speed
terms of the series are multiples of each other. As it
changing from zone to zone and the magnitude thereof
is convenient to give the frequency of the fundamental
in each zone being determined by the speed of response
term the frequency of the mains, Le. 60 c./s., the cycle of
of the servomechanisms which position the cutting tool
this term which is the period of the series is too small 45 with respect to the workpiece.
for allowing the machine tool driving servomechanisms
The invention further effects determination along the
to operate. The invention provides means for deriving
from the synthesizer sampling pulses having a recurrence
parallel curve of zones within each of which the cutting
speed is determined to avoid vibration of the workpiece.
frequency slightly differing from the series period and
The invention will now be described in detail by ref
thus allowing the function representative of a coordinate 50 erence to the accompanying drawings in which:
to be stroboscopically sampled.
FIGS. 1 to 3 are diagrams of right sections or pro?les
The invention further provides control apparatus of
the type described which makes it possible for the ma
chine tool to operate with a substantially constant out
of diiferent types of turbine blades, showing in conjunc
tion therewith the curves parallel to these pro?les traced
out by the axis of the cutting tool, indicating the geo
put of chips selected to be suitable to the cutting tool and 55 metrical signi?cance of the angular parameter as a func
to the workpiece.
tion of which the pro?le coordinates are developed.
The invention also provides control apparatus which
FIGS. 4 and 5 are diagrams for comparison of a curve
makes it possible for the machine to operate without , parallel to the theoretically desired pro?le section of a
blade with the curve obtained by means of the apparatus
For the purpose of achieving a constant rate of cut 60 of the invention, with one ‘degree of accuracy in FIG.
and smooth operation the cutting speed and the rate of
4 and another degree of accuracy in FIG. 5.
advance of the tool with respect to the work are con
FIG. 6 is a block diagram. of the apparatus of the in
trolled at each point of the motion of the cutting tool
with respect to the remainder of the machine to optimum
FIG. 7 is a schematic diagram of the interpolation
‘values which result in efficient operation of the machine. 65 harmonic analyzer.
According to the invention, the curve parallel to the
pro?le is de?ned by points equidistant therealong, i.e.
such that the curvilinear abscissa separating two succes
sive points is constant, and the coordinates of these points
are expressed as limited Fourier series as functions of a 70
?rst angular parameter‘ proportional to the curvilinear
abscissa and to time.
In these conditions, the movement
FIG. 8 is a diagram of an alternate form of harmonic
analyzer according to the invention.
FIG. 9 is a diagram of .a ?rst type of harmonic syn
FIG. 10 is a diagram of an alternate form of harmonic
FIG. 11 is a schematic diagram of the servomechanism
for positioning of the cutting tool when the curve paral
uniform if the servomechanism operating on the trav
lel to the desired pro?le is de?ned in polar parametric
erse elements between the cutting tool and the remainder 75 coordinates.
of, the cutting tool along the parallel curve would be
For the case of curve ‘4 of FIG. 2 (X0=22.50 mm.)
FIG. 1&2 is a functional diagram of the harmonic
Y=i+5 .48
+l107-26 cos a
‘-2.97 sin at
FIG. 13 is a diagram of a blade pro?le and of the
+5.85 cos 20c
-— 14.66 sin 20a
curve parallel thereto useful in explaining the operation
of the apparatus of the invention when the speed of ad
--7.»13 cos 30:
+0.09 00s 40:
+009 sin 3a
~—3.25 sin 40;
vance of the cutting tool is to be constant.
FIG. 14 is a schematic diagram, partly in block form,
of the apparatus of the invention as arranged for constant
+6.44 cos 5a
+0.21 sin 50:
-0.28 cos 60:
In the case of the curve 6 of FIG. 3, which is sym
speed of advance of the cutting tool.
FIGS. 15 and 16 are curves representing the deriva 10 metric with respect to the axis OY of FIG. 3 (X0=55
tives with respect to time of the Cartesian coordinates as
a function of the angular parameter.
+73.7-3 cos a
FIG. 17 is a diagram similar to FIG. 13 but illustrat
+:10.15 cos 2a
ing a different distribution of the data points to be ana
-8.73 cos 30c
lyzed in order to reduce over certain zones of the pro?le 15
+2.53 cos 4::
to be cut the speed of response required of the servo or
+2.32 cos 502
other machine tool driving elements.
+080 cos 6a
‘FIG. ‘18 is a curve giving as a function of the an
The development of the radius vector p of the curve 2
gular parameter the desired cutting speed (speed of ro
tation of the cutter) and the cutting speed actually ob 20 as a function of the same parameter on is
tained by means of the invention.
‘FIG. 19 is a schematic diagram, partly in block form,
of apparatus according to the invention for control of
the speed of rotation of the cutting tool by means of
~—4.l‘6 cos a
— 148.20 cos 2a
-~l.89 sin a
‘+0.76 sin 2a
v--3.31 cos 3m
—2.57 sin 300
-—24.40 cos 4a
a harmonic synthesizer.
FIG. 20 is a block diagram of a third type of har
+0.22 cos 5a
~0l82 sin 5a
monic synthesizer associated with sampling pulse generat
More generally it is possible to write
-—7.30 cos ‘60c
ing means and FIG. 21 shows the sampling pulse wave
FIG. 22 is a diagram useful in explaining the apparatus 30
of FIG. 23.
FIG. 23 is a block diagram useful in explaining the
operation of the apparatus of the invention in a machin
ing operation involving fractional passes or cuts and in
In the foregoing the coe?icients in the limited expan
which the harmonic analysis and synthesis of the desired 35 sions have been calculated. They may however be ob
parallel curve are made for successive sectors.
tained automatically by means of the apparatus of (the in
:FIGS. 1, 2 and 3 illustrate at reference characters 1,
vention from input data for the coordinates of 12‘ points
3 and 5 three light sections of blades and, at reference
(p=2n). The interpolation harmonic analyzer solves (for
characters 2, 4 and 6, respectively, the curves parallel
the case of the ordinate) the following equations:
to these pro?les traced out by the axis of a rotating cut 40
ter 7.
This cutter may for purposes of a concrete ex
Y1=a0+a1 cos a1+ . . . +0,n cos nal
ample only be assumed to have a diameter of 5 mm. The
pro?le 1 of FIG. 1 is a loukowski pro?le, that of FIG.
2 is of the type identi?ed as No. 64~1~212 of the Na
+171 sin (11+ . . . +bn_1 Sin (n—-1)d1
Y1=ao+a1 cos aj+> . . . +an cos not;
tional Advisory Committee for Aeronautics and the pro?le
of FIG. 3 is one comprising two circular arcs respectively
of 313 and of 461.4 cm.
One may start from the statement:
in which Y1, Y1 and Y1) are the values of the ordinates
of known points on the desired parallel curve corre
sponding to values a1, a1 and up of the parameter. The
2n unknowns are the coe?icients a0 to an and b1 to bn_1.
X=Xo Sill a
When the angular intervals between the parameter values
of the known parallel curve points are equal or in other
In this expression X0 is the half distance between two
tangents to the parallel curve 2, 4 or ‘6, these tangents
being parallel to each other and containing bet-ween them
the complete parallel curve as indicated in FIGS. 1-3. The
angle a then represents in these ?gures the polar angle
words when the known points correspond to equally
spaced values of the angular parameter, the coe?icients
are given by the relations:
of a point m on the circle tangent to the two parallel
tangents just mentioned, the center of this circle being 60
at the origin of coordinates. The point m has the same
abscissa as the running point M on the parallel curve.
Under these conditionsdhe developments in a Fourier
series limited to six terms (rt-=6) for the ordinate of
these same curves are:
With respect to the curve 2 of FIG. 1 (Xo=39.025
Since as a result of the equal spacing of the parameter
values otj=joz1, the preceding relations assume the form:
+8616 cos a
v—0.34 sin a
+18144 cos 2a
-—‘12.58 cos 3a
+2.50 cos 4a
—9.13 sin 20:
+0.90 sin 3a
‘—2.69 sin 4m
‘+3.81 cos 50:
+047 sin 50:
- 1.24 cos 6a
will be represented simply by means of a rectangle 76
having ‘four electrical terminals ‘109412, corresponding
to those shown in FIG. 6.
The interpolation harmonic analyzer 81 is shown in
FIG. 7. It includes at least p linear potentiometers num
For example FIGS. 4 and 5 show in dashed lines the
bered 17, to 17p, energized from a D.C. source 8 of
theoretical or desired pro?le I1 ‘and the corresponding
voltage U.
theoretical parallel curve 2, and in full lines the inter
On the potentiometer 17, there is set up the
value of the ordinate Y, corresponding to the parameter
value 0a,. Similarly potentiometer 17p is set to the value
polated parallel curves 2’ ‘and 2" for the case‘ where the
parametric equations of the parallel curves are limited 10 of the ordinate Yp corresponding to the parameter value
polated pro?les 1’ and 1" and the corresponding inter
respectively to ?ve and to eight terms. It will be observed
that even with the parametric equations limited to ?ve
The wipers 16, and 18, of potentiometer 17, are re
spectively connected to terminals 10, and 12, of rotary
terms the coincidence between the interpolated parallel
curve 2’ and the theoretical curve 2 is very good and that
switches 9 and 11. These terminals are inclined to the
are limited to eight terms.
is Y,. Similarly the wipers 16,, and 18p of potentiometer
it is nearly perfect between the interpolated curve 2" and 15 rest position of the switches at an angle a, which is equal
to the value of the parameter a for which the ordinate
the theoretical curve 2 when the parametric equations
17p are respectively connected to terminals 10,, and 12,,
Referring to FIG. 6 the invention comprises an inter
of switches 9 and 11, and these terminals are oriented
stituted Iby ordinates of sample points on the curve par 20 with respect to the rest position of the switches at an
angle 11,, which is equal to the value of the parameter
allel to the pro?le and of the parameters corresponding
polation harmonic analyzer 81 which, from data con
to these ordinates, develops the values of the coeflicients
in a limited Fourier development of the ordinate, these
coe?icients ac to an and b,, to b,,_, taking for example
the form of rotations of shafts 340 to 34,, and 41, to 25
for which the ordinate is Y,,. As previously stated, the
values on, to 0a,, of the parameter are equidistant, i.e.
ocj=joc1=j21r/p and the ‘data are the values of the ordi~
nate Y ‘for these linearly spaced values of the parameter
(2. Consequently the terminals on switches 9 and 11 are
equ-iangularly spaced. The arms 19 and 21 of switches
9 ‘and 11 are coupled together and are connected to the
voltage supply terminals of n sine and cosine multiplying
mentary input data a variable angle a in the form of a
rotation of the shaft 48 representing the output of a speed 30 potentiometers 15, to 15,,. The number n of sine po~
tentiometers is equal to the number p of linear po
reducer 47 driven by a motor 46. The synthesizer also
tentiometers. The cosine wipers 20, to 20,, of the sinu
receives as input data the quantity X0, again in the form
soidal potentiometers and the perpendicular sine wipers
of a shaft rotation. Of course, if the abscissa X of the
22, to 22,,_, (potentiometer 15,, has no sine wiper) are
running point M on the curve parallel to the pro?le, in
stead of being a simple sinusoidal function is like the 35 mechanically coupled by a tangent screw 13 which en
gages with the worm wheels 14, to 14,,. The diameters
ordinate a limited Fourier series, the terms for this series
of these worm wheels are such that when the wipers of
would ‘be obtained by means of a second interpolation
potentiometer 15, rotate through an angle as the wipers
harmonic analyzer identical to the analyzer 81.
of potentiometer 152 will rotate through an angle 20:
Output shafts 340 'to 34,, and 41, to 41,,_, from inter
polation harmonic analyzer 81 and input shafts of the 40 and so on with the wipers of potentiometer 15,, rotating
through an angle n06. Lastly the arms 19 and 21 of
same reference numerals for harmonic synthesizer 82 are
switches 9 ‘and 11 are connected via a coupling 23 to the
connected through clutches 90 in order to allow inde
worm wheel 14, ‘and hence rotate through the same angle
pendent reset of said shafts.
as wipers 20, and 221.
The synthesizer 82 develops the values of X and Y as
Shaft 13 may be turned by hand or some similar means
functions of oz, for example in the form of voltages. 45
‘and is designed to stop at predetermined angles 0a,,
These voltages are applied respectively to abscissa and
206, . . ‘ noq.
ordinate servomechanisms 87 ‘and 88 which control
When the tangent screw 13‘ is inclined at the angle a,
screws 64 and 74 for positioning the cutting tool with re
to its rest position potentiometers 15, to 15,, are ener
spect to the workpiece in orthogonal directions of a hori
gized with a voltage representative of Y, and their cosine
zontal plane.
The carriage 59 which supports the cutting tool is sub
wipers supply to the terminals of ‘load resistors 24, to
jected to two orthogonal motions with respect to the
24,, voltages
workpiece 77. The ?rst motion is derived from a motor
Y, cos a,
60 which is coupled to a speed reducer 61 and then to
Y, cos 211,
carriage '69 via lead screw 64. The motor 60 is energized 55
via ampli?er 62, by an error voltage which appears across
Y, cos not,
resistor 63 and which is equal to the diiference between
The rotations of these shafts thus constitute input data
to the harmonic synthesizer 82, which receives as supple
the voltage representative of X(oc) and the output voltage
and their sine wipers supply to the terminals of load
from potentiometer 65 ‘whose winding is ?xed to the
resistors 25, to 25,,_,, voltages
frame of the machine tool 76 and whose slider is ?xed
to the carriage 69.
The second motion is applied directly to the carriage
59 which supports the cutter itself. This motion is de
veloped by motor 70 which drives lead screw 74 through
Y, sin (Ir-1)“,
a speed reducer 71. The carriage 59 is of course coupled 65
to the lead screw by means of a nut.
When the tangent screw is inclined to its rest position
Motor 70 is energized Via ampli?er 72 from an error
at an angle foal, the potentiometers 15, to 15,, are ener
‘voltage which appears across resistor 73 and which is
gized with \a voltage representative of Y,- and apply to the
equal to the difference between the voltage representative
of Y(a) and the output voltage of potentiometer 75, 70 load resistors 24, to 24,, voltages
whose winding is ?xed to the main carriage 69 and whose
slider is ?xed to the secondarycarriage 59 which sup
ports the cutting tool itself.
For simplicity, in some of the succeeding ?gures of
the drawing, the machine tool with its servomechanisms
Y,- cos jot,
Y,- cos 2111,
Y, cos I'ljoc,
cupies the position fool) is applied to a servo-mechanism
and, ‘to the load resistors 25, to 25,,_1, voltages
including ampli?er 31, servomotor 30, speed reducer 32
Yj rSlIl jul
Yj S111 2j0t1
and lead screw 34 which drives a reference potentiometer
28—29. The potentiometer winding 28 is mounted on a
support 33 coupled to screw 34 while the wiper 29 is
stationary and is connected to resistor 102. It may be
seen here that the addition of the various products which
Servomechanisms 260 to 26,, and 27, to 27,,_1 have
their volt-age input terminals connected respectively to
the resistors 240 to» 241, and 251 to 25,,_,.
make up a given coe?icient in the Fourier series is no
Each servomechanism includes a reference potentiom
eter whose winding 28 is energized from source 8 and
each potentiometer also includes a wiper 29. For each
servomechanism the winding 28 is mounted on a support
33 coupled to lead screw 341. This screw is driven,
through a reducing mechanism 32, by a motor 30 which 15
is energized via an ampli?er 31 by means of a voltage
equal to the difference between the output voltage avail
able at wiper 20‘, and the output voltage of potentiometer
longer made eifeotive in a succession of operations but
takes place instead in the course of a single operation,
the same shaft 34 thus delivering the coe?icients a1—an
as the shaft 13 successively takes up the discrete angular
positions a, to mxl.
Of course a second group of p
sinusoidal potentiometers would on another single shaft
develop the coef?cients £11 to b,,_,.
FIG. 9 illustrates diagrammatically a ?rst synthesizer.
It comprises (n+1) ‘linear potentiometers 420 . . . 42,,
having parallel doulble windings and parallel double wipers
whose wipers are coupled to shafts 34o . . . 34,, of the
The wiper 29 is mounted on a support 38 which is
coupled to lead screw 39.~ T-his screw is driven, via a
interpolation harmonic analyzer and an additional set of
speed reducer 37, by ‘a motor 35 which is energized, via
parallel double windings and parallel double wipers whose
ampli?er 36, by the output voltage of potentiometer 28.
wipers are driven by shafts 41, . . . 41,,_1.
n—1 linear potentiometers 431 . . . 43,,_, having also
All of these
potentiometers are energized from the DC. source 8
A switch ‘40 makes it possible to apply the output
voltage of potentiometer 28 (i.e. that taken at tap 29) 25 whose voltage is designated U. The terminals of the two
windings on the same side of the potentiometers are con
to the terminals of resistor 24, or to the input of ampli
nected to opposite polarity terminals of source 8 so that
?er 36, these two positions for switch 40 being identi?ed
potentials symmetrical with respect to ground will appear
as 40’ and 40", respectively.
at the two wipers.
With now the tangent screw 13 inclined to its rest
position at an angle a1 and with switch 40 in position 30 The output voltages from potentiometers 421 . . . 42,,
energize the sinusoidal potentiometers 441. . . 4%. Like
40’, the screw 34 rotates through an angle proportional
wise the output voltages of potentiometers 431 . . . 43,,_1
to Y, cos 0:1. The switch 40‘ is then shifted to position
energize the sinusoidal potentiometers 451 . . . 45,,_,.
40" and the screw 39 turns through an ‘angle equal to
The wipers of the sine potentiometers are driven by
-Y, cos 0:1. The potentiometer 28——29‘ is then restored
to its rest position. The tangent screw 13 is then caused 35 a motor 46 through a reducer 47, a tangent screw 48
and worm wheels 491 to 49,, having diameters such that
to shift to a position inclined to its rest position at an
when the wipers of sine potentiometers 441 and 451 rotate
angle 2:11, and screw 34 rotates through an angle pro
through an angle at, those of potentiometers 44, and 45,
portional to Y2 cos 20:1. When the switches 9‘ and 11
rotate through ia, the wiper of potentiometer 44,, rotat
have scanned all of their terminals 10' and 12, the shaft
40 ing through I106.
341 gives an indication proportional to
The output voltages of the sine potentiometers which
are representative of the terms a1 cos ice and b, sin ioc are
applied to resistor 50‘ at the same time as the output
ZEY, cos joq
i.e. proportional to a, in accordance with Equation 1.
In the same fashion it may be seen that the servo
voltage of potentiometer 440 which is representative of
45 a0.
Thus one obtains at the terminals of resistor 50‘ a
voltage representative of Y(ec) .
rnechanism having its input terminals connected to re
A second type of synthesizer is diagrammatically illus
sistor 251 produces on its shaft 411 an indication pro
trated in FIG. 10. It includes as set forth in the publica
portional to [1,. In general terms, the servo connected
tion of Azaroff previously cited n auto/transformers 1851
to resistance 24, gives on its shaft 34, an indication pro
portional to a,, and the servo connected to resistor 25, 50 to 185,, which are energized by an AC. source 190 and
which in turn energize the rotors of resolvers 1871 to
gives on its shaft 41, an indication proportional to b,.
1871, via 11 phase selector switches 1861 to 186,,. Let it
The coe?icients a0 to an and b1 to b,,_, of the Fourier
be supposed now that the Fourier developments are known
development of the ordinate Y limited to the order n
are thus available on shafts 340 to 34,1 and 41, to 41,,_1. 55 in amplitude and phase rather than in terms of the co
e?icients of the sine and cosine terms. Each autotrans
FIG. 8 represents a modi?cation of the analyzer of
FIG. 7.
former is positioned with respect to a zero position at an
In this ?gure 991 . . . 99,, represent sinusoidal
potentiometers of a number [2 equal to the number p of
of points whose data de?ne the curve. This number is
at least equal to twice the number of terms in the Fourier
development employed to represent the curve to the de 60
sired degree of accuracy.
1001 . . . 1001) are control
tentiometers supply voltages proportional to Y, . . . Y,,.
The wipers of the potentiometers are positioned by opera
ing relative drive ratios of one, two . . . p.
1011 . . .
with respect to a zero direction at an angle equal to the
phase of the term represented.
The rotors are driven
by a shaft rotating according to the angle oz and by gearing
or coupling systems not shown such that when the ?rst
rotor turns at one speed the second turns at double speed
rheostats which make it possible to apply to these poe
tion of a shaft 13 which drives pinions 141 . . . 14p hav
elongation proportional to the amplitude of the term in
this series which it represents. Each rotor is positioned
and so on.
The ?rst stator windings of the various resolvers are
connected in parallel to the terminals of a resistor 188
at Whose terminals the voltage Y(O6) appears. Likewise
1019 are mixing resistors, and 102 is a common adding
the second stator windings of the resolvers are connected
in parallel to the terminals of a resistor 189, at which
The voltage appearing at the terminals of the resistor
70 terminals likewise there appears the voltage Y(oc) but
102 whose value is
with a quadrature phase relation with respect to the volt
age available across resistor 188‘.
(i.e. which is proportional to a, when the shaft 13 oc
Synthesizers of the ?rst and second types respectively
illustrated in FIGS. 9 and 10 will hereinafter be diagram
matically indicated as shown in FIG. 1S2 in the form of
a block including at the top thereof terminals to which
ers 311]-c are coupled to the phase of the output voltage
from phase shifter 3111c and the phase shifters 311,-, are
coupled to the phase of the output voltage of phase shifter
3121s. A single one of these phase comparators 306 is
shown in FIG. 20, subjecting the-output voltage of phase
are applied voltages a0, a1 . . . an representing the cosine
terms, lower terminals to which are applied the voltages
b1 . . . bn_1 representative of the sine terms, an input
shaft rotating according to a and an output terminal on
shifter 31126 to the same phase as that of the output
which appears‘ the voltage Y(oc) or, more generally, the
voltage of phase shifter 3111c.
quantity represented by the Fourier series under considera
In order to extend the cycle of the function Y from
the inadequate time Z'Ir/w out to a suitable value, the
A third type of synthesizer 300 is illustrated in FIG. 20. 10 said function is sampled by sampling pulses having a
This synthesizer includes a plurality of devices to generate
recurrence frequency which differs slightly from the fre
harmonic A.C. voltages whose amplitudes are propor
quency of Y and a phase rl/ which is slowly variable. The
tional to the coefficients of the Fourier terms, together
sampling pulse train is represented by expression:
with means for adding these voltages. Otherwise stated,
each term of the Fourier developments which, in the 15
synthesizer of the ?rst and second types hitherto described
was represented by a DC. or A.C. voltage of amplitude
proportional both to the value of that term and to the
where 1,0(t) =3”, \l/ being small with respect to w. The
sine or cosine of a variable angle is in the third type of
synthesizer represented by an A.C. current of amplitude 20 expression for the sampling pulse train is a sum of co
sines or arcs in arithmetical progression. It is known that
only proportional to the value of the coe?icient. The
this sum may be written
angular parameter a in terms of which the coordinates
are developed in limited Fourier series is here represented
by the phase of a reference A.C. current, i.e. the product
wt of the angular frequency w and the time I.
sin (n+1) 2
Since the value of the angular frequency of the A.C.
current which represents the fundamental in the Fourier
developments is too great to permit control, by means
of the voltages delivered by the synthesizer, of the
servomechanisms which drive the machine tool elements 30
for positioning the cutting tool with respect to the work
piece, a special device is provided which extends for
a longer time the cycle of the voltages which represent
the coordinates. If X(wt) and Y(wt) are the Cartesian
This function is shown in FIG. 21. It comprises pulses
315 of high amplitude having a recurrence angular fre
quency of (w—r//) separated by pulses of minor am
The quantity (4) is obtained 'by means of a synthesizer
coordinates of a point on the parallel curve and having a 35 .303 represented by the symbol of FIG. 12 receiving at
shaft 304- a slow input rotation 1p and energized with
period 21r/w of the order of %0 of a second at most, the
device for extending the duration of the cycle transforms
these coordinates into X(r//) and YO/J) where \l/ is an
angular parameter variable with time and having the
desired period for example of several minutes for con
voltages representative of cos wt . . . cos Itwt and sin
wt . . . sin not which are derived from the outputs of
phase-shifters 31110 to 311110 and 3111s to 311ns of syn
thesizer 300.
The sampling operation which, as well known, is a
trol of the servomechanisms.
In FIG. 20 307 designates a generator of sinusoidal sig
nals having for example a frequency of the order of 1000
c.p.s. The generator 307 energizes a frequency doubler
308 and a rectangular wave generator 309.
The output of the frequency doubler leads to a ?lter
3132 which passes only the frequency 2000 cps. The out
put of the rectangular Wave generator 309 leads to a
multiplication of the function to be sampled by the sam
pling pulses, is performed in multiplying circuit 305.
Multiplying circuits suitable in A.C. analog computers
45 are Well known in the art.
They are for example de
scribed at page v673» and illustrated in FIG. 19.9 of “Wave
forms” by Chance, Hughes, McNichol, Sayre and Wil
liams, McGraw-Hill Book Company, Inc., New York 1949.
The pulse modulated voltage issued from 305 is de
series of parallel connected ?lters 3133 to 313n which O! O
modulated in a pulse demodulator which is symbolized in
pass respectively the frequencies 3000 c.p.s., 4000 c.p.s
FIG. 20 by recti?er 314 and is used for controlling the
. . . . 1000n c.p.s.
Y co-ordinate drive of the machine.
Each ?lter is followed by two phase shifters, identi?ed
If the parametric equations of the curve parallel to the
for the ?lter 313,,1 at 311no and 311ns which provide
pro?le are in polar rather than Cartesian coordinates
quadrature outputs and which consequently produce sig
(FIG. 11) the voltage representative of X0 sin a is de
nals representative of sine and cosine terms.
veloped in a simple sine potentiometer and the voltage
The phase shifter 31111,.’ is followed by an ampli?er
representative of [3(a) is developed in a harmonic syn
31A1c having a gain proportional to an, and the phase
thesizer. It Will hereinafter be assumed that the voltage
shifter 311118 is followed by an ampli?er 312n5 having a
representative of X0 sin a is available across resistor 93
60 and that the voltage representative of p(ot) is available
gain proportional to bu.
All of the ampli?er outputs are added in a load re~
across resistor 98. The cutting tool carriage 59 is then
sistor 301, together With a voltage proportional to an
positioned in terms of the coordinate p just as it was
taken from a linear potentiometer 302. Across resistor
301 there appears the voltage Y(wl‘).
It has been assumed that the voltages at; cos 1' wt‘ for
example, in which 1' assumes the values from zero to n,
are in phase, so that the zeros of the ?rst voltage (i=1)
in terms of the coordinate Y in the embodiment of FIG.
6, reference characteristics 50 and 70 to 75 having in
FIG. 11 the same meaning as in FIG. 6.
The work piece table 92 in FIG. 11 is however mounted
for rotation and is'driven by a motor 95 through a speed
coincide with the zeros of the others. The parasitic phase
reducer 94. The output shaft of the reducer drives not
shifts introduced by the various circuits such as the ?lters
70 only the table 92 but also the slider of a sine potentiometer
may falsify this assumption. To this end the phase shift
ers for the various harmonics of the fundamental are
variable and are coupled together to produce harmonic
voltages in phase with the fundamental. By means of
97 whose winding is energized with the voltage repre
sentative of p(oc). Thus it is possible to pick oif from
potentiometer 97 a voltage representative of the product
p sin S2 in which the S2 is the polar angle through which
phase comparison circuits of known type, the phase shift 75 the table 92. isrotated. For the particularchoice made
\In FIG. 14, 105 and 106 are two synthesizers of the
type of FIG. 9 or FIG. 10. Both receive as inputs rota
for the perimeter a in the examples herein discussed this
If consequently the voltages ,0 sin 0 and X0 sin or are com
tions from a shaft 104 whose angular position represents
the parameter 7. The synthesizer 105 receives at its input
voltage terminals voltages which are representative respec
pared in resistor 93 one obtains an error voltage which,
tively of the coefficients k0, k1 . . . kn and I1 . . . ln_1
suitably ampli?ed in ampli?er 96 drives the servomotor 95.
in the development of y(7), and its output voltage is pro
p sin Q=Xo sin a
A description will now be given of those features of
portional to 7(7). The second synthesizer 106 receives
the invention which make it possible to obtain a constant
on its input voltage terminals voltages which are respec
speed of advance for the cutting tool and a substantially 10 tively representative of the coe?icients e0, e1 . . . en and
constant ?ow of chips. Referring to FIGS. 13 and 17, the
f1 . . . fn_1 of the development of X(7), and its output
pro?le 201 to be machined is assumed for the purposes
voltage is proportional to X(7).
v 7
of a concrete example to be of the type of Joukowsky.
The rotation on of a shaft 133 is however coupled to the
rotation of shaft 104 in accordance with relation (7), in
202 is the parallel curve to be followed by the axis of
the cutting tool for example a rotating miller, and 203 15 the following manner. 1'34 represents a motor and 135
a speed reducer whose output shaft 133 drives a sine
is the outline, in the plane of the pro?le 201, of the cast
potentiometer 1136 whose two perpendicular sliders are
ing to be machined, reinforced in the vicinity of the
connected to an adding network 137. This network also
leading and trailing edges of the blade for reasons already
receives from synthesizers 105 and 106 voltages propor
described. Let X(Ot) and Y(oc) represent ‘generally the
coordinates of a point M on the parallel curve 202 as 20 tional to the quantities X ( 7) and y(7) . The output signal
from network 137 is a voltage on conductor 138 which is
functions of the angular parameter a and let X(7) and
a linear function of the four quantities X(7), 311(7), cos cc
Y(7) 'be the coordinates of the same point as functions
and sin a, and this voltage on conductor 138 goes to zero
‘of an angular parameter 7 which is proportional to the
when the relation (7) is satis?ed. The error voltage from
curvilinear abscissa s of M measured from a point P of
zero abscissa and positive ordinate, this curvilinear ab 25 network 137 is ampli?ed in an ampli?er 139 and as so
ampli?ed energizes the motor 134.
scissa itself being proportional to time as already ex
Shaft 133 provides a mechanical input to two further
plained. 7 and s are related by
synthesizers .107 and 108, the ?rst of Which receives, as
electrical inputs, voltages representative respectively of
"Y— s
in which S is the total length of the curve 202. Finally
co, c1 . . . on and of d1 . . . d,,_1.
The second synthe
sizer 108 receives as electrical inputs voltages representa
tive respectively of a0, a1 . . . an and of b1 . . . bn_1.
let y(7) be the ordinate of m as a function of the
parameter 7. X(¢x) and Y(0L) are as before de?ned by
These two synthesizers respectively develop output volt
ages representative of X(a) and of Y(OL). The output
Equation 1. Similarly X(7), Y(7) and y(7) can be set
voltages from synthesizers 107 and 108 are applied respec
in the form
tively to subtraction networks 63 and 73 and the resulting
error voltages are applied to the terminals 109 and 110
and 111 and 112 of the machine tool 76 of FIG. 6, termi
nals 109 and 110 being input terminals for the X drive and
40 terminals 111 and 12 ‘being those for the Y drive.
Correct operation of the servomechanisms for the X
and Y drives of FIG. 6 comprising for the X drive ampli
21(1) =Zgkr
00597 +211
sin j'Y
For example, there will now be given the development in
series of X(7) and of Y(7) for the case where the curve
202 is harmonically analyzed for twelve points M1 to M12
which divide it into twelve equal parts:
(5) X(7)=—2.12
+4.10 00$ 7
+39‘5.72 sin 7
'—1.79 cos 27
—2.16 cos 37
+062 cos 47
+0.21 sin 27
—35.08 sin 37
—0.64 sin 47
+1.06 cos 57
+0129 008 67
+9.19 sin 57
?er 62, motor 60 and speed reducer 61 and for the Y drive
ampli?er 72, motor '70‘ and speed reducer 71 requires that
the speed of variation of the corresponding Cartesian co
ordinate X(a) or Y(<x) shall not exceed a speci?ed value
beyond which the error in machining will exceed a selected
Speeds X’(7) and Y’(7) with which the coordinates
X(7) and Y(7) vary are given by the following expres
sions which are derived respectively from relations (5)
and ( 6) by reference to 7.
(8) X'(7)=
(6) Y(7)=+16.91
+149.96 cos 7
‘>+17.83 cos 27
+3083 cos 37
+1.83 cos 47
+7.36 cos 57
—2.46 sin 7
60 (9) Y’(7)‘:
\—2.08 cos 67
The cutter will ‘be uniformly translated along the paral
lel curve 202 if the angular parameter a is coupled to the
angular parameter 7 instead of varying linearly with time
and being represented by the uniform rotation of a shaft
—4.1 sin 7
+395.72 cos 7
+2.58 sin 27
+6.48 sin 37
+0.42 cos 27
--105.24 cos 37
+2.48 sin 47
—5.30 sin 57
—1.74 sin 67
—2.56 cos 47
+45.95 cos 57
—149.96 sin 7
+3566 sin 27
—2.46 cos 7
--18.76 cos 27
+92.49 sin 37
—7.32 sin 47
—36.80 sin 57
+4.98 cos 37
—4.04 cos 47
+0.65 cos 57
+12.48 sin 67
The maximum permitted speed for the servos are in
dicated in FIGS. 15 and 16 by the horizontal lines of
constant ordinate vmax. It is seen that the speed X'(7),
represented in absolute value by the curve 214 in FIG. 15,
coupled to a constant speed motor. 7 itself is linear with 70 is everywhere below the maximum permitted value where
time and is represented by the uniform rotation of a
as the speed Y’(7) which is represented in absolute value
shaft, but 7 and 0: are related by the relation
by the curve 205 in FIG. ‘16 exceeds the maximum per
mitted value by some 29% in the vicinity of a 7 value of
(7) 75 90° and by some ‘34% in the vicinity of a 7 value of 270°.
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