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Aug. 28, 1962 G. G. FAYARD 3,051,389 MACHINE CONTROL 13 Sheets-Sheet 1 Filed Nov. 7, 1955 FIGJL. TYPU ,. INVENTOR Georges G. Fuyo rd Aug. 28, 1962 G. G. FAYARD 3,051,389 MACHINE CONTROL Filed Nov. 7, 1955 D1 . 13 Sheets-Sheet 2 I)", / ‘ 147 INVENTOR Georges G. Fuyurd BY M MIMIBW (1,174?» ATTQ R NEYS Aug. 28, 1962 3,051,389 G. G. FAYARD MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 3 lNVENTOR Georges G. Fuyord _ BY //M M Magnum/1% ATTORNEYS Aug. 28, 1962 G. G. FAYARD 3,051,389 MACHINE CONTROL 13 Sheets-Sheet 4 Filed Nov. 7, 1955 7:72A OHVA in 8G INVENTOR. GEORGES G.FAYARD O a BY ' /%m)ATTORNEY Aug. 28, 1962 G. G. FAYARD 3,051,389 MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 5 FIG.7 mvsmon Georges G. Fuyurd ATTORNEYS Aug. 28, 1962 3,051,389 G. G. FAYARD MACHINE CONTROL .n Filed Nov. 7, 1955 mFormM “M if LU m m s INVENTOR Georges G. Fayord av ATTORNEYS Aug. 28, 1962 G. G. FAYARD 3,051,389 MACHINE CONTROL 13 Sheets-Sheet 8 Filed Nov. '7, 1955 F I G -11 (35in .f\... 1 1 X0 sine‘ 96 93 95 94 97 75 : \JL 5 92 ?ak) ‘F e1 ' r“ F l 6. l2 Ya) L ,l/bn-1 51 INVENTOR Gseerges G. Foyurd M M/MMMJ?“ ATTORNEYS Aug- 28, 1962 G. G. FAYARD 3,051,389 MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 9 a 9D 12D 90 120 15D 18D 210 ‘I40 | Q70 . 30 6D 150 180 2IU 24m 30 l . 270 500 550 as lfindcgrees INVENTOR Georges G. Fcyurd . BY MWWQu-,$MPT% ATTORNEYS Aug. 28, 1962 e. G. FAYARD 3,051,389 MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet l0 ~73 FIG.14 co 134 l l 139 INVENTOR Georges G. Fuyord BY 1% MMIBM F7275,’ _ ATTORNEYS Aug- 28, 1962 G. G. FAYARD 3,051,389 MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 11 77"“ FIG -18 | 3D 5D 9D 120 150 180 l 210 | 240 270 SUD 55B 360 d INVENTOR Georges G. Foyurd BY fuw'q MIMI/law“ v-flybx AT TORNVEYS Aug. 28, 1962 G. G. FAYARD 3,051,389 MACHINE CONTROL Filed Nov. 7, 1955 13 Sheets-Sheet 12 in F23%505Il_0. 5 09L. mu-“.7524 .6g \Iil. mld uwmxrl? nr$. 25 5:9. E5 ,55,, _55m,. INVENI'OR H a GEORGES G FAYARD IB mom ./ W Unite St .tes atnt ‘ice _ 3,®5l,339 Patented Aug. 28, 1962 2 I 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 3,051,389 MACI-HNE CONTROL Georges G. Fayard, Paris, France, assignor to O?ice National d’Etudes et de Recherches Aéronautiqnes, Chatillon-sous-Fagneux, France, a French body cor porate Filed Nov. 7, 1955, Ser. No. 545,397 Claims priority, application France Nov. 6, 1954 Claims. (Cl. 235—180) 10 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 points with the theoretical curve. A better degree of 25 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 other. 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 35 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 sections. ly 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 faithfully. 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. of 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: 1) or of its parallel curve through Fourier analysis and syn EYi 008mm) thesis. The parameter chosen has been, so far as I am i=1 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) i=1 (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 at n-l having good convergence and a small number of terms. 2a,- cos ya-I-Eb; sin jOt 65 The invention is based on this fact. For a given curve i=0 i=1 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 3,051,389 3 4 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 uniform. 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 vibration. 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 vention. 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 thesizer. FIG. 10 is a diagram of an alternate form of harmonic synthesizer. 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 3,051,389 g 1 For the case of curve ‘4 of FIG. 2 (X0=22.50 mm.) FIG. 1&2 is a functional diagram of the harmonic synthesizer. 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 mm.) tives with respect to time of the Cartesian coordinates as a function of the angular parameter. Y=—|—7.02 +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. p=-+262 ‘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 form. 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: 65 With respect to the curve 2 of FIG. 1 (Xo=39.025 Since as a result of the equal spacing of the parameter mm.) Y=-—20.26 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 70 75 3,051,389 8 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 up. 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 4111-1- 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 60 frame of the machine tool 76 and whose slider is ?xed to the carriage 69. Y,sina, The second motion is applied directly to the carriage 171811121X]. 59 which supports the cutter itself. This motion is de cocoa-001i 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, 75 Y, cos I'ljoc, ‘3,051,389 9 10 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 28. 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 p 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 j=1 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. 65 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 resistor. 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 3,051,389 11 12 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 tion. 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 1 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 ll 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 cot-Ill 25 wt of the angular frequency w and the time I. sin (n+1) 2 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 (4) 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 plitude. 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 3,051,389 is 14 \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 means: 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 30 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 35 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 ‘IL 12-1 21(1) =Zgkr 00597 +211 sin j'Y j= j=1 45 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: 50 (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 tolerance. 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)= 55 (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 ‘—938 +1.66 —1.01 +0.13 sin sin sin sin 60 (9) Y’(7)‘: 27 37 47 57 \—2.08 cos 67 65 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 35(7) yo) 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°.