Solution to Lerbs propeller pitch correction for a propeller with any number of blades by means of an electronic digital computer

MECHANICAL ENGINEERING REPORT

MB-239

OLI ION O L:B

PROPELLER PITCRR

FOR A

PROPELLER WITH

ANY NUMBER OF BLADES

BY MEANS OF AN ELECTRONIC DIGITAL COMPUTER

ay

C. STRASZ

DIVISION OF MECHANICAL ENGINEERING

OTTAWA

SEPTEMBER 1961

THIS REPORT MAY NOT BE PUBLISHED IN WHOLE OR IN PART WITHOUT THE WRITTEN CONSENT OF

THE NATIONAL RESEARCH COUNCIL

Prepared by: Submitted by:

REPORT

Division of Mechanical Engineering Ship Laboratory

Pages - Preface - 2 Report: MB-239

Text - 16 Date: 5 September 1961

Figures - 2 Lab. Order: 13435A

File: M2-3-7 Internal

SOLUTION TO LERBS' PROPELLER PITCH CORRECTION FOR A PROPELLER WITH ANY NUMBER OF BLADES BY MEANS OF AN ELECTRONIC DIGITAL COMPUTER

J.S.C. Straszak S.T. Mathews Head Ship Laboratory E.S. Turner Head Hydrodynamics Section Approved by: D.C. MacPhail

Dire ctor

SUMMARY

A detailed description is given of the program for calculating the part of Lerbs' propeller pitch correction arising from lifting surface effect using the Bendix G-15D digital computer. The method is applicable for a propeller with any number of blades.

For:

TABLE OF CONTENTS

Paqe

SUMMARY (i)

1.0 INTRODUCTION 1

2.0 THEORY i

3.0 NOTE ON DERIVATION OF G FOR A FREE-RUNNING PROPELLER 3 G

4.0 INTEGRATION FORMULA FOR INTEGRATION OF X TERMS 4

5.0 COMPUTER 5

6.0 PROCEDURE o

6.1 Flow Diagram 6

6.2 Allocation of Channels 6

6.3 Storage of Constants and Program 6

6.4 Punched Paper Tape 7

6.5 Specimen Calculation Sheet from Computer

and Check on Accuracy 7

7.0 OPERATING INSTRUCTIONS 7 8.0 REFERENCES 9 SHEETS NO. i to 7 LIST OF ILLUSTRATIONS Fiqure Flow Diagram i

1.0 INTRODUCTION

Experience has shown that propellers designed from the theory of a moderately loaded propeller are uriderpitched. The reason for this discrepancy between the theory and experience was traced and developed by H.W. Lerbs (Ref. 1), and an

expres-sion for an approximate pitch correction was obtained by him.

To the best of the present writers knowledge, this approximate

pitch correction was first applied by M.K. Eckhardt and W.B. Morgan in a comprehensive method of propeller design developed by them in Reference 2. This propeller design method, together with the Lerbs' pitch correction, is used

occasionally by the N.R.C. Ship Laboratory in the form out-lined in Reference 2 but the evaluation of the pitch correc-tion by ordinary mathematical means was found to be somewhat laborious and prone to human error. For that reason it was decided to program the most cumbersome part of it for calcula-tion on an electronic computer.

2.0 THEORY

For the derivation and method of application of the pitch correction formula the reader is referred to References 1 and 2. This report will consider the theory only to the extent necessary for understanding the significance of

b - the term for the calculation of which the electronic computer was

programmed.

The total pitch angle correction is of the form SOLUTION TO LERBS' PROPELLER PITCH CORRECTION

FOR A PROPELLER WITH ANY NUMBER OF BLADES BY MEANS OF AN ELECTRONIC DIGITAL COMPUTER

(a1 + a2), but we shall only be concerned with a2 = (ab + a. - a ), and in particular with the term a

i o b

The remaining angles can be comparatively simply evaluated in accordance with Reference 2.

In the expression for ab which follows below, the calculations are made for a blade in the 90-degree position, and the effect of the other blades is determined on the bound vortex of this blade. Thus, for example, for a four-bladed propeller, calculations are made with i equal to 90, 180, 270 and 360 degrees. The additional pitch angle ab is then equal to the sum of the contributions for each of these four blade positions.

It will be noted, however, that the present machine solution is programmed for any required number of blades and as long as the correct value of z is read into the computer

(together with the correct values of ,

and G, of course), the correct solution will automatically follow.

In the following the values of and functions of at 0.7 radius are used, where they occur, and i is the

angular position of the blade in degrees. x

₍

) is the non-dimensional radius of any propeller blade section.

sin

O7

a

_b07=

2 _0.7

s1n_0.7cosicos)

1.0 _{G dx} radians

02

X,

P-(g3

RI p-, X [

r2

LX 2xk or G X Z + 0.49 - 2 cos p. cos 7

D07

+ 0.7 sin p.) xl

which can be written as

(X2 + A - B)3"2

p-, X

where A is a constant for any propeller and B varies with the blade position p. for any propeller and with x.

wt 2xk x

G = -i--- -- non-dimensional circulation per blade for a 3/2 A free-running propeller wt X 1 2V _(l-w S _X

for a wake-adapted propeller.

3.0 NOTE ON DERIVATION OF G FOR A FREE-RUNNING PROPELLER

Frequently, in the course of the propeller design it is more convenient to use the following expression for the value

of G, derived from the propeller theory.

where

0.7

sin p. - 0.7 cos _{cos p.)} C,

CL' VR 27rD V

where is normally available from theory at an early stage of propeller design

cos _(13f

_-

_P)

sin ₁₃

where and ¡3 are at the required propeller blade radii x.

VR

-t- and - are available for the range of values of x between

L _VA

0.2 and 1.0, and thus the required values of non-dimensional circulation can be easily obtained.

G

4.0 INTEGRATION FORMULA FOR INTEGRATION OF X TERMS A and VR V A G

For integration of

₍₃

terms, the fol±owing

nine-point integration formula, obtained from Reference 4, was used:

Area under the curve

=

I

ydx = 28350 (989y0 + 5888v -± 10496 - 4540 + 10496 - 928 y3 y4 y5 y6 + 5888 + 989 2368 y10h y7 y8 ₄₆₇₇₇₅

where x0 x8 represent values from 0.2 1.0 at h = 0.1 common interval, and y represents values of at the

appropriate propeller blade sections. The last term on the right is an error term and can be neglected as insignificant.

The last column is then read into the channel 08 of the computer, and punched on the paper tape to form a permanent part of the program.

5.0 COMPUTER

The machine used for the calculation was the Bendix G15D digital computer for which the Intercom 1000 single preci-sion coding system was used. The coding system is fully

explained in Reference 3.

Using h = 0.1 and working out the multipliers for each ordinate we obtain the following:

Partial Final X y Constant Multiplier x 102 0.2 _Yo 989 2.7908 0.3 y' 5888 16.6151 0.4 -928 -2.6187 0.5 _Y3 10496 29.6182 0.6 Y4 -4540 -12.8112 0.7 _Y5 10496 29.6182 0.8 y6 -928 -2.6187 0.9 y7 5888 16.6151 1.0 _Y8 989 2.7908

6.0 PROCEDURE 6.1 Flow Diagram

In order to facilitate coding a block diagram showing the sequence of all arithmetic operations was prepared. It is appended in Figure 1.

6.2 Allocation of Channels

Orders are stored in the computer's memory in

addresses from 0700 to 1899 (where the first two digits signify the "channel" and the remaining the "word position") and are executed from sequential locations unless a "transfer" order is encountered. The choice of channels to be used is left to the discretion of the programmer, and the following were chosen in this particular

case:-program itself is in channels 07 and 08 square root sub-routine is in channel 13

('c) sin and cos sub-routine in channel 14

L

(d) variable data z,

f3,

and values of Gx are entered

in channel 11 of the computer's memory

(a), (b) and (c) are the permanent part of the program and are punched on the paper tape as indicated later in this report. 6.3 Storage of Constants and Program

Sheets 1 and 2 give the addresses used for the storage of constants and also symbolic or numerical contents of each location used for this purpose. In order to perform the calculation for

01b

for any propeller, the symbols contained in locations from 1100 through to 1111 (sheet 1) are replaced by their numerical values and are read into the computer's memory as shown in Section 7, "Operating Instructions". The

constants and their locations enumerated in sheet 2 are a permanent part of the program and remain unchanged.

The program itself is set out in sheets 3 to 7 and remains unchanged.

6.4 Punched Paper Tape

Commands of the program were first manually punched on a paper tape using the Flexowriter and read into the computer together with the sub-routines required, in accordance with

instructions contained in Reference 3.

After checking and correcting, a final paper tape was punched out by the computer and is now available in the Ship Laboratory for the performance of this calculation.

6.5 Specimen Calculation Sheet from Computer and Check on Accuracy

A copy of a specimen calculation sheet typed out by the computer is appended, together with some explanatory notes added by hand, in Figure 2. Calculations in Figure 2 refer to Propeller No. 24, for which the same computation has been

carried out previously by ordinary mathematical means. In this, as well as in the other propellers for which the machine calcula-tian constituted a check, the agreement was found ta be excellent. Small discrepancies which were found in individual values of

r i3/2

L j and consequently in ab are attributed to the human errar

in manual calculation.

7.0 OPERATING INSTRUCTIONS

Put "Intercom 1000 Single" magazine, rewound in photo reader. Punch switch off, computer switch off, enable switch on.

Enable switch off. Compute switch to Go. Computer is now in manual operating mode.

Remove Intercom magazine and replace it by punched tape in photo reader. Take care that tape is rewound and that its narrow side faces the machine.

Type 55 0700 "tab S". Computer will now read the contents of channel 07 into the memory.

Type 55 0800 "tab S". Computer will now read the contents of channel 08 into the memory.

Type 55 1300 "tab S". Computer will now read the contents of channel 13 into the memory.

Type 55 1400 "tab S". Computer will now read the contents of channel 14 into the memory.

Type 51 1100 "tab S". Computer is now ready for typing into the channel 11 of its memory the variable fixed point data, the locations and symbols for which are outlined in sheet 1. At this stage the computer automatically assumes the next successive address and types it out so that it is only necessary to type into it the appropriate figure for the location set out by computer. It must be remembered to use /

("slash") for a decimal point and to follow each figure by "tab S".

(lo) Having completed typing of fixed point data at location 1111, the computer types 1112 (the next assumed location). Type 0690700 // tab S ("slash, slash, tab S") to start automatic computation.

(11) Computer automatically types a column of [

j3/2 values beginning at x = 0.2 (and ending at x = 1.0) for

blades without a break. This is followed by three carriage returns and the answer in RADS. , a break, and an answer in DEGS. to the right on the same level. REMOVE tape from computer and REWIND it for future use.

8.0 REFERENCES

1. Lerbs, H.W. Propeller Pitch Correction Arising from

Lifting Surface Effect.

D.T.M.B. Report No. 942, February 1955.

2. Eckhardt, M.K. A Propeller Design Method.

Morgan, W.B. Trans. S.N.A.M.E., Vol. 63, 1955,

pp. 325-361, 372-374.

3. Baxter, D.C. The Intercom 1000 Programming System for the Bendix G-l5D Digital Computer.

N.R.C. Mech. Eng. Report MK-3, May 1959.

4. Milne, W.E. Numerical Calculus, p. 124, Eq. 8.

Princeton University Press, 1949.

STORAGE OF CONSTANTS

Intercom 1000

Locn

K OP ADDR Accumulator

iioo z No. of blades

1101 degrees 1102 1103 G for X = 0.2 1104 G for X 0.3 1105 G for X 0.4 1106 G for X = 0.5 1107 G for X = 0.6 1108 G for X = 0.7 1109 G for X = 0.8 1110 G for X = 0.9 1111 G for X = 1.0

N.B. The constants shown in this sheet in their assigned locations must be replaced by their numerical values and then read into the computer's memory as explained in para. 7.

STORAGE OF CONSTANTS

Intercom 1000 Loc'n K OP ADDR LOCATION ACCUMULATOR 1120 0890 2.0 1121 0891 0.49 1122 0892 360° 1123 0893 9Q0 1124 0894 0 1125 0895 0.2 1126 0896 0.7 1127 0897 0.1 1128 0898 449° 1129 0899 57.296° 1130 1131 0880 0.027908 1132 0881 0.166151 1133 0882 -0.026187 1134 0883 0.296182 1135 0884 -0.128112 1136 0885 0.296182 0886 -0.026187 0887 0.166151 0888 0.027908 COS

(2

+ 0.49 = A Aji = degrees ji degrees S X sin ji cos ji sin

ji

0.7 cos cos

ji

sin - 0.7 cos cos

ji

= C cos cos ji

2x rO.7 sin

ji

+ cos

ji

cos B

(X2 + A - B)

(X2 + A - B)3'2 Accumulator

OP Accumulator 1 degrees sin sin ¡3. 2 l3 degrees cos + 0.49 360° 3600 z 0

5=0

Intercom 1000 Loc'n K ADDR

sin 0700 0 42 1101 2 0701 0 08 1439 0702 0 48 0890 0703 O 49 1120 cos 0704 0 42 1101 0705 0 08 1423 0706 O 49 1121 + 0.49 A 0707 0 42 1102 0708 0 44 2101 0709 0 43 0891 0710 O 49 1122 Calculate A 0711 0 42 0892 0712 O 48 1100 0713 0 49 1123 Set p 900 0714 0 42 0893 0715 0 49 1124 Set I = o 0716

0420894

0717 0 49 1125 Set $ O 0718

0491126

L 2

i Loc'n K OP ADDR Accumulator

Set X 0.2 0719 0 42 0895 X = 0.2

0720 0 49 1127 Assign word difference 0721 1 71 0001 Assign word limit 0722 1 72 0008 Assign channel base 0723 1 73 0000

Assign word base 0724 1 70 0000

sin p 0725 0 42 1124 degrees 0726 0 08 1439 sin i 0727 0 49 1128 cos p fl728 0 42 1124 p degrees 0729 0 08 1423 cos p. 0730 O 49 1129

sin - 0.7 cos cos C 0731 0 42 1102

0732 0 44 1128 sin 0733 0 49 1130 0734 0 42 0896 0.7 0735 0 44 1121 0.7 cos 0736 0 44 1129 0.7 cos cos 0737 0 49 1131 0738 0 42 1130

L.

sin p. 0739 0 41 1131 L sin p. - 0.7 cos . cos p. i 0740 0 49 1132

Intercom 1000 Loc'n 1K op ADDR Accumulator

r

L2X cas p. cos 3. 0741 0 42 1102 L + 0.7 sin B 0742 0 44 1121 cos 0743 0 44 1129 cas Ç3. cas p. 0744 0 49 1133 0745 0 42 0896 0.7 0746 0 44 1128 0.7 sin p.

0747 0 43 1133 0.7 sin p. + cas p. cos

0748 0 44 0890 0.7 sin p. + cos p. cas 0749 0 44 1127

_{2<0.7 sin}

p. + cas p. cos 0750 0 49 1134 [X2 + A -

B]32

0751 0 42 1127 X 0752 0 44 2101 X2 0753 0 43 1122 X2 + A 0754 0 41 1134 X2 + A - B 0755 0 49 1135 0756 0 08 1397 (X2 + A -

B)'2

0757 0 49 1135 (X2 + A -

B)32

0758 0 49 1136 0759 0 38 2101 Type out (X2 + A

-r G J ( )3/2 X X + 0.1 New X Increment W.B. by i r G J ( )3/2 G C J )3/ New tipi new p. -449° Transfer, if - ve Total sin 3. 2 3 carriage returns Intercom 1000 Loc'n K OP ADDR

Calculate G 0760 1 47 1103 0761 1 44 0880 0762 0 43 1126 0763 0 49 1126 0764 0 42 1127 0765 0 43 0897 0766 O 49 1127 0767 1 76 0745 Calculation of cumulative 0768 0 42 1126 L 0769 0 44 1132 0770 0 43 1125 0771 _{0 49 1125} Increment blade ji 0772 0 42 1124 0773 0 43 1123 0774 0 49 1124 Limit operations to required 0775 0 41 0898 number of blades 0776 0 22 0790 0777 O 42 1125

0778 0441120

0779 0 30 0003 Accumulator G ( )3/2

Intercom 1000 Loc'n K OP ADDR Accumulator

Answer in RADS. 0780 0 33 2101 b in radians

0781 0 44 0899 times 57.296 Answer in DEGS. 0782 0 38 2101 h in degrees

0783 0 67 0000 end

0790 0 42 0894 0

0791 0 49 1126 r 0792 0 29 071g

SET COUNTER TO p9O0 [i.e. TO BLADE#1) SET O SET j SET x :0.2 V 4 +

SET WORD BASE: O

SET WORD DIFFERENCEZI

SET WORD LIMIT=8 SET CHANNEL BASE:O

CALCULATE & STORE SINF a COS

EVALUATE,9(% FOR x

4

INCREMENT WORD BASE

OBTAIN

f$

/2 FOR , :900

INCREMENT BLADE

OBTAIN [-h- SINL -0.7 COS$1 C0S/h]f3/2 FORp9O°

ADD TO NO MULTIPLY BY ¿ TYPE OUT ab N RADS. 8 DEGREES 8 HALT 8 TIMES CALCULATE 8 STORE SINß1 8 COSß

FLOW DIAGRAM

CALCULATE & STORE

[)2O.49] [sIN,_o.7

_{COSßI COS/J.] [2x (} COSi COSß1+O.7SINL)]

YES IF BLADE TRANSFER

511100

silO0

'Y

s 51.40000

1101 33/2 1102 /49091 1103 /01314 1104 /02121 1105 /02311 1106 /03315 1107 /03657 1103 /03696 1109 /03470 1110 /02763 0.2 0. 3

0.4

0.5

0.6

0.7

0.8 0.9 .0 0.2 0.3 0.4 0.5

0.6

0. 7 O. 8

0.9

0.2 0. 3 0.4 0.5 0.6 0. 7 0.8

0.9

1.0 .9045 1.1029 1. 3469 1.6 L 19 1.9934

2.4071

2.3335 3 .4436 4 . 0731 0775 L. 3 .325 1.7473 1795 2.6333 2653 3.9315

4.6373

5.5402 4725 4354 .4217 .4305 4625 .519 1 .6030 .7 176 .3672

/L:90°

: 1800

.0143 RADS = .8207 DEG. : ANSWER

PROPELLER NO. 24

s 52.33200

s 50.49091

:/3.

L

s 49.13140

s s 49.21210 49.23110

T

VALUES 0F G

VARIABLE DATA READ

INTO COMPUTER SPECIFIC

s 49.33150

_{AT DIFFERENT PROR} FOR THE PROPELLER

s

s

49.36570

49.36960 BLADE RADII X. UNDER CONSIDERATION.

s 49.34700

s 49.27630

1111

/0

s .00000 1112

0690700//

s 3/ X

[

f2 0.2

.34/0

0.3 .2539 0.4 .1904

0.5

.1439 0.6 .1257 o.? .1133 0.8 .1257

0.9

.1439 1.0 .1904

270°

:36