ARCHIEF
lab. y.
SchéepsbouwkunJe
Technische
Hogeschool
Mech. Eng. R
65/3Deift
UNIVERSITY OF ADELAIDE
DEPARTMENT OF MECHANICAL ENGINEERING
RECTANGULAR MACHINING COORDINATES
FOR
A N ARBITRAR Y IMPELLER DESIGN
by
M. R. HALE
March, 1965
RXFERENCES TABLES FIG.URES TABLE OF C0NTETTS Page ABSTRACT i 1.0 INTRODUCTION 3
2,0 THE DEFINITION 0F BLADE SHAPE
¿I2,].
2,22.3
Restriction on blade shape Blade shape equations
The blade shape of. the Hrthojet Impeller
5
9
3,0
THE METhOD OP COMPUTING TH MACHININ COORDINATES 103.1 Computing Stages 10
3.2 Points, Reference Planes and Axes 10
3.3
Computation of Surface Points 153.4
Direction Cosine at a Point 18¿.O STABILITY OF CALCULATION 21
¿i.1 Convergence in computing boundary points 21
4.2 Convergence in computing points on surface 22
500 INPtJT/OUTPUT INSTRUCTIONS AND EXECUTION TIME .23
5.1 Iñput Instructions 23
5.2
Output Instructions 25lo COMPUTATION OF
RECTANGULAR MACHINING COORDINATES FOR AN ARBITRARY IMPELLER DESIGN
by M.R. Hale
ABSTRACT
A digitalcomputer programme in FORTRAN has been developed to calculate the coordinates of points on the surface of an impeller (or propeller) blade of arbitrary shape from a series of specifying equations, The programme then computes the, coordinates of points on the
locussurface of the centre of a spherical cutter which would machine the prescribed. surface. All the points calculated lie on a rectangular grid,
Although the impeller shape can be arbitrary, it must be possible to define the blade shape by a series of equations of given form, Certain minor restriotions on impeller geometry must also be Satisfied.
The bladeshape equations
*hich
are required are those which define thefollowing:Radial chord distribution.
Maximumsectionalthickness distribution, Maximumordinate of mean line dis tribut ion.,
Bladé angle distribution,
Täkness form of the blade sections. Profile öf the mean line.
The restriction placed on the impeller geometry are the following:(i) The blade profiles at all sections must have a similar basic
form, superimposed on the mean line,
(2) The mean lines must be similar at ail sections., but may have
(3)
The centroid.s of all sections must lie n a straight radial line, i0e,, the blades must have no rake or skew0The number of computed surface points is limited. by the storage
capacity of the computer, With the IBM 709/7090 II computer, having
a total storage capacity of 32,%1 words (about 28,000 usuable words) a matM of points 53 by 120 can be calculated for each side of the blade,
The output of the program _{is a typewritten list of machining } c9ordinates and instructions in a forni suitable for the operator of _{a}
handoperated vertical ml1.er,
The programme can be modified to cope with impeller or propeller designs which do not satisfy the aboyé restrictions, It can also be
extended to generate machining inatruòtionz for an automatic miller with magnetic tape control,
i.o
INTRODUCTIONThe most common method of machining an impeller or propeller is to use a cutter which moves (relative to the blade) on a cylindrical path with centre on the axis of the impeller0 The cutter motion is controlled by
a follower moving over a series of templates which may be either cylindrical or expandedcylindrical sections depending on the mechanism used. to
convert movement of the folloler to movement of the cutter0
A±iother thethod, which is suited for the majority of milling machines,
is based on spotmachining of points on the blade surface using either a polar or a rectangular
grid0
This method requires considerable computation to transform an object naturally defined. by polar coordinatesinto one defined by cartesian coordinates if the latter are used. In all the machining techniques commonly employed it is usual to use only a small number of sections to define the completé blade shape0 Hand machining is then used to "fair in" between the machined regions0
Disadvantages of the above methods
are:The time con.suined in handfairing between the accurately
machined sections.
The blade surface can only be as accurate as the templates. () The accuracy is dependent on the size of the cutter. It is
not usual to attempt to define the locus of a cutter moving over the blade surface at the desired section0 As the
templaté shape is usually that of a blade section, an appreciable error is thus introduced unless the cutter has a cuttingedge radius small in comparison with the radius of curvature of the blade surface at the point being considereda
Perhaps the ideal method of machining an impeller would be to make use of an automaticallycontrolled milling machine using a magnetic tape
as an input medium for all machining instructions and contröl. The
technique described in the paper does not satisfy this ideal òompletely but is a step towards the fina], objective0
This paper outlines a method of determining the coordinates of points on the locussurface of the centre of a spherical cutter, such that the surface being machined would be the surface of an arbitrarily defined impeller blade The machined points are obtained in a
rectangular grid, pattern0
The use of the programme is illustrated by considering the design of impeller which is being used in a model stud7 of a Hydrojet Propulsion Unit (Ref. i). The 8 inch model impeller (scale ratio of 22.5) was
machined in a Pantògraph copying machine from a threedimensional master template. This master template which was four times larger than
the
model, was machined in a universal miller to d.emensions calculated bythis programme. The intermediate stage of using a master template was
introduced to reduce hand.fairing errors, and. to reduce the number of
machining coox'dìnates necessary to obtain the required accuracy,
2,0 TI DEFINITION OF BLADE SHAPE
2.1 Restrictions on blade shape.
The programme as described. in the following sect4.ons restricts the
blade shape by requiring zero rake and skew, and requiring that the centroids of all sections lie on a straight radial line. This restriction cOuld easily be removed by inserting extra equations defining these parameters in terms of the radius,
5,
shapes0 Corresponding ordinates of blade sections at all radii from tip
to boss must be proportional to the maximum thickness of the sections, The, mean line shape must also be similar at all sections i,e, correspondiÌig
ördinates of the mean line must be proportional to the maximum displacements of the mean lines, These restrictions simplify the problem of defining
the whole impeller blade by a series of related equations,
2,2 Blade shape equations
The following shape parameters must be defined in terms of equations: (i) Chordwidth distribution with respect to radius,
Maximum sectional thickness with respect to radius,
Maximum ordinate of the mean line with réspect to radius, Blade angle distribution with respéct to radius,
Thickness form of the blade section with respect to unit thickness and unit chord width.
Profile of the mean line with respect to unitj chord width.
The form of these equations expressing the blade shape parameters is not restricted in any way, If any of the equations differ from those
used. for the Hydrojet impeller, as expressed below, then these equations
must replace .the corresponding equations in the programme as written in
Table ' 2 The majority of statements which refer to the shape
paraI
meters are fouid between statemén numbered 600 and 660.
In the case of the Hydrojet impeller some of these relationshirs were expressed in the form of polynomial expansions, A previous programme
(Ref. i) had carried out the design of the impeller, om its
output tabular values defining the above distributions (See Table _{i)}
were taken .For some of the bladeshape distributions polynomials were fitted
6
polynomial coefficients were calculated. with the aid of IBM Library
Programme No. 7.00O2, This uses a modified Gaussian elimination method to solve equations representing the condition for leastsquare deviation, the polynomial coefficients then being calculated by back substitution0
The equations representing the design data for the impeller under consideration are given below. Reference should be made to Table
_{5}
for the nomenclature used.(i) Chord.\ width: The impeller design chosen has a linear
chord distribution and. this was represented by the
following relationship between the chord at the tip COD, and the chord at the boss CID, for varying radii RX
CX = COD  (coD  CID) * (RADM  Rx) / (PADM (i,o  RATIO)
..
1This assumes a hub to diameter ratio of RATIO.
(2) Maximum Section Thickness: The thickness distribution with
respect to radius was expressed by three different equations,
each being applicable over a certain region0
Between the radius fraction
0.7
and 1.0, where the maximum thickness TX of the section was determined b7 minimum profile drag conditions, the relationship wasTX = O.0)+5 CX
Over the interval between radius fraction
0.65
and0.7
where the working stress is made equal to the allowable Stress, but where the change in design thickness is small, the thickness to be used. in the programme was assumed constant0i
Por the region between radius fraction _{O.,2 and.}
_{O65}
_{where}the thickness is determined by the design _{stress, a }
thirdorderpolynomial expression was used to specify the thickness to within an accuracy of 0.00012 ft., or 0.,3%.,
TX =
()i
_{OQ}3.,
where
i=
_{0,1,2,3}
and. IR = radius fractionMaximum meanline ordinate:
This distribution was defined by a fourthorder polynomial to an _{accuracy of 0.,}_{5%,} _{the equation}
being
AM=
A(flt)1
..,o ¿f.,
where J. = 0, 1, 2, 3, _{4.,}
Blade angle: _{The impeller was designed. to have the}
chord line
at a].]. sections lying on a helical _{surface.,} _{The blade angle}
was defined as:
ANG = Aretan
Thickness form of the blade section: The biadê section used.
for the impeller was a NACA  16 pröfile. (Ref., _{2, p155).} Because of the large change in curvature from the leading edge to the trailing edge, an equation, _{differing in form from}
a
pOlynomial, had. to _{be found to describe the shape} _{of the section.}
An equation was first chosen to give the correct curvature on the leading edge., _{The profile was assumed}
in this region to
be given (See Ref., _{3)} _{by a parabola of the form}
y2
= 2Kx
where y is the y ordinate at _{a fractional} distance x along the chord.,
Since rd.ius of curvature p =
which can be approximated by a polynomial expansion. for the profile thus has the following
form:yo
= (i  xx)[0/o.98879xX] +
Y.where i = 0, 1, ..., 9
..
6.
The final equation
0 0.0 9,
A tenthorder polynomial was necessary if the maximum percentage error was to be limited to O.l. The overall accuracy of this method for defining such an aerofoil section can be judged from the deviations
shown in Fig. 1.
The ideal trailing edge thickness of the blade is zero. As this is impractical, and in arr case is impossible to machine, the edge
thinkness of the model impeller was made equal to the smallest machinable dimension, Hence the blade section ordinates on the trailing edge
half of the blade were increased above the calculated theoretical value then at the L,E, where
y0
= 0, x O 000 7p=K
For this profile p = K may be calculated, at the L.E. as
O.4888 of chord for unit thickness
At the T.Ee where x = 10 the profile equation must satisfy
Yo 09 X = 1.0
To satisfy this condition the parabolic equation is modified. to
y0(1x) J2Xx'
000 8.The ordinates given by equation 8, if subtracted from the
Y0 for this region to the value YQ' given by
YoQ _{= Yo + ALTE} _{ScALM.}
_{(xx}
_{0,5) /(scALE * Tx)} _{000}_{io,}
This equation applies to the model in question, which has a scale ratio of SCÁLM (1:22,5), The allowable edge thiclaieas AL'1 waschosen to be 0.020 of an inch. The factor SCALE in equation
is the scale ratio óf the rnaster template to the prötotype,
(6) Profile of bean line: The NACA a = .1,0 mean line was used. in. the
impeller design. The equation representing the shape of the mean
line
is (Ref. ii., equation ¿+.26)= C1 _{[(i  x) log} _{(1  x) + x loge} _{X]} _{eee}
_{11.}
C
whére = YM = y Ordinate of mean line
C11 = Ideal lift coefficient for nonviscous fluid
c = Chord width
= xx
= Fractional distance along chord,but f
=0,05515 C1i
where
_{f = AM = Max,}
_{ordinate of mean}_{line}
thus _{(0,05515) [(1 }_{xx)}
_{log} _{(i}A ¿it e

xx) + xx loge xx]
o..
13.
2.3 The Blade Shape of the Hydrojet ImpellerThe dimensions of the Hyd.rojet impeller are gien together with the estimated operating characteristiçs of the impeller in Table
L
From this data equations representing the blade shape parameters were determined.in
the
form described in Section 2,2 These equations formed part of the input to the machining coordinate programmeoTHE hTHOD OF COtJTIN THE MACHINING COORDINA.TES 3.1 Computing St
The programme can be d.ivid.ed. into two major
stagès:(i) From the blade shapè equations previously disòussed,
the coordinates Of the following surface points are calculated:
(i) Points on the leading and trailing edges of the
blade surface.
(ij) Points on the face and back surfaces of the blade.
(2) The direction cosines of the normal to the surface at each
of the above points are determined. By makIng the length
of each normal equal to the radius of the spherical cutter,
the extrasurface points ön the required locussurface described by the cutter centre are obtained.. For convenience
of the machinist the coordinates of the bottom of the cutter are specified in the programme output0
A flow diagram representing the importat steps in the computation is given in Fig. 2.
The complete programme listing is given iii Table 20
32 Points
Reference Planes and AxesTo decÍ'ibe the position of any point in space four sets of axes were used. These can be divided into two groups one groups being
used in Stage (1) of Section 3.1 and the other in Stage (2) of the same Section.
3.2.1 Surface POints
FOr the calculation of surface points (Stage i) the following pairs of reference axes are used (See Fig.
3):(i) A set of righthanded cartesian axes X,Y,Z, aligned so that X ].ies.alòng the liné of centroids of the blade séctiona and Z along the axis of the impeller0
(2) A set of righthanded cartesian axes XB, YB, ZB, aligned _{parallel}
to X, Y, Z, and. with YB colinear with Y, as shown in Fig. 3 The distance between the X, Z planeandlB, ZB plane is _{denoted}
YD.
To simplify the oomputation,the points chosen to define the blade
surface were those resulting from
the
_{intersection of a twodimensional} fundamental mesh (Ref. 6) with the blade surface0 The planesforming the fundamental mesh were aligned respectively parallel to X, Z axes and. the Y, Z axes.
Pianes parallel to X, Z axes are called column planes and _{planes}
parallel to Y, Z, axes are called row planes or stations,, _{The }
distance between column planes is not equal (See below). _{The distance}
between row planès is equal, and has a value DX.
The row pianes are identified as I = 1, 2, 3, .... as shown in
Fig.
_{5.}
It was necessar7 to select I = i outside the blade surface tosimplify
the computing routine0 Thus the row plane passing through thé tip of the impeller on the X axes (see Fig.5)
waschosen to be I 5 row plane.
The column planes associated with each row plane are named
separately (Fig.
5)
and by two different methods:(i) Nomenclature of column planes  Method 1.The plane containing the axes X, Z, is chosen as the datum plane N = O and. planes on either side are named by N = 1, 2, _{3}
O...,
an additional identification of M = i or 2 being requiredto indicate which side of the datum is being considered0 This
distance between planes is (up to a certain distance from N o)
equal, and. of value SPACX. At a predetermined distance from the
N O plan the spacing between the column pianes is changed to
SPACN (one fifth of SPACX) and. the identification of the planes
continued. nuñierically without further interruption.. The
predetermined distance is chosen to be equal to or greater than a given fractional distance of the value of the Y ordinate of the point on the leading or trailing edge on the same row plane. This
fiactional value PCX is given as an input instruction to the
programme. The reason for using planes with a smaller spacing
(SPACN) over part of thé blade is to define more accurately the
blade surface in the region where the surface curvature is increasing rapidly near the blade edges.
The numerical value of N associated with the column planes where
the spacing distance changes from SPACX. to SPACN is recorded as
the value of LEJE (I) and of MTE (i) where these corréspond to the
leading and trailing edge respectively at a given row plane I.
The total number of N planes on either side of X, Z plane are recorded as NLE (i) and ITE (L) for the leading and trailing edge side at a given row plane I (Fig.
5).
It should be noted that column pianes axe not equally spaced over the entire blade surface, and. also that points on the same plane
parallel to the X, Z axes but in different row planes may not have
the tame numerical value of N.
Assoôiated with are point on the blade surface a curved. plane YR.,
ZR. (Pigs. 3, 5) passing through the point can be defined as a
portion of a cylindrical surface with its axLe coincident with the
J.).
The above identification of ooluffln p1anes applies only to the
computation in Stage (i). The following nomenclature was used.
in. the programme to store values associated with these
planes:(2) NOmenclature of column planes  Meihod 2.
The column planes N are identified in. another way by being num
bared. consecutively from the trailing edge to the leading ed.ge
with the identifier (subscript) of NNN. The values of NI1N are
chosen 5o that the numerical value of the N1N plane containing the X, Z axes (i.e. N = O plane) is made equal to a specified value NN. The value of NN must be òhosen so that the identifier
NIIN for any column planes on the blade suz'faoe is positive and. at
least greater than N1'IN =
_{5}
The chosen value of must be specified as an input statement. For the Hydro jet impeller, thevalue of 11 was chosen to be 60.
It should be noted that although the column plane N are numbered consecutively their spacing is not constant and that for different row planes the column planes which hOld. the same
value of NN are not necessarily coplanar.
A point on the blade surface can be identified. by the numerical values
of the row and. column planes which pass through lt. The additional
subscript J = i indicates back Or face Surface of the blade. Thus a point
(i, NNN, J) = (6, 13, 2) would be a point on the face surfaöe of the blade at I = 6 whose identifier has a value of NI"lN = 13.
 The coordinates of points a5sociated with the axes XB, YB, ZB are given
by m(i, ISNN, J), YB(I, N, J) and ZB(I, NNN, j). For example
Before the direction cosines of the normai to the surface at a point can be calculated, a number of neighbouring points must be located from the complete array of points which was located. in the store of the computer.
Tó aid the identification of points in this section of the progra.e, each point is now renamed according to its distance from the N = O plane0
Consider a series of planes parallel to N = O plane, which have a spacing interval of SPACN. On the trailingedge side of the axes X,
Z a column plane is chosen as a datum plane I'IK = O and ali the column
planes spaced SFACN are identified consecutively from the datum as = 1, 2, 3, .., (F.go
5)0
The numerical value of the IIK plane containing the X, Y axes (i.e. N = O plane) is represented br NKN in the programme. The value of NKXN is controlled by an input statement and.its value is further discussed in Section 5.1.
Thus in the centre of the blade surface where the spacing of the
surface points is SPACX, every fifth .I plane will be an N plane. Points
which hold the same value of the identifier I'IK will be pbysically on the
same column plane, and hence are easily distinguished from the complete array of points.
The 16 points which surround a given mesh point from which a normal is to be erected are located using the NK identification above. These
points are then reidentified by a separate nomenclature KA. The mesh
point is chosen. to be the point KA
= 5.
The points on the row plane passing through KA_{= 5}
are identified. as L = 1 and KA = I to _{9} in thesame sense ae the positive direction of NNN (See Fig.
_{5)0}
The points on a column. plane thràugh KA= 5
are denoted b7 L = 2 and KA = I to9 in
the same sense as the positive X direction, Each of these points KA = I¿50
3.2,2
Extra  Surface PointsTo identify points outside the blade surface in the section of the programme which computes the direction cosines of the normals, and the machining còordinates, (iee, Stage (2) Section _{3i) the following} pair of reference axes are used (Pig, _{4),}
(i) _{A set of righthanded cartesian axes XN, YN, ZN which} are
parallel to the X, Y, Z ares but with _{XN displaced a distance}
1!D from X, and with YN _{displaced a distance ZD from Y (See}
Fig. _{4).} _{This set of axes is only used}
to identify the
machining points associated with _{the face surface.}
(2) A set of right.handed.
cartesian axes XM, YM, ZM obtained by rotating bodily the set of _{axes XN, YN, ZN about the X axis}
as shown in Fig, _{4.,} _{This set of axes identifies}
machining points associated with _{the back surface,}
The coordinates of points _{related to these axes are}
specified as Xxx(J), YYY(j), ZZZ(J) _{where J = 1 or 2 refers}
to the back or face and hence _{identifies which set of}
axes is
being considered,
3.3
Computation of Surface PointsThe computation of the coordinates of points on the blade _{surface} from the blade shape equations _{is divided into two }
sections:(i) _{Computation of points on the boundary edge of the blade} surface for each row plane,
(2) _{Computation of points on the blade surface at the intersection} of each row and column plane0
3.3.1
Computation of Boundary Pointsgiven row plane are calculated and stored so that th _{calculation of} points in the following section 332 can be confined to only the blade surface (See 2/1).
The iterative convergence procedure called "Regular Falsi" is used.
tö compute thèse boundary points (See Fig. _{2/13).} Here the
ludeperìd.ant variableradius RX,ia adjusted to make the dependent variable,
1engthof..the..arc CA equal to CB a 1iown proportion öf the chord at
the Íad.ial distance RX. _{These distances are shown in Fig0}
_{5.}
Before entering the "Regular Falsi" routine the distance Rl (See Fig 2) was approximated byRI = RADM * P.CR + X
0
where X = X ordinate of the row plane
Te value of PKR.CR must be selected to producé CA just greater than CB for all row planes of the impeller.
The Regular Falsi procedure is as fo11ows.
The value of RX is progressivèi..y aecxeased until the difference
between CA and CB changes its numerical sign. At this stage,. the
d.esired value of RX is between the two previous values0 The
calculation is then repeated for a value of RX midway between these two values. The values of RX whieh surround the desired. value are
again selected from the preceed.ing two values of RX and the currént value.
The convergence procedure is continued until the required. accuracy is
obtained0
At eaöh statiòn, the lading and then trailing edge coordinatés are computed and then stored as subscripte& variable in I (the row plane number).
The computation in
this
section is shown diagra.matically in the flow diagrams Fig. 2/1 and _{2/13.}3.3,2
Computation of Surx'ace PoixtsBefore entering the computing routine for the surface points, the subscripted variable YB(I,IUN) (i.e0 the Y ordinate) for each point
on the twodimensional fundamental mesh which covers the blade surface is equated to zerO. This is a necessary condition for the
calcu].ation stage to be described in Section
3i.
The geometry of each blade section is specified by three variables CX, TX, AM and by equations for the thickness form and. mean line
profile (See Fig.
6),
From this data the Z ordinate (of the surface) at a given surface point is computed by an iterative procedure (See Fige, 2/2 to 2/6). By considering a cylindricalblade section passing through a point (i, N ) an initial approximation of the distance along the chord line to the desired surface point at (i, N ) is made by assuming this distance to be equal to XC the distance between E and Q (See Fig,
6),
At this cylindrïòal distance XC from the leading edge, the coordinates for the points on the blade surface are compute&an& compared with the desired values0 The value of XC is then progressively
adjusted using the "Regular Falsi" iterative convergence technique until the coordinates are determined to within the desired accuracy,
The calculated coordinates of the chosen points on the surface are designated by the suhscx'ipted. variable (i, NNN). The subscripted.
index INN refers to the numerical order of the points in the positive YB direction from the trailing edge to the leading edge, and is centred about a point on the X, Z plane where N1N = NN. This mèthod. of
subscripting is in contrast with the more usual method of naming points according to their distance for a reference axis, The former method was chosen because the available storage locations ±n the main store of
18 the IBM
_{709/7090}
II computer were limitéd. _{The execution time for} this programme would have been greatly increased if these variable had to be placed ona mgnetio tape store0 _{These factors necessitated} keèpixig 'a numericâl count of the number of points _{on each row plane}(i) which had' spacing intervals SPACX and SPACN0 A biowledge of this numerical count was also necessary before the coordinates _{of points}
surrounding any given point could be selected from the _{stored variables}
specifying the given point0
34
Direction Cosine at a PointSince the mesh of points set up to óover the blade surface _{i}
rectangular, some of the mesh points lie in a region outside the blade area and are therefore disregarded in the following computational routine, _{These points are easily distinguished because their YB}
ordinates have previously been equated to _{zero (Section} _{33,2) }
whereas all other points on the surface have values of YB(I,NTN, J) _{greater}
than zero (Fig0 2/7)
In turn, at each surface mesh points (I, NN, J) the _{coordi.nats}
of sixteen points arranged about the given mesh points as shown in Fig, _{5 are selected from the complete store of points (See Fig,}
2/7 to 2/12). _{The spacing distance between points in the} _{row plane}
and the column plane must be equal, _{These points are reidentified.}
as KA = 1 to
_{9}
on both the row plane and the cOlumn plane, in both cases being centred about the given mesh point (KA_{= 5),}
_{'om these}
sixteen 'selected points, the most symmetrical array of eight points surrounding the given mesh point on the blade surface is selectedThe interseátion of the row plane through the mesh point, and.
the surface of the. blade, determines a space curve through the point
the "row space curve". Similarly the intersection of the column
piane through the mesh point, and the blade surface, determines the "column space curve". The gradient (i.e. differential coefficient)
of the row space cürve at the. mesh point, can now be found by
substituting the ordinates of the four selected points on the row plane plus the ordinate ôf the mesh point into the appropriate equation for a fivepoint GregozyNerton differentiation0 Similarly, the gradient of the
column space curve at the mesh point can be föund. Each gradient is of òourse, equal to the slope of the tangent to the curve,
By combining the gradients of the two space curves through the mesh point,. the direction cosines of the normal to the surface is
ca].culate. as folloWs0
Equation of.tangent to row space curve at mesh point XB, YB, ZB is given by,
z = a(y  YB) + ZB . OO 15,
where a .= (i,J) = Slope of tangent to surface in YB direction
Similarly the equation to column space öurve at mesh point is given
z=b(xXB)+ZB
,,
16Where b =. GR.AD(2,J) = Slope of tangent to surface in XB directiOn
Hence eq.uation of plane containing both tangents is given by,
20.
The direction cosines _{COSA,} _{COSB,}COSC of the outward _{pointing}
normal
to the back surface_{áre}

G.RAD(2,J), _{ c.AD(I,J),}_{+ ].}
,2(2J)
_{BAD2(IJ)}
+
Note
that:Direction cosines of
_{normalI}
Direction, cosines of normal to back surface
) to face surface
... 19
The position of the _{centre of the milling cutter along the normal} to the given
surface
at the mesh point is now calculated,. and specified in terms of the
_{machining}
_{axes (Fig0}4.) The information
required
by the machinist is not the blade surface
_{dimensions,}
butthe motions of
the three lead
_{screws neces3ary to traverse}
_{the}_{cutter from the known}
origin of
theaxes to
_{the}_{point in question0}
Hence it Was necessaiy to
convert each coordinate of
_{the cutter centre into}
_{an equivalent number}
of complete turns and
_{parts of a turn (j.}
_{e. thousandths of an inch) of}
the appropriate lead_{screw0}
_{Since the machinist}
can easily check the
positionof the bottom of
_{the}_{cutter, it is more}
useful for the position
of this point to be
_{specified than the position Of thé cutter centre0}
Thus the fina]. print...out
_{is in terms of the}
_{position of the bottom of}
the cutter.
Since the blade surface
hasso far only been determined
accurately
up to but not at the blade edge, further points are required
_{to define}
the actual blade edge.
_{These points are}
_{calculated by a similar}
differentiation technique to
_{that}
used.for the points
on the surface.
These edge points
_{are then specified with reference}
to the sane
machining axes described abOve
_{and illustrated in Fig.}
_{4..}The corresponding
positions Of th cütter are also calculated0 The blade shape is ñirther
d.efinedby specifying the radius of the leading edge (iT.i(I)) at each
row plane0
.O STABILITY OF CALCULATIONS
Then applying this computing technique to an impeller or rOpel1er design, care must be taken to ensure that the iterative routines incorporated. in the Solution do in fact converge0 The rate and
accuracy of the convergence must also be checked0
The two iterätive prôcedures in the prograe are considered separateIr in, the following,
Li.,]. Convergence in computing boundary points, (Ref0 Section
3.3.1)
This routine is convergent provided that the initial value assumed for RI is greater than the actual value of RI at a given row plane. This is governed by statement number (170 + 0002) Table 2,
RI = X + RADM * FCR ... 20
The value of PERCR. can be adjusted for each impeller design and is
submitted to the programme by an input statement0 The rate of convergence is governed by statement
230 RI = RI.  STEP]. ,,. 2].
The accuracy of the calculation of the boundary point is determined by t'e test statement
I'(ABsF(cA  CB)  BEVNL)270, 270, 260 ,.,. 22
This terminates the calculation of CB when an accuracy of less than + or  (DEvN1) of an inch is obtained.,
k.2 Convergenòe in mputing points.on surfaôe (Ref. Section 3.3.2)
(i) n Stage 1 of the òomputation the N plane on which the surface
point is located. is progressively moved closer to the trailing
edge of the blade. The position of the surface point n an
N plane near the trailing edge needs careful investigation0 This stirface pOint must be located on the actual blade surface9
i0e0 the face or back surface9 and not on the square region of thè edge (See détail A. FIg0
_{6)}
In statement (4.3o + 000i) Table 2, the Y ordinate of the
th
N plane is tested to determine whether the plane cuts the
blade n the blade surface or not0 The significant variable
in this expression is YYTE. This variable représents the
value of the distance, parallel to the X, Y plane, from the intersection of the chord line and the trai ng edge to the
intersection of the face surface of the blade and the trailing edge, at a given row plane0 The value of .YYTE is derived
from the allowable edge thickness ALTE, and the actual biade thickness YO, at XC = lO. This value. YO is referred to
in the programme as DEVN2. (See statement 328, Table 2)..
DEVN2 exists because of the inaccuracies in the equation defining the thickness forme The actual value of YO at
XC = 1.0 for the NACA  16 profile is zero0 The extra factor of i00O]. in statement 34.8 increases the. distance TITE to allow for accunulated errors in the computation0
if another thickness form is used in place of the NACA  16,
the value of DEVN2 submitted as input to this programme shoi1d be a zero or positive value although its actual value may be
negative0
(2) By choosing an imitial vlue of XC as given in statement
(65Z ± 000i.) (Table 2) a solution is always possible provided.
o XX 1,0 where XX = XC/CX
23.
i.e0 XX must be within the blade section,
If during the. iteration the value of XX (See statements _{790,} 810) falls outside the above range, it is foròed to hold one of the limiting values0
A with the first convergence routine, the rate of convergence and. accuracy are governed by statements _{790,} 810, and 820, (Table 2) and. values of STEP2 and DEVN3O
5,0 INPUT/OUTPUT INSTRUCTIONS AND EXECUTION TIME
5.1 Input instructions
The programme as written (Table 2) will accept d.ata for any
iìnpéller or propeller satisfying the stated. conditions and whose
blade shape is capable of being expressed by equations of the form given in Section 2,O
An example of the input data, taken from the HYDRO  1 impeller, is shown in Table
3,
The values of most the variable have dimensions in feet and apply to the prototype impeller (See Table5).
The following items in the input data require. further discussion,
me válues of the constants YD and ZD must be chosen to maintain all values of YN(I,NNN), PTY(J) and PTZ(J) positive. This places
the origin of the axes XB, YB, ZB and .J, YN, ZN, and XM, YM, ZM outside the fundamentál mesh which covers the blade surface,
number of points to be calculated in any one row plane on the traîlin edge side of the N = O plane section. The value Of N1 must be
divisible by five
_{(5)}
and equál to or greater than the maximum value ofN(I) + 4S IvITE(I) + _{4.}
..
24.The maximum value of this expression occurs where the projected blade area on the X Y plane has the maximum Y dimension.
The dimensión of YB (i,NNN) in the X d.irection i.e0 I, must be
at least 8 more than the total number Of row planes required0 The
other subscripted, variables referring to row planes R.XLE(i), iLE(I)
etc0 must have the subscript greater thàn the total number of row
planés, by at least four (4).
The dimension of YB(i,NNN) and
ZB(I,N,J)
in 'the YB direction j.é. NN, muà't be 'at least tWice the value of NN0The machining dimensions can be calculated for points between any twò row planes, The input variables controlling this are the values
of IP and IPATO The value of' IP selects the beginning point for the
calculation and is the (i 4.)th row plane. The value of IPAT
determines the end row plane with I = (IPAT + 14) where the calculation
ceases, If the (ip + i) and (IPAT + 4.) row planes lie within the
blade surface area and. are not on the edge of this surfaóe then the
vaMablés associated with the first two and. the last two row 'planes have small errors due to the differentiation in Section 3.4.
Care muet be taken when choosing the value of DEVN2 and. reference
should be made to Section 4.? for a discussion on its value0
As prev.ous].y mentioned in Section 4.O the tnitial value of
Falsi" routines9 must b carefully selected0
If the leading and trailing edges are to be adequately defined then the value of PCX must be less than 090 This variable PCX d.etermines the point where the spacing of column planes change from
SPACX to the srnaller spacing of SPACNO
If a progressive printout of aU major calculations and decisions in the programme is needed NTEST should be set to a positive number,
otherwise it should be zero or negative0
The' programme as written occupied approximately 25,300 words in
the store of an IBM 709/7090 II computer with main storage capacity of
32,561
words052 Qutput instructions
The machining technique governs the form of the output instructions. The output for the Hydrojet impeller was chosen to be punched card.s
which were later listed. These output instructions are in a form
suitable for the machinist of a handoperated vertical milling machine0 The machì,i instructions for the back and. face surfaces of the
blade could not be separated in the computer without increasing storage and. running time0 Each alternate card. of output thus refers to the
same blade surfce0 The deck of output cards may be later processed by an IBM Collator, to separate the alternate cards0
In the output listing the row planes are called stations and are numbered from one (i) at the blade tip0 Thus each station is actually
 th
the
(I
+ row plane.Aesociated. with each point on the blade surface and. its machining
coordinates there is a reference number. This reference number is
actually the number of the th column plane which passes through the point. All surface points with the same reference number IK are
26.
ph'sically on the
sameóoiumn
plane0The
xample of the machining instructions given iii Table
1
_{Is}
fo±theHDRO 
i
impoller
at a statioi onthe back surface.
Fig. _{7}shoWs a plot óf computed points on the back
and face siface ofthe
Impeller at selected sections.
503
:Exeoution Time
The average time taken by the IBM 709/7090 to calculate the machiM ng
coordinates for 100 mesh points (i.e. 100 points on both the back and.
face surface)
Was approximately 21 seconds.
Thus the execution time
requiredfor the 39666 mesh points of the impeller HYDRO
1 was_{765}
Ref e reircese
REFERENCES
McGraw Hill. 1955.
1. HALE, M.R:  The Des±gn of Ducted Impellers using a Vòrtex
line Analysis and. an Optimizing Compuier
Technique0
University of Adelaide, Dept0 of Mech. &ig, Mech0 Eng
_{R65/}
2. 0 'BRIEN, T.P: _{The Design of Marine Screw Propellers.}
Hutchinson & Co. Ltd. London. 1962.
3.
HLSTING, C.J: Approimtions for Digital Computers. Princeton University Press0 1955.4
ABBOTT, I.L and Theory of Wing Sections  including a VON DOENHOFF, A0E:Summary of Aerofoil Data0
Dover Publications, Inc., New Yörk, also NACA Rep0 824  1945.
CHORD
Table 1. _{Impeller Dimeflion8 ana Characteristics}
DESIGN PARAMETERS
_{HYDROJET IMPELLER}
DESIGN ASSUMPTIONS
BETZ MIN ENERGY CONDITION
CONSTANT AXIAL VELOCITY
NACA PROFiLE
_{SHOCK FREE ENTRY}RADIUS
_{VELOCITY}
_{ROTATION} _{BLADES}_{IDEAL EFF}
TOTAL EFF
THRUST SHP7.50
_{19.30}
_{1.50}
_{4}_{0,9733}
_{0.8699}
44800.
1807.
BOSS
_{TIP}
STRESS
CHORD
_{BOSS RATIO}
_{IMMERSION} _{ALLOWABLE}TIP
5.000
RAD FRAC CHORD
3,000
_{0,200}
_{11.50}
THI CKNESS
MEAN LINE
ANGLE B.!7250.
LiFT COEF
DRAG/L IFT
1.00
5.000
0.2250
0.092 &
_{15.67}
_{0.249}
_{0,030}
0.95
4.875
_{0,2194}
O .0 876_{16.45}
_{0.241}
_{0.031}
0.90
4.750
0. 2137
_{0.0826}
_{17.31}
_{0.233}
_{0.032}
0.85
4.625
0.2081
0.0776
18 26
r'.y.
_{0.033}
0.80
4500
0.2025
_{0.0726}
_{19.32}
_{0.217}
_{0.034}
0.75
4.375
0.1969
_{0.0675}
_{20.51}
_{0.201}
_{0.036}
0.70
_{4.250}
_{0,1912}
_{0,0625}
_{21 .84}
_{0.197}
0.038
0.65
4.125
0.2035
0.0574
23 o 3k_{0.187}
_{0.040}
0.60
4.000
0.2302
_{0.0,23}
_{25.06}
_{0.175}
_{0.044}
0055
3.875
0.2 562
_{0,0471}
_{27.02}
_{0.163}
_{0.049}
0050
_{3.750}
_{0.2813}
_{000419}
_{29.29}
_{0, 1O}
0.056
0.45
3.625
0.3056
_{0.0367}
_{31.94}
_{0.136}
_{0.066}
O 40
_{3.500}
0. 3289
_{0.0315}
_{35.04}
_{Oc, 1.21}_{0.080}
0.35
3.375
0.3508
0.0263
38.71
0.104
0.100
0.30
3.250
_{0.3723}
_{0.0211}
_{43.08}
_{0.081}
_{0.130}
0.25
3.125
00391.5
_{0.0160}
_{48.29}
_{0.069}
_{0.178}
0,20
_{3.000}
_{0,4056}
_{0,0112}
_{54.51}
_{0.050}
0.264
Table
1.
Continued0
RAD FRAC
ST F
ST B
ST E
ST N
CAVI S
CAVI P
THRUST GRAO TORQUE GRAD1.00
o,
0.
0G0,
0.555
0.347
00181E 06
0.425E 06
0.95
090
505.
140.
'127L,
385e
113.
113.
386.
0.68
0.615
0.340
0.333
0.155E
0.132E:
06
0'
0.363E
0.308E
06060.85
1149.
133C834.
834.
_{03(66}
0.325
0.111E 06
o259E 06
 80
2089. 2476e
1444. 1441.0.861
0317
0.922E 05
0, 216E 060.75
3392. 4061,
2229. 2216.0973
0,3090.759E 05
0.171E 06
0.7Ó
5095. 6113. 3168. 3139. 1.1060.300
0.616E 05
0a144E 06
o 65
6204. 7246.
3300. 3246. 1.265 0.3030.492E 05
Oo 115E 060.60
6454. 7248.
2?58 2673.1.457
0.3180.386E 05
090,t 05
0.55 6657e '7247. 2330e 2201. 1,690 0.334
0.296E 05
o699E 05
0.5Ò
6829.
7248.
1989.
1800.
1.975
0.349
0.221E 05
0.528E 05
0.45
6976. 7249.
171?. 1446.¿.327
0,365
0.159E 05
0.387E 05
o 40
7101. 7247, 15O3 1122.2.762
0.3810.110E 05
o 274E 05
0.35
721107241.
1346. 816.3303
0.397
0.719E 04
0.185E 05
0.30
7247. 7184.
12 36 507.3.969
0.415
0.433E 04
0.118E 05
0.25
7247. 7099.
1191. 194.4.779
0.4320.228E 04
o 695E 04
Table
2.  Programme Listing.
C
**1330 M R HALE/FOtLER MECH ENG DEPT U OF A TEL 461**
C
**RECT COORDINATES
MACHINING DLMENSIONS4H*
C
**PROGRAMME NO 0022/7090**
010 FORMAT(].H4,42H
PATTERt COORDINATES OF HYDROJET IMPELLER
015 FORMAT(1HO,47H MACHINING COORDINATES UNIFORM RECTANGULAR GRID)
020.FORMAT(1HOD15H MODEL SCALE
F703).021 FORMAT(1H ,15H MODEL RADIUS
F703v3HINS
025 FORMAT(1H0920H
REFERENCE DISTANCES)030 rORMAT(lHo,41H
REF PT TO LINE OF CENTROID
N Y DtCN
o35 FORMATtÌH ,41H
REF PT TO LINE OF CENTROID
N Z DIRCN. ,F7.3)040 EORMAT(1H19J3,1ÌH STATION NOo9X16HaACK COORDIÑATS
041 FORMAT(1H1,13,1IH STATION NO,9J16HFACE COORFflNATES)
045 FORMAT(1HO,30H
RADIAL DISTANCE TO STAflON
F8.3)048 FORMAT(1HO,11X922HMACHIMING
COORDNATES916X16HßOTTOM 0F CUTTER)
049 FORMAT(1H5,6X17HPØINTS ON SURFACE)
050 FORMAT(1H ,9X1HXr12X,1HY,12X9 1HZ12X1HX,8Xv1HY,8X,1HZ)
051 FORMAT t 1H5,5X91HX8X91HY,8X,1HZ)
055 FORMAT(1H 939H
NO REV THOU
REV THOU
REV THOU)
060 FORMAT( 1H
062 FORMAT( 1H ,4XF90394X,F90394XF90fl
065 FORMAT11HO18H
ST CHORD WIDTH =9F8o37XG11HLE RADIUS
,F6.3)066 FORMAT(1H53F93,X6)
067 FORMAT(1H 943X93F903)
. r068 FORMAT(ÌH ,I4)
070 FORMAT(4E14e7/517/5E1407/3E14r7/5E14.7/4E14.7)
080 FORMAT(3E.14.7/4E1407/5E1407/5E1407/3E1407,5E1407,5E1407)
.090 FORMAT(5F10059F303,15)
091 FORMAT(4F1003D18)
DIMENSIONRXLE(57)9RXTE(57),YLE(57),YTE(57),RLE(57),NLE57),MLE(57)
2GRAD(2,2),PTX(2)9pTy(2)pTZ(2)9yyy(2ZZZt2)
COMMONYB
100lDX,SPACX,SPACNPCX9PERCR9YDZD,ALTE,STEP1,STEP2DEVN1ODEVN2,DEVN
Table 2.  Continued..
2CUTR,XMILR,YMILR,ZMILR
READINPUTTAPE2,O8Q,COD,CID,CX79TO,T1,T2T3,AO,A1,A2,A3,A4,AMO,
1AMI9ANGo,ANG1,PD,HBC,YBCT,RADLEiYOY1Y2,V39Y4iY5,Y6,Y7,Y8aY9
UNIT12.O/SCALE
RADMRAD*UNIT
PUNCHO 10PUNCHOÎO
PUNCHO15
PUNCHO 15PUNCHO2O,SCALE
PUNCHO2 O, SCAL E.PUNCHO21,RADM
PUNCHO21 ,RADMPUNCHO25
PUNCHO25
PUNCHO3O , YD PUNCHO3O , YD PUNCHO3 5 ZD PUNCI1035 ,ZDc
**COMPUTING END POINTS AT SPECIFIED RADIAL DISTANCES**
013.1415927
P I TCH=PD*2 O*RADM COD=COD*tjNI TCIDCID*UNIT
AMO=AMO*UN I TAMIAMI*UNI T
YR(1.O_HBC)*COD*COSF(ANGO)+YBCT*AMO*SINF(ANGÓ)
XOL=RADM*COSF (YR/RADM)
YR(1.OHBC)*CID*COSF(ANGI )YBCT*AMI*SINF(ANGI)
XILRATIO*RADM*COSF(YR/tRAT1O*RADM
YRHBC*COD*COSF ( ANGOHYBCT*AMO*SI NF CANGO)
XOTRADM*COSF (YR/RADM)
YRHBC*C1D*COSF(ANGI)+YBCT*AMI*SINFANGI)
IPPIP+4
I 1IPAT+4
H p, H eTable
2.  Continued..
A15
XRADMDX*A
D03.10M1
92IFU4Ii1QOO11Oß14O
110 1F(XXOL)13O910g120
120 RXLE(I.)RAIM
RLE( I)0e0
YLE( ISQRTFtRADM*RADP4X*X
GÓTO31O
130 IF(XXIL)10009170917O
.140 1FtXXOT)16O916O9150
150 RX1T(t)RADM
YTEC I )=SQRTF(RADM4'RADMXX)
OT01O
j,0 IF(XXIT)100091709170
170 RB00
RA00
RXX+RADM*PERCR
180 CA=RX*ATANF(SQRTF(RX*RX.X*X)/X)
IF(M1 )1000,19Oß200190 K1
GOTO600
195 C.B=(100H8C)*CX*COSF(ANG+YBCT*AM*SINF(ANG
GOTO21O200 K=2
GOTO6001205 CBHBC*CX*CO5F (ANG)YBCTAM*SINF (ANG)
210 IF(CACB)240270,220
220 RARX
IF(RB)10009230ß250.
230 RX=RXSTEP1
GOTO10
240 RBRX
250 IF(ABSF(CACB)DEVN1)27092709260
_760 RX=O,5*(RA+RB)
N
Table
2.  Continued..
GOTO 180
270 ¡F(M1)1000,280,300
280 RXLEU)=0.5*(RA+RB)
YLEC I )=SORTF(RXLE( I )*RXLE( T )X*X)
RXRXIE(I)
K3
GOT0600
290 RIEf I )RADLE*TX*TX/CX
GOTO31O
300 RXTE(I)0.5*(RA+RB)
'(TEl T )=SORTF(RXTE( I )*RXTE( I )XX)
310 CONTINUE
!F(NTEST) 330,330,320
320 WRITE OUTPUT TAPE 3,090,X,RXLE(I),RXTE(I),YLEtI),YTE(I),RLE(I),1
330 CONTINUE
**CAICULATING COORDINATES OF POINTS ON SURFACE**
ç
**UNTFORM GRID SPACTNG**
IT!TPAT+8
NN2=NN*2
D03401=].,ITI D0340N=j ,NN2340 YB(I,N)=0.0
D0550!IPP,II
A=I5
)X=RADMDX*A
1Ff 15) 1000,344,346344 NLEII)=0
I) =0 NIEl I )=0MIEf I)0
G0T0515
46 NIE(1)10*NN
NTE(1)=10*NN
RX=RXTE (I)60T0600
Table 2.  Continued. 4O
D0510M1,2
v=o.0
005 1ON=1 ,PIN
rr(M]. ) 1000,350,400
350 iF(NNLE(I)1)360,5l0,510
360 IF(YPCX*YLE(
I)370,380,380
370 (=Y+SPACX
MLE( I )NG010450
380 YY+SPACN
IFtYYLE( I) )450,390,390390 NLE(I)N1
GÓ10510
400 IF(NNTE( I)1)410,510,510
410 IF(YPCX*YTE(1) )430,430,420
420 Y=YSPACX
MTE(I)=N
G0T0450
430 YYSPACN
¡f(Y(yTE( I )+YVTE))440,440,450
440 NTEI)=N1
GOTO5]0
450 RXSQRTF(X*X+V*Y)
IF
(RXRAT I 0*RADM ) 460,470.470460 Z(1)=0.0
Z(2)0.0
YMYD
G010480
470 K4
YA=Y
IF(M1 ) 1000,47.1,472411 YR+RX*ATANF (SORTF t RX*RXX*X ) IX)
$
GOTO600
472 YR=RX*AIANF ( SQRTF t RX*RXX*X ) /X)
Table
2.  Continued..
480 j'FtÑi.1O00,49Ó,495
490 NNNNN+N
ÓTO46
495
NNN=:NN.496ZB(1vMNN1)Z(i)
ZB(I,NNI1,2)Z(2)
YB(I,NN4)=YA+.YD
1F (NTEST)5i0,510,00
500 WRITE OUTPUT TAPE 3,09i,ZaIóNNN,1),YB(I,NNN),ZE(jNMN,2),X,NNN
.510 CONTINUE
515 RXX
YR2O.O.
K=5IFCRXRÁT10*RAOM)52O,6Ô0,60ô
520 Z(1)0.0
¿(2)20.0
YR=VD
530 ZB(t',MN,i)=Z(l)
ZBiI,NN,2)Z(2)
YB(I,NN)=YR+YD,
IF( NTE&T ) 550 , 550, 540
540 WRITE OUTPUT TAPE 3,091,ZBU.,NN,1),YB(I,NN),ZB.(t,NN,2),X,NN
550 CONI INUE
**DETERMINATION OF DiRETION COSiNES AT A POINT**
**HENCE CORRECTING COORDINATES FOR THOSE AT BOTTOM 0F CUTTER**
003Ô00I=IPP,I I
AI5
IA=I4
X=RADM.DX*A
PUNCHO4O, LAPUNCHO4I,IA
PUNCHO45,x
PUNCHO45,X PUNCHO48 PUNCHO48 PUNCHO49Table
2.  Continued.
ÚÑHÖ49
PUNCHO5O
PUNCHOS O PUNCHOS iPUNCHO51
PUNCHO 55PUNCHOS5
*TRAILKNG EDGE COORDINATES*
SPX120O*DX
IF(X+2o0*DX'XOT114O91.80911iQ.
1110 IF(X+DXX0T)1130p11OlU5
1115 IF(.XOT)11201120140O
1120 GDT(300*YTE(I+4)16.0*YTEU+3)+360*YTE(I+2)48o0*YTEU+1)+
120*YTE( I) )ISPX
G0101200
1130 i) /SPX GOTO 12001140 1F(X2o0*DXXIT)11501.l8Og118O
1150 IF(XDXXIT)117091160116O
1160 GDT(_3OeYTE(I+1)l00O*VTE(I)+18o0*VTE(i1)6oO*YTE(2)+Y1E(I3)
I ) /.SPXGOTOI200
1170 GDT(_25O*YTE(1)+4800*YTE(I1)_36o0*YTE(I2)+16o0*VTE(13)
13.0*YTE(14))/SPX
GOb 12 00
1180 GDT=CYTE(1+2)800*YTE( I+1)+800*YTE(11)YTE(12fl/SPX
1200 RX=RXTE(I)
K=6
G010600
1210 ANGLEATANF(GDI)
PZ148C*CX*S1NF (ANG)VBCT*AM*COSF (ANG)
BZZDPZ+CUTR
FZZD+PZ+CUTR
FY=YD+YTE I i ) CUTR*COSF (ANGLE)
Table
2.  Continued0
PXX+CUTR*S ¡NE (ANGLE)
PUMCHO67,PX BYB
PUNCHO67,PX,FYFZ
c
*pØ9T5 ¡N BLAE AREARENUMBERNG POINTS A0UT LINE OF CENTRÓID
c
** AS N=NN
HENCE SELECTING
4 PIS SURROUNDING THE PT ON EACH S!D
1400 NANNNTE(I)
NB=NNMTE (I) NC=NN+MLE (I) ND=NN+NLE C I)002600N=NA9NO
IF(YB(19N).)1O00926O0 1410
1410 IF(NNBi1430çl42O,1420
1420 IF(NNC)1550,155091650
1h30 NK=NKKN4*MTE( I )NN+N
SP AC SPAC NDO154OKA=1,9
NKKNK5+KA
NNN=NKKNKKN+4*MTE( I +NN
IF (NNNNB) 14809 l48O [47O1470 YC(1KA)000
G0101490
1480 YC( 19KA)=YB(1 9NNN) PT (1 9KA1 )=ZB( J 9NNN ,1)PT(1,KA92)Z.B(
1490 IIII+5KA
NNNNKNKKN+4*WrE( I
I +ÑN1500 IFt(ÑK+k)/NK/)100091520911Ò
1510 YC(2,KA)=000
GOTO140
1520 NNN= (NKNKKN) /5+NÑ
1530 IF(NNN)1535153591534
1534 ¡FCNNNNN2)1536ç1536c1535
1535 YC(2KA)=O00
G0101540
1536 YC(2cKA)=YB(III9NNN)
Table
.2.  Còntinued.
P1I29KA,1 =ZB 111 ,NNN1
..P1(2,KA,2)ZB(II:I,NNN,2
1.40. COÑTINUE GO 102000150 NKNKKN5*(NN44)
SPAC=SPACX,DO1640KA=1,9
NKK=NK25+5*KA
NNN=i NKKNKKN) /5.+NNIF(NNNNB156Q,1560,157O
i 560 N.NN= NKKNKKN+4*M1 E (I.) +NN IF(NNN)158:5,,1585,159O1579 IF(NNNNC)1590,i.580,158O
1:5.8:0 NNN=NKKNKKN6*MLE( 1)+NN
1.F(NNN44t42)i.590,1585..1585
1585 YC(1,KA=0.0
GO 1015951590 YC(1,KA)=VB(1,NNN)
195 PT(1,KA,1)=ZB(1 ,NNN1)
PT(1,KA,2)=ZB(I,NNN,2)
1.1 1=1+5KA NNN= (NKNKKN) /5 +NN1F(NNN(NNMTE( 11.1))
11600,1600,1610
1600 NÑN=NNKKN+4*MTE(I1I)+NN
GOT01630
161.0 IF.(NNN(NN+MLE(I.ij),))1630,1620,l620 1620 NNN=NKNKKN4*ML,E (1.1 I.)+NN1630 1F(NNN)1635,i635,1634
1634
I F (NNNNN2 ) 1636 1636 ,35 1635 YC(2,.KA)=0.0 .., GOTO]..640.. . .., .16.36 VC(2,KA)=YBUII ,NNN)
PT(2.,KA,1)Zß(1l1,NNN,1)
PT. 2,KA,2..)
B(11i:,t4tt42
1640 CONTINUE.
..GOTO2000
. .Table 2.  Continued.
1650 NK=NKKN+4*MLE t
SPACSPACN
DO1I3OKA=1,9
NKKNK5+KA
NMN=NKNKKN4*NLEt! liNN
1F(NNNNC) 1660,1670,1670.
1660 VC(1,KA)0.0
G0T01680
1670 YC(1,KA)YB(I,NNN)
PT(l,KA,1)=ZB(I,P4NN.1)PT(1,KA,2)ZB(I,NNN,2)
1680 111=1+5KA
NNN=NKNKKN4*MLE(11I )+NN
1F1NNN(NN+NLE(1I1)))1690,1720,l12O
1590 1FUNK+4)/5NK/5) 1000,1710,1700
1700 YC(2,KA}0.0
GOTO 1730 1710 NNN= (NKNKKN) /5+NN1720 IF(NNN1725,1725,1724
1724 1F(NNNNN2)1726,17261725
1725 YC(2,KA)=O.0
GOTO17301726 YC(2,KA)=YB(I1i,NNN)
PT(2,KA,1 )=ZB(1 Il ,NNN,i)
PT(2,KA,2)ZB( 1.11 ,NNN,2)
1730 CONTINUE
C
**ROUTINE FOR DETERMINING COORDINATES OF CUTTER AT EACH XYZ
y*
2000 DO2S7OJ=1,2
NHNtO
D02480L=1,2
1r(L11ooO,2O1o,202O
2010 SP12.0*SPAC
G0102030
2o20 SP=12.0*DX
2030 IF(NNN1)1000,2480,2040
2040 IF(YC(L,3) )l000,2350,23a0
Table 2.  Continued1
230 IF(YC(L,4))1000,2360,237Ö
2360 IF(YC(L,9))1000,2420,2430
2370 IF(YC(L,8))1000,2420,21.40
2380 IF'(YC(L,7))1000,2390,2450
2390 IF(YC(L,6))1000,2400,2410
2400 lr(YC(L,1) 1000,2420,2470
2410 IF(YC(L,2))1000,2420,2460
2420 NNN=1
GO TO 24802430 IF(YC(L,6)*YC(L,7)*YC(L,8))1000,2420,2435
2435 GRAD(L,J)( 25.0*PT(L,5,J)+48.0*PT(L,6,j).36.0*PT(L,7,j)+16.0*PT(L
1,8,j)3.O*PT(L,9,J) )/SP
G O T 024802440 IF(YC(L,6)*YC(L,7) )1000,2420,2445
2445 GRAD(L,J)(3.0*PT(L,4,J)i0.0*PT(L,5,J)+18.O*PT(L,6,J)6.0*PT(L,7
1 ,J)+PT(L,ß,J ) /SP GOTO 2480 2450 IF(YC(L,4)*YC(L ,6) ) 1000,2420,24552455 GRADCL,J)(PT(L,3,J8.0*PT(L,4,J)+8.0*PT(L,6,J)=PT(L,7,J)
I/SPG0T02480
2460 IF(YC(L,4fl1000,242092465
2465 GRAD(L,J)=(PT(L,2,j)+6.0*PT(L
_{,3,J)18.0*PT(L,4,J)10.0*PT(L,5,J)+}
13.0*PTIL,6,J) )/SPG0T02480
2470 IF(YC(L,2)*YC(L,4) )1000,2420,2475
2475 GRAD(L,J)(3.0*PT (L,1 ,J)l6.O*PT(L,2,J)+36.0*PT(L,3,J)48,O*PT(L,4
1, J) +25 0*PT (L ,5 ,J) /SP 24.80 CONTINUE IF (J1 ) 1000,2485,24902485 YYY(J)=2.0*YDYB(I,N)
ZZ2(JZDZB( I
,N,J
G0102494
2490 YVY(J)Vf3(j,N
ZZZtJ)ZD+ZB( I ,N,J)24.94 IF(NNN1)1000,2495,2500
2495 PUNCHO68,NK
Table
2.  Continued.
G0T02570
2500 DEN=SQRTFUO+GRAD(19J)*GRAD(19J)+GRAD(29J)*GRAD(2J))
IF(J1)100Ó,251092520
2510 C0SA=GRAD(2,J/DEN
COSD=GRAD(1,J)/DEN
COSC= 1.0/DEN C PX= X +C UTR *C0 SACPY=YB(IN)+CUTR*CO5B
CPZ=ZB(I.N,J)+CUTR*C0SC
PTX(J)CPX
fPTY(J)200*YDCPY
PTZ(J»ZDCPZ+CUTR
GOTO2 56022O COSMGRAD(2,J)/DEN
COSB=GRAD(1,J) /DEN
COSC1.Q/DEN
CPXX+CUTR*C0SA
CPYYB( 19N)+CUTR*C0Sß
CPZZB( 19N,J)+CUIR*COSC
P1X(J)CX
P TY t J ) PIZ (J) =ZD+CPZ+CUTR2560 B=PTX(J/XMILR
KX B
BKX
C=B*XMILR
KXX'1OO.O.0.*(PTXu C)+0e5BPTY(J)/YMtLR
KYB
BKY
CB*YMILR
KYY=1000.O*(PTY (J)C)+0.5
BPTZ(J)/ZM1LR
KZ=L3B=KZ
C=B*ZMI LRTable
2.  Continued.
z10009o(pyzJLd.oÓ;
PUNCH060,NK,KX,KXX,KY9KYY,KZ9KZZ9PTXLJ),PTVtJhPTztJ
2570 CONTINUE
PUNCHO66,X,YYY(fl9ZZZ( 1) NK
PUNCH066,X,YYY(2 9ZZZ( 2) ,NK2600 CONTINUE
C
**LEADING EDGE COORDINAIES**
IF (.X+2 0*DXXOL ) 26409268092610
2610 IF(X+DXXOL260,263092.15
2615 IF(XXOL)2620,2620,3000
2620 GDT
( 3.0*YLE( 1+4 16.0*VLE( 1+3 H36. 0*YLE( 1+2 48.0*YLE( 1+1) +2500*YLE( !/SPX
GO TO 2 002630 GDT(YLE(L+3)+6e0*YLE(I+2)18.0*YLE(I+1I+10o0YLE(I+e0*YLE(I1)
, /spxG0T02700
2640 IF (X2.0*DXXIL2650 9268092680
2650 IF(XDXXIL)2670,2660,2660
2660
1) /SPxG0T02700
2670 GDT(25.0*VLE(I )+48e0*YLEt11)_36.0*YLE(J2}+16.0*VLE(I3)_300*YL
IE( I4)) /sPXG0T0270Ô
2680 GDT=(YLE( 1+2)8.0*YLE( I+1)+8.0*YLE( I1)YLE(12})/SPX
2700 RX=RXLE(I)
K= 7GOTO600
2710 ANGLE=ATANF(GDT)
PX=XCUTR'SINF (ANGLE)
FY=YD+YLE( I )iCUTR*CÒSF(ANGLEBY=YDYLE (I) CUTR*C0SF (ANGLE)
PZ(1.0HßC)*CX*SINF(ANG)YBCT*AMC0SF(ANG)
8ZZbP2+CUÌR
FZ=ZD+PZ4CUTR
Table 2,  Continued.
PUNCH06?9PX,FYFZ
CXSTVLE( I )YT(j)
PUNCHO65,CXST,RLEtfl
PUNCHO65,CXST,RLE( I)3000 CONTINUE
GO TO 100C
**ROUTINE FOR CALCULATING PROPERTIES OF BLADE SECTIONS
**
C
**
ALSO z ORDINATE FOR A GIVEN X,Y COORDINATE
**
600 CX=COD(CODCID)*(RADMRX)/(RADM*(100RATIO))
XRRX/RADM
IF(RXO.7*RADM)6209610,610
610 TX=0.045*CX
GOTO6SO
620 IF(RX*0.65*RADM64O,60p630
630 TXO.O45*CX7*UNIT
G0T0650
640 TX=(T0+T1*xR+T2*(XR**2)T3*(XR**3) )*UNIT
650 ANG=ATANF(PITCH/(2.O*Pi*RX)
AM=(AO+A1*XR+A2*(XR**2)+A3*(XR**)+A4*(XR**4))*UNtT
GÓT0(195,205,29O,654,654ç121O,27109348 9K
654 D0840J=1,2
MTEST=0
XCA=O.O
XCBO.0XCt(.1.O+IBC)*CXYR/COSF(ANG)
655 XXXC/CX
660 MTESTMTEST+1
IF(MIEST30 669 96709610
670 Z(iO.O
Z(2O.o
YA'YD
G0T0840
669 IF (XX)661 ,&61,662661 XXO.O
G0T0664
662 IFIXX1.0)665,6639663
Table
.2.  Continued0
.3kxiö
664 v'0.O3Q.
YMO.O
6tÖ666
665
YO=(.1.O_XX)*O..98879*SQRTF(.xX)+y0+Y1*XX+Y2*(XX**2)3*4**3
Y0Y0+Y9* (XX*9)
666 ttXXO.5:668,668,66
66l. YOY0+ÄLTE*5CALM* (XXO
i I 66.8 XC=X.X*C.XIF(J1i00O,6909700
690 YDC=(YM_YBCT)*AM+YO*X
6010710
700 YDC=(YMYBC.T)*AMYO*TX
710 GL(1.0HBC)*CXXC
GPSQRTF (.Y.DC*YDC+GL*GL) IF (GL) 720 7309 760 720 GAMMA=ATAÑF (YDC./GL.)+P I GO TO? 70 73.0 I:F(YDC.)740,1000 975.0740 GAMMA=PI:290
G010770
750 GAMMAPI/2,0
G0T077O
760 GAMMA=ATANF t YDC/.GL)770 DELTMGAMMA+ANG
Z(J)GP*SiNF(DELTA)
y.YRGP*(OSF( DELTA) If (YRYYR)8PO,840,780780 XCMXC
IF (.XCB).1000.79O,820790 XXXX.5TEP2
ÓTÓ 60
800 XCB=XC
IF (XCA) 1000,810,820Table
2.._{Continüed.}
RIO XX=XX+S.TP2
G010660
820 IF(XCAXCBDEVN384o9.84O9a3O
830 XC(XCA+XCB)*005
G0T0655
840 CONTINUE
1000 CALL EXiT
END
Table
_{ Output Machining Instructions.}
49 STATION NO
_{BACK COORDINATES}RADIAL DISTANCE TO STATION
_{4.000}
MACHINING _{COORDiNATES}
X
_{y}
_{z}NO REV THOU
_{REV THOU} _{REV THOU}BOTTOM OF CUTTER
X Y Z405
3.764
8995
7.114
406
407
408
409
410
415
15
48
68108
49118
_{3.798}
_{8.608}
_{6e243}
420
15
64
_{67}
_{108}
_{48} _{60}_{3.814}
_{8.483}
6.060
425
430
1515
7994
66
65
109
109
47
45
81
7_{3.844}
3.829
_{8.234}
8.359
5.882
_{5.706}
435
440
15109
64
109
44
34
30859
8.109
5.34
445
450
1515
124
139
6362 106106 4241115
74
3.874
_{3.889}
7.983
_{7.856}
5.365
_{5.199}
15154
61104
40 37_{3.904}
_{7o729}
_{5,037}
455
15169
_{60}
_{102}
_{39} _{4}_{3.919}
_{7.602}
4.879
460
15183
59
100
37
99_{3.933}
_{7.475}
_{4724}
465
15
198
58
98
36
72
_{3.948}
_{7.348}
_{4.572}
470
475
15
1213
227
57
_{56} _{93}96 35_{34}
48
_{27}
_{3.977}
3963
7.221
442
7.093
4e277
480
15
241
5590
33 83.991
_{6.965}
4.i
485
490
161620
6 545390
88
3130119
103
4.006
_{4.020}
6.840
_{6.713}
3o94
_{3.853}
495
500
16
1634
52
84
29
88
4.034
6.584
3713
505
16
46
61._{50}
51 78_{73} 28_{27}_{64}
74
4,048
_{4.061}
_{6.323}
6.453
3.574
3.43
510
1674
49
66
_{26}
56_{4.074}
_{6.191}
_{3.06}
Table
L
S1
16
86
520 16 98 525 16 10 530 16 120 53516
130540
16
140
545 16 149 55016
157
555 16 164 560 16 170 565 16 174 570 1.6 177575
16178
580 16 174 585591
592
ST CHORD WIDTH
= Continued46
59 47 52 46 44 4535
4426
43
17
42
6 40 21939
106 38 9137
74
3653
35
2733
116
4.717 25 52 2451
23 53 22 5 2167
20
80
19 96 18116
1 14 17 42 16 75 15 111 15 27 11+74
LE RADIUS =4Ó86
4.096
41O9
4120
43130
4240
4249
43157
4.164
417O
4.174
4l77
4.178
4.174
3.776
0.046
6ò59
5..27
5.794
53660
5526
5.392
5.256
5.119
4.981
4.8414.699
4o553
4.402
4.241
3.116
3,ò5i
292.8
269
280
2.366
2.264
2.167
2.075
1.986 1.902l824
231
Table
3Input Data for
HYDR.0 I Impel].er0
.7500000E+01
.20Q0O0E+G0
.5625000E+01
.22!O000E+O2
1 53 60 00 10