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projected area-ratio is

0.1648 Xo. 2 -7-0.4 = 0.08241.

Conversely, the width-ratio corresponding to any given area-

ratio may be found by multiplying by 0.4 and dividing by 0.1648.

Thus a blade having the area-ratio 0.08 will have the width-ratio

0.08X0.4-^0.1648=0.1942.

The blade-area computed by this method is very nearly

correct for propellers which have spherical hubs; if the hub is

barrel shaped and the blade is narrow there may be an error of

one per cent, a quantity which has no appreciable effect.

The total projected area-ratio for any propeller is found by

multiplying the area-ratio for one blade by the number of blades.

Factor for Blade-angle. In drawing the standard projected

blade contour it is convenient to lay off the angle eof, Fig. 12,

by aid of the dimension ef.

Turning to the circular blade contour of Fig. 13, we have for

that case

ef=eo tanc0/ = o.5 tan 41 48' = 0.44721.

For any other blade the factor may be made to depend on

the width-ratio, or the projected area-ratio. By projection, the

width-ratios and the dimensions ef are proportional. But the

area-ratios are proportional to the width-ratios, so that the dimen-

sions ef are proportional to the area-ratios. Thus the area-ratio

0.08 has the width-ratio 0.1942 as computed. The factor for ej

is therefore

ef= 0.4472X0.1942^0.4 = 0.21 71,

or /=o.4472Xo.8-7-o. 1648 = 0.2171.

Axial Dimension. Turning to Fig. n it will be remembered

that the blades there subtend 60, and have consequently one-

TO DRAW PROJECTIONS 43

sixth of a turn of the screw; the axial width shown by Fig. 10

is therefore one-sixth of the pitch. In the same way the axial

dimension of the blade in Fig. 12 will have the same ratio to the

pitch that the angle tot' has to 360. The laying-down table gives

the dimension ef, and ef divided by oe gives the tangent of the-

angle eof\ this is the half-angle and is to be divided by 180. There

the factor for the blade-angle is 0.2171 for an area-ratio of 0.08,

and the axial dimension factor is computed as follows:

0.2171-^0. 5 =0.4342 = tan 23 28' = tan 2347;

23.47-^-180 = 0.1304.

To Draw Projections. Since all the dimensions and propor-

tions can readily be computed for the standard projected contour,

the designer will follow his judgment and habit whether he will

make a drawing of the propeller or trust that to the makers.

The following method will be found rapid and accurate. After

the diameter and the projected area-ratio of the blade of a pro-

peller have been determined by methods to be given later, the

projections can be drawn as shown in Figs. 14, 15, and 16.

Let it be assumed that the propeller has four blades, a diameter

of 10 feet, a pitch of 20 feet, and a projected area-ratio of 0.075

for one blade. By interpolation in the laying-down table the

following dimensions can be found.

Width-ratio 0.1820; width 10X0.1820 = 1.820 ft. = 21.84 in.;

Axial factor 0.1230; axial dimension 20X0.1230 = 2.46 ft. = 29.52111.

In Fig. 14 the length is laid off equal to 5 ft., scale i in. =

i ft.; and the radius of the hub is made 0^ = 0.2X10^2 = 1 ft.

The line we is bisected at h and the width 21.84 m - is laid off

from x to y. An ellipse is drawn with we and xy as the axes.

The dimension ef is computed as follows: after the factor 0.2036

is found in the laying-down table,

ef= 10 X 0.2036 = 2.036 ft. = 24.43 i n ->

and is laid off on Fig. 14 and the line of is drawn; it is tangent

to the ellipse at t and locates the straight-edge ut of the blade

44

PROPELLERS

FIG. 14.

TO DRAW PROJECTIONS

45

FIG. 15.

46 PROPELLERS

contour. The line u f t f laid off on the other side of the blade com-

pletes the contour. In Fig. 14 the hub is drawn cylindrical,

as shown by the arc uu' '.

From the centre o the arc ge is drawn and divided accurately

into ten parts by spacing with dividers; the arc on the other side

of the blade gives a symmetrical construction and is therefore

omitted.

The axial dimension uu' in Fig. 15 is laid off equal to 29.52 in.

(scale i in. = i ft.) and is divided accurately into twenty

equal parts and numbered consecutively from the right-hand

or forward edge. This propeller, like that shown by Fig. 10, is

right-handed and is represented as driving the ship toward the

right. The left-hand surface or face is to be a true helical surface.

The point a on Fig. 14 is the intersection of the No. 5 radial

line with the elliptical contour. On Fig. 15 this point is projected

onto the No. 5 ordinate; the symmetrical point a' of Fig. 14 is

projected onto the No. 15 ordinate. In like manner a sufficient

number of points like a may be located and the contour may be

drawn through them. It is now evident why the angle eog is laid

off and divided with precision. The point / is accurately located

by drawing st at 0.16667 of the diameter from the centre; in

this case

t

os = 10X0.16667 = 1.667 ft. = 20 in.

The straight line edges ut and u't' of Fig. 14 appear as boundary

elements ut and u't' in Fig. 15. Since the hub is shown as a

cylinder the root line of the blade is shown as a helical curve

uwu f ' 7 a point like z is found by projecting z of Fig. 14 onto the

corresponding ordinate; in the case shown the ordinate is No. 5.

The contour of a blade at right angles to that just described

is shown by u"a"oa'"u' n on Fig. 15. The point a" is located

on the fifth ordinate by making 50" equal to ba of Fig. 14; the

point a'" is symmetrical with a" on the fifteenth ordinate. The

bounding elements are u"t" and u' f 't'" and the root line u"ou' n

is a part of a helix.

The drawing of the propeller for the inlormation of the designer

and the pattern-maker should be accurately drawn to a large

INTERSECTION AT HUB-PLANE SECTION 47

scale, if not full size. All lines should be drawn with a steel

straight-edge; the axis of the ellipse and the ordinates should be

laid off at right angles by a geometric method instead of depending

on a triangle. The division of eg of Fig. 14 and of the axial

dimension of Fig. 15 should be by spacing or some equally

accurate method. A line through e parallel to the axis of the

shaft should be laid off accurately and the divisions of the axial

dimension should be transferred to it, so that the ordinates may

be accurately located. The projection of a point, like a from

Fig. 14 to Fig. 15, should be made by measurement; thus 50

should be laid off equal to ob.

The thickness of the blade, which is all applied to the back

of the blade, is indicated by the line eid. The thickness od at the

axis divided by the diameter of the propeller is known as the

thickness ratio. In this case it is made equal to 0.02 of the diameter,

so that the thickness is 0.2 of a foot or 2.4 inches. The thickness

at the tip is 0.005 f tne diameter, which in this case is 0.6 of

an inch. Bronze blades are commonly made thinner at the tip;

the thickness at the hub is greater for narrower blades. Cast-

iron blades are much thicker.

Intersection at Hub. For simplicity the hub is represented

to be cylindrical and its intersection by the face of the blade

is a helix. The hub is always a surface of revolution so that

the intersection by an element of the face can be located by aid

of a plane through it and the axis, which plane is to be revolved

into the plane of the paper. The actual construction may be left

to the draughtsman who will work to a large scale. In practice

the blade joins the hub with rounded fillets cut by the pattern-

maker.

Plane Section. To show the form of the back of the blade

and for the instruction of the pattern-maker, it is customary to

give a number of sections like those shown on Fig. 14, where

mjhj'm'i is a plane section and vkhv'i is a developed cylindrical

section, to be explained in the next section.

A plane section perpendicular to the line oe, Fig. 14, cuts the

contour at xhy and in Fig. 15 at mhm' '; the points m and m' are

>rojected to / and I', and show the section of the blade contour

48 PROPELLERS

u"t"ot'"u" f , cut by a similarly placed plane parallel to the plane of

the paper and at the distance ob above it. The plane section

Ikok'l' is shown in its correct form; it will be found to be slightly

curved. To construct a point like k, draw the element ok 5 of

the helical surface on Fig. 14 and note where it cuts the line xhy

at the point &; this gives the correct transverse location of this

point. On Fig. 15 draw the corresponding element 5& and make

$k equal to hk of Fig. 14. The symmetrical point k' is located

by making 15^' equal to 5^. Having a sufficient number of points

like k and k r the section Ikok'l' can be drawn and transferred to

Fig. 14. The thickness of the blade is laid off equal to hi and

the back is drawn as the arc of a circle.

For cast-iron blades the edge cannot be so thin as this con-

struction gives; so some thickness is given at the edge and then

the back is rounded to the arc of a circle.

Very commonly the curvature of the line mjhj'mf is ignored in

drawing plane sections of a blade because it is slight. The curva-

ture is, however, important and must be allowed for, when sections

are made to be employed for sweeping up the mould of a propeller

on the floor of the foundry.

Developed Cylindrical Section. Suppose that a cylindrical

surface is constructed by revolving the line mhm', Fig 15, about

the axis of the shaft; it will cut the surface of the blade in a helix

shown by nhn' and by the arc nhn' of Fig. 14. If the cylinder

is developed into a plane the helix becomes a straight line. The

development of the cylinder can be made in Fig. 14 by laying

off the line hr equal in length to the arc hn f . The fore-and-aft

dimension of the helix hn f of Fig. 15 is pn f . If this be laid off at

hq, Fig. 14, the diagonal qr will give the half-width of the developed

helicoidal section. This dimension is laid off at hv and hv', and

the back is drawn through v, v' ', and i\ for this purpose an arc of

a circle may be used, though this is not quite correct if the plane

section is constructed with the back rounded to the arc of a circle.

Sections like those discussed in this and in the previous section

are drawn at intervals for the instruction of the pattern-maker;

the choice of section depends on how the pattern is made. The

draughtsman should have a practical knowledge of the making

BLADES WITH A RAKE 49

of propeller patterns; there should be a competent person charged

with the responsibility for the correct making of patterns and

for maintaining them in correct form.

Blades with a Rake. Fig. 16 shows the projection of the

propeller of Fig. 14, but with 15 rake aft. The ordinates are

now drawn with that inclination; the radius is measured per-

pendicular to the axis. In order to locate the helicoidal elements

the helix e'ee" must be constructed and then the elements like

o,e' and 2o,e" can be drawn. The points a and a' of Fig. 14

may now be projected onto the proper elements at a and a' on

Fig. 1 6. The contour of the edge of the blade u"t"a"a"'t"'u" f

can be drawn by the usual method of projections from Fig. 14

and the contour utaa't'u'-, then the point a" can be located on

the vertical line aa" at a distance b"a" below the axis, this distance

being equal to ba of Fi<*. 14. The thickness is laid off at right

angles to the line io,e.

The cylindrical section vkhv'i of Fig. 14 will be constructed

as before, and will differ only in that the dimension hi will be

slightly larger, because it is measured on a line inclined to the

axis 10, e of the blade.

As for the form of the plane section, it will depend on how it

is taken. If the plane is parallel to the axis of the shaft, the sec-

tion will differ very little from that shown in Fig. 14, and that con-

struction can be accepted for pattern-making or for sweeping up

blades in the foundry; the sections in the foundry must in such

case be set vertical, the blade being inclined at the angle of the

rake from the horizontal. But if the section is perpendicular to

the element io,e as shown by nhn f of Fig. 16, the form will be

materially different; it can be drawn by the ordinary methods of

descriptive geometry, but the construction is omitted to avoid

prolixity.

Helicoidal Area. The true or helicoidal area of the blade of

a propeller can be determined by aid of developed cylindrical

sections, such as that which gives the line vhv' of Fig. 14; a number

of such lines can be constructed at intervals from w to e, and a

contour or bounding line can be drawn; the area of that figure

will be the true area of the face of the blade. When the design

50

PROPELLERS

\ 1

\

\

\

1

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

^ j

\ t

""'

\

\

\

\

\

\

\

\

\

\

\

\l

\l

1

t'

^_

r\

\

FIG. 16.

DEVELOPED CONTOUR 51

of a propeller is based on the projected area-ratio there is little

reason for dealing with the area of the blade.

Developed Contour. The surface of a screw-propeller is a ruled

surface which cannot be developed, but there are conventional

methods of constructing a plane figure which has nearly the same

surface as a blade. These methods are called developing the

blade, and the figure is called the developed contour.

The development of the blade of a propeller, and the inverse

process of constructing the projections from the developed contour

have an importance, because (i) certain propeller theories are

based on the developed contour, (2) nearly all the experimental

propellers tested in model basins have been designed from the

developed contours, (3) and the results of such experiments sys-

tematized in tables and diagrams are stated in the same terms.

In consequence engineers and designers are accustomed to working

with the developed contour, and for that reason, if no other,

the methods of drawing developed contour must be understood.

In Fig. 17 there is drawn half a turn of a helix gabch and the

development bf of half a turn of the helix beginning at b. A quarter

turn of the helical surface is shown by nabcp, comparable to the

quarter turn shown on Figs. 3 and 4. The line bf is tangent

to the helix at 6; the deviation of the tangent at s from the helix

at c, for an eighth of a turn is small; for less than an eighth the

deviation is insignificant. Propeller blades seldom if ever are so

wide as would be given by a quarter of a turn. x

The conventional development of the blade of a propeller

depends on the substitution of the straight line bs in place of the

helical arc be. The tangent bs is most conveniently located by

drawing the triangle tbu in which tu is computed by the proportion

be : ef::bt : tu.

But be = ^d, bt = %d, and ej=\p, where d is the diameter of the

propeller and p is the pitch. Substituting and solving for tu,

52

PEOPELLERS

In Fig. 1 8 one-sixth turn of the helical surface is represented

approximately by nabcp, in which abc is the tangent line in place

of the true helical curve. Let a plane perpendicular to the plane

of the paper be passed through the cylinder at lbm\ it will cut

an elliptical section of the cylinder which can be rotated into the

plane of the paper, as shown on Fig. 19 by ea fr b'c"f. The elliptical

FIG. 17.

arc a"b r c" is considered to be the development of the helical arc

shown in projections by abc, Fig. 18, and a'b'c', Fig. 19. Two

other cylinders are represented by l\m\ and l^m^ in Fig. 18, with

approximate helical surfaces a\bc\ and azbcz', elliptical sections by

planes through the line l\m\ and hni2, are revolved into the plane

of the paper in Fig. 19, thus locating the elliptical arc* a\'c\ r

and 02 'W. A curved contour is drawn through a' r a\" , a 2 "ao

DEVELOPED CONTOUR

53

and another through c n c\'c^'c^ The points OQ and Co are located

by making

oao = oco = bp (of Fig. 18).

The ellipses are all drawn from the foci o\ and 03, which may

be located in the usual way; that is, by drawing arcs from b'

with radii 0'0i and b'oz each equal to oe. Or since the triangles

P

FIG. 1 8.

/0/o of Fig. 1 8 and b'ooi of Fig. 19 are equal to each other, the

points 0i and 02 can be located by making

001=002 = 0/0 (Fig. 1 8)=,

because 6/0 in Fig. 18 corresponds to tu of Fig. 17.

Another and simpler method is to take the lines ac, aid, and

#2^2 of Fig. 1 8 and lay them off at kj, k\j\ and 2/2 on Fig. 20,

and then draw the contour aokjco for the developed contour of

the blade. The contour Ooa"ftVco is repeated for comparison.

54

PROPELLERS

In designing propellers the developed contour is frequently

drawn first and the projected contour is then constructed by

reversing the methods just explained.

As an example we may refer to Fig. 25, page 58, which is

given primarily to show the construction of a propeller with

G" J

FIG. 19.

FIG. 20.

separable blades. The developed contour is shown by the dotted

ellipse. OA is laid off equal to ^-f-2x to find the focus of the

elliptical section, the point A corresponding to 02 of Fig. 19.

Choosing a point B we draw through it a circular arc EB from

the centre O and an elliptical arc DB, with OB and AB for the

semi-minor and semi-major axes. Through the intersection D

STANDARD DEVELOPED CONTOUR

55

of the elliptical arc with the dotted contour, a horizontal line DF

is drawn, which cuts the circular arc at E\ this is a point of the

projected contour. A comparison with Fig. 19 will justify this

construction. A more precise method of locating points like F

will be given in the description of Fig. 25.

Standard Developed Contour. A form of developed contour

for propeller blades which was first proposed by Wm. Froude and

which is known as the Admiralty blade, is shown by Fig. 21. It

0.8

0.3

\

0.954

ADM I R ALT/

BLADE

FIG. 21.

^"

0.047 ^^

/

0.787 \

/

0.869 \

0.8

0.922

0.956

0.7

0.978

0.991

0.6

0.997

1

0.5

1

0.995

0.4

0.980

0.954

0.3

0.916 /

0.866 /

/

TAYL

BL>

OR'S

,DE

FIG. 22.

STRAIGHT-EDGE

BL|DE

FIG. 23.

is an ellipse with the radius of the propeller as the major axis,

and the minor axis is 0.2 of the propeller diameter. The diameter

of the hub is 0.22 of the propeller diameter, and the contour is

shown cut off by a straight line. More correctly the development

of the root should be a curve depending on the form of the hub.

With the advent of high-power and high-speed ships, especially

turbine ships, the elliptical contour has been increased in width

till it approaches a circle.

Fig. 22 shows a contour proposed by Naval Constructor D. W.

Taylor, U.S.N., and used by him for many experimental propellers.

56

PROPELLERS

Its form is sufficiently determined by the ratios of the widths to

the maximum width. Fig. 23 is put in to show the relative form

of a straight-edged blade having about the same area, and slightly

rounded corners.

In order to show the comparison of the proposed projected

contour with the Admiralty blade, two developments by the

conventional method are given by Fig. 24. The contour ertml

FIG. 24.

is the proposed circular projected contour, the radius cr being

0.2 of the propeller diameter so 'that the projected width is 0.4

of the diameter. The hub is 0.2 of the diameter and the contour

at the hub is completed by a circular arc. The contour errffm' is

drawn with the width equal to 0.2 of the diameter. The developed

contours are drawn by the method of page 53, for a pitch-ratio

of unity, that is, with the pitch equal to the diameter; a different

pitch-ratio would have but little effect on the conclusion that

AREA OF THE ADMIRALTY BLADE 57

can be drawn from the figure. The dotted ellipses are drawn

through the points h and hi on the line gh at the middle of the

radius; they are the developed contours of the corresponding

Admiralty blades. The developed contours shown by the full

lines are wider at the tips and narrower at the hub; the arealsf

somewhat less. Our design of propeller will be based on projected

area-ratio which will set aside questions of width and area, but

minor variations of either property have no appreciable influence.

Area of the Admiralty Blade. The importance that is attached

to the Admiralty blade makes it desirable to give ready means

of determining both the developed and the projected areas.

The developed contour being an ellipse its area will be pro-

portional to its width. If its width were half the diameter of

the propeller, its area, neglecting the hub, would be 0.25 that

of the disk or circle having the propeller diameter. The hub

may be assumed to take away a segment having a rise 0.2 of the

diameter of the circular contour; the segmental area is 0.1424 of

that of the circle; consequently the net area is

0.25(1 -0.1424) =0.214,

that of the disk. For any other width the area will be pro-

portional; then for a width 0.2 of the diameter of the propeller

the developed area is 0.0856 of the disk.

Barnaby gives the following rule for the projected area:

developed area

Projected area =

+0.425 (pitch-ratio)

This rule will give approximate results for other oval projected

contours.

Construction Drawings. The construction drawings for a four-

bladed propeller with separable blades are shown by Figs. 25 to 28.

The projection on a transverse plane looking forward is shown by

Fig. 25, which gives also the developed contour on which the

design is based. As previously explained OA is laid off equal to

P-T-ZK and is the focus for the ellipses used in the development

of the blade; or in the construction of the projected contour.

58

PROPELLERS

FIG. 25.

CONSTRUCTION DRAWINGS

59

FIG. 26.

60 PEOPELLEES

A point B is chosen and through it an elliptical arc is drawn; also

a circular arc from the center 0. A horizontal line DF locates

the point E of the projected contour. Fig. 26 shows the projection

of two blades, without rake, on a plane, through the axis of the

shaft. 05 is equal to OA of the preceding figure and SR is equal

to OB', 01 gives the projection of EF in its true length. Drawing

// perpendicular to 05 gives O/, the proper half-breadth FE of

the contour EG. It also gives // the proper length of EF of

Fig. 25, and this is a more precise way of locating that point than

that previously given.

The points I and / are points on the contour of a blade which

presents its tip to the observer. The thickness of the blade and

other details are omitted to avoid complexity.

Fig. 27 gives the projections of a blade with a rake, together

with the effect of thickness of blade on the configuration of the

tip and the root. The blades are fastened to the hub by flanges

and bolts. Fig. 28 gives a longitudinal section of the blade and

hub and shows details of construction.

Propeller Experiments. The first systematic propeller experi-

ments were made by the Froudes, father and son, at the Admi-

ralty experimental tank, all being of the Admiralty type with the

width of blade equal to 0.2 of the diameter. Mr. R. E. Froude

has reported later experiments with various widths of blades.

Probably the most satisfactory tests are those by Naval Con-

structor D. W. Taylor, U.S.N., made at the model basin at

Washington. The tables in this book are derived from these

tests with the permission of Mr. Taylor. It has been shown that

the tests by the Froudes and by Mr. Taylor are in substantial

accord, so that both series of experiments may be claimed as the

basis of the tables given in this book.

The tables for three-bladed propellers are based directly on

an extensive set of experiments made on propellers of the Admi-

ralty type with various widths, thicknesses, and pitch-ratios.

The tables for four-bladed propellers were deduced from a com-

parison of tests on thin-bladed propellers of the type shown by

Fig. 22 (some with three and some with four blades) with the

tests on the Admiralty type. A table for two-bladed propellers

PROPELLER EXPERIMENTS

61

FIG. 27.

62

PROPELLEKS

METHOD OF EXPERIMENTS 63

was deduced in like manner from tests of thin bladed propellers

on that type.

Method of Experiments. In making experiments in a model

basin, the model propeller is placed at the front end of a shaft

which is suspended from the towing carriage. The shaft atrthe

rear extends into a boat-shaped box which contains the driving

gear on the propeller shaft. The towing carriage is propelled at

0.1648 Xo. 2 -7-0.4 = 0.08241.

Conversely, the width-ratio corresponding to any given area-

ratio may be found by multiplying by 0.4 and dividing by 0.1648.

Thus a blade having the area-ratio 0.08 will have the width-ratio

0.08X0.4-^0.1648=0.1942.

The blade-area computed by this method is very nearly

correct for propellers which have spherical hubs; if the hub is

barrel shaped and the blade is narrow there may be an error of

one per cent, a quantity which has no appreciable effect.

The total projected area-ratio for any propeller is found by

multiplying the area-ratio for one blade by the number of blades.

Factor for Blade-angle. In drawing the standard projected

blade contour it is convenient to lay off the angle eof, Fig. 12,

by aid of the dimension ef.

Turning to the circular blade contour of Fig. 13, we have for

that case

ef=eo tanc0/ = o.5 tan 41 48' = 0.44721.

For any other blade the factor may be made to depend on

the width-ratio, or the projected area-ratio. By projection, the

width-ratios and the dimensions ef are proportional. But the

area-ratios are proportional to the width-ratios, so that the dimen-

sions ef are proportional to the area-ratios. Thus the area-ratio

0.08 has the width-ratio 0.1942 as computed. The factor for ej

is therefore

ef= 0.4472X0.1942^0.4 = 0.21 71,

or /=o.4472Xo.8-7-o. 1648 = 0.2171.

Axial Dimension. Turning to Fig. n it will be remembered

that the blades there subtend 60, and have consequently one-

TO DRAW PROJECTIONS 43

sixth of a turn of the screw; the axial width shown by Fig. 10

is therefore one-sixth of the pitch. In the same way the axial

dimension of the blade in Fig. 12 will have the same ratio to the

pitch that the angle tot' has to 360. The laying-down table gives

the dimension ef, and ef divided by oe gives the tangent of the-

angle eof\ this is the half-angle and is to be divided by 180. There

the factor for the blade-angle is 0.2171 for an area-ratio of 0.08,

and the axial dimension factor is computed as follows:

0.2171-^0. 5 =0.4342 = tan 23 28' = tan 2347;

23.47-^-180 = 0.1304.

To Draw Projections. Since all the dimensions and propor-

tions can readily be computed for the standard projected contour,

the designer will follow his judgment and habit whether he will

make a drawing of the propeller or trust that to the makers.

The following method will be found rapid and accurate. After

the diameter and the projected area-ratio of the blade of a pro-

peller have been determined by methods to be given later, the

projections can be drawn as shown in Figs. 14, 15, and 16.

Let it be assumed that the propeller has four blades, a diameter

of 10 feet, a pitch of 20 feet, and a projected area-ratio of 0.075

for one blade. By interpolation in the laying-down table the

following dimensions can be found.

Width-ratio 0.1820; width 10X0.1820 = 1.820 ft. = 21.84 in.;

Axial factor 0.1230; axial dimension 20X0.1230 = 2.46 ft. = 29.52111.

In Fig. 14 the length is laid off equal to 5 ft., scale i in. =

i ft.; and the radius of the hub is made 0^ = 0.2X10^2 = 1 ft.

The line we is bisected at h and the width 21.84 m - is laid off

from x to y. An ellipse is drawn with we and xy as the axes.

The dimension ef is computed as follows: after the factor 0.2036

is found in the laying-down table,

ef= 10 X 0.2036 = 2.036 ft. = 24.43 i n ->

and is laid off on Fig. 14 and the line of is drawn; it is tangent

to the ellipse at t and locates the straight-edge ut of the blade

44

PROPELLERS

FIG. 14.

TO DRAW PROJECTIONS

45

FIG. 15.

46 PROPELLERS

contour. The line u f t f laid off on the other side of the blade com-

pletes the contour. In Fig. 14 the hub is drawn cylindrical,

as shown by the arc uu' '.

From the centre o the arc ge is drawn and divided accurately

into ten parts by spacing with dividers; the arc on the other side

of the blade gives a symmetrical construction and is therefore

omitted.

The axial dimension uu' in Fig. 15 is laid off equal to 29.52 in.

(scale i in. = i ft.) and is divided accurately into twenty

equal parts and numbered consecutively from the right-hand

or forward edge. This propeller, like that shown by Fig. 10, is

right-handed and is represented as driving the ship toward the

right. The left-hand surface or face is to be a true helical surface.

The point a on Fig. 14 is the intersection of the No. 5 radial

line with the elliptical contour. On Fig. 15 this point is projected

onto the No. 5 ordinate; the symmetrical point a' of Fig. 14 is

projected onto the No. 15 ordinate. In like manner a sufficient

number of points like a may be located and the contour may be

drawn through them. It is now evident why the angle eog is laid

off and divided with precision. The point / is accurately located

by drawing st at 0.16667 of the diameter from the centre; in

this case

t

os = 10X0.16667 = 1.667 ft. = 20 in.

The straight line edges ut and u't' of Fig. 14 appear as boundary

elements ut and u't' in Fig. 15. Since the hub is shown as a

cylinder the root line of the blade is shown as a helical curve

uwu f ' 7 a point like z is found by projecting z of Fig. 14 onto the

corresponding ordinate; in the case shown the ordinate is No. 5.

The contour of a blade at right angles to that just described

is shown by u"a"oa'"u' n on Fig. 15. The point a" is located

on the fifth ordinate by making 50" equal to ba of Fig. 14; the

point a'" is symmetrical with a" on the fifteenth ordinate. The

bounding elements are u"t" and u' f 't'" and the root line u"ou' n

is a part of a helix.

The drawing of the propeller for the inlormation of the designer

and the pattern-maker should be accurately drawn to a large

INTERSECTION AT HUB-PLANE SECTION 47

scale, if not full size. All lines should be drawn with a steel

straight-edge; the axis of the ellipse and the ordinates should be

laid off at right angles by a geometric method instead of depending

on a triangle. The division of eg of Fig. 14 and of the axial

dimension of Fig. 15 should be by spacing or some equally

accurate method. A line through e parallel to the axis of the

shaft should be laid off accurately and the divisions of the axial

dimension should be transferred to it, so that the ordinates may

be accurately located. The projection of a point, like a from

Fig. 14 to Fig. 15, should be made by measurement; thus 50

should be laid off equal to ob.

The thickness of the blade, which is all applied to the back

of the blade, is indicated by the line eid. The thickness od at the

axis divided by the diameter of the propeller is known as the

thickness ratio. In this case it is made equal to 0.02 of the diameter,

so that the thickness is 0.2 of a foot or 2.4 inches. The thickness

at the tip is 0.005 f tne diameter, which in this case is 0.6 of

an inch. Bronze blades are commonly made thinner at the tip;

the thickness at the hub is greater for narrower blades. Cast-

iron blades are much thicker.

Intersection at Hub. For simplicity the hub is represented

to be cylindrical and its intersection by the face of the blade

is a helix. The hub is always a surface of revolution so that

the intersection by an element of the face can be located by aid

of a plane through it and the axis, which plane is to be revolved

into the plane of the paper. The actual construction may be left

to the draughtsman who will work to a large scale. In practice

the blade joins the hub with rounded fillets cut by the pattern-

maker.

Plane Section. To show the form of the back of the blade

and for the instruction of the pattern-maker, it is customary to

give a number of sections like those shown on Fig. 14, where

mjhj'm'i is a plane section and vkhv'i is a developed cylindrical

section, to be explained in the next section.

A plane section perpendicular to the line oe, Fig. 14, cuts the

contour at xhy and in Fig. 15 at mhm' '; the points m and m' are

>rojected to / and I', and show the section of the blade contour

48 PROPELLERS

u"t"ot'"u" f , cut by a similarly placed plane parallel to the plane of

the paper and at the distance ob above it. The plane section

Ikok'l' is shown in its correct form; it will be found to be slightly

curved. To construct a point like k, draw the element ok 5 of

the helical surface on Fig. 14 and note where it cuts the line xhy

at the point &; this gives the correct transverse location of this

point. On Fig. 15 draw the corresponding element 5& and make

$k equal to hk of Fig. 14. The symmetrical point k' is located

by making 15^' equal to 5^. Having a sufficient number of points

like k and k r the section Ikok'l' can be drawn and transferred to

Fig. 14. The thickness of the blade is laid off equal to hi and

the back is drawn as the arc of a circle.

For cast-iron blades the edge cannot be so thin as this con-

struction gives; so some thickness is given at the edge and then

the back is rounded to the arc of a circle.

Very commonly the curvature of the line mjhj'mf is ignored in

drawing plane sections of a blade because it is slight. The curva-

ture is, however, important and must be allowed for, when sections

are made to be employed for sweeping up the mould of a propeller

on the floor of the foundry.

Developed Cylindrical Section. Suppose that a cylindrical

surface is constructed by revolving the line mhm', Fig 15, about

the axis of the shaft; it will cut the surface of the blade in a helix

shown by nhn' and by the arc nhn' of Fig. 14. If the cylinder

is developed into a plane the helix becomes a straight line. The

development of the cylinder can be made in Fig. 14 by laying

off the line hr equal in length to the arc hn f . The fore-and-aft

dimension of the helix hn f of Fig. 15 is pn f . If this be laid off at

hq, Fig. 14, the diagonal qr will give the half-width of the developed

helicoidal section. This dimension is laid off at hv and hv', and

the back is drawn through v, v' ', and i\ for this purpose an arc of

a circle may be used, though this is not quite correct if the plane

section is constructed with the back rounded to the arc of a circle.

Sections like those discussed in this and in the previous section

are drawn at intervals for the instruction of the pattern-maker;

the choice of section depends on how the pattern is made. The

draughtsman should have a practical knowledge of the making

BLADES WITH A RAKE 49

of propeller patterns; there should be a competent person charged

with the responsibility for the correct making of patterns and

for maintaining them in correct form.

Blades with a Rake. Fig. 16 shows the projection of the

propeller of Fig. 14, but with 15 rake aft. The ordinates are

now drawn with that inclination; the radius is measured per-

pendicular to the axis. In order to locate the helicoidal elements

the helix e'ee" must be constructed and then the elements like

o,e' and 2o,e" can be drawn. The points a and a' of Fig. 14

may now be projected onto the proper elements at a and a' on

Fig. 1 6. The contour of the edge of the blade u"t"a"a"'t"'u" f

can be drawn by the usual method of projections from Fig. 14

and the contour utaa't'u'-, then the point a" can be located on

the vertical line aa" at a distance b"a" below the axis, this distance

being equal to ba of Fi<*. 14. The thickness is laid off at right

angles to the line io,e.

The cylindrical section vkhv'i of Fig. 14 will be constructed

as before, and will differ only in that the dimension hi will be

slightly larger, because it is measured on a line inclined to the

axis 10, e of the blade.

As for the form of the plane section, it will depend on how it

is taken. If the plane is parallel to the axis of the shaft, the sec-

tion will differ very little from that shown in Fig. 14, and that con-

struction can be accepted for pattern-making or for sweeping up

blades in the foundry; the sections in the foundry must in such

case be set vertical, the blade being inclined at the angle of the

rake from the horizontal. But if the section is perpendicular to

the element io,e as shown by nhn f of Fig. 16, the form will be

materially different; it can be drawn by the ordinary methods of

descriptive geometry, but the construction is omitted to avoid

prolixity.

Helicoidal Area. The true or helicoidal area of the blade of

a propeller can be determined by aid of developed cylindrical

sections, such as that which gives the line vhv' of Fig. 14; a number

of such lines can be constructed at intervals from w to e, and a

contour or bounding line can be drawn; the area of that figure

will be the true area of the face of the blade. When the design

50

PROPELLERS

\ 1

\

\

\

1

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

^ j

\ t

""'

\

\

\

\

\

\

\

\

\

\

\

\l

\l

1

t'

^_

r\

\

FIG. 16.

DEVELOPED CONTOUR 51

of a propeller is based on the projected area-ratio there is little

reason for dealing with the area of the blade.

Developed Contour. The surface of a screw-propeller is a ruled

surface which cannot be developed, but there are conventional

methods of constructing a plane figure which has nearly the same

surface as a blade. These methods are called developing the

blade, and the figure is called the developed contour.

The development of the blade of a propeller, and the inverse

process of constructing the projections from the developed contour

have an importance, because (i) certain propeller theories are

based on the developed contour, (2) nearly all the experimental

propellers tested in model basins have been designed from the

developed contours, (3) and the results of such experiments sys-

tematized in tables and diagrams are stated in the same terms.

In consequence engineers and designers are accustomed to working

with the developed contour, and for that reason, if no other,

the methods of drawing developed contour must be understood.

In Fig. 17 there is drawn half a turn of a helix gabch and the

development bf of half a turn of the helix beginning at b. A quarter

turn of the helical surface is shown by nabcp, comparable to the

quarter turn shown on Figs. 3 and 4. The line bf is tangent

to the helix at 6; the deviation of the tangent at s from the helix

at c, for an eighth of a turn is small; for less than an eighth the

deviation is insignificant. Propeller blades seldom if ever are so

wide as would be given by a quarter of a turn. x

The conventional development of the blade of a propeller

depends on the substitution of the straight line bs in place of the

helical arc be. The tangent bs is most conveniently located by

drawing the triangle tbu in which tu is computed by the proportion

be : ef::bt : tu.

But be = ^d, bt = %d, and ej=\p, where d is the diameter of the

propeller and p is the pitch. Substituting and solving for tu,

52

PEOPELLERS

In Fig. 1 8 one-sixth turn of the helical surface is represented

approximately by nabcp, in which abc is the tangent line in place

of the true helical curve. Let a plane perpendicular to the plane

of the paper be passed through the cylinder at lbm\ it will cut

an elliptical section of the cylinder which can be rotated into the

plane of the paper, as shown on Fig. 19 by ea fr b'c"f. The elliptical

FIG. 17.

arc a"b r c" is considered to be the development of the helical arc

shown in projections by abc, Fig. 18, and a'b'c', Fig. 19. Two

other cylinders are represented by l\m\ and l^m^ in Fig. 18, with

approximate helical surfaces a\bc\ and azbcz', elliptical sections by

planes through the line l\m\ and hni2, are revolved into the plane

of the paper in Fig. 19, thus locating the elliptical arc* a\'c\ r

and 02 'W. A curved contour is drawn through a' r a\" , a 2 "ao

DEVELOPED CONTOUR

53

and another through c n c\'c^'c^ The points OQ and Co are located

by making

oao = oco = bp (of Fig. 18).

The ellipses are all drawn from the foci o\ and 03, which may

be located in the usual way; that is, by drawing arcs from b'

with radii 0'0i and b'oz each equal to oe. Or since the triangles

P

FIG. 1 8.

/0/o of Fig. 1 8 and b'ooi of Fig. 19 are equal to each other, the

points 0i and 02 can be located by making

001=002 = 0/0 (Fig. 1 8)=,

because 6/0 in Fig. 18 corresponds to tu of Fig. 17.

Another and simpler method is to take the lines ac, aid, and

#2^2 of Fig. 1 8 and lay them off at kj, k\j\ and 2/2 on Fig. 20,

and then draw the contour aokjco for the developed contour of

the blade. The contour Ooa"ftVco is repeated for comparison.

54

PROPELLERS

In designing propellers the developed contour is frequently

drawn first and the projected contour is then constructed by

reversing the methods just explained.

As an example we may refer to Fig. 25, page 58, which is

given primarily to show the construction of a propeller with

G" J

FIG. 19.

FIG. 20.

separable blades. The developed contour is shown by the dotted

ellipse. OA is laid off equal to ^-f-2x to find the focus of the

elliptical section, the point A corresponding to 02 of Fig. 19.

Choosing a point B we draw through it a circular arc EB from

the centre O and an elliptical arc DB, with OB and AB for the

semi-minor and semi-major axes. Through the intersection D

STANDARD DEVELOPED CONTOUR

55

of the elliptical arc with the dotted contour, a horizontal line DF

is drawn, which cuts the circular arc at E\ this is a point of the

projected contour. A comparison with Fig. 19 will justify this

construction. A more precise method of locating points like F

will be given in the description of Fig. 25.

Standard Developed Contour. A form of developed contour

for propeller blades which was first proposed by Wm. Froude and

which is known as the Admiralty blade, is shown by Fig. 21. It

0.8

0.3

\

0.954

ADM I R ALT/

BLADE

FIG. 21.

^"

0.047 ^^

/

0.787 \

/

0.869 \

0.8

0.922

0.956

0.7

0.978

0.991

0.6

0.997

1

0.5

1

0.995

0.4

0.980

0.954

0.3

0.916 /

0.866 /

/

TAYL

BL>

OR'S

,DE

FIG. 22.

STRAIGHT-EDGE

BL|DE

FIG. 23.

is an ellipse with the radius of the propeller as the major axis,

and the minor axis is 0.2 of the propeller diameter. The diameter

of the hub is 0.22 of the propeller diameter, and the contour is

shown cut off by a straight line. More correctly the development

of the root should be a curve depending on the form of the hub.

With the advent of high-power and high-speed ships, especially

turbine ships, the elliptical contour has been increased in width

till it approaches a circle.

Fig. 22 shows a contour proposed by Naval Constructor D. W.

Taylor, U.S.N., and used by him for many experimental propellers.

56

PROPELLERS

Its form is sufficiently determined by the ratios of the widths to

the maximum width. Fig. 23 is put in to show the relative form

of a straight-edged blade having about the same area, and slightly

rounded corners.

In order to show the comparison of the proposed projected

contour with the Admiralty blade, two developments by the

conventional method are given by Fig. 24. The contour ertml

FIG. 24.

is the proposed circular projected contour, the radius cr being

0.2 of the propeller diameter so 'that the projected width is 0.4

of the diameter. The hub is 0.2 of the diameter and the contour

at the hub is completed by a circular arc. The contour errffm' is

drawn with the width equal to 0.2 of the diameter. The developed

contours are drawn by the method of page 53, for a pitch-ratio

of unity, that is, with the pitch equal to the diameter; a different

pitch-ratio would have but little effect on the conclusion that

AREA OF THE ADMIRALTY BLADE 57

can be drawn from the figure. The dotted ellipses are drawn

through the points h and hi on the line gh at the middle of the

radius; they are the developed contours of the corresponding

Admiralty blades. The developed contours shown by the full

lines are wider at the tips and narrower at the hub; the arealsf

somewhat less. Our design of propeller will be based on projected

area-ratio which will set aside questions of width and area, but

minor variations of either property have no appreciable influence.

Area of the Admiralty Blade. The importance that is attached

to the Admiralty blade makes it desirable to give ready means

of determining both the developed and the projected areas.

The developed contour being an ellipse its area will be pro-

portional to its width. If its width were half the diameter of

the propeller, its area, neglecting the hub, would be 0.25 that

of the disk or circle having the propeller diameter. The hub

may be assumed to take away a segment having a rise 0.2 of the

diameter of the circular contour; the segmental area is 0.1424 of

that of the circle; consequently the net area is

0.25(1 -0.1424) =0.214,

that of the disk. For any other width the area will be pro-

portional; then for a width 0.2 of the diameter of the propeller

the developed area is 0.0856 of the disk.

Barnaby gives the following rule for the projected area:

developed area

Projected area =

+0.425 (pitch-ratio)

This rule will give approximate results for other oval projected

contours.

Construction Drawings. The construction drawings for a four-

bladed propeller with separable blades are shown by Figs. 25 to 28.

The projection on a transverse plane looking forward is shown by

Fig. 25, which gives also the developed contour on which the

design is based. As previously explained OA is laid off equal to

P-T-ZK and is the focus for the ellipses used in the development

of the blade; or in the construction of the projected contour.

58

PROPELLERS

FIG. 25.

CONSTRUCTION DRAWINGS

59

FIG. 26.

60 PEOPELLEES

A point B is chosen and through it an elliptical arc is drawn; also

a circular arc from the center 0. A horizontal line DF locates

the point E of the projected contour. Fig. 26 shows the projection

of two blades, without rake, on a plane, through the axis of the

shaft. 05 is equal to OA of the preceding figure and SR is equal

to OB', 01 gives the projection of EF in its true length. Drawing

// perpendicular to 05 gives O/, the proper half-breadth FE of

the contour EG. It also gives // the proper length of EF of

Fig. 25, and this is a more precise way of locating that point than

that previously given.

The points I and / are points on the contour of a blade which

presents its tip to the observer. The thickness of the blade and

other details are omitted to avoid complexity.

Fig. 27 gives the projections of a blade with a rake, together

with the effect of thickness of blade on the configuration of the

tip and the root. The blades are fastened to the hub by flanges

and bolts. Fig. 28 gives a longitudinal section of the blade and

hub and shows details of construction.

Propeller Experiments. The first systematic propeller experi-

ments were made by the Froudes, father and son, at the Admi-

ralty experimental tank, all being of the Admiralty type with the

width of blade equal to 0.2 of the diameter. Mr. R. E. Froude

has reported later experiments with various widths of blades.

Probably the most satisfactory tests are those by Naval Con-

structor D. W. Taylor, U.S.N., made at the model basin at

Washington. The tables in this book are derived from these

tests with the permission of Mr. Taylor. It has been shown that

the tests by the Froudes and by Mr. Taylor are in substantial

accord, so that both series of experiments may be claimed as the

basis of the tables given in this book.

The tables for three-bladed propellers are based directly on

an extensive set of experiments made on propellers of the Admi-

ralty type with various widths, thicknesses, and pitch-ratios.

The tables for four-bladed propellers were deduced from a com-

parison of tests on thin-bladed propellers of the type shown by

Fig. 22 (some with three and some with four blades) with the

tests on the Admiralty type. A table for two-bladed propellers

PROPELLER EXPERIMENTS

61

FIG. 27.

62

PROPELLEKS

METHOD OF EXPERIMENTS 63

was deduced in like manner from tests of thin bladed propellers

on that type.

Method of Experiments. In making experiments in a model

basin, the model propeller is placed at the front end of a shaft

which is suspended from the towing carriage. The shaft atrthe

rear extends into a boat-shaped box which contains the driving

gear on the propeller shaft. The towing carriage is propelled at