# Патент USA US3059860

код для вставкиOct. 23, 1962 G. G. FAYARD 3,059,852 CONTROL UNIT FOR MACHINE TOOL Filed D80. 5, 1957 6 Sheets-Sheet 1 Oct. 23, 1962 G. G. FAYARD 3,059,852 CONTROL UNIT FOR MACHINE TOOL Filed Dec. 5, 1957 6 Sheets-Sheet 2 FIG. 7 2G 4"“ ‘ i ‘__ _ FIG. 2b m . ____>. Oct. 23, 1962 G, G. FAYARD ' 3,059,852 CONTROL UNIT FOR MACHINE TOOL Filed Dec‘ 3, 1957 Y,‘ : 0.75 Acosq-OJ Ajcos 3x + 0,2OAsin4u 6 Sheets-Sheet 3 Oct. 23, 1962 e. G. FAYARD 3,059,852 CONTROL UNIT FOR MACHINE TOOL Filed Dec. 3, 1957 6 Sheets-Sheet 4 ? 4 mmRm Q Oct. 23, 1962 e. G. FAYARD 3,059,852 CONTROL UNIT FOR MACHINE TOOL Filed Dec. 5, 1957 6 Sheets-Sheet 5 United States Patent O??ce 3,959,852 Patented Oct. 23, l 962 1 2 $359,852 Georges G. Fayard, Paris, France, assignor to Office Na longer equal to the parameter, but is itself ‘a limited Fourier series of the same. It results that machining operations can no longer be performed by rotating the workpiece support in terms of the parameter and by con CONTROL UNIT FOR MACE-{WE TOQL tional d’Etudes et de Recherches, Aeronautiques, Chan tillon-sous-B-agncux, France, a French body corporate Filed Dec. 3, 1957, Ser. No. 700,368 2 Claims. (Ci. 235-180) trolling a radial tool in terms of the radius vector value. Rather in accordance with the invention the machining operation is controlled in terms of rectangular Cartesian co-ordinates, no rotational movement being imparted to The present invention relates to control units for auto— the workpiece support. matically controlled machines for the shaping of tur 10 I have disclosed, in my copending application Ser. No. bine, compressor or propeller blades, wings and fuselages 545,397, now Patent No. 2,930,084, ?led November 7, of airplane models and templates and the like. 1955 and assigned to the same assignee as the present The invention provides a control unit for a machine application, harmonic synthesizer means which, receiv tool of this type which dispenses with memory devices ing as input data the numerical values of the Fourier co for the shapes of patterns such as cams to be followed 15 e?icients, supplies in the form of voltages the instanta by a feeler, or magnetic or perforated tapes which re cord a function of space for reproduction by the ma chine. neous values of the cartesian coordinates to servomech anisrns adjusting the position of the cutter of a machine tool with respect to the workpiece to be machined. The object of the invention is to provide a control unit I have further found that, due, and only due, to the for a machine-tool requiring only that the pro?le to be 20 geometrical signi?cance of the chosen parameter, the followed by the cutting tool of the machine be known by geometrical data usually of dominant in?uence in the the values of geometrical data of said pro?le. design of turbine blades or aerodynamic pro?les or their A further object of the invention is to provide means square root are linear functions of the coe?icients of the for producing machined pro?les or models in which a sine and cosine terms in the developments of the co certain number of geometrical data remain constant from 25 ordinates and consequently that said coe?icients are sample to sample, whereas one or more other geometrical given, from said data, by linear simultaneous algebraic values vary from ‘sample to sample. This makes it pos equations. This property will be explained later. sible to determine experimentally the in?uence of such The control unit of the invention comprises a comput particular elements of data on the performance of the ing machine transforming the geometrical data of the ?nished piece or model. 30 pro?le into coe?icients of the Fourier series representa It is known that the pro?les of turbine blades, wings tive of the coordinates and a harmonic synthesizer effect and fuselages of airplane models are closed curves that ing the summation of the series, said computing machine can usually be de?ned in Cartesian or polar parametric and synthesizer being suitably interconnected as to allow coordinates which can be developed in the form of the latter to contribute to the computation. Fourier series of an angular parameter limited to a cer 35 The geometrical data which are particularly important in the case of hydrodynamic and aerodynamic pro?les pro?le is obtained by causing the angular parameter to are for example, the sagitta of the median line of the vary between zero and 211-. Prior attempts have been pro?le and eventually the abscissa of the location of said made to control machine tools by control units generat sagitta, the ordinates and the radii of curvature of the ing signals representative of the abscissa and the ordinate leading and trailing edges and the maximum thickness of the pro?le to be machined through Fourier analysis of the pro?le and its location. Other data may be con and synthesis. The parameter chosen has been, so far sidered according to the general scope of the invention, as I am aware, the polar angle of a point on the pro?le. as for example the location of a point of in?ection, etc. This has the advantage of making it possible to drive by The invention will now be described in detail by refer the same movement the pattern and workpiece supports 45 ence to the accompanying drawings in which: and the analyzer and synthesizer of the control unit. FIG. 1 is a diagram of a right section or pro?le of a tain number of terms. A complete description of the But practice has shown that, for aerodynamic pro?les, turbine blade, indicating the geometrical signi?cance of the convergence of the Fourier series in terms of such the angular parameter as a function of which the pro?le a parameter was rather poor. I have found that, by a coordinates are developed; proper choice of the angular parameter in terms of which 50 FIGS. 2a and 2g represent well known aerodynamic the Cartesian or polar coordinates of the pro?le are ex pressed, it is possible, for a given limit of accuracy, to obtain developments having good convergence and a pro?les and indicate the corresponding parametric de velopments. Under each pro?le, only the development in Fourier series of the ordinate is written, the abscissa be small number of terms. The geometrical signi?cance of ing always taken as a simple sinusoidal function; 55 the parameter chosen according to the invention will be FIG. 3 is a schematic representation of the succession given hereinafter. Though I do not desire that the in of operations involved in machining a pro?le from the vention as set forth in the appended claims depend upon geometric data values identi?ed in FIG. 1; such an explanation, I am of mind that the reduction of the terms of the Fourier series of the coordinate de FIG. 4 is a diagram of the control unit of the inven tion, showing the computing machine and the synthe— velopments obtained with the parameter I choose is due 60 sizer cooperating therewith, together with the servomech to the fact that two points respectively located on the anisms for positioning the cutting tool of the machine, extrados and the intrados of the pro?le at the same ab scissa along the general extension of the same correspond to simply interrleated values of the parameter. These are the two values of the parameter relative to said points are 65 supplementary angles, while in the case where the param eter is the polar angle of the running point of the pro?le, there is not any simple relationship between these two and FIG. 5 is a particular view of the matrix of the ma» chine. FIG. 1 illustrates at reference character 1 the right section or pro?le of a blade and at reference character 2 the curve traced out by the axis of a rotating cutter 3. The path followed by the axis of the cutter will here values. With my choice of parameter, the abscissa is inafter be referred to as “the parallel curve.” The an then always a sine function of the parameter. Unfortu 70 gular parameter in terms of which are developed the nately, the polar angle of a point of the pro?le is no Cartesian coordinates of the pro?le or of the parallel 3,059,852 3 1-i the leading edge and trailing edge of the same (in FIG. The abscissa of said maximum thickness, to be called XT; The sagitta of the median line, to be called Sp; The abscissa of said sagitta, to be called XS. ‘Of course, certain of these quantities, such as Y1, Yt, 1, 4’—5' are tangent to the parallel curve 2) are traced Rt, Sp, . . . may be made equal to zero. out and their distance represents the length 2X0 of the parallel curve or of the pro?le in their direction of major extension. These quantities may be written directly as functions of a1! b1! a2’ 172! a3, b3: a4: thus: curve is de?ned in the following manner. Two tangents 4——5 (or 4’-—5’) parallel to each other and bordering either the pro?le l itself or the parallel curve 2 nearest (1°)—Ordinate of the Leading Edge Y1 The angle on taken as a parameter represents, in FIG. 1, the polar angle of a point m on a circle 6 tangent to 10 Y1=1/2[Y(a1)+Y(r—ai)l the two parallel tangents 4—5 (or of a point m’ on a where a1 is the value of the parameter corresponding to circle 6’ tangent to the two parallel tangents 4'—5’) just the leading edge which is approximately equal to 31r/ 2. mentioned, the center 0 of this circle taken as the origin of the coordinates being located inside the blade at a Y1=—b1-'a2+bs+as (1) point the location of which is a matter of choice, but 15 (2° )-—Radias of Curvature of the Leading Edge R1 is preferably on the median line 7 of the blade. The § point in or m’ have the same abscissa as the point M’ ( X11 + yrs ) 2 on the parallel curve and the point M on the pro?le. Under these conditions, I have found that, the abscissa of the points M (or M’) being taken equal to a sinusoidal function of a, the ordinate may be developed in a Fourier RPM for series of a, which, for the usual pro?les known in hydro dynamics and aerodynamics, has a good convergence and comprises four terms or less. Theoretically, the machining of a model requires the knowledge of the parametric equation of the parallel curve. However, means are known from my abovemen 31r . a1=—2— (approximately) where X’ and Y’ are the ?rst derivatives and X" and Y" are the second derivatives of X and Y respectively with respect to a. Calculation gives: (2) tioned copending application for automatically machin ing models from the knowledge of the Fourier develop ments of the pro?le itself. Consequently, there will be 30 considered only in the following the developments in Fourier series of the pro?les and not those of the parallel where at is the value of the parameter corresponding to the trailing edge which is approximately equal to 1r/2. curves. FIGS. 2a to 2g represent well known aerodynamic pro Yt=b1—a2'—b3+a4 (3) (4° )——-Radius of Curvature of the Trailing Edge Rt ?les and the corresponding parametric developments. In all of these pro?les, the coordinate X is taken equal to X0 sin on. That of FIG. 2a is a pro?le by Joukowsky. That of FIG. 2b is a pro?le of Legendre. FIG. 26 repre R _ (X/2+Y/2)a/2 sents a pro?le identi?ed as No. 64-1-212 of the National Advisory Committee for Aeronautics (NACA). That of FIG. 2d is a fuselage pro?le of the type of the Gloster for oa=g (approximately) “Meteor,” whereas FIG. 2e is that of a super G. Constel~ lation. 'FIG. 2]‘ represents a pro?le of a fuselage show ing in dashed lines wing junctions with recesses on the fuselage according to the proposals of Whitcomb. This 45 Formulae 1 to 4 show that, due to the geometrical signi? pro?le terminates in an ogive. FIG. 2g represents a wing pro?le, NACA No. 64-A—O~O6. It will be seen that of the seven examples shown in FIGS. 2a to 2g four, those of FIGS. 2a, 20, 2d and 2g are pro?les obtained from developments which include as regards the coordinate Y, at most one term in cos a, one term in cos 20:, one term in sin 20c, and one term in cos 304. The other pro?les of FIGS. 2b, 2e and 2]‘ are developed respectively in six, four and three terms. cance of the chosen parameter which involves that the parameter values of the leading and trailing edges are re— spectively integral quarters of 21r, namely 31r/2 and 1r/2, Y1, \/R1, Yt, \/R, are linear functions of the coet?cients of the Fourier series development of Y considered as un knowns and reversedly the unknowns a1—a4, b1—b.;= are linear functions of Y1, \/R;, Y,, \/Rt, the said linear functions having constant numerical coef?cients. These pro?les and the developments by which they may be reproduced underscore the utility of the inven tion as hereinabove already described inasmuch as it Tm=2(a1 cos am+b2 sin 2am+a3 cos Scam-H74 sin 4%,) clearly appears that a very small number of coefficients su?ices to reproduce any one of these known pro?les. (5) It will be assumed in the following that the ordinate may 60 where am and (qr-um) are the parameters of the points be limited to the fourth order and written in the form: of the pro?le between which extends the maximum thick ness parallel to the axis OY. 0cm Must satisfy the condi tion that the derivative of Tm with respect to am shall 65 be Zero, i.e.: The geometrical data which de?ne the blade or more —a1 sin am+2b2 cos ham-3113 sin 3um+4b4 cos 4am=0 generally the pro?le are shown in FIG. 1. They are: The ordinate of the leading edge, to be designated by Y1; The radius of curvature of the leading edge, to be called R1; The ordinate of the trailing edge to be designated by Y,; The radius of curvature of the trailing edge, to be called Rt; The maximum thickness, to be called Tm; (6) The value of . X am=S1IF1 Y: results from Equation 6’ infra. In the case where XT is not imposed, Equations 5 and 6 between which am is to 75 be eliminated are equivalent to one relation between the ‘3,059,852 6 coe?icients a—b. In the case where XT and consequently am is imposed, these equations are equivalent to two rela tions between the coefficients a-b. b1 sin ap-l-az cos Zap-H23 sin 3ap+a4 cos 40£p=Sp (15) (6°)~—Abscissa 0f the Maximum Thickness XT b1 cos ap—2a2 sin 2ap+3b3 cos Barf-4a‘; sin 4ap=0 (16) XT=X0 sin am whence It may be noted that all these equations in which the Fourier’s series coefficients a1, a2, a3, a4, b1, b2, b3, b4 are the unknowns. are of the general type: 10 0 (7°)—Sagitra of the Median Line Sp 1 . SD=§[Y(ap)-|—Y(1r—a1,)]=b1 S111 ap+ag C08 201;. where K is a constant which may be equal to zero, :11 de +223 sin 3010+ (14 cos 4% notes particular values of the angular parameter and e1 and q; are factors equal to unity or respectively to j or k according to the equation concerned. When in the fac tor multiplying the unknown aj (or bk), ej (or ck) is equal to unity the said factor will be called in the ap (7) where up and (1r—¢xp) are the parameters of the points at the intersection of the pro?le and of the straight line parallel to OY passing through the point of upper ordi nate of the median line. up Must satisfy the condition 20 pended claims “unweighted trigonometric function”; when that the derivative of Sp with respect to up shall be zero, Ej (or 6k) is equal to j (or k), the corresponding factor 1.e.: will be called “weighted trigonometric function.” All the ?rst members of the equations, Equations 9 to b1 cos cap-M2 sin Zap-+3123 cos 3zxp—4a4 sin 4ap=O 12 included, are limited Fourier series of speci?c values 25 of the parameter. For example the ?rst member of (8) The value of Equation 9 is equal to the ?rst member of Equation 15 with ap=37r/2; the ?rst member of Equation 11 is equal is to the ?rst member of Equation 15 with ap=1r/2; the 0 ?rst member of Equation 10 is equal to the ?rst member results from Equation 8' infra. In the case where X5 is 30 of Equation 14 with ozm=31r/2 and the ?rst member of not imposed, Equations 7 and 8 between which up is to Equation 12 is equal to the ?rst member of Equation 14 be eliminated are equivalent to one relation between the With otm=7r/2. coe?icients a——b. In the case where XS and consequently The synthesizer comprised in the apparatus of the in up is imposed, these equations are equivalent to two rela vention has two modes of operations: (i) receiving as in 35 put data the coe?icients of the Fourier series develop tions between the coe?icients a—b. ment of the ordinate and a continuously varying param (8°)—Abscissa of the Sagitta of the Median Line Xs eter value, it produces a continuously varying output sig XS=XO sin up nal representing the said ordinate; (ii) receiving as input whence data approximative value of the coefficients of the Fourier 40 series development of the ordinate and a continuously Xs , up = sin-1 varying parameter value, it is allowed to stop when its X0 (8) output signal is equal to a predetermined value, whereby Formulae 5, 6, 6', 7, 8, 8' show that, due to the geometri a corresponding particular value of the parameter is cal signi?cance of the chosen parameter which involves known and is used for deriving more precise values of that the two points limiting the maximum thickness and 45 the coef?cients of the Fourier series. the two points at which the sagitta of the median line These relationships may be written in the matrical cross the pro?le have supplementary parameter values, form: Yr (R1X0)1/2 Y; (R¢Xo)m Tin/2 0 1 O = —1 0 1 —1 0 -—1 1 0 0 0 -—2 0 2 Cos am 0 0 sin 2am XT ~S1n am _ 0 0 Sp X5 0 0 Sin an 005 up Cos 211p —2 Sm 2a,, 0 —3 0 —3 Cos 302m 2 Cos 262m —3 Sin 30am 0 0 0 0 1 0 —1 0 1 O 1 0 —4 a1 b1 8,2 b1 0 0 Sin 4am at 0 0 4 Cos 4am b3 0 0 84 b4 Sin 3a,, Cos 4a,, 3 Cos 3a,, -4 Sin 40:; 0 4 0 ( 17) Tm, XT, Sp, XS are linear functions of the coe?icients of the Fourier series developments of Y considered as The question of the sign on the radical of Equations unknowns and reversedly the unknowns a1—a4, 121-114 10 and 12 is unimportant. To change from 1+ to — is are linear functions of Tm, XT, Sp, Xs the said linear equivalent to changing from a to 1r——0c. The leading edge functions having constant trigonometrical coe?icients. and the trailing edge are thus interchanged. It results from the preceding equations that the knowl 60 Computation of the unknowns (ll-b1 to ?it-b4 from edge of the values of a certain nurnber of geometrical the last eight equations may be performed. by means of data of the pro?le, namely Y1, R1, Yt, Rt, Tm, XT, Sp, XS an analog or digital machine adapted to solve systems impose between the coe?icients a and b of the develop of linear equations to the input of which are applied ment of Y the following relationships the geometrical data. Electrical voltages representative 65 of the values of said unknowns are ‘applied to the har monic synthesizer of the control unit which produces driving voltages for the cutter of the machine. It is to ‘be stated that the computation of the trigonometric coef?cients of Equations 13 to 16 is performed by the 70 synthesizer of the control unit. a1 cos am+b2 sin 2am+a3 cos 3ocm+b4 sin 4am=Tm/2 Referring to FIG. 3 of the drawings, 8 represents the plug board for the geometric data. These data are ap (13) plied by means of knobs 9--16 cooperating with dials to the transducer or data translator 17 which transforms (14) 75 them into analog electrical voltages. Of course the dials 3,059,852 8 assumes the angular displacement am (or up) thereby solving Equation 6' or Equation 8' above. During the cooperating with knobs 10 and 12 are graduated in square root proportion of the corresponding radius of curva ture and the dial cooperating with knob 13 is graduated in half the maximum thickness. The translator 17 trans mits the data in turn to the matrix 18 which develops operation ‘an electromagnetic clutch 69 disconnects shaft 20 from shaft 77 and connects this latter shaft to the shaft '78. The synthesizer comprises in the embodiment disclosed eight resolvers 441 to 444 and 451 to 454 whose rotors are respectively energized by A.C. voltages available at in the form of electrical voltages the coefficients al to (:4 respectively at terminals 51 to 54 and the coe?icients b1 to b4 respectively at terminals 55 to 58. Computers or matrices for solving systems of 11 linear equations with n unknowns are well known in the art and it is not nec terminals 51—58 and proportional to al . . . £14, £11 . . . 10 b4. Only one stator winding is necessary in each re solver and the stator windings of resolvers 441-444 are respectively perpendicular to the stator windings of re essary to disclose them in full. The values of the coe?icients are applied to a harmonic solvers 451—454. The rotors of the resolvers are driven by a tangent screw 79‘ through the shaft 77 driven itself through clutches and bevel gears to a main shaft 77, a 15 either by shaft 20 or by shaft 78. Four worm wheels 481 to 4%.; drive directly the rotors and they have di branching secondary shaft 773 and several other branch ameters such that, when the rotor of resolvers 441 and ing secondary shafts 771, 772, 78, 79 to‘ be disclosed 451 rotate through an angle 0:, those of resolvers 44, and hereinafter in FIGS. 4 and 5. The rotation of these synthesizer 19 driven by a motor 32 through a speed reducer 46. The shaft 20 of this speed reducer is coupled shafts represents the parameter a. This synthesizer gives 451 (z' integer from 1 to 4) rotate through ioc. The out between the voltage representative of X(0L) and the output voltage ‘from potentiometer 30 whose winding is which comprise voltage multipliers, phase inverters and slider is ?xed to the carriage 26. of are well known in the art. the ordinate Y(oc). The shaft 773 drives a resolver 21 20 put voltages of the stator windings of the ?rst set of resolvers which represent the cosine terms of the Fourier with one rotor and a single stationary winding which series are individually available at terminals 61—64 and gives the abscissa X(e). Resolver 21 may be considered are applied together in parallel to resistor 39. In‘ the ‘as a second synthesizer in the case where the Fourier same manner, the output voltages of the stator windings series development of the coordinate to be obtained has 25 of the second set of resolvers which represent the sine a single sine term. terms of the series are individually available at terminals The carriage 22 which supports the cutting tool is 65-68 and are also applied in parallel to resistor 3%. subjected to two orthogonal motions with respect to the From output 61-68, there are taken off the necessary workpiece 23. The ?rst motion is derived from a motor voltages to‘ produce by combination a voltage represen 24 which is coupled to a speed reducer 25 and then to carriage 26 via lead screw 27. The motor 24 is energized 30 tative of the ?rst member of Equation 14 and similarly a voltage representative of the ?rst member of Equation via ampli?er 28, by an error voltage which appears 16. The combination is effected in devices 5% and 60 across resistor 29 and which is equal to the difference adding means. Devices of this kind adapted to linearly ?xed to the frame of the machine-tool 31 and whose 35 combine a plurality of voltages and the multiples there The output voltages of devices 50 and 60 ‘are applied, through switch 59, to the motor 32 which is stopped when the controlling 22 which supports the cutter itself. This motion is voltage is zero. developed by motor 34 which drives lead screw 37 through If the motor 32 is controlled by the device 50, the a speed reducer 35. The carriage 22 is, of course, 40 shaft Zil when the motor is stopped represents the angle coupled to the lead screw by means of a nut. am. If the motor 32 is controlled by device 60, the Motor 34 is energized via ampli?er ‘38 from an error shaft 20 when the motor is stopped represents the angle voltage which appears across resistor 39 and which is equal to the difference between the voltage representative In order to enter into the matrix 18, the trigonometrical of Y(oc) and the output voltage of potentiometer 40, functions in am referred to in the ?fth and sixth lines whose winding is ?xed to the main carriage 26 and whose of the square matrix of Equation 17, there are provided slider is ?xed to the secondary carriage 22 which supports (FIG. 5) resolvers 711, 714 715 and 718 whose rotors the cutting tool itself. are driven by secondary shaft 771 separated from the The coefficients of the matrix 18 are numerical coef main shaft 77 by a clutch 70 and are energized by a ?cients as regards Equations 9 to 12 and trigonometrical given voltage representative of the unit of length. There coei?cients as regards Equations 13 to 16. Although are also provided resolvers 721, 724, 725 and 728 whose these trigonometrical coefficients can be obtained by com rotors are driven by the same shaft 771 and are energized putation, they could also be obtained by the control unit by voltages respectively representative of one time, two of the invention itself, whether the abscissa of the times, three times and four times the length unit taken maximum ‘thickness and of the median line sagitta be respectively from sliders 81—84 of potentiometer 80. imposed or not. Generally speaking, this computation A second set of resolvers comprising resolvers 732, is performed by the synthesizers 19 and 21 which re 733, 736, 737, 742, 743, 746, 747 is provided to enter the ceives input data respectively constituted by the value trigonometrical functions in up referred to in the seventh of the coefficient X0 of the sine function representing the abscissa, provisional values of the coel?cients (ll-a4, (if) ‘and eighth lines of the square matrix of Equation 17. The second motion is applied directly to the carriage b1—b4 of the Fourier series development representing the ordinate, ‘and a continuously varying angular value rep resenting the parameter and are stopped when their out put signal is equal to a predetermined value. When X119 (or X5) is known, Equation 6’__‘[‘or (8')] giv They are driven by secondary shaft 772 separated from the main shaft 77' by a clutch 75. Their energization voltage values results obviously from the last‘ line of matrix 17 and the necessary voltages are picked up at a. 423 D taps 81—84 of potentiometer ‘80 according to whether they ing the corresponding value of the parameter is solved represent one, two, three or four times the length unit. for am (or up) with one resolver 33 or any other logo meter device comprising an electro~mechanical servo included in known manner in a feed-back loop. The The operation of the apparatus is the following: The geometrical data are ?rst entered into the plug resolver 33 (FIG. 4) comprises essentially a ?rst stator winding 41 energized by a voltage proportion to XTQ (or XS) picked up at the output of analog translator 17 through switch 76, a second stator winding 42 energized by a current source '33’ giving a voltage proportional to board. If the geometrical data XT or XS, or both, are imposed, the shaft 77 is disconnected by clutch 69 from shaft 20 and connected to shaft 73. The stator winding 42 of the resolver being always energized by a voltage representative of X0, the stator winding 41 is succes sively energized by voltages representative of XT and When XT is applied to X0, and a rotor Winding 43 journaled on a shaft 78 which 75 X5 by means of switch 76. 3,059,852 1-0 stator winding 41, the clutch 70 is operative and the stored therein, and means for deriving from said data and clutch 75 unoperative. Conversely, when XS is applied unweighted and weighted trigonometric functions the to resistor 41, the clutch 75 is operative and the clutch values of said constant coe?icients, and harmonic syn '70‘ unoperative. thesizer means for developing from said coei?cient values 11f the geometrical data XT or XS are not imposed, the and said continuously varying angular parameter the value shaft 77 remains connected to shaft 20‘ of speed reducer of said ordinate continuously varying with said parameter. 46. The clutch 70 being engaged and the switch 59 2. Apparatus for producing a continuously varying being on the position towards '50, the shaft ‘771 stops at signal representative of the ordinate of a closed aerody a position representative of 0am and the trigonometrical namic pro?le, said ordinate being expressed in the form functions of said angle are entered into the matrix 13. 10 of a limited Fourier series in terms of a parameter equal, Then the clutch 75 being engaged and the switch 5? being ‘for a given point of the pro?le, to the polar angle of on the position towards 60, the shaft 772 stops at a posi a corresponding point on a circle circumscribing said tion representative of up and the trigonometrical functions pro?le and tangent to two parallel straight lines tangent of said angle are entered into the matrix. Finally, both to the pro?le near the leading and trailing edges of the clutches 70* and 75 being disengaged and all the data 15 same, said corresponding point on said circle having the being entered into the matrix, the machining operation same abscissa as the point on said pro?le, said pro?le may start. being de?ned by geometrical data belonging to a group comprising the ordinate and the square root of the radius of curvature of the leading edge of said pro?le, the While the invention has been described herein in terms of a number of preferred embodiments, numerous modi?cations and variations, particularly in the synthe sizer structure (use of sine potentiometers instead of resolvers) may be made therein without departing from the scope of the invention itself which is set forth in the appended claims. What I claim is: 1. Apparatus for producing continuously varying sig ordinate and the square root of the radius of curvature of the trailing edge, the maximum thickness of the pro ?le, and the sagitta of the median line of the pro?le, said data being related to the coe?icients of said limited Fourier series by a set of simultaneous linear equations 25 in which said coef?cients are the unknowns, the ?rst nals representative of the coordinates of a closed aero dynamic pro?le in terms of a continuously varying angu lar parameter equal, for a given point of the pro?le, to members are in the form of limited Fourier series in terms of particular values of the parameter and of the multiples thereof, at least one of said particular values being unknown, and the second members are quantities the polar angle of a corresponding point on a circle 30 depending upon the data, said apparatus comprising circumscribing said pro?le and tangent to two parallel matrix means having as input data signals representative straight lines tangent to the pro?le near the leading and of said geometrical data and of supplementary data con trailing edges of the same, said corresponding point on stituted by at least the unknown particular value of the said circle having the same abscissa as the point on said parameter, said matrix means being adapted to derive pro?le, said pro?le being de?ned by geometrical data 35 from said data-representative signals a plurality of co belonging to a group comprising the ordinate and the existing signals representative of the coe?icients of said square root of the radius of curvature of the leading limited Fourier series, means to generate a variable cycli edge of said pro?le, the ordinate and the square root cal signal representative of the parameter, main harmonic of the ‘radius of curvature of the trailing edge, the maxi synthesizer means connected to said matrix means hav mum thickness of the pro?le and the abscissa of the 40 ing as input data said coexisting signals and said cyclical maximum thickness, the sagitta of the median line of the pro?le and the abscissa of said sagitta, whereby the signal, said main synthesizer means being adapted to derive from the input data thereto the value of the ordi abscissa of a point on the pro?le is a sine function of said nate, a secondary harmonic synthesizer means simulat angular parameter and the ordinate is a limited Fourier ing the particular linear equation containing the un series of the parameter the coe?‘icients of which are re 45 known particular value of said parameter, said secondary lated to said data by a set of simultaneous linear equa synthesizer means being connected to said matrix means, tions in which said coef?cients are the unknowns and having as input data said coexisting signals and said are each multiplied by non-weighted trigonometric func cyclical signal and being adapted to derive from the input tions of particular values of the parameter and multiples data thereto the value of the ?rst member of said par thereof and weighted trigonometric functions equal to 50 ticular equation varying with said cyclical signal, and the products of said non-weighted functions of particular means to stop the variable cyclical signal generating values of the parameter and of the multiples thereof means When said ?rst member varying value is equal by a weight equal to said multiple, said apparatus com to the second member of said equation depending upon prising linear equation solving means having as input the data, whereby the value of the cyclical signal when data signals representative of said geometrical data and 55 the generating means is stopped constitutes the supple including, ?rst sine potentiometer means having its output mentary data. connecting to a tool following the pro?le for storing said unweighted trigonometric functions, second sine References Cited in the ?le of this patent potentiometer means having a slider ?xed to a support for UNITED STATES PATENTS the cutting tool for storing said weighted trigonometric 60 second potentiometer means by multiples of said reference 2,478,973‘ 2,660,700 2,808,989 Younkin _____________ __ Oct. 8, 1957 voltage equal to the weight of the trigonometric functions 2,883,110 Spencer et a1. ________ __ Apr. 21, 1959 functions, means for energizing said ?rst potentiometer means by a reference voltage, means for energizing said Mahren ____________ __ Aug. 16, 1949 Gates ______________ _. Nov. 24, 1953

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