An alternate approach to the manufacture of ship propellers.

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OF SHIP PROPELLERS

by

Stephen Derin

B.S.M.E., The City College of New York (1967)

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

IN SHIPPING AND SHIPBUILDING MANAGEMENT

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

February 1976

Signature Redacted

Signature of Author . ,.- . ., . ..

I

Department of Ocean Engineering

Signature Redacted

February 1976

Certified by Certified by Accepted by Thesis Co-Supervisor

Signature Redacted

. . . . . . . m .. . .. . . . Thesis Co-Supervisor

Signature Redacted

WRCHIV Chairman, Departmental Committee on Graduate Students

APR

151976

iRAW8

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by

Stephen Derin

Submitted to the Department of Ocean Engineering on February 21,

1976, in partial fulfillment of the requirements for the degree

of Master of Science in Shipping and Shipbuilding Management.

ABSTRACT

Present propeller manufacturing processes define the

propeller geometry in terms of blade sections at given blade

radii.

This definition of propeller geometry is used since

it is consistent with the propeller definition as specified

on a typical propeller drawing and as given by the propeller

designer based on hydrodynamic considerations.

Hence, the

entire manufacture, checking and repair of the propeller is

based upon setting up and monitoring the blade surface contours

at specified blade radii.

Considering the geometry of modern

ship propellers with airfoil type sections, varying pitch

lines, and varying degrees of rake and skew, the use of blade

radii in defining blade geometry for manufacturing purposes

has become a rather burdensome and perhaps costly concept

which the propeller manufacturer has been forced to accept.

The purpose of this paper is to develop an alternate method

of defining the propeller blade geometry which may enable

the propeller manufacturer to produce an inherently more

accurate propeller and at the same time reduce the costs

associated with the manufacture of ship propellers.

Thesis supervisors:

Justin Elliot Kerwin, Ph.D.

Professor of Naval Architecture

Henry Stuart Marcus, D.B.A.

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ACKNOWLEDGEMENTS

The author wishes to express his appreciation to several people for their assistance in preparing this paper:

Professors Justin Kerwin and Henry Marcus for their supervision of this thesis project. Mr. S. D. Lewis for his helpful

comments and suggestions. My sister, Jewel, for her grammatical assistance. I would also like to express my

gratitude to my wife, Amy, for her patience and understanding during the preparation of this document, and to my children Alicia and Rachel, who helped in their own way.

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TABLE OF CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . 6

Derivation of Planar Sections . . . . . . . . . . . 13

Program Documentation . . . . . . . . . . . . . . . 23

Program Verification . . . . . . . . . . . . . . . 26

Economic Considerations . . . . . . . . . . . . . . 28

Conclusions and Recommendations . . . . . . . . . . 34

References . . . 35

Appendices A. Propeller blade tolerances . . . . . . . . . . 37

B. Fitting data with a cubic spline . . . . . . . 39

C. Solution of cubic equation . . . . . . . . . . 45

D. Planar section program listing . . . . . . . . 48

E. Program output listing . . . . .6

I.

II.

III.

IV.

V.

VI.

VII.

VIII.

. . . 64

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LIST OF ILLUSTRATIONS

1. Planar Blade Section . . . . 7

2. Cylindrical Blade Section . . . . 8

3. Development of Planar Section . . . 14

4. Expanded View of Propeller Section . . . . . . . . . . 16

5. Model Propeller-Planar Sections . . . 27

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I. INTRODUCTION

The intent of this paper is to develop an alternate

propeller blade geometry definition that may prove beneficial from both a manufacturing and economic point of view. The proposed blade geometry definition consists of defining the blade surface contours in terms of planar sections taken perpendicular to the blade center axis as shown in Figure 1. This method differs significantly from the typical method of defining the blade surface contours based on cylindrical sections as shown in Figure 2. It is readily apparent that a reduction in the complexity of blade surface definition is achieved by the use of planar sections. The planar section is two-dimensional as opposed to the cylindrical section, which is three-dimensional. This factor of dimensionality may not seem too important from theoretical considerations since a complete blade surface definition exists in either case. However, for manufacturing purposes the use of planar sections significantly reduces the problems associated with accurately defining, measuring, and machining the propeller blade surfaces.

The use of cylindrical sections in the manufacturing process is an outgrowth of the propeller design method. The propeller designer defines the propeller geometry in terms

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PLANAR BLADE SECTION

(CONSTANT Y PLANE)

y

V

.,BLADE CENTER AXIS

SLAE _PLANAR SECTION PLIANAR SUfFr SHAFT AKTI

Figure

1

PROPELLER t hcE

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CYLINDRICAL BLADE SECTION

CYLTtI.DtrICAL SECTrON PRoEU-ER BLADE CYLINP9CAL SuiFPACE -- PITC w GL.IX 7-AXT S

Figure 2

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of cylindrical sections based on hydrodynamic considerations.

This data is then given to the propeller manufacturer in the form of a propeller drawing where the blade attributes are all referenced to the appropriate cylindrical blade sections. Final propeller acceptance by the shipowner requires that the propeller conform to the drawing requirements. Thus the manufacturing procedures were oriented toward developing acceptable cylindrical sections. This presented no great difficulty in manufacturing propellers where the pressure face of the blade was coincident with the generating line since a reference surface (the pressure face of the blade) was readily available from which the blade geometry could

be easily measured and assessed for acceptability. References [1] and [2] provide the manufacturing details associated

with the production of this type of propeller. With the

advent and use of airfoil type sections in marine propellers, the reference surface from which the blade section shape was determined no longer existed, and the propeller manufacturers were well aware of the manufacturing difficulties that would have to be overcome as noted by the following quote from reference [1]:

The first essential for the production of an

accurately finished blade is the machining of the correct face-pitch surface, from which the back thickness

dimensions, etc., may be measured. This is not always appreciated by the designers, who frequently specify a variation of pitch from root to tip which is quite aribtrary in form, and may be associated with a curved rake, and blade sections having a rounded or hollow

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face; so that the manufacturer can find no datum line or surface from which to make the necessary measurements. Such a blade is extremely difficult to make, and it is obvious that unless complete face templates are fitted at each radius, which is, to say the least, a most expensive procedure, the accuracy of the finished propeller must be less than that of one which is so designed to take account of the methods of definition and machining which can readily be applied.

The airfoil type sections were here to stay and the propeller manufacturer was compelled to use a complete set of cylindrical gages to define the blade surface contours. Application procedures were developed to incorporate the use of these cylindrical gages in the manufacturing

processes. Reference [31 provides a description of the

manufacture and application of cylindrical gages to a marine propeller. It should be noted that the use of cylindrical gages is not the only means of generating the desired surface contours. Another method is to take measurements of thick-ness on the blade casting and then drill holes to the desired depth on the blade surfaces and machine down to the bottom of these holes. References [4] and [5] provide additional details of the implementation of this method. This method only gives point definition to the blade surface and unless sufficient measurements are taken, the resulting blade

geometry may be deficient. In addition to the complex blade shape, the propeller manufacturer was also faced with the problem of the need for tighter tolerances, that were

being imposed by the propeller designer. This was to fully realize the predicted performance characteristics associated with the particular propeller design. Appendix A presents

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a summary of tolerances for ships' propellers. Thus, the propeller manufacturer was required to produce a complex propeller geometry to fairly rigid specifications, using

cylindrical sections as the basis for propeller blade surface definition.

The fundamental reasons for the use of cylindrical sections in the manufacture of marine propellers has been given in the previous paragraphs. Now let us consider the manufacturing aspects associated with the production of marine propellers.

The marine propeller is the end product of a casting and machining process. References [1], [2], and [4] through

[7] provide details of the manufacturing process. The manufacturer is concerned with the problems of producing a complex casting, and being able to machine the casting to the final propeller dimensions. In order to achieve this goal the propeller manufacturer should be provided with a complete propeller geometry description. This geometry definition must be one which can readily be applied to the manufacturing process and also must have the characteristics

of simplicty and ease of measurement. The marine propeller may vary in size from several feet to over thirty feet in

diameter and propellers have had cast weight exceeding ninety tons (reference [8]). It should be realized that having to reject a casting of this dimension because of

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improper blade geometry would be a very costly proposition to the propeller manufacturer. Therefore, it is essential that any definition of propeller geometry must be readily measurable in the foundry, and must also be kept as simple as possible to insure against measuring errors during

production. It is for these reasons that the use of planar sections to define the propeller blade geometry is being presented in this paper. A planar section is more readily measurable and definable in the foundry and is also a simpler

system to work with compared to using cylindrical blade sections for defining the blade geometry. In addition to these manufacturing advantages, the use of planar sections provides many areas of possible cost reduction in the manu-facturing process. These possible cost benefits will be presented in more detail in Section V. Section II will provide the mathematical formulation necessary to generate a planar section. Sections III and IV will give a descrip-tion of the program and the verificadescrip-tion procedure used to check out the program. Concluding comments and recommenda-tions will be presented in Section VI.

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II. DERIVATION OF PLANAR SECTIONS

The basic concepts involved in generating a planar section can be stated as follows.

1. For both the pressure and suction blade faces, a set of spanwise curves is generated, where each curve consists of the offset points corresponding to a

given percent chord varying from blade root to tip as shown in Figure 3.

2. Place the planar section at the desired elevation and perpendicular to the blade center axis.

3. For each spanwise curve, on both the pressure and suction faces, determine the coordinates of the intercept point between the planar section and the spanwise curve.

4. If a plot is now made of all these intercept points the desired planar section can be established.

The above four concepts are easily understood. Their implementation in developing planar sections for actual marine propellers is just as easy.

In order to develop the spanwise curves it is first necessary to define a coordinate system in space and then

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DEVELOPMENT OF PLANAR SECTION

y

A.

SFANWISE, Coti'6P.NT

PE-CENT CHoRP, CURVES

TPL

APR

SECT10oN YPLAWE BLADE TIP f OINTS IRE FACE pOINTS

PROJECTED VIEW

(VIEWED FROM AFT LOOKING FORWARD)

Figure 3

SET INTE&CePT 4 OFF ?RESS

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The system to be used is the rectangular coordinate system. The x-axis is the shaft axis, the positive direction going

from aft, forward. The y-axis is the blade center axis when in the vertical position and perpendicular to the shaft

axis. The positive direction is up. The origin is the

intercept of the x-axis and the y-axis. The z-axis is in the athwartship direction, the positive direction chosen to form a right-handed system as shown in Figure 2.

Having defined the coordinate system, the next step is to transform the blade offset data, as given on the

propeller drawing, from cylindrical offsets to rectangular coordinates. This transformation can be accomplished with the aid of Figures 2 and 4. If the reader is not familiar with propeller geometry terminology, a review as presented in reference [9] may prove beneficial at this time.

Figure 4 gives an expanded view of the propeller

section for a specific blade sectional radius. In a view of this type, the propeller section has the true airfoil shape and a definite position in the expanded coordinate system defined by the x-axis and the z'-axis. The z1-axis lies on the surface of the cylindrical section in question and is located in the y-z plane. In order to develop the expanded view it is necessary to "unroll" the cylinder under

consideration. The geometry associated with typical airfoil type sections is given in reference [10]. The definitions of the variables depicted on Figure 4 are as follows:

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EXPANDED VIEW OF PROPELLER SECTION

FACE K=MDCO P = Or F SET D POIWT POINT SUCTION FACE RAKE

Figure 4

-6N

s:y,

.40 PITCH tl:NE MtP

Q

L

5 1--10

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Rake is the fore and aft slope of the generating line. For a given blade section this is equivalent in magnitude to the x-coordinate of the intercept point "Q".

By convention rake is considered positive when it has a negative intercept value.

Skew is the offset of the blade midchord point from the intercept point "Q" measured along the pitch line. By convention, skew is considered positive when z' is negative.

0

= pitch angle = tn

2.r

P = section pitch r = section radius

ETA is the value of the offset corresponding to a point "P" on the section surface when measured perpendicular to the pitch line. The positive direction for ETA is as shown in Figure 4.

PC is the percent chord corresponding to a given ET.

The leading edge of the blade section is the 0 percent chord point, while the trailing edge represents the 100 percent chord point. The midchord point "K" has a PC of 50 percent.

CI is the distance between the midchord point "K" and the point "L" as measured along the pitch line. CI is positive if PC is less than 50 percent. Otherwise, it is considered negative.

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The coordinates for the midchord point, K(z',x K) are determined by the following equations:

XV, = - rcike - (s kew)(. Csin 03

Z.'K

= - ( sk ew)(

cos

0)

The coordinates for any offset point "P" are determined from:

X,=

XK+

(c-)(sin0)

+ (F~~i)(

cos 0)

' ' + C. o(Cosi) -n

where (

From Figure 2 it is seen that z' represents an arc length along the circumference of a circle of radius "r" and the following relationships follow:

yp

r cose

Thus for any blade offset point "P" the corresponding rectangular coordinates can be defined as given below:

x,= 4+

(Eci)( S in

#)+

(ETr-A)(. Cs CO

)

yp

r

cos

e

This completes the initial phase of defining a coordinate system and locating the blade offsets in this coordinate

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system. It is now possible to generate the desired planar sections by either graphical or analytical techniques. Both methods will be described but emphasis will be placed on

the analytical technique since the graphical approach must sacrifice accuracy by virtue of graphical construction. Graphical approach

1. For each constant percent chord, construct a plot of x vs. y and z vs. y.

2. Using a spline, construct a smooth curve through these points.

3. On each of these curves, locate a line of constant y corresponding to the desired planar section.

4. From these plots read off the intercept values of x and z for the y plane under consideration.

5. The planar section can now be constructed by plotting the intercepts obtained from steps 1 through 4 for

each spanwise curve, and placing a smooth curve through these points.

6. By repeating steps 1 through 5 it is possible to obtain as many planar sections as desired.

Analytical approach

In formulating the analytical approach it was realized that a high-speed computer would be required in order to

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carry out the numerous calculations that would be required. In order to minimize computer costs a mathematical formula-tion was developed that is definitive in form and does not require iterative methods for finding the required solutions to the equations used in the formulation.

The first step in the mathematical formulation was the selection of a curve fitting routine that would have the following characteristics:

a. The curve should pass through each of the input data points.

b. The curve should be single valued at each point. c. At any point on the curve the slope and curvature

must be identical when approached from either direction. That is to say, the first and second derivatives must be continuous for all points on the curve.

d. For a given value of the dependent variable a definitive solution for the independent variable should exist that does not employ iterative

methods.

The above conditions can be satisfied by fitting a chain of cubic equations to the input points as is done in

reference [11] and reprinted in Appendix B. A method for finding the solutions to a cubic equation is given in

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reference [12] and is reprinted in Appendix C. Having selected the curve fitting routine, the next step was to define the spanwise curves in terms of a single valued

function. This was accomplished by selecting "r" the local radius as the independent variable. For each constant

percent chord a chain of cubics for x, y, and z in terms of r is determined using the offsets associated with this

constant percent chord according to the formulation given in Appendix B. The resulting cubic chains are as follows:

all ri + 6(

r

-r)

-v- c, i(

r-

rz)

4

(1

y

t's

30-"c

(-)

J. (2)

~r

ji~rr)

6-~~rc)--

3

(-

1

4

(3)

For a given planar section that is to be generated

(i.e., a given y value), the corresponding y value is placed into Eq. (2) and the required value of r is found by the method given in Appendix C. With r determined it is now possible to solve for x and z by using Eqs. (1) and (3). The value of z can also be determined by

Z = r-.y- (4)

but this does not always yield the correct sign. In the program that was developed to determine the planar sections Equation (4) was used to determine the magnitude for z and Equation (3) was used for determining the sign. Therefore, the values of x and z for the intercept point between the

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planar section and the constant percent chord line,

consisting of the offset points for the given percent chord, have been determined. This procedure is repeated for the remaining constant percent chord spanwise curves for both the suction and pressure faces of the blade. The resulting intercept points are then plotted to obtain the desired planar section. A program was developed that uses the

above methodology for finding planar sections and is presented in Section III.

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III. PROGRAM DOCUMENTATION

This program uses the methodology outlined in Section II to generate planar sections for defining the propeller blade surface contours. A listing of the program and the

necessary subroutines is given in Appendix D. The input variables to the program are as follows:

Variable

KK NR NCP NCS ZDIAM SCALE ROTATE Description

The number of planar section to be

developed (1 through 50).

The number of blade sectional radii for which input data will be given (7 through

25).

The number of input chord stations to be used for the pressure face (25 maximum). The number of input chord stations to be used for the suction face (25 maximum). The propeller blade diameter, inches. Permits the output data to be scaled. A scale of 1.0 represents full scale.

Permits a rotation of the planar section output from the design condition, degrees. A positive angle is used to increase pitch from the design condition. A value of 0.0 corresponds to the design condition.

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Variable

Yo(KK) XR (NR) PITCH(NR) CHD (NR) RAKE (NR) SKEW (NR) PCP (NCP) PCS (NCS) ETAP(NRNCP) ETAS(NRNCS) Description

The y value for each planar section, arranged in increasing order, inches. The input blade fractional radii going

from root to tip.

Pitch for each fractional radii, inches Chord length for each fractional radii,

inches.

Section rake for each fractional radii, inches.

Section skew for each fractional radii, inches.

The percent chord values to be input for the pressure face of the blade. The leading edge is the 0.0 percent chord and the trailing edge is the 100.0 percent chord.

The percent chord values to be input for the suction face of the blade.

Offset data for each percent chord, for each fractional radii, for the blade pressure face, inches.

Offset data for each percent chord, for each fractional radii, for the blade suction face, inches.

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The above variables and a title card are entered into the program in the formats specified in the program listing given in Appendix D.

The output from the program contains the following information:

1. A listing of the input data

2. A listing of the blade offset data translated to Cartesian coordinates

3. A listing of the x,z coordinates for each planar section.

The program was run for verification purposes and a listing of this output is given in Appendix E.

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IV. PROGRAM VERIFICATION

A model propeller was used for the purpose of verifying the program. The propeller model was No. 4389 M per reference

[13]. The program was run and a listing of the output is given in Appendix E. A set of gages was then constructed for two of the planar sections corresponding to y values of 1.49 and 2.98 inches. The planar section contours for these

sections is shown in Figure 5. The gages were next placed

on the model and set at the required y values. The gages were then checked to insure that their location in the x and z directions were correct.

After these checks were completed, the gages, if properly generated, should be oriented so as to conform

precisely to the blade surface. Based on visual observations by several persons it was concluded that the gages did in fact match the blade surface. It was therefore concluded that the program methodology was correct.

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MODEL PROPELLER PLANAR SECTIONS

ox

'4

Figure 5

Y=

Z.98 INCHES

~~jIN

5-IH ES

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V. ECONOMIC CONSIDERATIONS

The manufacturing of marine propellers can be described by the following sequence of operations:

1. Pattern making

2. Pouring system preparation 3. Mould assembly

4. Melting and pouring 5. Fettling

6. Machining 7. Measuring 8. Grinding

9. Final inspection

The areas where planar section can be used to advantage are pattern making, machining, measuring, and final inspection.

Pattern making.--In using planar sections a pattern can readily be developed by stacking planar sections as shown in Figure 6. These planar sections would have incorporated in them the necessary allowances for shrinkage and machining. The blade pattern could be made outside of the mould and checked for accuracy prior to locating pattern in the mould. A visit was made to F. W. Dixon Co., a pattern maker with previous experience in making propeller patterns. They indicated that potential cost reductions in the order of

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PLANAR PATTERN PRODUCTION

y

I

I sCfktSff LtNE - FOR Of Op.r1TAITION e'U.?ROS$S

I

p

1)

Locate base

2) Locate and glue down metal templates and wood

filler material

3) Remove wood until metal templates are reached

Figure 6

METAL ,PLAWAR rE MPLA TE S SAS

I

\W 00O D I

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25 percent of the total pattern cost were probable, if planar

templates were used.

Machining.--In machining operations using planar sections,

the cutting tool requires motion along two of the coordinate

axes for any planar cut.

In machining with cylindrical

sections, either a rotational positioning of the propeller in

conjunction with a single coordinate motion of the cutting

tool must be employed, or the cutting tool must be required to

have motion along all three of the coordinate axes.

In either

case the cylindrical sections require the use of machines of

greater complexity than if planar sections were used.

Several

cost-related benefits are therefore possible for machining

planar sections, namely:

1. Planar sections can be machined on less expensive

machinery.

2. More possible machinery sources exist for planar

sections.

3. Programming machines for cutting planar sections is

less intricate.

4. Planar sections can be generated at any spacing that

may prove beneficial in the machining process.

Measuring and final inspection.--The application of

cylindrical gages to the propeller blade is a time-consuming

and expensive procedure, but is usually necessary to insure

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the propeller geometry is of the correct form. One objection to the use of cylindrical gages is that the latitude permitted in applying these gages is such that an out-of-tolerance blade can be checked and still found to be acceptable. In applying cylindrical gages to a blade, the entire gage must lie on the surface of a cylinder, with a radius corresponding to the blade section radius being checked. If any part of the gage is outside of this surface the entire gage shape is changed. Gages six feet in length are commonplace and if these gages are to be applied properly, they must be accurately fixed and held in space by an expensive gage-supporting system. A different supporting system is required for each radius that is to be gaged, since a different curvature must be applied to each gage. If planar gages are used, the measuring problem is reduced considerably. The amount of supporting structure required to insure that the planar section remain in the

plane when applied to the blade is a minimum. A flat mounting plate located under the gage would suffice. The plate can be designed with clamps to locate and 'position the planar gages. This mounting plate could then be placed on a vertical column and elevated to the appropriate measuring station. This type of measuring system is much less susceptible to errors in both positioning and locating the gage on the blade and the

following benefits result.

1. A significant reduction in gage application time and a corresponding decrease in the time-related

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cost

variables such as machine time, labor hours,

etc.

2. A more accurate verification of the blade geometry, and a reduction in the possibility that an out-of-tolerance blade will be accepted. The propeller design as

specified on the propeller drawing is assumed to have optimal operating characteristics for its intended use, and departures from the specified geometry will result in a suboptimization of the propeller's operating characteristics. Most propellers designed today are optimized based on some cost-related function, usually propeller efficiency, and departures from this

optimal design imply an increase in costs associated with the propeller's operational characteristics.

3. The use of planar gages permits rapid incorporation of design changes in propeller geometry that may occur during the manufacturing process.

4. The addition of shrinkage allowance and machining tolerances can be added in the direction of the coordinate axis which is standard practice in the foundry industry. This results in better control of the pattern geometry and consequent reduction in

excess material on the cast blade surfaces, resulting in lower machining costs, since less material is to be removed.

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The above cost advantages are all possible if planar gages are used and the author believes that the use of planar gages for manufacturing marine propellers will be the rule rather than the exception in future years.

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VI. CONCLUSIONS AND RECOMMENDATIONS

The use of planar sections for defining the propeller blade geometry constitutes an acceptable alternative to using cylindrical sections. The planar sections provide many areas in which cost reductions are possible, and the propeller

manufacturer should have a higher degree of confidence that the final propeller, upon passing inspection, actually

conforms to the drawing requirements.

The author suggests that a pilot program be developed with a propeller manufacturer, to produce a propeller, using planar sections. The program should incorporate a control propeller that would be manufactured using cylindrical gages. Both propellers should be carefully checked with sensitive measuring equipment, to determine the degree of conformance with the drawing requirements. The production costs for each propeller should be carefully monitored, in order to provide realistic cost estimates for the production process.

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REFERENCES

[1] Burrill, L. C., "On propeller manufacture," Transactions of the Institute of Naval Architects, Vol. 96, 1954. [2] Langham, J. M., "Founding of marine propellers,"

Proceedings of the Institute of British Foundrymen, Vol. XLVII, 1954.

[3] Halpin, F. W., "Manual of instructions for design and application of propeller blade gages," NAVSHIPS

0987-011-2000, November 1964.

[4] Langham, J. M., "The manufacture of marine propellers with particular reference to the foundry," Published by Stone Manganese Marine Limited, Issue No. 1,

February 1964.

[5] Gunsteren, L. A. van, and Hall, A. F. van, "Propeller production conceptions," International Shipbuilding Progress, March 1971.

[6] Meyne, K., "Propeller manufacture--propeller materials--propeller strength," International Shipbuilding Progress, March 1975.

[7] Bosman, L., and Haanstra, J., "The manufacture of

bronze propellers from a metallurgical point of view," International Shipbuilding Progress, February 1971. [8] "Production of large SMM propellers," Shipping World

and Shipbuilder, August 1973.

[9] Principles of Naval Architecture, The Society of Naval Architects and Marine Engineers, 1967.

[10] Abott, I. H. and Von Doenhoff, A. E., Theory of Wing Sections, New York, Dover Publications, 1959.

[11] Gerald, C. F., Applied Numerical Analysis, Addison-Wesley, 1970.

[12] Adams, E. P., Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Washington, D.C.,

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[13]

Lewis, S. D., and Tsao, S. S., "Propeller cavitation

tests for modified propeller on APL C6-S-85b and Farrell

Lines C6-S-85a class vessels," MIT Department of Ocean

Engineering, September 17, 1975.

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APPENDIX A

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ACCORDING TO ISO-R484 1966

CLASSIFICATION

Class Accuracy of manufacturing Normal use

S high precision propellers of superior

quality for special purposes

I and I medium precision for majority of merchant

vessels

Ill large to'erances for vessels without special characteristics, in general only applied to cast iron propellers.

PITCH

Class Description Local pitch 1.5% t 2% 30/% with a minimum of 15 mm 20 mm 30 mm

Mean pitch per blade at 1% 1.5% 20/ 5/

any radius

with a minimum of 10 mm 15 mm 20 mm 50 mm

Mean pitch per blade 0 75% 1% 1.5% t 4%

with a minimum of 7.5 mm 10 mm 15 mm 40 mm

Total pitch 0.6250/ 0.75% t1 3%

with a minimum of 6 mm 7.5 mm 10 mm 30 mm

Expressed as percentages of the design pitch at corresponding radius.

Tolerances for sections at 0,2 R, 0,3 R and 0,4 R should be increased by 50/%.

RADIUS

Class Description

S 1 11 111

Upper and lower 0.25% 0.50/ 0.5% 0.50/

deviations

with a minimum of 2 mm 3 mm 3 m 5 mm

Expressed as percentages of the radius of the propeller.

THICKNESS

Class Description Upper deviations + 2% + 30/ + 4% + 8% with a minimum of 2 mm 2.5 mm 3 mm 6 mm Lower deviations - 1% - 1.5% - 2% - 4% with a minimum of 1 mm 1.5 mm 2 mm 4 mm

Expressed as percentages of the maximum thickness of the corresponding, blade-section.

PRESSURE SIDE OF HOLLOW SECTIONS

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Class

Description _l

Upper and lower deviations 1/o t 10/ 1.5% 20%.

with a minimum of 1.5 mm 5 mm 10 mm 10 mm

Expressed as percentages of the length of the corresponding blade section. Tolerances for sections at 0.2 R, 0,3 R and 0,4 R should be increased by 50/.

DESIGN MEDIAN LINE

Class

Description

I

if IIl

Upper and lower deviations 0.25/a t 0.5% 0.75%c 1%

with a minimum of 5 mm 10 mm 15 mm 20 mm

Expressed as percentages of the propeller diameter.

LONGITUDINAL POSITION OF

THE

PROPELLER BLADE

Class

Description _{_}

Upper and lower deviations of generator line at radius 0,3 R and

0,95 R (or the innermost and outermost

radii respectively) 0.50/ 10/ 1.5% 3%

with a minimum of 5 mm 10 mm 15 mm 30 mm

Expressed as percentages of the propeller diameter.

SURFACE FINISH

Class

S I I

Maximum values of the mean height of roughness, Hm in Ym

3 9 19

STATICAL BALANCING

When finished, all screw propellers should be statically balanced.

The balancing mass g (in kilogrammes) at the tip of the propeller blade is defined by

G

For n < 160 revolutions per minute c = ck g = CR

For n > 160 revolutions per minute c = ck 2

where

g = balancing mass at blade tip, in kilogrammes,

c and ck = factors depending on the classification of the screw propeller, defined as follows:

Class S I 11 111

ck 1.2 1.8 2.7 5.4

g max. 1.1 x ck x D kg

g min. 0.07 D2 + 0,02 kg

G = mass of the screw propeller, in tons (1000 kg)

0

R = D = radius of blade tip, in metres

n = _{number of revolutions per minute of the screw propeller,}

D = diameter of the screw propeller, in metres.

Note:

The accuracy of measurement of the balancing-apparatus should be at least half of the balancing mass g.

(40)

APPENDIX B

FITTING DATA WITH A CUBIC SPLINE

The conditions for a cubic spline fit are that we pass a set of cubics through the points, using a new cubic in each interval. To correspond to the idea of the draftsman's spline, we require that both the slope and the curvature be the same for the pair of cubics that join at each point. We now develop the equations subject to these conditions.

Write the cubic for the ith interval, which lies between the points (x ,y ) and (x i+'yi+) in the form

Y

=C1C2-~x -- x +- k~iX-Xi)o+ cj(%x-xi) i-cki(11

Since it fits at the two end points of the interval,

yZ

i-x

*-

x'-

xi)

bxc

4 y,.x.

-(1.2)

(1.3) In the last equation, we use h for ax in the ith

interval. We need the first and second derivatives to relate the slopes and curvatures of the joining polynomials, so

we differentiate Eq. (1.1):

-3

c

(k

+

-~ (.-X , (1.4)

(41)

The mathematical procedure is simplified if we write the equations in terms of the second derivatives of the

interpolating cubics. Let s. represent the second derivative of the point (x ,y ) and si+1 at the point (xi+,' ₁ i+1I

From Eq. (1.5) we have

Hence we can write

b

=

(1.6)

~

(1.7)

We substitute the relations for a., b., d. given in Eqs.

S1 1

(1.2), (1.6), and (1.7) into Eq. (1.3) and then solve for c .

-Ylj+1 + 2S.-s~h +

C-ik.

+ Y )

V'i -Y

s6

We now invoke the condition that the slopes of the two cubics that join at (x ,iy) are the same. For the equation in the ith interval, Eq. (1.4) becomes, with x = x.,

(42)

In the previous interval, from x _ to x₁, the slope at its right end will be

Equating these, and substituting for a, b, c, d their relationships in terms of s and y, we get

/Z

Yi+

1 - . 'L~t

66

6

On simplifying this equation we get

hg

st,

+ (z

* Z..hj)SZ +

bi;s

(Y+i

Y

-

Y.

k.)

(1.8)

Equation (1.8) applies at each internal point, from i = 2 to

i = n-1, n being the total number of points. This gives n - 2 equations relating the n values of S i. We get two additional equations involving s1 and sn when we specify conditions

pertaining to the end intervals of the whole curve. To some extent these end conditions are arbitrary. If we take that s is a linear extrapolation from s2 and s3' with analogous linearity of sn, sn-l and sn-2' we find that, for a set of data which are fitted throughout by a single cubic, the spline

(43)

curve is the same cubic. Alternate end conditions do not have this property.

We shall assume linearity, so

s s _S-

_s

ss___

_sn..-s-

₎

_he

_{,S_(hI.+}

.

₀

It is convenient to write the set of equations in matrix form:

K

-(f~,+~,

_2~

)

z

(hK+h

₃

)

0-0

113 2. (hiv b)

0 X3 X

YX

2 h2

Y4 -

Yq

h3

h-3

0

e

0

0 K2

0

a a -S,

S

3 S24

S.'

> b S, h+ 2 s is= 0>

h,

b-.,

h ;h ..

,_

(44)

After the s. values are calculated, the values of a.,

b., c., and d. are obtained, which gives the y equations.

(45)

APPENDIX C

SOLUTION OF CUBIC EQUATION

Cubic equations of the following form are used to define the spanwise, constant percent chord curves

y

=

4; (r-

r-)

i+

b; (r- rj)

+

ci (

r--ri) +

ci

where r and y are the only unknown variables in the equation. For a given value of "y", it is possible to determine the real values for "r" by the following procedure.

Expanding the above equation and expressing in terms of

3

2

r , r , and r results in

4,r+ (bc-3ci

t%)r'

2

+,

(cl-a

6irc

+sc~ir'L)v-Incorporating y in the constant term and dividing through by ai results in 3

6r

where

6 b C

~Ir+3i

+-

bo-r-~~~

(46)

The solution to this form of the cubic equation as given in reference [12] is as follows:

r + ar + b - + C

0

(1) Substitute

y

- (2) Resulting in

y

3 -3 - 0 (3) where 46 Roots of (3):

If

p

>

>0)

2. cosh

#

y

1

=

2 j

cosh (0/6)

YZ=

-

(y,

/?)

+

if/eV' sirh

($13)

If < O

cosh

#

-y,=

-

2 .j7osn

(0/-)

(47)

If p < O )

S~n~

9,

y, =

2 J

szn)

(s/

?-) + i

-s

'

y

3

= -

ky

I

If A I?

Cos

cosh (

0/5)

-

.3p 'cosh

;

fT

cos

(#/3)

Y=

-(Y,/a)

+

39 T

Ys=

-

(Y,

/)

-sin

(0/3)

Sh-V(0/3)

Roots of (1) :

r

3 ~

y~

3z

3 sFp

'

(48)

APPENDIX D

(49)

&PITCHt25),CH)(25),RAKE(25),SKEW (25),pCP(25),PCS(25)

&ETAP(25,25),ETAS(25,25),PHI(25),ZWK (25),RAD(25),XP(25,25), &YP (25,25) ,XS(25,25) ,YS 25,25) ,ZS (25,25) ,ZP 25,25)

&ICOUNT (50,2) ,ICKY(50)

,XI (25)

,YI (25) ,ZI(25) ,YQ(25),

SAE25),BEF25),CE(25),DE(25),YIN(25), YMAX(25),RADA(153), &JCOUNT(25) ,XO(150) ,ZO(150),XF(50,75,2) ,ZF(50,75,2),

&ZOP(150) READ:5,10) TITLE 10 FORMAT(18A4) READ (5,11) KK, NR, NCSNCP, ZDIAM 11 FORMATt413,2XF15.5) READ(5,12) READf5,12) 12 FORMAT(7F1 READ(5,13) 13 FORMAT(5F1 READ :5,12) READ(5,12) DO 20 I=1, READ:5,12) 20 READ(5,12) C SCALE,ROTATE YO, K),FK=1,.KK) 0.3) (XR I) ,PITCH #I) CHD 0.5) (PCP (J) ,J=1, NCP) (PCS(J) ,J=1,NCS) NR "ETAP I,J) ,J=1,NCP) (ETAS(I,J) ,J=1, NCS)

C ** CONVERT OFFSETS TO CARTESION COORDINATES ** C PI=3. 14159265 DELP=57.295779 PH IP=ROTATE/DELP APP=COS (PHI P) BPP=SIN:PHIP) DO 22 K=1,KK 22 YF :K)=Yo K)*SCALE DO 27 I=1,NR RAD(I)=XR (I)*ZDIAM/2.0 PHI(I)=ATAN:PITCH:I)/PI*2.0*RAU (I))) PLANO003 PLAN0004 PLANC005 PLAN0006 PLANO007 PLAN008 PLAN0009 PLAN0010 PLANO011 PLAN0312 PLAN0013 PLAN0914 PLAN0015 PLAN316 PLAN0017, PLAN00180 PLANCO19 PLANO020 PLANO021 PLANO022 PLAND023 PLAN0024 PLAN0025 PLAN0026 PLAN0027 PL&N0028 PLAN0029 PLANO)3C PLAN0031 PLANO332 PLAN0033 PLANO034 PLAN0035 PLAN0036 PA3E 1 tI),rRAKE fI),SaKEW!I),rI=lNR)

(50)

DO 25 J=1,NCP

C1='%"HDII)*(O.5-PCP{J)/100.0)

ZWP=ZWK (I) +C1 *COS (PHI(I)) -ETAP (IJ) *SIN (PHI (I)) THEP=ZWP/RAD (I)

XP (I ,J)

_=-RAKE

(I) +C 1*SIN (PHI (I)) +ETAP (IJ) *C'S (PHI (I)) +iX K YP 'IJ ) =RA D fI) *COS fTHEP)

25 ZP (IJ) =RAD (I) *SIN (THEP) DO 27 J=1,NCS

Cl=CHD(I)*t(.5-PCS(J)/100.0)

ZWP=ZWK (I)+C*COS(PHI(I))-ETAS(IJ)$SIN(PHI(I)) THEP=ZWP/RAD (I)

XS (I,J)=-RAKE(I)+C1*SIN(PHI(I))+ETAS(IJ)*COS(PHI(I))+X1K YS (I,J) =RAD (I) *COS "THEP)

27 ZS(IJ)=RAD(I)*SIN(THEP) WRITE (6,30)TITLE 30 FORMAT(1H1,18A4//) WRITE (6,32)ZDIAM 32 FORMAT(1HOvBLADE DIAMETERINCHES = ',F10.5) WRITE (6,34) KK

34 FORMAT(1H ,'N. OF PLANAR SECIIONSKK =',I5)

WRITE(6,36)NR

36 FORMAT(1H ,'N3. OF BLADE FRACTIONAL RADIINR =',I5) WRITEf6,38)NCP

38 FORMAT(1H ,'NJ. OF PRESSURE FACE )FFSET STATIONS,NCP =',15) WRITE(6,40)NCS

40 FORNAT(lH ,'N3. OF SUCTION FACE OFFSET STATIONSNCS =',15) WRITE (6,44) SCALE, ROTATE

44 FORIAT 1HO,'BLADE SCALE=",F1O.5,5X,'BLADE ROTATIONDE3=',F10.5,5X, &' (POSITIVE FOR INCREASING PITCH)')

WRITE f6,50)

50 FORMAT(1HO,' R/R RADIUS(IN) PITCH (IN) CORD(IN) RAKE(I

&N) SKEW(IN)f)

DO 52 I=1,NR

52 WRITE (6,54) XR (I) ,RAD (I),PITCH (I) ,CiID (I) ,RAKE (I) ,SKEW (I)

PLAN0039 PLAND040 PLAN0041 PLANO342 PLAN0043 PLAN0044 PLAN0045 PLAN0046 PLAN0047 PLkN0048 PLAN0049 PLANO050 PLAN0051 PLAN0052 PLAN00530 PLAN0054o PLUNO055 PLAN0056 PLANOD57 PLAN0058 PLAN0059 PLAN0060 PLAN0061 PLAN0062 PLAN0063 PLAN0064 PLANO065 PLAN0066 PLAN0067 PLAN0068 PLAN0069 PLANO070 PLAN0071 PLAN0072 PAGE 2

(51)

56 FORMAT(1H0,'PERCENT CHORDS FO PRESSURE FACE$)

WRITE(6,58) (PCP(J) ,J=1,NCP) 58 FORMAT(1H ,3X,5F12.6)

WRIT!(6,60)

s0 FORMAT(1HO,'PERCENT CHORDS F03 SUCIICN FACE') WRITEf6,58) PCS J) ,J=1,NCS)

WRITE(6,62)

62 FORMAT(1H1,'*** BLADE OFFSET DATA ***') DO 68 I=1,NR

WRITE (6,64) XR (I)

64 F0R1IAT(iHO,'R/R=',F12.6,5X,'PRESSURE FACE') WRITE(6,58) (ETAPfIJ) ,J=1,NCP)

WRITE (6,66) XR :1)

66 FORMAT(1H ,'R/R=',F12.6,5X,'SUCTION FACE') 68 WRITE(6,58) (ETAS(IJ) ,J=1,NCS)

WRITE6,70)

70 FORMAT(1H1,W*** BLADE OFFSET )ATA B3ANSLATED TO CARTESIA &TES ****)

WRITE(6,72)

72 FORMAT(1HO,#PRESSURE FACE') WRITE :6,74)

74 FORMAT(IHO,' % CHORD',7X,'R/R ,12X,'-X-',12X,'-Y-',12X,' DO 76 J=1,NCP 76 WRITE(6,78) (PCP(J),XR(I),XP(IJ),YP(I,J),ZP[I,J),I=1,NR) 78 FORMAT(1H ,F10.6,F12.6,2X,3F14.6) WRITE :6,70) WRITE(6,80) 80 FCRMAT(1H0,'SUCTION FACE') WRITE(6,74) DO 82 J=1,NCS 92 WRITE(6,78) (PCS(J),XR(I),XS(I,J),YS(I,J),ZS(IJ),I=1,NR) N COORDINA

FOR THE PRESSURE FACE "N=1"

PLAN0075 PLAN0076 PLAN0077 PLAN0078 PLINO079 PLANO080 PLAN0081 PLAN0082 PLANO083 PLkN0084 PLAN0035 PLAN0086 PLAN0037 PLANO088 PLANDO389 PLANO090 PLANO091 PLAN0092 PLAN0093 PLANO094 PLAN0095 PLAND096 PLANO097 PLAN0098 PLA N3 099 PLAN0100 PLAND131 PLAN0102 PLAN3133 PLAN0104 PLAN0105 PLAN0106 PLAN0107 PLAN0108 PASE 3 U, H C C C - Z- I)

(52)

NCA=NCP 88 DO 90 K=1,KK 90 IOUNT(K,N)=0 DO 500 J=1,NCA DO 92 K=1,KK 92 ICKY (K)=0 INTA=0 IF (N.EQ.2) GO 1o 96 DO 94 1=1,NR XI ()=XP(IJ) YI[I)

=

yp I,#J) 94 ZI (I)=ZP(I,J) GO TO 100 96 DO 98 I=1,NR XI (I)=S (IJ) YI (I)=YS (I, J) 98 ZI (I)=ZS(IJ) 100 CONTINUE CALL CONCUB(RADYIRADYQAE,EE,CE,DiNRNR) C

C CALCULATE "YNAXII)" AND "YRINJI)" C

FJR EACH CHORD INTERVAL DO 150 I=1,NRM1

A=AE (I)

B=BE (I) -3.0*AE (I)*RAD(I)

C=CE(I)-2.0*BE(I)*RAD(I)+3.0*AE1I)*RAD :I)**2

D=DE (I) -CE (I) *RAD(I) +BE(I) *RAD (I)**2-AE (I) *RAD (I) **3 E=-B/3.0*A) EE=E*E FF=C/ 3. 0*A) IF(FF.GT.EE)GO TO 116 F=SQRT (E**2-C/ (3.0*A)) R1=E+F R2=E-F PLANO 111 PLANO112 PLAND 113 PLANO 114 PLAN0115 PLA NO 116 PLAN0117 PLANO 118 PLAN0119 PLAND120 PLAN0121 PLAN0122 PLAN0123 PLAN0124 PLA NO 125 PLAN0126 PLA NO 127 PLAN0128 PLAN0129 PLAN0130 PLANO131 PLAN0132 PLAN0133 PLAND134 PLAN0135 PLAN0136 PLAN0137 PLAN0138 PLAN0139 PLANO140 PLAN0141 PLLN0142 PLAN0143 PLAN0144 PA3E 4 Ln

(53)

Y1=A*R1**3+B*R 1**2+C*R1+D

**Y2=A*R2**3+B*R2**2 +C*R2+D**

YKIN I)

=Y1

RMIN=1

YMAX (I) =Y2

RMAI=R2 Yi PP=6.0*A*R1+2.0* B Y2PP=6.0*A*R2+2.0*B IF(Y1PP.GE.0)GO TO 105 YMAX (I)=Yl

RMAX=R1

YMIN(I)=Y2

RMIN=R2

105 CONTINUE IMI=0 IMA=O

IF((RAD(I) .LE.RMAX) .AND. (RAD(I+1).GT.3MAX))IMA=1 ,IF [RAD :I).eLE.RHIN).AND.(fRAD CI+1). Gl.MIN) )IMI=1

IF((IKA+IMI).EQ.2)GO TO 145 IFCIMA.EQ.1)GO TO 107 GO TO 110 107 YMIN(I)=NIN1(YI(I) ,YI(I+1)) GO TO 150 110 IF(IMI.EQ.1)G0 TO 114 GO TO 116 114 YMAX(I)=MkX1(YI(I),YI(I+1))

GO TO 150

116 YMIN(I)=HIN1(YI(I),YI(I+1)) YMAX (I)=MAX1(YI(I),YI(I+1)) GO TO 150 145 WRITE(6,147)I,J,N

147 FORMAT(1H ,'***MINIMA AND MAXIMA ARE IN INTERVAL*** I=',I4,4X,'J=

*,4,r~4X,'N=4 ,I4)

WRITE(6,148) RMAXYMAX (I) ,RfIN,Y3IN (I)

148 FORMAT(1H ,'RMAX=',F1O.4.2X,'YMAX(I)=',F10.4,2X,"RMIN=',FlO.4,2X,I &YMINfI)=IF10.4) PLAN 145 PLAN0146 PLAND147 PLAN0148 PLAND149 PLAN0150 PLAN3151 PLANO 152 PLAN0153 PLAND 15 4 PLAN0155 PLAN0156 PLAN0157 PLAN0158 PL&NO159 PLAN0160 PLAN3161 PLAN0162 PLAN0163 PLAN0164 PLAN0165 PLAN0166 PLAN0167 PLAN3168 PLAN0169 PLAN0170 PLAN0171 PLAN0172 PL&NO173 PLAN0174 PLAN0175 PLANO 176 PLAN0177 PLANO178 PLAN0179 PLANO180 PASE 5 uL w

(54)

150 CONTINUE C

C C

FOR EACH INTERVAL ALONG CHORD DETERMINE IF YO"K) IS BETWEEN YKIN(1) AND YKAX(I). IF IT IS THEN CALCULATE ASSOCIATED RADII.

DO 400 I=1,NRM1 DO 395 K=1,KK

IF(Y3(K).GT.YAAX(I))GO TO 400 IF(YO(K).LT.YMIN(I))GO TO 395 ACUB=AE I)

BJUB=BE (I) -3.0*AE (I)*RAD (I)

CCUB=CE (I) -2.0*BEI)*RADI)+3.0*AE(I)*RAD(I) **2

DCUB=DE (I) -CE (I) *RAD (I) +BE (I) #RAD (I) **2-AE (I) *RAD (I)**3-YO (K) CALL CUBSOL(ACUBBCUB,CCUBDCB,R1R2,rB3,ITEST)

IF (R1.GE.RAD (I)).AND. fR1.LT.RAD (I+1)))GO TO 160

152 IF((R2.GE.RAD(I)).AND.(R2.LT.RAD(I+1)))GO TO 170 IF "ITEST.EQ.2) GO TO 395 156 IF((R3.*E.RAD(I)).AND.(R3.LT.RAD (1+1)))GO TO 180 GO TO 395 160 ICKY(K)=ICKY(K)+1 INTA=INTA+l1 RADA fINTA)=R1 IF(ITEST.EQ.1)GO TO 395 GO TO 152 170 ICKY(K)=ICKY(K)+1 INTA=INTA+1 RADA(INTA)=R2 IF(ITEST.EQ.2)GO TO 395 GO TO 156 180 ICKY (K)=ICKY(K)+1 INTA=INTA+l1 RADA (INTA) =R3 395 CONTINUE 400 CONTINUE PLAN0183 PLAN0184 PLAN0185 PLAN0186 PLAN0187 PLAN0188 PLAN0189 PLAN0190 PLAN0191 PLAN0192 PLAND193 PLAN0194 PLAN0195 PLAN0196 PLAN0197 PLANO198 PLAN0199 PLANO200 PLhNO201 PLANO202 PLAN0203 PLANO204 PLANO205 PLANO206 PLANO207 PLAN0208 PLANO209 PLANO210 PLANO211 PLANO212 PLANO213 PLANO214 PLANO215 PLANO216 PA3E 6 uL_ob om

(55)

CALL CONCUB(RADXIRADA,X0,AE,BE,CE,DENR,NJ) CALL CONCUB(RADZIRADAZO,AEBE,CE,DENRNJ) I SUM =0 D3 450 K=1,KK ICKI=ICKY [K) DO 445 IK=1,ICKI ICK=ICOUNT [KN) +IK ISUM=ISN+1 SIGN=1.0

IF (Z3 (ISU) . LT.O .0) SIGN=-SIGN

IF (YO(K) .GT.RADA (ISUM) ) RADA(ISU)= YO (K) ZOP FISUM) =SQRT (RADA (ISUM) **2-10 K) **-)*SIGN ZF (KICK, N) = (ZOP (ISUM) *APP-XO (ISUM) *EPP) *SCALE XF 'KICK, N)= (ZOP (ISUM) *BPP+X3 (ISSUN) *AP P) *SCALE

445 CONTINUE

ICOUNT (KN) =ICOUNT (K, N) +ICKI 450 CONTINUE 500 CONTINUE IFN.EQ.2)GO TO 510 N=2 NCA=NCS GO TO 88 510 CONTINUE WRITE (6,30) TITLE WRITE (6,44) SCALEROTATE DO 600 K=1,KK WRITE(6,515)YF(K)

515 FORKAT'1HO0,'*** PLAK&R SLICE hY"='v,F1.5,2X,'* DO 595 N=1,2

IFfN.EQ.2)GO TO 530 WRITE(6,517)

517 FORMAT (1H0,'PRESSURE FACE PLANAR OFFSEIS')

518 WRITE#6,520) 520 FORMAT(1HO,' NJ ,7X,#-X-',12X,'-Y-', 12X,'-Z-'., **',

//)

PLANO219 PLANO220 PLANO221 PLANO222 PLANO223 PLANO224 PLANO225 PLANO226 PLhNO227 PLAN0228 PLAN0229 PLANO230 PLANO231 PLANO232 PLANO233 PLAN0234 PLANO235 PLANO236 PLANO237 PLAN3238 PLANO239 PLANO240 PLANO241 PLANO242 PLA NO243 PLANO244 PLANO245 PLAN0246 PLANO247 PLANO248 PLAN0249 PLANO250 PLANO251 PLANO252 PAGE 7 12X,' -R-', 1) U. U.'

(56)

IP=ICOUNT(K,.N) DO 528 ICK=1,IP

RF=SQRT (ZF (K, ICKN) **2+YF (K) **2)

WRITE f6,525) ICKoXF (KICK, N) YF (K), ZF (K,ICK, ) , RF

525 FORMAT(1H ,13,4F15.6) 528 CONTINUE

GO TO 595 530 WRITE(6,532)

532 FORMATIHO,'SUCTION FACE PLANAR OFFS3IS') O TO 518 595 CONTINUE 500 CONTINUE STOP END PLANO253 PLANO254 PLANO255 PLAN0256 PLANO257 PLkN0258 PLANO259 PLANO260 PLANO261 PLANO262 PLAN0263 PLANO264 PLANO265 PLAN0266 U, C.' PA3E 8

(57)

SUBRUTINE CONCUB(XIYIlO,YO,AEi3BC3,DE,NN,JJ)

DIMENSION XI(NN),YI(NN),XO(JJ),Y(JJ),A(25,25),Ht25),AE(NN),

&CE (NN)

,DE

(NN) ,B (25) ,X(25) ,BE NN) ,AK (625) EQUIVALENCE (B(1),X(1)) DO 11 I=1,25 H (I) =0.0 B(I) =0.0 X(I)=0.0 DO 11 J=1,25 A I,J) =0.0 11 CONTINUE NMI=NN-1 12 DO 15 I=1,NMI 15 H(I)=XI(I+1)-XI(I) A (1,1) =H(2) A (1,2)=- (H (1) +H(2)) A (1,3)=H (1) A (NNNN) H (NN-2) A(NN,NN-1) =f-(NN-2) +H :NN-1)) A (NNNN-2) =H (NN-1) ITP3=NN-2 DO 20 I=1,ITP3 20 A(I+1,I)=HI(I) ITP1=NN-1 DO 25 I=2,ITP1

A (II) =2.0* PH :I-1) +HEI)) A (II+1)=H (I)

BP=tYIII+1)-YI (I) )/H 'I) - YIFI) -YI I- 1))/H*I- 1) 25 B(I)=6.0*BP NQ=NN DO 26 J=1,NN DO 26 1=1,NN K=(J-1)*NN+I 26 AK(K)=A(IJ)

CALL SINQ [AKBNQ,KQ) IF(KQ.EQ.0)G3 TO 35 CONCOO01 CONCO002 CONC0003 CONC03 4 CONCO005 CONCO006 CONCO007 CONCO008 CONCO009 CONCO010 CONCO311 CONCO012 CONCO013 CONCO014 CONCO015 CONCO016 CONCO017 CONCO018 CONCO019 CONCO020 CONCO021

CONCO022

CONC0O23

CONCO024

CONCO025

CONCO026

CONCO027

CONC0028

CONCO029

CONC0030

CONCO031

CONCO032

CONCo033 CONCO034

CONCO035

CONCO036

PAGE 9 Lii_-j

(58)

35 CONTINUE NM2=NN-2 DO 42 I=1,NM2 AE (I)= (X (I+1)-X(I))/(6.0*H(I)) BE FI)=X I)/2. ) CEE1= (YI(I+1)-YI(I))/H(I)

CEE2= (2.0*H(rI)*X (I)+H(I)*XfI+1))/6.0 CE (I) =CEE1-CEE2

42 _{DE1I) =YI FI)}

GE=XI (2) -XI [1) AE(2)=AE(1)

BE (2) =3.0*AE(1)*GE+BE (1) CE (2)= (BE (2)+BE (1)) *GE+CE (1)

DE 2) =AE 11)*GE**3+BE 1)*GE**2+CE :1)*GE+DE GE=XI (NN-1) -XI (NN-2)

AE :NN-1)=AE(NN-2)

BE (NN-1)

**=3.0AE(NN-2)GE+BE**

NI-2)

CE (NN-1)=(BE(NN-1)+BE(NN-2))*GE+CE(NN-2) DE (NN-1) =AE (NN-2) *GE**3+BE (NN-2) *G;**2+CE IF(JJ.LT.1)GO TO 100

DO 90 J=1,JJ DO 85 I=1,NMI

IF (XO (J) .GE.I(I)) .AND. (XOJ) .LE.XI (I+1)

GO TO 85 45 _{H1=XO FJ) -XI(II)} H2=H1*H1 H3=H2*H1 YO(J)=AE(I)*H3+BE(I)*H2+CEI):*H1+DdII) GO TO 90 85 CONTINUE 90 CONTINUE 100 CONTINUE RETURN END 'NN-2) *GE+DE :NN-2) ))GO TO 45 CONCD 039 C9NCO040 CONCO041 CONCO042 CONCO043 CONCO044 CONCO045 CONCO046 CONCO047 CONCCO48 CONCO049 CONCO050 CONCO051 CONC0052 CONCO053 CONC0O54 CONC0O55 CONCO056 CONCO057 CONCO058 CONC0O59 CONCO060 CONCO061 CONCO062 CONCO063 CONC0064 CONC0O65 CONCO066 CONC0O67 CONCO068 CONCO069 CONCO070 CONCO071 CONCO072 PA3E 10 uL

(59)

TOL=0.0

KS=0 JJ=- N DO 65 J=1,N JY=J+1 JJ=JJ+N+1 BIGk=0 IT=JJ-J DO 30 I=J,N IJ=IT+I IF(kBS(BIGA)-ABS(A(IJ))) 20,30,30 20 BIGk=A(IJ) IMAX=I 30 CONTINUE IFtABS(BIGA)-TOL) 35,35, 40 35 KS=1 RETURN 40 I1=J+N*(J-2) IT=IAI-J DO 50 K=J,N I1=11+N 12=I1+IT SAVE=At(1) A(I1)=A(I2) A(12)=SAVE 50 A(I1)=A(I1)/BIGA SAVB=B IMAX) B(IMAX)=B(J) B CJ) =SAVE/BIGA IF(J-N) 55,70,55 55 IQS=N*(J-1) DO 65 IX=JY,N IXJ=IQS+IX IT=J-IX