Computer aided design and manufacture of marine propellers

9 FEB.

9984

ARCHIEF

COMPUTER AIDED DESIGN AND MANUFACTURE OF MARINE PROPELLERS

T. J. Langan, R. Bhattacharyya, P. Rodriguez, U. S. Naval Academy, Annapolis, Maryland N. R. Fuller, Jr., U. S. Coast Guard, Headquarters, Washington, D. C.

1. INTRODUCTION

This paper presents a computer aided design and manufacturing method for marine propellers that can be executed on

a

micro-computer based system and can be adapted to individual, designer's particular requirements. Conceptually, we approached the development of the method from the point of view of the designer and not that of a researcher or a computer enthusiast. The computer is used for calculations, data retrieval, plotting routines, and controlling machine tools. It

does not replace the designer but provides him with the information necessary for design

decisions. Rather than use the most eloquent mathematical treatment of the prOpeller, we have chosen models which are adequate for design purposes and Which should be familiar to designers in some form. Although the individual programs depend on details of the methods, the computer techniques are indepen-dent Of the design method and computer hard-ware. A designer can utilize them with any

method

or

computer hardware avialable to him. During the Design Phase the paper dis-cusses the Various sources of data from which to develop the initial propeller:. In addition, methods for modifying the sections and the hydrodynamic basis for the Modifications are given. A brief overview of structural consid-erations is also presented and a discussion in some depth is presented On several mathematical surface representation techniques. Following a brief discussion of the cOmputational hard-ware utilized for the work described herein is a description of the numerically controlled process used to create the example propeller.

2. PROPELLER DESIGN

2.1. Preliminary Design

One always approaches the preliminary design of a propeller from either the point of view of the naval architect or that of the marine engineer (Oosterveldtl, Sabit2, Troost3). In either case there have always been different paths through the design process, and the choice of which path to use depends on the . initial data available. To implement either approach on the computer and cover all possible paths is possible but not necessary. _{A single}

Lab.

_{v. Scheepsbouwkunde}

Technische

_Hogeschool

Delft

. .

scheme has been developed which. permits the designer to start With any data and rigorously develop a preliminary design. The computer assumes values for parameters which are either not provided by. the designer or cannot be cal-culated from the data provided. _{These values} together with the Ones provided by the designer are used for an iterative scheme that develops the preliminary design.

The methodical series propeller charts form the basis of this preliminary design scheme. _{Figure 1 is a flow diagram showing} how the computer proceeds through the calcu-lations, and each step of the process is repre-sented by a box. _{The principal items for the} design of a propeller are the data needed on the hull, engine, and specific constraints. Each of the first five boxes of this figure contains a list of variables and a set of equations. Some of these boxes give values for variables below a dashed line; the computer assumes these data for the variables If they were not specified by the designer or if the

computer has not been able to compute them from one of the equations. _{Through a series} of questions the computer asks the designer to

specify the data for the variables appearing in the box. If enough variables are specified to compute _a value for one Of the variables from a formula in the box, the computation is

made and the designer is not asked the value for this variable. When the computer com-pletes its inquiry and computation for the values specified at each step of this process, it proceeds to use the assumed values for variables not known and to fill in computed values of other variables not yet determined. At the end of this prodess it continues _{to the}

next Step leaving undetermined variables not determined by the process thus far. At the completion of the first five steps it again tries to determine undefined values, and then delivers a list Of all variables together with

their values. The list indicates whether the designer specified the value, it was computed, or the computer assumed it. _{The designer can} make changes in values, and the computer will check for the consistency of these values with those specified for other variables. _{If at} .

this point

in

_{the procedure the blade area} ratio, the delivered horsepower, and the Speed

5

1..

of advance have not been specified, there is not enough information to proceed with the design and the process stops.

The Troost B-series charts are used as a basis to iterate for the toque or delivered horsepower required to develop the specified

thrust. If the diameter is not given, this iteration is carried out by the computer along the curve of maximum efficiency, and the compu-ter decompu-termines pitch, diamecompu-ter, propulsive efficiency, and delivered horsepower. When the diameter is given the iteration is carried out along a constant 6-curve; this process deter-mines the same variables with the exception of the specified diameter. The iterative scheme for a fixed diameter always converges for each value of the effective area ratio, since the 6-constant curves are convex functions of Bp. The maximum efficiency curves are not all con-vex functions, to for some values of the

effec-tive area ratio the compUter may need help from the designer to force convergence of the

itera-tive scheme.

If only the minimal data was specified, there is no need for

an

iterative scheme. The B-series charts are then used to determine the pitch, diameter, propulsive efficiency, and the

thrust horsepower.

After each of these calculations the derived propeller must be checked for cavitation since the design charts of the methodical series are meant only for non-cavitating conditions:. A Cavitation Chart as that of Burril is used for this purpose. If the propeller cavitates the designer can specify an increase in the expanded ratio, the number of blades, or both and repeat the iterative process for the new values. The Troost B-series charts may be exhausted without finding a subcavitating pro-peller. In this case either the propeller is

too heavily loaded for the series to work or there is no subcavitating propeller to satisfy the design criterion. Since the latter could not be verified at this point, the designer

should follow the procedures for heavily loaded propellers. Should the method converge to a series propeller, the delineated propeller should be considered as a possible candidate, for modification to the B-series propeller can often be made to improve overall performance.

We have classified propellers into four categories for the purpose of ,deciding which design procedures to follow in,finalizing the design. These categories follow:

Library - a propeller meeting the conditions has already been designed, and the design is stored in a computer or drawing library.

Series - the propeller derived from the series design is subcavitating and its design parameters put it to the left and below the maximum efficiency curve on the design charts.

If the propeller satisfies the criterion for a series propeller and if the designer is satisfied that the series propeller will do the job in an optimum manner and also satisfy any off design requirements, the design is com-pleted.

(3) Near-series :7. It is often necessary to design a propeller having some

of

the

basic characteristics somewhat different from those of the standard series. In order to design the propeller or to predict the per-formance of the modified version, correction.

factors, based On theoretical analysis,

experimental verification and full-scale trial results are applied. Some of the necessary modifications from the standard series may be:

blade area ratio blade thickness ratio blade section shape or blade outline

hub diameter, etc. Since the design and performance characteristics of the propeller have been originally derived from the standard series data, the detailed calculation method outlined below may also be applied for all the necessary changes in the design as modified from the series.

(4) Detailed Calculation Method - In the design of

a

propeller from the methodical ! series data only the average flow, conditions at the propeller plane are taken into account and so the actual wake field is not considered. Also when the design requirements are rather unusual the performance characteristics can only be predicted adequately by detailed

cal-culatiodmethoda. The prinicpal advantages of this method are the proper pitch distribu-tion'to suit the radial variation of the wake field and the Selection of blade sections in order to minimize cavitation.

It should, however, be eMPhatited that even when the detailed calculation method is adopted for the design it is essential to

estimate the propeller performance first with the help of the methodical series data.

In this method of calculation, strength analysis is also made including stress deter-mination considering both rake and skew of

the blade.

Since the detailed calculation method has been evolved through a continuous develop-ment

of

the circulation theory and a thorough

verification with the experimental results, a comprehensive data bank subject to continuous updating has been compiled.

The logical steps of the detailed method of calculation are based on the loading characteristics of the propeller as defined below:

Lightly loaded - blade area ratio is less than

0.45,

and it is to be fitted to coasters and small vessels requiring low power and operating speeds.

.Moderately loaded - expanded area ratio is between

0.45

and

0.65,

and it is to be fitted to cargo vessels, tankers, and traw-lers in free-running conditions.

Heavily loaded - expanded area ratio more than

0.65,

used for passenger liners, fast cargo ships, naval vessels, and container ships.

Towing. duty ,similar to heavily loaded, but the propeller is to be fitted to

tug boats, ice breakers,- or trawlers for trawling. The advance coefficient is small due to low speed

of

advance and so a high

thrust coefficient is to be expected.

Also

the blade sections of such a pro-peller operate at a considerably larger angle of incidence and with high lift coefficients. Although the detailed calculation method is applied for the final design, the charts available from experimental results are to be stored in the data bank for designing such a propeller satisfactorily.

2.2. Modification of Individual Section Charaeteristics

The purpose of the calculations des-cribed in this section is to modify a previous-ly.obtained propeller shape to improve its overall performance. This Shape may be the . B series propeller determined by the prelimin-ary design, a librprelimin-ary propeller, A propeller determined by a lifting line analysis, or one

the designer wishes to check out. In any case, the procedure is an iterative scheme that seeks to optimize the propulsive efficiency of each individual section yet maintain a cavity free blade. The degree of improvement depends ultimately on the service to which the pro-peller is to be applied.

The calculations are performed for each section starting, with the tip and working in toward the root. The zero lift angle is com-puted by .any standard method such as the one due to Glauert or Pankhurst, and the pressure distribution is computed by any of the exist-ing methods.

If the propeller blade fails to produce the proper lift to drag ratio, propulsive efficiency, or if the pressure distribution indicates cavitation on the blade, the section shapes, the chord length and thickness distri-butions can be changed. The changes are not

straightforward and design experience is important at this state of the design:

The selection of the blade Section requires a great deal of experience and is important if an efficient design is to occur.. For moderately loaded propellers it is desire= able to use composite sections with NACA16

thickness distribution and an NACA-65 para-bolic mean line. Since the inner sections are

not susceptible to cavitation, the maximum - thickness and mean line ordinate should be

positioned at about 35 per cent of the chord from the leading edge for the section at

r/R 0-2 to

50

per cent for sections between r/R = 0.7 and the blade tip. These

modifica-tions to the NACA-16 thickness distribution

and NACA-65 mean line should result

in

the ideal incidence occurring at a small positive

value. This small value helps to insure that the inter sections will not operate at a negative angle of incidence in the confused

flow in. this region, which results from the flow interference between the blades and between the blades and the hub. For the same reason it is desirable to use a 1° to 2° angle of incidence and a 0.10 to 0.15 camber-thickness ratio for the blade section at r/R

= 0.2. This section is then faired into the section at r/R =

0.4:

In the outer region r/R > 0.7 the blade sections should be designed for shock free entry, that is for ideal

inci-dence. This requires a higher camber and results in a concave blade face. The concave blade face would result in face cavitation and should be checked, or it might also not meet manufacturing specifications for a flat face. O'Brien 4 suggests several alternatives if a concave face is not acceptable.

Shouldmodifying the sections fail to

Achieve a cavitation free blade, the present spanwise load distribution needs to be modified. One could abandon the present detailed calcu-lations and follow the method recommended for heavily loaded propellers; however, an alter-native approach' isto redistribute the present spanwise loading and to optimize this loading' under the restriction that the blade be cavi-tation free. Such a solution is possible, only if, when each section is loaded to the point of cavitation inception or to a pre-determined value of minimum pressure, the total lift force exceeds the required thrust.

Let us assume that the condition Is satisfied. A system of discrete horseshoe vortices are used to represent the propeller blades (Figure 2); these vortices are distri-buted along a line as with lifting line theory. The unbound training vortices trail off to

infinity along a helical surface- The value Of the circulation for each vortex is to be determined through a minimization of the . induced drag subject to the following con-straints: the total thrust produced by this system of vortices equals the, desired thrust, and the local lift produceddoes not result7

in cavitation on the section. This problem can be expressed as an optimization problem with inequality constraints and a quadratic drag function. 5 Since the drag function is quadratic, the optimal solution is obtained with an exterior point penalty method as. 6

explained by Reighter,-Phillips, and Wilde-Once the optimal distribution of cir-culation is obtained, the computer uses this distribution and the detailed section.compu-tation to establish the blade shape. The designer need only check the structural. pro-perties of the foil and any off design criter-ion to complete the design of the moderately. loaded propeller.

3. . STRUCTURAL ANALYSIS

The sturctural analysis is presently based on beam theory and is performed after the first iteration of the hydrodynamic design. If there is a potential structural failure, the blade sections are modified to increase the strength, and the hydrodynamic analysis is. repeated. Klein and Viswanathan 7 incorporate. structural requirements into the optimization of spanwise lift distribution for wings and found that the optimal distribution was not the classical elliptic distribution. The proper way to perforn the structural analysis may therefore be in conjunction with the hydro-dynamic analysis and not after it. Our future plans. are to investigate the optimization pro-blem for the propeller loading, when structural

requirements are incorporated into the optimi-zation process. If there is a significant difference between propellers designed with the present method and those with structural-

-hydrodynamic optimization, then the beam theory will be integrated into the detailed calculations for moderately loaded propellers. This will be done along the lines used in the wing analysis. A similar analysis for heavily

loaded propellers would

follow.

Although the 'section through which the blade fails is a plane section parallel to the axis of rotation, our structural analysis is based on. the expanded area properties rather than the developed area. Our present method for mathematically fairing the blade is to fair sections during the hydrodynamic design and after completion of the design to do a surface fairing. The sections that are faired are the expanded sections, and their properties can easily be computed with the faired

func-tions. The moment of inertia-neutral axis,

and section modulus for the developed areas are approximately the same as those forthe.-developed sections. However, any point of extreme stress should be investigated with a detailed finite element analysis.

The structural analysis does account for rake, twist, and skew. We calculate from the pressure distribution the moments about

the principal axes passing through the

cen-roid of the section. These forces are

assumed to be typical of those acting on the portion of the blade halfway between the next inner expanded section and the next Outer one. The loading on this portion of the blade is then the section load times the span of this portion of the blade. These discrete loads are used to compute the shear and moments on the sections.

Before a large or expensive propeller is manufactured a detailed finite element structural analysis and a vibration analysis of the'propeller should be executed. While we do not have such programs running in the micro-computer, they are available on large mainframe computers;

4. PROPELLER DELINEATION

Let us first consider the Troost B-series propeller for the purpose of

illustra-tion. The leading and trailing edge radii for all sections of .r/R > 0.7 are the same. This radius is not the leading or trailing edge radius of the expanded blade section; but is a radius normal to the edge of the

blade. It is designed to be machined into the blade with a single tool; a router which is run around the blade from the leading edge at r/R = 0.7 around the tip to the trailing edge at r/R - 0.7. From the manufacturing point of view this is efficient and prevents

tool induced material failure at a sharp edge, and from the hydrodynamics point of view it has performed satisfactorily. For r/R

< 0.7 both the leading and trailing edges are sufficiently thick for milling. In this region the edge radii are the radii for the expanded section.

In general every blade with a sharp edge segment should .have this segment shaped with a routing tool. The thicker segments with larger radii can be milled after the

face and back have been shaped, or at the same time.

The manufacturing process gives a clue as to how to fair the propeller blades. Except near the leading and trailing edges the face and back of--a propeller are smooth surfaces with continuous curvature in the spanwise and chordwise directions. Near the leading and trailing edges the thickness dis-tribution and the shape.are defined by the edge radius. Although the blade surface has a continuous normal and curvature near. the edges, the surface slope as a function of chord position goes to infinity. Neither a polynomial nor a spline approximation can approximate a function with.an infinite slope; however, away from the edges the surface can be approximated with. polynomials or splines.

We have used both a polynomial approxi-mation and spline surfaces to represent the thickness distribution and Meanline of the blade away from the edges. The thickness is '

faired into the edge radius by trial with the polynomial Approximation and by forcing contin-uity of curvature with the spline fits. Each of

the approximations is continued beyond the blade edge with a constant chprdwise slope, resulting

in a blade with attapozoldal shape near the

edges after cutting. In manufacturing, this shape is left along those segments to be shaped

with a touter; along the othet segments the desired shape is machined. For hydrodynamic and structural analyses the expanded sections are rounded with a constant radius equal to

the edge radius..

Since all

blade analysis is done numerically, there will be no significant error from any approximation neat the edges. Away from the sharp segments the appropriate edge radius is used to connect the face and back, and one is dealing with the actual

surface.

The polyhominal approximations are more efficient than the spline approximations in

that they require Conetants Which are less by an order of magnitude; e.g., fifteen to about 800 for the spline. At present the program for determining the coefficients

for

the

nominal runs in a large mainframe computer, whereas the spline surfaces are computed in a Tektronix 4052. We have hot tried to adapt the polynominal approximation program to run in the 4052, but do use the coefficients deter-mined on the Mainframe computer in working with the Surface in the 4052. The polynominal

approximation may not be accurate enoughin all

cases of interest, or it could develop surface oscillations; however, the sOline should main-tain accuracy for all but some pathological blade surface. At present the degree Of accu-racy needed is not known, and the designer must use his judgment whether to use the polynominal approximation or the spline. Future study is needed to Check whether one or the other

approx-imation should be used and under what circum-stances. This study should consider the manufacturing process and any effect on pro-peller performance that might arise from the use of one or the other.

For the faired representation of the thickness distribution a polynomial of the Sate degree in 6 is fitted to the thickness distri-bution for each expanded section at chosen values of r where e is the angle between the generating line and a radial line through the . surface point. The coefficients of these poly-nomials are then faired in the radial direction with a polynomial in r. The order of the poly-nomial is the same for each of the coefficients. Kuo 8 programmed the fairing methods to.deter-ment the coefficients aij of the resulting polynomial surface. His program is a least, squares method which uses ForsythApolynomials and is based on Forsyth's method.' As an example we fitted a modified Kaplan series Ka 3-75 propeller with a fifth order poly-nomial in 6 and a third order in r. The ex-panded sections are shown in Figure 3. The resulting polynomial for the thickness and mean. line distributions has the form

t(r,6) = 411 a21r 431r2 a41r3 a21e

+ 422r8 + a32r2 e + e42r3e + a1382

+

+ a46=30

See Table I for the coefficients a4 _{for the} thickness distribution surface as 0111 as those for the faired meanIiMe surface. There

is

no apparent scientific reason for using a fifth order chordwise and third order radially; however, the fit shown

in

Figure 4 is reason-ably satisfactory. In Table II the actual numerical values for selected blade elements are compared both for the thickness

distribu-tion and the mean line. Another propeller would have a different set of coefficients, and in order to achieve a good fairing it could require a higher order polynomial.

Ahlberg, Nelson, and Walsh10 discuss spline on spline approximations, and we have followed this method for computing spline approximations for the mean line surface and for the-thickness distribution surface. In the case of the mean line surface we first fit a spline to the mean line of a section. The points that are fitted are called the control points, and the second derivative of the spline evaluated at the corres-ponding values for the control points are referred

to as the moments. The spline is a function of 6 the control points, and the moments. To achieve

the spline on spline fitting the spline func-tion is differentiated and evaluated at the control point 8-values. A spline is then fitted to these values, differentiated, and ,

the derivative

of

this spline is evaluated for the same values of e. These newly computed values Should be the second derivative of the original spline function. Therefore, we use these values in place of the moments in the first spline approximation to obtain a new spline representative of the camber curve. This new spline has a continuous third deriva-tive, which for a physical spline, would mean that the ducks are not exerting a force on the spline normal to the curve This is the effect one strives for when lifting and replacing ducks on a physical spline, until the curve does not move. This spline representation of the mean

line is used

in

the hydrodynamic calculations as well as a spline on spline representation of the thickness distribution. If there is need to Change the mean line or thickness distribu-tion, a new spline representation is calculated

After all the section splines have been determined the moments are fitted with a spline on spline representation as a function of the

radius. The mean line surface is constructed chordwise from the moments determined by the radial second derivative at each value of r for the mean line. As noted earlier, a similar approximation is carried out for the thickness distribution.

With either the polynomial or spline representation of the thickness distribution and mean line, the blade is constructed in

the same way. First, at any r the blade face is constructed by subtracting the thickness distribution from the mean line and the back 135 adding thickness distribution. The pitch for the value.of r is then determined and the ex-panded section is rotated through the pitch angle. Next, the plane containing the expanded section is wrapped on a cylinder of radius r and the z coordinate along the axis of the pro-peller is adjusted by an amount -r*, where * is the rake angle. Coordinates of the face and back are converted into the 2t,

y,

z, coordinates' used in the milling machine. At present the total transformation calculation is carried out with software; however, at some future time it could be hardwired.

The hub dimensions are fitted by a spline on spline approximation with respect to z. Fillet radii are fitted to the blade by forcing the spline curvature to correspond to a radius equal to the fillet radius. With the polynomial representation it is performed more or less emperically.

5. THE COMPUTER

In our View the computer is used to Cal-culate, retrieve and store data, draw, and cOn trol machine tools. It is not intended to replace the designer, but is rather a tool to provide him with the information necessary for making rational decisions. This information

should be presented in a format that is clear, precise; usable, and alternatives and their

consequences should be indicated where possible. The software and hardware should be easy to use after an initial familiarization period and Should be modular so as.. not to restrict the designer from using other methods.

All programs have been run on a

Tektronix 4054 microcomputer with the exception of the program used to determine the coeffi-cients of the polynimial approximations for the thickness distribution and the mean line. These programs can be run on any other micro-computer of the Tektronix 4050 Series, but we do not recommend using the 4051 due to the significantly longer run time. It is to be noted that none of the special features of the 4052/54 have been used, thus 'facilitating the use of other machines. Besides a joystick some of the programs require use of one of the graphics tablets or the interactive digital plotters to enter graphic data. Although no actual drawings are needed a plotter would be necessary if drawings are desired. We have also used the 4907 File Manager to take advantage

of the speed of a disk versus tape, but all software is capable of successful execution using only tape.

Our choice of the Tektronix hardware was based on availability and the high quality of its graphics. There are numerous other -microcomputers that could just as easily be 'used, particularly if one of the Tektronix

graphids terminals is used with it.

The software described above is Primarily written

in

basic, since this is the only

lan-guage the Tektronix terminals use. The two types of software utilized are general and special purpose. The general purpose programs are used for storing graphic data or charts, plotting, iterating, and fairing. Special purpose programs are the initial data and preliminary design program, the program for

computing the section properties, the optimi-zation routine for Optimizing lift with the cavitation restriction applied to some' sec-tions, lifting line program, structural pro-gram, delineation prograt, and the program for numerical control of ,the milling machine.

The uniqueness of these programs lies in their graphic presentation of information to the designer. For instance, when the iterative process of the preliminary design is being executed, the progress is displayed on the terminal screen. The curve along

which the iteration is to progress is displayed, and each step is indicated on this curve by a. symbol when it is completed. At the comple-tion of the preliminary design the series chart is displayed and the design point is clearly marked. With this information the .

designer can determine whether 'he should use a. series propeller, modify it, or go to the' heavily loaded propeller design method.

6. THE MANUFALluRINGTROCESS

The propeller is manufactured on a Pratt and Whitney Trimac XV Numerical Control Mach-ining Center in a manner that requires a mini-mum of manual labor to finish the blade. This machine provides three axes contouring with rotation about one of the axes. The controller for the milling machine is a minicomputer, which is designed to receive instructions from a paper tape. This controller has been modified with a simple custom designed interface to a Tektronix 4052 microcomputer, and through this

interface the microcomputer pastes cutting

1

instructions to the controller.1 A BASIC pro-gram running in the 4052 converts the final surface data into tool offsets, generates the necessary computer code, and sends it to the

controller to drive the machine tool. Since the program works from the faired surfaces in their mathematical form, there is no need in the manufacturing process for design drawings. During the machining, the computer provides a graphic display of the cutting operation that proceeds the actual cutting by several steps; the operator can observe this display and stop 'themachine before serious problems can occur with the cut. The program has an option to run only this display without actual con-nection to the machine tool, and with this option the cutting paths can be checked

before-hand. Upon completion of the machining pro-cess the machinist uses hand finishing and polishing to finish the propeller.

For discussion purposes we introduce a fixed right handed coordinate system with the

vertical axis denoted by z, see Figure 5. The cutting tool moves up-and down along this vertical axis; it has no Other degree of

free-dom. In the machining process the piece, the

eventual propeller, is mounted to the faceof

a vertical table. This table is free to rotate around an axis perpendicular to the face of the table; this axis of rotation will coincide with the propeller axis. The vertical table support is fixed to the top of a horizontal table. The x-axis is parallel to the axis of rotation and is in the same horizontal plane; the y-axis is perpendicular to the x and z axes. The horizon-tal.table is free to translate in both the x and y direction. Rotational movements of' the vertical table are controllable to

+ 0.001 degrees and All translationai'move-meats of the horizontal table and of the

verti-cal tool are controllable to 4 tolerance of + 0.0001 inch.

In our first attempt to cut a propeller we used cutting patha that were a conatant per-centage Of the chord from the leading edge. The resulting blade was too tough for its design purpose, Which was for accouetical aPplicatione. To finish the blade to this design tolerance it required excessive hand finishing.

The present cutting paths lie On con-centric cylinders around the propeller axis and thus far have resulted in smoother surfaces than the above mentioned paths, and the blades have required a minimal amount of hand finishing.

At present the smallest radial incre-ment used between the cylinders is 0.1 inches, and this was on a 15 inch diameter propeller. The cutting path on each cylinder is derived directly from the mathematical surfaces repre-senting the expanded propeller blade. Both sides of the blade must be cut as the blade shapling, proceeds from the tip to the root; for otherwise, the blade may vibrate or break off while the back is being cut if the entire face was cut before the backside. Although both sides are cut on a given cylindrical

surface before preceeding to the next inner concentric circle, the face cut and the back cut are considered as two distinctive cuts. The y-position of the propeller axis ye is held constant during each cut, and the piece is ro-tated around the propeller axis and translated in the x-direction to obtain the shape as a function x(0) of the angle of rotation 6. With the tool restricted to a movement along

the z- axis, the y-position yc of the propeller axis for each cut must be chosen so that the

tool can cut all points on the path x(6) and not undercut the blade at other points. It

is also desirable to choose ye so that the tool is not forced to cut too much material at one time. Three or Lour experimental

itera-tions with the computer graphics are usually sufficient to determine ye.

The cutting tool is a ball mill which has the shape of a drill with a hemispherical head of radius r . During the cut, the

hemispherical surface is tangent to the cutting path, and it is the part.of.the tool used to shape the work.- In our tool environment the choice of re is:presently left to the Machinist on the floor At the time of cutting: Since the

cut instructions to the tool controller for each cutting path are generated just prior to making the cut, this has posed no problem,

Ideally, the radius should be chosen based on . tool blade 'clearances, blade shape curvatures, and fillet radii, but we do not have the experience to make this choice' and

rely

on

the machinist,

feeling for

picking the tool

size.

Because of its shape and the restriction that it can only move vertically, the tool can-not cut any point on the path x(0) for which the z-component of the normal is less than

zero. Ideally, the normal should be determined from the analytical surface representation, however cuts based on this method have not been attempted. -Instead,-two rows of points a distance Ar apart are prescribed with a distance of rA6 between the points in the respected row, where r is the radius of the respected rows from the propeller axis. One row lies on the first data cylinder and the second row is along the second data cylinder, Figure 6. A second surface S on which the cutter center travels between 6 = 61 and 62 is constructed above the blade as follows: a

normal of length re is constructed at (r/, 01) using the vectors (r2 -r1, 0) and (0,r1(62-61)) a second normal of the same length rt

is constructed at (r2,61) using the vectors (r2-r1, 0) and (0, (r2(62-61))4a third vector of the same length is constructed at (r2,62)

using (r2-r1, 0) and (0,

r2(62-00;

and a

-final vector of the same length is constructed at (r1,62) using (r2-r1, 0) and (0, r/(62-61); The tips of the normals at (r1,61) and (r2,61) are connected by a line and those at (r1,82) and (r2,62) are connected by a second line. The surface S is the surface generated by a line passing through these two lines at points intersected by the same cylinder. The

radial projection of the curve (r,6,z(6)) onto this surface is the curve along which

the cutter center must travel during the

cut. In other words, the controller must adjust the z-position of the center of the tool as a function of z(6) of the angle 6 and the x-position of the horizontal table x(6) during the cut; so that, as the piece rotates through the angle 6 the tool center follows the projected curve over each incre-mental of the surface S. Thus far the smallest value of Ae used has

been-(e "

eTE)/N for N = 100.

LE

6LE is the value of 6 at the leading edge and uTE is the value at the trailing edge. Using N = 26 has provided a smooth cut, and N = 100 produces what appears to be a continu-ous cut although the hydrodynamic effect on torque has not been cheated, If y is properly chosen before the cut, there will be no hidden faces, that is a face with the:z-component of

the normal vector less than zero, and the tool will be able. to complete the cut without Under-cutting the blade at some point. Several

cut-ting paths may be cut on cylindrical surfaces between rl and T2; the number depends on Ar . and the resulting smoothness of. the

These are primarily cleaning

up

cuts..

The calculations to determine z(6); x(6), and yc are done on the tool floor just . prior to each cut. A simulation of the cutting is run in the 4052 to insure no problems With_ hidden planes will occur at the time of. cutting.

After completion of the first blade the entire piece

is

rotated through an angle eb = 360/Z, where Z is numberof blades. The second blade is then cut in the same manner as the first; since the y's, z's, and x's can be atored the first time, there

is

no need to recompute them for each cut on the new blade. At the completion of the second blade the piece

is again rotated Ob degrees, and the third. blade cut; this is repeated until all-Z. blades are tut.

The location of the propeller axis is critical to obtaining a correct blade shape and a balanced propeller. To locate the axis relative to the center of the

hemispher-ical tool center, the machinist makes a circu-lar cut around the piece. with a diameter

larger than the propeller diameter. ,He then .

measures the diameter parallel to the hori-zontal table. Half this diameter should equal the distance from the cutter center to the axis minus rc. If this is correct the center is at the center of the xyz-coordinate system; otherwise, a y-adjustment is made, and the circular cut is repeated until the rotation axis and the x-axis

coin-cide.

Finally, the propeller requires two additional steps-to be finished. The first is to use a router to round off the blade edge from the leading edge to the trailing edge at the specified value of r/R. Smooth.: ing and polishing the face. is the second step needed to complete the propeller.

REFERENCES

1 OOSTERVELDi 141.1.C.., VAN OOSSANEN, P.

'Presentation of Propeller Characteristics Suitable for Preliminary Ship Design. Studies' Proc. ICCAS,

TOvo.,

1973.

2 SABIT, A.S.

'Optimum Efficiency Equations of One NSMB Propeller Series 4 and 5 Blades'

ISP, 1976

3 TROUST,

'Open-Water Tests with Modern Propeller.

Forms!.

NECIES, 1951

4 O'BRIEN, T.P.

-'The Design of Marine Screw Propellers' .

- Hutchinson Scientific and Technical Press, London, 19626

NECIES, 1957

5 LANGAN, T.J., BHATTACHARYTA, R. 'Optimal Design of Moderately Loaded Marine Propellers' (to be published).

BEIGHTLER, C.S., PHILLIPS, D.T., Wilde, D.J. 'Foundations of Optimization' (Second

Edition), 1979.

7 KLEIN, A., VISWANATHANi S.P.

'Approximate. Solution, for Mimimnm Induced Drag td Wings with Goren Structural Weight'

J. of Aircraft, 1975...

8 KUO, C.

'Details of a Subroutine on the Least-Squares Curve Fitting Techniques using Orthogonal Polynomials'

DTMB HydromeChanics Technical Note No. 54,-1966.

9 FORSYTHE, G.E.

'Generation and Use of Orthogonal Poly-nomials for Data Fitting with a Digital Computer'

J. Soo: of Industrial Applied.Mathematics,

1957.

10 AHLBERG, J.H., NELSON, E.N, WALSH, J.L.

'The Theory of Splines and their Appli-cations'

Academic Press, 1967

11 RODRIGUEZ, F. A:, ROM's, D. F.

'Interactive Graphica and the DNC Production of

Complex

Three-Dimensional Slopes'

Proc,

National Computer Graphics Assoc.,

'Values of a

for the Thickness

Distribution Surface

Values of aii for the Faired Meanilne Surfate,

TABLE I 1 _{2 4}

0.1347x10o

0.1177x100.

0.1483x10o

-0.6448x10-1

-0.1304x10-1

0.4774x10-1

2

0.5315x10

-1

-0.9492x10-

-0.2732x10°

0.8305x10-

1

0.2551x10-

-0.7900x10-1

3

-0.4452x10-1

0.2324x10-1

_0.9741x10-1

_-0.2334x10-1

_-0.1843x10-

1

_6.1i§gx10-1

.0.6058X10-

-0.1790x10

-2

-0.1003x10

-1

0.2627x10

-2

0.1429x10

-2

-0.6016x10-2

1 4

0:1369k100

0.5133X10-1

-0.4404k].0

-2

0.6015x10

2

6.1267x10°

-0.1080x100

-0.2365x10-2

3

0.1170x10°'

$1.2445#16°

1

0.8923x10--2

-0.9294x10

4

-0.5634x10-1

0.9293x10--O. 3048x10-1

-2

-.3658x10

0.1653x10-1

-0.2674x10-1

0.3529x10-2

-2

-0.1239x10

0 6322x10-1

-0.1141x10°

0.4782x10-1

=0.8008x10-2

Input Data Versus Calculated Surface Data for Thickness Distribution

Angular Measure

in

Radians from Blade generating Line

Input Data Versus Calculated Surface Data for Faired Meinline

Angular Measure Inlladiana from Blade Generating Line

Input 0.3R 0.0821 0.1103 0.1306 0.1490 0.1480 0.1377 Calculated 0.3R 0.0815 0.1105 0.1309 0.1466 0.1427 0.1288 Input 0.7A 0.0 0.0365 0.1309 0 1466 0.1427 0.1288 Calculated 0.7R 0.0015 0.0371 0.0548 0.0530 0.0384 0.0007 Input 1.0R 0.0054 0.0192 0.0188 0.0072 Calculated 1.0R 0.0060 0.0194 0.0195 0.0091 TABLE II -0.75 -0.50 -6.25 0.25 0.50 6.75 Input 0.3R 0.0811 0.1069 0.1301 0.1494 0.1468 0.1331 Calculated 0.3R 0.0815 0.1082 0.1310 0.1467 0.1421 0.1260 Input 0.7R 0.0 0.0365 0.0547 0.0534 0.0386 0.0 Calculated 0.7R 0.0015 _0.0373 0.0549 0.0530 0.0385 0.0007 Input 1.0R 0.0054 0.0192 0.0188 0.0072 Calculated 1.0R 6.0059 0.0194 0.0195 0.0089

ID

= YIP()

= Shaft efficiency

ID =

'°*-7

= 0:97 if engine.

isatidships

F Number of BladeS

= Number of Revolutions per minute = Density

= Diameter = Disc Area = Expanded Area

/Ao =

Expanded Area -Ratio

= DHP N .PD Y

is

THP _ no*Dlip

9'

RT

=

Ship Resistance Vs = Ship Velocity RT Vs EHP 550 T = Thrust Speed of Advance. = Wake fraction CB = Block Coefficient t = Thrust Deduction Function

k = Function of hull form

TVa

THP -550 = = 0.23 CB + 0.115 R,r Vs (1,-w) DHP no if

no is not known, use

DHP

= n

*SHP if

no

is known, use -DHP =

TBP/no

D known VPQ -Yes Yes

Check cavitation and calculate

missing valueswhere possible

V

121!Li51ed_I

xxxxxxxx\N1/4\,`Ita\V

4.

Fig.2. Discrete Vortex Distribution

0.25*

0.20

0.10 0.05

0.00

Fig.5. Cutting Tool and Propeller Orientation Showing

cutter radhl

1st data

cylindgr

cutter cente path

.2nd data

eialtrider.

Principal

Axes

Fig.6. _{Parameters Used in Numerically Controlled Propeller Blade Manufacturing}

-3.0 -2.0 1.0 2.0 3.0