ARCHt
THE GEOMETRY OF THE
SCREW-PRO PELLER
BY
C. P. HOLST
Late Professor of Mechanical Engineering in the Technical University of Deift (HoIlaI3d); Hon. Doctor of Technical Science, Deift.
LErnEN - E. J. BRILL Ltd.
Publishers and Printers
1924.
kb. y. Scheepsbouwktrntle
Technische Hogesthool
THE GEOMETRY OF THE
SCREWPROPELLER
BY
C. P. HOLST
Late Professor of Mechanical Engineering in the Technical University of Delft (Holland); Hon. Doctor of Technical Science, Delft.
Lrn - E. J. BRILL Ltd.
Publishers and Printers
PRINTED 13V E. J. DRILL, LEIDEN (HOLLAND).
A careful examination of the working-drawings of screw-propellers and of the methbds employed in their execution, soon shows, at least in a vast majority of cases, an unsatisfactory state of things
with regard to the geometry of the design.
Very often there is no certainty that the. blades, as shown in
the drawing, will
fit the boss; and very often too it is fully
certain, that the cross-sections will not fit the helicoidal surface, intended to be the pressure-side of the blade. Almost always the
exact relation between the "developed" or "expanded" blade,
and the projection of the blade on a plane perpendicular to the centre line of the shaft, is left undefined. Nearly always an
approximation is made use of, to simplify the process of drawing
itself, but on close examination it becomes apparent, that with the very same simplifying assumption, an even better agreement to the truth may be obtained with much less work. Moreover, a further investigation reveals the unexpected fact, that by far the easiest and most simple mode of proceeding consists in discarding the assumption alluded to, altogether and returning to the true
geometry of the helical line. Then too, approximations are to
be resorted to, but these are now of a quite different character:
the error incurred being entirely free from uncertain consequences,
perfectly known and hardly perceptible.
In this treatise, the author has endeavoured to lay down the
2
of the screw-propeller. For a good deal they date back to the
days of his employment as a draughtsman in a large engine-works'
drawing-office; the syste,matic application however, of one or
another graphical constructiòn for the rectification of circular arcs,
has only been found in his DelfI days. Indeed, this intended
application was one of his inducements for the investigation into
DESCRIBING-LINE CONSTRUCTIONS.
In strict accordance with its title, this study has only to deal
with the blades of the screw and their position on the boss of the propeller, or rather with one blade, since the number of
blades is quite indifferent for the present purpose.
The same holds good as to the distinction between a
right-handed and a left-right-handed propeller. A blue-print, for instance, either of the one or the other, may be obtained from the same
tracing; the result depending only on the question which side of the tracing is exposed to the source of light. Hence it is quite sufficient, to describe and to illustrate the method of drawing for one type only. To prevent confusion, exclusively right-handed
'screw-propellers will be dealt with in this treatise.
As to the boss, its longitudinal section, in outline only, is
wanted to define completely the surface of revolution, whereupon
the blade has to find its seat. Whether cast solid, or loose and
bolted on by means of its flange of spherical shape, or
dove-tailed
in a groove and keyed on to the boss, the, mode of
attachment has no real. influence upon the geometry" of theblade.;
at the most it has something to do with the radii of
rounding off the corners at its root.
The blade of a screw-propeller is a solid, limited by, or enclosed
between two curved surfaces, sitting astride on a third: the sur-face of the boss. The two first mentioned, the sur-faces of the blade,
may be distinguished as its pressure-side and its suction-side.
Their line of intersection is the outline of the blade; in reality
its more or less sharp edge. On drawings however, for practical
reasons, it is customary and advisable too, to show this edge
sharply, as a well defined line, it being one of the data for the shape of the blade.
In some cases of recent practice however, the leading edge is
considerably thickened and completely rounded off; the following
edge being left fine and sh3rp. Under such circumstances, it will
be found of advantage, to assume a suitable line to represnt
the missing edge: preferably the line passing through the Ïeading
points of the thick leading shoulders, thus completing, for these cases too, the outline of the blade.
This outline may be given: either by its projection on a plane perpendicular to the axis of revolution, as the "projected" blade;
or as a plane curve, the "developed" or "expanded" blade, arrived
at by a process to be described and examined in the sequel..
The areas enclosed by these different curves bear the same names: the "projected" surface and the "developed" surface,
and either of these may be given for the size of the blade, being
an indirect statement of the widths or tangential dimensions,
required for the desired area.
The maximum radial dimension of the 'blade and its outline, is given by the diameter of the screw, being that of its
circum-scribed, and of course co-axial, cylinder.
The shape of the pressure-side surface of the blade is always given directly by its geometrical data; that of the other,
suction-side face always indirectly, by the thickness of. the blade at
different points of its surface; the latter being thus made dependent
The geometrical data referred to, are the describing line and
the pitch.. The first is the line of intersection of the blade-surface
with a radial plane, i. e. a plane passing through the axis or the.
centre line of the shaft. The helical srface is
described, or,,generated" by it, if it is turned round about the axis, and at
the same time displaced, or "advanced".along.it; the angle
be-tween the axis and the describing line being kept constant during
the motion. If the translation is proportional. to the rotation, the advance corresponding to a complete turn of 3600,. is the pitch
of the screw.
The describing line may be a straight line, either perpendicular
to the axis or sloping;., or it may be a curve, convex as well
as concave.
The pitch may be uniform, having the same value at all points
of the surface; sometimes however, it is made increasing
tangen-tially from the leading towards the following edge, or the pitch is made different in a radial diiection, either increasing or even decreasiúg from. the root of the blade towards the tip.
it first the pitch will be assumed to be uniform all over the blade
The suction-side face of the blade is given by means of a set of parallel cross-sections, plane or cylindrical, and in this last case shown by the drawing either in projection, or developed into a plane figure.
Almost always a "longitudinal section" through the blade is
added to complete the drawing. This is no reäl section at all, but, a figure showing the axis and the describing line in their
common plane, the describing line being "backed" by a thickness of material, . equal to the maximum thickness of the blade at the
same distance from the axis: a convenient diagram for inter-polation purposes, but nothing else. . .
6
It is evident, that a propeller-blade may be defined by the
set of cross-sections alone, if these are sufficient in number and
suitably placed: the first, near to the boss, just beyond the
rounded-off corners at the root; the last, close to the tip of the blade; the other, intermediate ones, preferably at equal distances
apart and from these two. The "latitudes" of their site being thus fixed, the "longitudes" are referred to an arbitrary "first
meridian" as a datum-line. If the outline of the blade is a sym-metrical curve in its projection on a plane normal to the centre
line of the shaft, then of course the axis of symmetry of this
projection is the best datum-line, and the radial plane containing it, is in such a case really a first meridian: a zero-plane.
With unsymmetrical blade-projections however, the choice is not so dictated by the circumstances of the case. The best plan. seems, to make this first meridian to bisect the angle, enclosed between the two radial planes that touch the outline of the blade, one i.n the foremost point of the leading edge, the other in the extreme point of the following edge. The. object of this preference
is to minimize the error, introduced by the assumption, that the parts of the helical lines as limited by the outline of the blade, are sensibly plane curves; an error rapidly increasiñg with. the,
half-width of the blade. If this assumption is not applied,, the
choice is really a free one and the zero-line may be taken for
instance, passing through the middle of the width of the blade's
root, or it may be made to coincide'with the projection of the describing line, that cuts the surface of the boss in a point of
its maximum cross-section.
If now a blade is moulded in the usual way, the helical
pressure-side face being mechanically "constructed" by the motin of its
placed with respect to the zero-line, would be quite sufficient for the manufacturing of the blade! For a working-drawing however, something more is wanted:' the outline of the blade ought to be
shown, iñ projection and in its developed form as well, to enable
the designer to compute the two corresponding, blade-areas. Lastly,
in sorne cases it may be necessary to show the complete propeller
even in three projections, viz, an eñd-view or elevation.: this
being the projection repeatedly referred to; a side-view: the
longitudinal elevation; and a top-view or plan:. For ä four-blâded
propeller, these latter two are identical of course.
The first mentioned, the "end-view", is the only one required for the design and, though not strictly necessary, for the work-shop.
If in the sequel of this treatise, the single name "projection" is used, it always refers to this view: the projectiÓñ on a plane at right anglçs to the centre liñe of the 5haft.
The root of the blade cannot be so wide of coUrse, as to extend
beyond the boss -if cast solid with it, or beyond the circular
base-edge of its segmental flange, if made loose. On the other
hand, it should be as wide as possible, occupying t:he full length
available, in order to reduce the working stress in the material for a' given thickness, or conversely: the thickness and the resistance
due to it for .a given greatest. stress.
To ensure the fulfilment of these conditions, the first thing to be done is to draw the line of intersection of the helical surface, of which the blade's face 'is a part, with the surface of the boss. The two systems of lines, to be used here to represent the two surfaces, are the describing line in different positions and the circular
Fig. i b. 3' 2' i' 0 ¡
2 3
8
The simple construction is shown by, fig. i ;-.i a being the
end-view, i b the side-view and i e the top-view r plan; Though
the boss is illustrated as spherical, the fo1lowng descriptioi is
perfecIy general, and may be applied t any surface of
revolu-Fig. i c.
tion. The same holds good for the describing line: though being illustrated in the drawing as a straight sloping line, it may have another form, or anòther slope, as well.
The end-elevation, i a, shows the traces of a set of radiaiplanes,
drawn at equal angular intervals; 74-° or 90 beiñg chosen for
right-handed screw: the zero-line, as usual, being the vertical, the
positions I, 2, 3, etc. towards the leading edge and i', 2', 3', etc.
towards the trailing side, corresponding to the numbering shown
in thé starboard-side view, fig. i b: from o in the middle, z, 2,
etc. forwards and 1', f, 3',etc. backwards. If now these radial
planes, each with its own describing line in it, are swung round
about the axis, so as to make them coincide with the vertical
zero-plane, then all the describing lines will occupy the numbered
positions shown in fig. z b, being parallels cutting the axis in
points, remained stationary at equal mutual distances, of - in
this case
=
L
of the given uniform pitch. But thelongi-360 48
tudinal sections of the surface of the boss in the different radial pianes, now coincide with the vertical section in the zero-plane,
that shown in fig. i b; hçnce: the points of intersection of the parallels with the outline of the boss in i b, swung back about
the axis, with their radial 'planes into their original positions
again, are the points required, where the surface of revolution is "pierced" by the describing lines of the helical surface: their locus is the line of intersection of the blade's face with the boss. The' cross-sections on which these points are situated, are fully
given by fig. z b; their projections in that view being the vertical
lines, passing through the points where 'the parallels enter into,
the boss; their projections in za are concentric circles, whose
radii are the lengths of 'these vertical lines, measured from the
points on the outline to the axis of the boss; their projections
in z c of course, are identical with those in z b.
If the describing line is straight and at right angles to the axis, there is a very slight simplification, but in fig. z b only:
the "parallels" there, then becoming vertical lines, and at the
Io
same time, the projections of the corresponding cross-sections. On the other hand: a curved describing line does not entail the slightest complication, nor any additional work in drawing: the part of the line, coming into play here is simply a fiction,
apper-taming only to the two triangular parts of the blade's face,
enclosed between the boss and its co-axial circumscribed cylinder.
Hence their fundamental shape is completely merged into the
materiâl of the rounded-off corners, and a straight sloping tangent
to the curved describing line is all that may reasonably be used for its
inner end, within the halfdiameter of the boss. The straight sloping
parallels in fig. i b, therefore really show the most general case.
The construction itself is easily and rapidly done. First the
radial lines in i a and the parallels in r b, are drawn in the corresponding projections of the boss, and numbered as required.
Then the distances to the axis of the points in ¡b, where the parallels cut the outline of the boss are set off in r a, as radii
from the centre, on the corresponding radial lines; the curve
passing through the points thus found, is the projection in i a, of the required line of intersection: the base-line of the face of the blade. To find its projection in ib, vertical lines are drawn
through the points already used on the outline of the boss, where
it
is cut by the parallels; the points in
Ia, being projectedhorizontally upon these vertical lines, their locus is the side-view
of the base-line. Lastly, the cross-sections of the boss, indicated by these vertical lines in i b, are shown by corresponding hori-zontal lines in the top-view: Ic, at the same distances from the
middle-section of the boss as in ¡b, of course; by projecting
vertically the points of the curve, in i a upon these horizontal lines, the points of the top-view, or plan, ¿f the base-line of the blade's facé, are found too.
II
If the limits only of the base-line are wanted, it will be almost always sufficient to draw the projection of tht line in the end-view i a, without even completing its other projections. For, if
the blade is
to be cast solid with the boss, the limits of its
breadth are given by two conceñtric circles: the outlines of the end-faces f the boss, as shown in that figure; and if the blade is loose, to be bOlted on by its flange, the edge of this face is
shown by a chord of the circular outline of the boss, and all
these lines are well defined and clearly shown in i a. Fig? I C too shows the base-line very favourably; fig. i b on the contrary very indistinctly.
The breadth at the root of a loose blade, fixed to the boss in a dove-tailed groove, is evidently given by the length of
that groove.
The outline of the blade may be drawn in the three projections shown by fig. i, too, by using thé corresponding projections of a sufficient number of describing lines. in the end-view I a, they
are by definition radial lines, and they ought to be drawn at
equal angular distances apart.: --, -- or of the circumference,
2436
4baccording to circumstances, being suitable values. If the describing
line is straight and perpendicular to the axis of revolution, then
in the side-view i b, its different positions are shown as equidistant
vertical lines; and in plan, i c, by a set of horizontal lines, at the same mutual distance, being .---, --- or of the uniform
24 36 48
pitch, in accordance with the angular interval, used for the
end-elevation I a.
The somewhat tedious task of preparing these three sets of lines, before the beginning of the actual drawing of the blade
12
itself, is complicated in the case of the describing line, though
being still straight, is not a perpendicular to the centre line of
the shaft; for now its projections in side.view and in plan show, wit,h respect to the axis a slope, varying with its distance from the zero-position. The complication cannot be avoided, but it is
reduced to a minimum, and ease and good accuracy are obtained
by the following process, shown by fig. 2, a and b. The radial lines in a, and the vertical ones in 2b, are drawn in exactly
Fig. 2b. Fig. 2a.
the same way, as previously described for the case of a straight
generating line, at right angles to the axis. Then, the sloping
describing line is drawn in 2 b, in its zero-position in the vertical
plane,
cutting one or more of the vertical lines öf the first
operation. If the slope be too slight, to furnish even one point of intersection within the limits of the drawing, then the chosen interval should be reduced. Now, one point of intersection o, is taken, preferably the first one beyond the half-diameter of the screw, or if this be inconvenient, the last point within; the objectbeing to get a good perpendicular distance from the axis, to
improve the accuracy attainable. With this distance of the chosen
point to the centre line as a radius, a circle is drawn in 2a about
the centre of the screw, and then its points of intersection with the radial lines, projected horizontally S upon the vertical lines
in 26; the pòint on the zero-radius of a now corresponding of
course, to the first chosen point o of 26. The points thus found
in the latter figure, are the projections of that point o of the
describing line in its different positions; hence, if joined to the equidistant division-points of the corresponding number on the axis, they give the required set of projections of the describing line itself; completing with this operation the "network of lines
of preparation".
It is evident, that exactly the same process might have been
applied to the top-view or plan, there being no intrinsic difference
between this and the longitudinal elevation or side-view: in both cases the plane of projection is parallel to the axis of revolution. Therefore it is quite superfluous to draw a preparatory network
of lines for more than one of these two projections, and it is
indifferent for which of both the work is done.
The complication becomes still worse, when the describing line
is a curve. Of course, the fundamental principle of representing the helical surface by its describing lines, may be adhered to, even in such a case; its application however, would involve the
drawing of a whole set of different curves, showing the projections
of the describing line under various angles of inclination of its plane with respect to the plane of the drawing. Considering that
in the end-elevation a radial line cuts the outline of the blade
in one or two points only, each curve contributes with but one
'4
the blade; but it required for its construction the determination of a good many more.
An improvement may be obtained by the use of an other
generatrix, better suited to the circumstances of this case. Any given helical surface of uniform pitch may be generated indeed, by any line whether plane or not, broken or continuous, dra.rn
upon that surface; and among this infinity of possibilities a better
"strickle" is offered, by the line of intersection of the helical surface with a plane ät right angles to the axis of revolution,
i. e. parallel to the plane of projection of the end-view. This line
of intersection is a curve too, but with this advantage, that in
its different positions, as a generatrix, its projections do not alter in shape, showing in end-view always its real form; whereas in the other views its projections are simply perpendiculars to the
axis of revolution.
If now this normal generatrix" were given, instead of the
usual "radial describing line", the preparatory work would consist
in drawing a set of equal and similar curves, round the centre
in the end-elevation, at equal angular distances of 3600 apart,
and in the, other views a set of perpendiculars to the axis at the
corresponding mutual distances of of the pitch. It is however
not very likely, that such a normal generatrix will be given, as one of the data for the design of a screw-propeller, intended to have a curved describing line in its radial planes; therefore it is desirable to devise a construction, allowing to draw the new section, if the more familiar other one is given.
To do this, a zero-plane is chosen at right angles to the axis, passing through one of the verticals of the alrèady "prepared"
side-view, or plan; the curved radial describing line being drawn
too in this view, in its tiue shape and position relatively to the
axis, say in its zero-position. Since this line, properly guided, describes the helical surface, it will cut, during its motion, the chosen zero-plane in its line of intersection with that surface, viz, in the required curve: the new generatrix. Any point of the zero-plane, where a point of the moving line passes through, is a point in that curve. Hence the following construction, shown
in fig. 3.
In the end-elevation 3 a, a set of radial lines, and in the other
projection 3 b, a corresponding set of perpendiculars to the axis, are drawn, constituting together the first stage of preparation.. This havin,g been done, the curved describing line, in its zero-position is drawn, and the chosen zero-plane indicated, in 3b; the zero-vertical in this figure bing taken to represent that plane. A certain number of verticals, oo included, cut the describing line n as many points at different distances from the axis; and with these distances as radii, concentric circles are drawn,' as
shown, in a. These circles, together with the radial lines, divide
that figure into a number of sectors of rings: four-sided figures, approaching to. a trapezoidal shape. Any continuous line, built
up of a chain of curvilinear "diagonals" of these pseudo-trapezoids,
is a projection of the new normal generatrix, and the complete
set of all such lines, that may possibly be drawn through the
maze-work, thus obtained in 3 a, corresponds to the complete set. of vertical lines, used in 3 b.
The whole process, though tedious, is so simple that no further
explanation is wanted, with one single exception, perhaps: the
choice between the two "diagonals of the trapezium". The origiflal numbering is, at first sight, even misleading, but the following
reasoning is correct and safe: the describing line as shown in
fig. 3b, is cut by the zero-plane in its extreme point at the root;
all its other points are situated aft of that plane; its extremity
at the tip of the blade most of ali. Therefore, in ordcr to .make
its points successively pass through the zero-plane, the describing
line must be turned "ahead", not for its extreme point at the root, but more or less for all its other points, and most of all
for its extremity at the tip of the blade. Concluding in other
words: the generatrix in a plane at right angles to the axis, if
curved forward, describes a helical surface with a radial describing
line, either curved or even straight, but always raking backward;
if curved the other way, the blade will show a
"lean-to".A straight sloping line
in 3b would answer to a spiral of
Archimedes in fig. 3a.
It is easily seen, that the construction as shown is perfectly reversible: allowing to find the usual describing line in a radial plane, corresponding to any proposed generatrix in a plane per pendicular to the axis.
If nothing else were wanted in the drawing of the blade, than the projections of its outline, then a considerable simplification
might be obtained, in the cases of the describing line being either
straight but sloping, or curved.
The projections of the blade's outline are first drawn, as for the simple case of a straight describing line at right angles. to the axis: the required preparation being limited to t.he drawing of the radial lines in the end.elevatioñ, and of the corresponding perpendiculars to the axis in the other views. Then, all points belonging to. the projections of the outline in these last, sheer or plan or both, are displaced parallel to the axis, by an amount equal to the rake, either backward or forward, of the real des-cribing line with respect to the first-assumed, straight perpendi-cular one; the amount of this rake or shifting, for each point to be measured at a radial distance from the axis, equal to that of
the point thus shifted. The end-elevation of course, does not
undergo any alteration at all.
It is obvious, that the results thus obtained by simpler means, than by the modes of operation illustrated by fig. 2 and fig. 3,
are exactly the same and quitè correct, but the method employed is often defective. Except in one single case only, it is incomplete, giving the outline of the blade, but not providing for the enclosed
curved surface itself; in other words; it does not always afford the indispensable means to draw the necessary cross-sections of
the blade; and the secondary additions required to fill this want, are
precisely the disagreeable secondary preparations, supposed to have been saved at the outset. And though a working-drawing of the blades of a screw-propeller may be very useful, without a longitudinal elevation and a plan of the outline of the blade,
i8
incomplete, and really defective, if it does not correctly show
the cross-sections wanted.
The exceptional case, just referred to, will be shown in ne
of the following paragraphs.
The cross-sections of the blade are either plane, or cylindrical.
If plane, they are taken at right angles to the radial zero-linç
O A, fig. a of the end-elevation, being thus fully determined by their distance O C, from the axis O, to which they are parallel.
In fig. a such a section is shown in end-view, by the straight
line P1 P1, perpendicular to O A. About O as a céntre, a circular arc P1 A P1 is struck, being the projection of a cylindrical
cross-section, passing with the plane section first mentioned, through
the same points P, of the outlin'e of the blade, of which the points P, are the projections. This of course, is possible with
symmetrical blade-projections only; with unsymmetrical önes,
the arc is drawn as before, but the plane section, still parallel
to the axis, then has a more or less sloping chord for its projection,
or: the two sides of the figure are dealt with separately, as if
each vere one-half of a symmetrical figure. Both "methods" are
indicated in fig. a, by dotted lines. Sound reasons however, for
such a wonderful mixing together of the data, belonging to two quite different kinds of cross-sections, are not yet known: the distance O C alone, fully determines the plane one, whereas the radius OA gives the only quantity wanted to define the
cylin-drical section.
The line of intersection of an arbitrary plane with the given helical surface, passes through the points in which that plane is cut by the describing lines of that surface; more generally "by
lines on that surface", but these are dependent on - and
secondary to - the former, at least in the problems referred to here. A plane section, such as P1 CP1 in fig. 4, being horizontal
itself, is best drawn in the plan; no further construction is then
required. The points in which P1 P1 is cut by the radial lines
in fig. a, are simply projected vertically upon the horizontal
Fig. 4 b.
perpendiculars to the axis, to which they correspond, in the case
of a straight describing line, at right angles to the centre line of the shaft; in the case ofa sloping one, the points are projected upon the corresponding lines of different slope' in the top-view.
20
P1 P1, to be projected, are given by the intersections of this line,
with the different curves shown in fig. 3 a; the lines upon which
their projections fall, are the corresponding equidistant
perpendi-culars to the axis, like in the simple first case.
None of all these lines of intersection of the helical surface of
uniform pitch with a plane, is a straight line. This is easily verified
by considering the curve of fig. 4b first, and then the effect of
a certain amount of rake, or lean-to, in altering its shape. The original curve shown in full lines, due to a straight strickle,
perpendicular to the axis, is cut by the vertical centre line of
the figure in its point of contrary flexure C, dividing it into two
exactly equal parts; the vertical ordinates of its successive points
being: d, 2d, 3d, etc. and their horizontal abscissae proportional
to tang , tang 2 , tang 3 , etc.; both positive for one half
and negative for the other half, if counted from C as origin; d stañding for the distance between two perpendiculars to the axis in 4b, and for the angle between two radial lines in a.
The conclusion is, that the line of intersection shown in 4b, cannot be a straight line under any circumstances, nor even have
a straight part; the curve being at both sides concave towards the axis of abscissae: the horizontal line passing through C. As to the influence of the rake of an arbitrary describing line, either
straight and sloping or curved, it is the same for all points at the same radial distance from the axis. Hence any two points
of the curve, having equal abscissae, but with opposite signs,
get equal amounts of displacements parallel to the axis, and both in the same direction as shown by the dotted line. The
consequence is, that a system of such displacements for the several
pairs of points of the curve, fit to diminish the initial curvature of one half of the line, straightening for instance the side to the
21
left of the centre line, increases on the contrary the curvature of the other half, the part to the right. Therefore, though it is
possible that, by chance, the rake neutralizes, for a certain plane,
the curvature of its line of intersection with the helical surface,
a part only of this line can be straight; such a part falling
wholly at one side of the centre line of thé section.
The curve of its pressure-side having béen constructed, the crosssection is completed by a second curve for its suction-side; this curve passing through the extreme points of the first one,
the points of the outline of the blade. A third point is given by the amount and the place of the greatest thiékness of the
blade in the plane of the section, and through the three points thus obtained, a curve is drawn, according to the taste and the intentions of the designer. These two cooperate as well in fixing
the place of maximum thickness, either at the centre of the
section or nearer to the leading edge, for instance; but as to the amount of this quantity, ordinary practice is not always
conse-quent or correct, in its ways of making use of the so-called
"longitudinal section" of the blade, to get the necessary infor-mation. This figure, besides showing the real radial section of the helical surface, is at the same time a diagram, showing the gradual decrease of the maximum thickness of the blade, from
its root to the tip, as dependent on its distance from the axis:
in fact a diagram with these rectangular coordinates. Hence for the section P, CPI, fig. a, the required maximum thickness is that, corresponding to the distance O C, or to some intermediate distance between O C and O P1; but certainly not that at O A, which is very often taken instead of the right one.
This last "approximation", combined with the other one, assuming
'22
line, gives the "mixed" cross-sections already alluded to. They
are perfectly independent of any "geometry" of the screw-propeller.
The cylindrical section is simple and rational: its intersection with the helical surface being a helical line, completely defined by the pitch, and by the radius of the cylinder, and consequently
wholly independent of the shape and the position of the describing
line. The maximum thickness is doubtless indicated: the terms 'radius" and "distance" being now synonymous for all points of
the section, so that it seems as if all ambiguity and all uncertainty are done away with, by the adoptio,n of this type of cross-section. But these good qualities are completely spoiled by projecting the
section on the plane CF1 P, thereby assuming again a straight line, CPm in fig. 4b, the tangent to the helix in A, to represent,
with a sufficient approximation, the curve of the pressure-side of
the section,' and thus giving, though by another way, a second instance of the "mixed type".
The term "developed blade" does not refer to the geometrical
definitiön of that expression, but simply to a convention. A helical
surface cannot be developed, in the sense as attached by the
mathematician to this word; hence the term is to a certain extent meaningless, if not explained anew by some conventional definition This is best given by merely describing how the "outline of the
developed blade" is obtained.
In fig. 4a, the line CF1 is produced, and on it from C, is-set off a length CF,, equal to the distance of the point C itself
on the surface of the blade, to the point P of the outline of
the blade. Then the locus of all points, such as P1, is the outline
of. the developed blade.
through A and P1, as shown, and on this tangent from A, the
length A Pm is set off, equal to the distance C)',, just referred
to. Then the locus of all points, such as Pm, is the outline of
the developed blade.
Or: in fig. .a, on the tangent just alluded to, from A a length
A P is set off, equal to the rectified length of that part of the helical line upon the blade's surface of which the arc A P1 is the projection. Then the locus of all points such as P, is the
outline óf the developed blade.
Thesè conventional definitions are framed so as to be applicable
to unsymmetrical blades as well as to symmetrical ones.
It will be observed that no allowance is made for the lengthening
of the blade in a radial direction, by the rake of a sloping or
a curved describing line; the points C and A being taken on the
projection of that line, and not on its
rectification, showingconclusively that no attempt is made to extend the process of
developing or expanding, to the radial dimensions too.
The indices of P,, Pm and P refer to the kind of
blade-section: plane, mixed or cylindrical, to which the developed blade
H.
STRAIGHT-LINE CONSTRUCTIONS.
The great simplicity of the cylindrical sections would, at first
sight, suggest the probability, that a helical surface may be represented and analyzed by the, aid of its helical lines, with
less trouble and complication than by means of its describing
line, placed in different positions.
In principle, the change of method involved, requires the
substitution of the set of planes, radial or perpendicular to the axis, as used before, by a set of concentric cylinders, sufficient
in number, but of quite arbitrary radius, cuttingthe helical surface
in the well-known spirals: the helical lines proposed. In the end-elevation, the projections of these lines are of course, those of the cylindrical surfaces: a set of concentric circles; in the other views however, a set of as many different sinoidal curves, requiring for their construction exactly the same troublesome and highly uninteresting preparation, that was intended to be avoided, and that is so completely sufficient to solve all problems connected with the projctions of the blade, that simply by its being drawn alone, the sinoidal curves themselves at once become wholly super-fluous: a useless complication, instead of the simplification expected.
This disappointing result however, should not be accepted
as a final one. Something like a compromise is still possible:.the
simplicity aimed at, may be obtained indeed; but it is paid for
approximations, the whole of the preparatory work may be
dispensed with.
In fig. 5 the projections are shown of the úpper half of a horizontal cylinder; 5 a being the elevation and 5 b the plan.
A helical line of uniform pitch is drawn upon its surface, and
its sinoidal projection constructed, by means of the «radial lines"
in 5 a, and the "perpendiculars to the ax4s" in b; that is: with
Fig. 56.
the usual preparations. These however are to be made super-fluous now: the curve should be drawn without their aid, and then if not correctly, with a good approximation at least.
The middle of the curve for the part shown, is the point A
of .its intersection with the centre line of the figure. This point
divides the curve into two exactly equal parts,' both concave
towards the axis. With A as origin, the ordinates are proportional 3
DL
26
to the angles of slope of the radial lines, the abscissae to the
sines of these angles only; hence for small angles, they are nearly,
but not yet quite, proportional to their corresponding ordinates,
and A is a point of inflection or of contrary flexure, in which ethe curve is cut by its tangent for that point.: its tangent of
minimum slope to the abscissae. But this slope is the real one
of the helical line itself at all its points, for the "central strip"
of the cylindrical surface is parallel to the plane.of 5 b;
approxi-mately for a narrow strip, and exactly for the central line. And since, by definition, a helical line climbs or advances one pitch-length at each full turn, its constant slope is simply: "pitch" to "circumference of cylinder", and this ratio is the goniometrical tangent of its angle of inclination, a. Consequently: the straight line, touching and cutting the true sinoidal projection in its point
of intersection with the centre line, is a fair approximation of
the central part of the curve; it may be drawn at once, since
its
slope a is known; lang a being equal to pitch:
circwn-ference.
This is the principle upon which drawings of screw-propellers
are nowadays almost exclusively made, and it is at the same
time its full justification. On the other hand, if in cases of rare occurrence, mathematical exactness is aimed at, the old sinoidal
curve makes its amusing show of indispensability, almost without
exception too; trying to make believe, that the grating of lines,
the w3rp to which its own appearance is due, is unfit to have
another pattern of line woven in.
Accepting the circles and the straight sloping lines discussed above, to represent the helical surface of the blade, its base-line on the boss ought to be constructed now by these means only;
the radial lines and the parallels employed in fig.
i, do not
belong to the new system.
The simple, straightforward construction is shown in fig. 6, a top-view of the boss, like that of fig. i c. A set of co-ax-ial
cylinders cuts the surface of the boss in a series of parallel-circles,
whose projections are shown in the figure, as a set of perpendi-culars to the axis at unequal, arbitrary intervals. They are best
taken wide apart where the diameter of the boss decreases slowly,
Fig. 6.
and at smaller mutual distances where this decrease becomes more and more rapid, and such that each diameter of parallel-circle is used twice: once at each side of the. greatest cross-section of the boss. At right angles to the axis a straight line
is drawn, at a distance from the centre of the blade, equal to
ßitch: 2 s =p, and on this line are set off all the radii of the
parallel-circles or of the cylinders employed; their common
zero-point being the zero-point of intersection with the axis. Lines joining their other extremity with the centre of the böss, have a slope
28
of pitch: 2 - to radius, that is of pitch lo circumference; hence,
by the approximation now adopted, these lines are the projections
of the several helical lines on the. cylinders, and on the blade itself. The parallel-circles of the boss too, are lines on the cylinders,
therefore the point wherein such a "helical" line cuts the parallel"
of its own cylinder, is a point of the surface of the blade and
on the surface of the boss, in other words: it is a point of the
required base-line.
It is almost superfluous to add, that the curve thus found first
in plan, may be drawn in the elevation - not shown - simply by projecting each point upon its corresponding circle: the
projection of that one of the concentric cylinders to which that
point belongs.
The construction is given for the case of a straight describing
line, perpendicular to the axis. The other cases will be considered further on.
It is not necessary to determine many points of this base-line; sometimes it will be sufficient even, to know its end-points only. The construction answers well to such simplifications; it being possible to find the required point directly for any given parallel-circle. The mathematical error is neither considerable,
nor troublesome in its consequences.
On the whole this construction is really a very satisfactory
one, and seeing its great simplicity, much. better than the exact
one of fig. I.
For the purposes of this treatise a working-drawing of a
screw-propeller is complete, if it shows:
O, the end-elevation of one blade, with the boss: the "projected"
2°, the outline of the "developed" blade;
30, the "longitudinal section" of the blade and, that of the boss; 40, the "cross-sections" of the blade, in sufficient number; and
sometimes:
50, the top-view or "plan" of the blade and the boss;
this last being not always necessary.
If then all the required data are known, and they include the projected blade-area, as against its developed surface, being not
given now, the drawing may be made as follows: the projections
of the boss, wanted for 10 and 3°, and if added: for 50, are
drawn; the section 0, being further an independent diagram,
showing the describing line and a "review of maximum thicknesses", is at once completed. Assuming that the clumsy exact method
is rejected, in favour of the approximating simplification, the
base-line of the blade is drawn in the view5°, according to the
construction shown in fig. 6, and then its projection determined
in the elevation z° too, as explained. The outline of the projected
blade may now be sketchçd in this view, the width at the root corrected, if needed, so as to suit the base-line, and finally its
area computed and checked. If this is found to be right, the
projection of the outline of the blade is definitively drawn, thereby
completing the elevation 1° itself; but in ordér to provide for
the data, still wanted for the rest of the drawing, 2° and 4°, a
set, of concentric circles is added to the figure, showing the
projections of as many helical lines on the surface of the blade. To draw the projections of these lines in the top-view ° too, use is made of exactly the same method, that of fig. 6, already employed for the construction of the top-view of the base-line. The radii of the concentric circles, or of the co-axial cylinders, are set off as previously, from the zero-point on the axis, upon
30
the horizontal line at a distance p = pitch : 2 r from the centre; and the points thus obtained joined with that centre by straight
lines, the approximations of the flat parts of the true curves wanted. The points of intersection of the outline of the blade and the circles in z°, have their projections in plan on these
corresponding lines in 50, hence these projections are points of the top-view of the outline, so that this line may now be drawn and the plan 5° completed.
All the necessary ingredients being now ready to hand, the temptation is really too strong not to indulge in mixing them
together. Indeed, referring to fig. 4 too, there are made cylin-drical sections through the blade, such as P1 A P1, but in plan the straight line Fm Pm only is shown in its full length, and this
line has but three points in common with the blade: Pm, C
and Pm. Still it is usual to take this line, to represent, in length
and in direction, the pressure-side boundary of the intended
cross-section, and this, combined with a maximum thickness that is to be found at some point of the cylindrical section P, A P1, corn-. pletes the mixtures z°. And as to 2°, the "developed blade", it is usual now, to set off the half-lengths CPm as measured in the plan 50, on their own projections in the elevation I or
some-times on the same principle in a separate figure, but not on the
tangent to the circle in A, fig. 4. Being set off on CPI, they
answer to the first of the three definitions given.: the "developed
blade" thus constructed, has an outline of the P,-type, properly belonging to a system of true plane sections.
If these same half-lengths however, are set off from. A on the tangent to the circular projection of their "own" cylinder, then the outline belongs to the Pm-class, of the second definition; it
general - another area of enclosed surface, but it is more natural,
being more closely related to the nature of the accompanying cross-sections.
This sounds somewhat like an arbitrary verdict, but its justi-fication is readily given by the simpl consideration, that the construction in this last case is perfectly reversible, without any
addition at all, whereas the first is not. In fact, if a developed outline of the Pm-type, according to the second definition be
given, any straight line, drawn across at right angles to the centre
line, is a tangent to a circle, whose radius is the distance of the
centre to that line; and this radius being known, the line "standing"
for the projection of a helical line upon the cylinder of that radius, may be drawn in plan. By setting off on this line the
half-breadths given by the developed outline, two points of the projection of the blade in 50 are found, and by projecting both
upon the circle, touching the first
line, drawn in i, the two
corresponding points of the outline of the projected blade are
determined.
If the same be tried with the usual developed outline of the
first definition, it appears at once, that without some suitable addition, this reversed problem cannot be solved; the several
points of the outline of this type having undergone a lowering equal to the versed sine of an unknown arc, forming part of a now unknown circle. It is owing to th influence of this versed
sine, that the intimate connection between the developed and
the projected outlines is.lost. Practice however, has been unwilling
to give up the lowering, but decided to make its amount known,
by graphical means, for any width of the blade at a given
cylindrical section. In other words: the circles belonging to the projected blade, are "developed" also, just like the outline, and
32
in order to simplify the work involved, with the aid of the
approximation already made use of.
The special problem to be solved here is this: given a
cylin-drical section of a helical surface of given uniform pitch, to find
the developed line of intersection, this developing to be performed,
according to the first definition.
The line of intersection is the helical line. By the adopted
approximation however, it is identified with a plane curve, since,
irrespective of its length, it is shown in plan by a straight line:
its own tangent A B, fig. 5 b, which is
the projection of an
ellipse, this being the plane section of a cylindrical surface. The
semi-minor axis of this ellipse, vertical in direction, is the radius
of the cylinder, QA or AD; its semi-major axis is equal to AB. In fact, the method of developing now prescribed, applied at all points of the circular arc A P1 in fig. 4, leaves the versed sines
unaltered, at the same time lengthening the half-chords all in
the same proportion, viz, in the ratio AD: A B or "projection"
to "projected line"; hence the result is a part of an ellipse as described. With x standing for the angle of inclination of the
helix, this ratio = cas z : i.
Its semi-axes being thus given, the curve may be drawn by
any of its well-known simple constructions; of course only the elliptic arc required to be actually shown in the drawing. In this case however, there is a coincidence, almost inviting to make use of the foci F, F', of the ellipse for its construction. Indeed,
since A F = O B,. fig. 7, the cosine of angle F A O is equal to
AO:AF or' OA:OB=cosa, and O,F=OA.tangx=
= O A .pitck : circumference; this circumference. being. equal to
O A. 2 ,. it appears that OF pitch : 2 r ==ft. Hence, this
of the method under consideration, is equal to half the distance
FF' of the foci apart, and this for any section, irrespective of
Fig. 7.
its distance from the axis. Therefore: all elliptical arcs, wanted
34
expanded outline of the first definition, may 'be constructed with
the same points F and F' as their common foci. But, remarkable though this property is, it does not follow that, the invitation
ought to be accepted; the simple lengthening of the half-chords
of the circular arc in the ratio cosX: ï or O A: O B, may be
preferred.
By the addition of these elliptical arcs, the necessary
reversi-bility of the construction is obtained; these arcs being for the
developed outline, equivalent to the circular arcs for the projected
outline: the points of intersection of the expanded arc with the expanded blade, are the points Pft in the prolongations of the
half-chords CF1, fig. a, for any breadth of the blade.
The simplified, approximating system as fully described in the
preceding paragraphs, has two 'weak points: the complication,
due to the drawing of a whole set of elliptical arcs; and the
rough approximation of the shape of the cross-sections. As to the complication, it may be observed, that it is not very
conse-quent 'to add an elaborate, exact construction of an ellipse of
only secondary importance, to a fundamental construction for
the delineation of a screw, simplified eveñ to the exteñt, that
the screw-liñe, the helix, is shown as a straight line. And as to the cross-sections, they are intended to be plane sections of the blade, judging by the way they are made use of in the moulding
of the blade; by the way they are drawn however, they are
really projections of cylindrical sections, found by means of the
ruling approximation. Hence they are plane, but they are no sections of the blade and cannot fit the helicoidal surface of
the mould.
the value of the simplified drawing, certainly justifies a closer
examination of the difficulties to be overcome.
Doing away with all initial approximations, the first question then is: how is the exact relation between the projected outline
and its true developed shape, according to the first definition: the Ft-line?
Let R stand for the radius OA; s for the versed sine A C;
b for the distance CF1, the projection of the half-width of the.
blade; w for the real half-width: CP= CF,; d for the distance FP1, all the points, P excepted, being taken in the plane of
projection; for the corresponding angle A OF1; and lastly, for brevity's sake: p for pi/ch : 2 r.
On A O produced, see fig. 7, a point Q is taken as the centre
of a circle, of radius p, passing through the points A and Fp; the distance O Q = pR, to be denoted by z. Then
in the circle about Q: s (2 p - s) = w2,
and in that about O: s (2 R - s) = b2;
hence, by their difference: 5.2 Z = W2 - b2. (I)
But in the horizontal, rectangular triangle P CF1 is
CF2 CF12=FP12,
or:
w2b2=d2,
and, since by the fundamental property of the helix,
d:pitch = :2
-, (in radians)d=p.cp, and d2=p2.2.
This gives, being substituted in (I), with the value for
s=R(icosc):
2Z.R(I-hence:
36
the exact expression for the distance O Q and for the differ-ence p - R.
At first sight, it is a disappointment to find this remarkable quantity to be a function of , i.e. still dependent of the width
of the blade. But the denominator: 2 (i - cos c), being expanded,
appears to be equal to:
- 4: ¡2
±
: 360 - etc.: a valueonly slightly less. than that of the numerator: 2; at least for angles not exceeding say 600, corresponding to z. Hence,
by a first approximation, the fraction containing , though always
somewhat more, may be taken as roughly equal to z, and (II) be read as:
z
2: R, (III)bearing in mind that this value is in reality too small.
With this approximate, simple expression for z, the raçlius
p=R+z is found:
cR
p2 R2+p2R0A2+0F2
AF2 OB2OA
- 0A
QAp being really somewhat more.
Now OB is the semi'major .axis, and O A the semi-minor axis,
of the ellipse of the usual approximation; hence its radius of curvature at the point A is equal to O B2: O A; the circle of
curvature of this radius touches the ellipse in A, and lies entirely
at its outside. But the circle of radius p, that would be wanted
for the exact constructiòn, is still somewhat larger, as shown
by (IV), touching the ellipe in A toc,, and falling with all its
other points beyond the circle of curvature. This last circle, lying
everywhere between the exact circle and its usual approximation,
the ellipse, therefore is a closer approximation and by far the simplest of the two. Its centre is found at once, by drawing FQ
at right angles to A F; for then OF2 OA .QQ, or ft2 = R. z,
Hence the use of the ellipse should be discarded: the circle,
with its centre in Q, of radius Q A, giving an even better approxi-mation, with much simpler means.
The second weak point is the construction of the cross-sections.
It has already been observed in a preceding paragraph that, as
determined by the usual method, these are in reality the projections of cylindrical sections, being found by applying the simplification
of drawing a straight tangent to represent, with the adopted
approximation, the true helix supposed to be wanted.
All this however is wrong: a plafle projection of a cylindrical
section is no plane "section"; and the tangent, really wanted,
is not that to the sinoidal projection of the helix on the cylinder of radius O A, but that touching and cutting the curve shown
in fig. 4, in its point of inflexion, and this curve is a "tangentoid",
quite different in shape from the wavy other one: the sinoid. The mistake of the customary process may be seen at once
by the following consideration. The line of intersection of a given
plane with a given curved surface, is of course perfectly
inde-pendent of any "inscription", that may be traced upon that surface. The outline of a propelIerblade, however, is nothing
else but a simple inscription, "an arbitrary scratch", on a helicoidal
field; therefore the length of the line PP cannot have the least influence on the shape or on the direction of the curve of
inter-section. But an other distance P - P, means another distance P1 - P1 too; hence, since O and C are given points, an other
cylinder-radius O A also, and therefore a different angle of
inch-nation x. The wonderful result is: that a modification of the
"inscription" has the effect of turning the cross-section, in its own plane, about its centre C!
38
The simple remedy is easily found by an inspection of the
true curve, shown in fig. 4. If this curve should be represented
by a straight line, with an approximation quite in accordance with the system now described, then here too, the tangent in
its point of inflection C, ought to be used by preference to all other straight lines Now the plane of the section P1 P1 touches
the cylinder of radius OC in a line of which the.pointC is the
projèction; hence the curves of intersçction of the helicalsurface with this plane and with this cylinder touch each other in that point. In other words: the true cross-section curve and the helix on the cylinder of radius O C', have a common tangent in C'.
Therefore: the pressure-side boundary of the cross-sections, if approximated to by a straight line, should be made to cOincide
with the tangent to the helix of radius O C, that is: with that helix itself, as drawn in ccordance with the simplified system of approximation. The use of the cylinder O A should be discarded altogether.
If the helical surface is generated by a sloping or by a curved
describing line, the contour of the blade is. Lthrown back" in the drawing, by the simple process, already referred to, of displacing
all its points parallel. to the axis, and this is quite correctly and consequently performed by shifting the straight lines, standing for the projections of the true sinoidal curves, parallel to them-selves by the required amount, measured in an axial direction,
proper to the corresponding cylinder-radius.
The simplified system does not take into account the ensuing
modification in the shape of the cross-sections, whose boundaries
change from the symmetrical full line, as shown by fig. 4, into the unsymmetrical dotted one. This defect is the direct result of
the mixed character of the usual process: a cylindrical cross-section does not alter by a rake of the describing line, and
therefore its plane projection is not affected by it either, but a
true plane section is certainly altered. The system however, does
not afford the means to show such a modification.
Still, though the true curve undergoes the change, illustrated by fig. 4, the slope of its tangent in the point C remains the same as before: that of the tangent in that point to the helix
HELICAL-LINE CONSTRUCTIONS.
The approximations, characterizing the simplified method
des-cribed in the preceding section, consist in the systematic substi-tution of a curve, by the tangent in its point of contrary flexure; and this expedient is applied to the curve of sines: the projection of the helical line, as well as to the curve of tangents: the plane
section of the helicoidal surface. It is certainly not without interest
to know exactly the amount of the errors thus introduced, for
any proposed section, and for any half-breadth of the blade. Both parts of this double problem require the computation of
the distances at several points of a given curve from a given
straight line, in this case the tangent to the curve in its point
of inflection. With the origin in this point: O, fig. 8b, the projection of the axis of revolution as the axis of ordinates, and the
abscissae-axis at right angles to this direction, the' simplest mode of operation is to find the difference between the ordinates of the curve and of the straight line, corresponding to any given abscissa-value; thereby stating the error of the approximation as a difference in
axial direction.
Let P denote the pitch of the helical surface, cut by a co-axial
cylinder of radius R, and the angle included between the
radial plane passing through the arbitrary point to be considered,
and the zero radial-plane, containing the point of the helix,
corresponding tó O.
Then the axial advance d that corresponds. to the angular
interval in radians, is:
P
d=P, or with =p, as before:d=ft.;
and this is the length of the ordinate of the true sinoidal curve
at the considered point. The abscissa belonging to it, being equal
Fig. 8a.
Fig. Sb.
to R sin , the ordinate d, for the straight line, passing through the origin O, is found by the simple proportion:
d3:Rsin=p:R, giving at once:
and the difference:error=dd=p(psincp),
(V)in the approximate projections of the helical lines.
The values of this quantity, for different angles, multiples of i5°,
are as follows: 4
for = 15°
3Q0 45° 600 750 900, error = 3.0 23.6 78.3 181.2 343.1 570.8 p : ¡000; o1 = 0.5 3.8 12.5 28.8 54.6 90.8 P: ¡000.42
Supposing that the true curve of a plane section, such as that shown by fig. 8, is by approximation identified with the tangent
in its point of inflection, then, for the section in the plane touching
the cylinder of radius R, already used above, and with the same
notation:
the ordinate of the point in which the plane is ut by the des-cribing line in the radial plane, at the angle from its
zero-position,
is again: d=p .
. Its abscissa however, is here equalto R tang ; hence the ordinate d of the tangent, by supposition
a line through the origin, including with the abscissa-axis an
angle , whose goniometrical tangent is equal to p : R,
d:RtangcP=p:R, giving: d1=p.tangd,,
and the difference:error =d d=p (tang - ),
(VI)in the approximate constructions of the plane sections of the
helicoidal surface.
The values of this quantity for the same angles as used above,
are as follows:
for = ¡50
300 45° 600 750 900,error = 6.2 53.8 214.6 684.8 2423. CiD : ¡000;
or
= ¡.0
8.6 342 ¡09.0 386.o P: ¡000.
In both cases the errors are independent of the radius R: they
appear to belong to a whole radial plane, rapidly increasing
with its angular distance from the "first meridián". This increase is, of course, much more with the second series than with the first but even for very small angles the errors of the cross-section
approximations are twice those of the first list. Indeed, the ratio
at
¡0,
as given by the two tables, viz. I to 2 (nearly) isgives
4:
6 for the first term of the series, whereas tang'-begins with : 3 already, the smaller value being still somewhat
too high, the othet one too low.
The property that the errors are simply fractions of the pitch,
constant for all points of the same radial plane, is of great
importance, in so far as it enables the designer to know from
beforehand, with perfect certainty, if in a given case the application
of the simplifying straight-line approximations will be justified
or not. A simple consultation of the two tables of errors, as
given above, or the direct computation of these errors from the
formulae (V) and (VI), with the aid of the tables of circular arcs,
sines and tangents, is all that is wanted to settle this question and to dispel any possible doubt. It will be found that in many
cases the error is not to be neglected, and that, even with a
describing line perpendicular to the axis, the cross-section error
(VI) is almost alwáys very considerable.
With the high disc-area ratios, now often met with, or with unsymmetrical outlines of blade, now seldom encountered, the "approximations" no longer deserve their name.
And so the question arises, as one of actual importance, if
there is no alternative and if the clumsy and tedious, but accurate
methods, described in the first section of this treatise, must be
adopkd, or re-adopted, in all those cases of increasing frequency, in which the labour-saving simplified method of the second section
fails for want of accuracy.
A third possibility will be examined in this third section.
What does necessitate these disagreeable preparations when exact
results are required? Why must the desired pattern be woven
44
the fundamental questioñ, and the answer upon it is, that. this network of lines is indispensable, as a means to lay down in the
drawing, the first principle of the genesis of the helical surface: the constant proportion of a rotation about an axis, to a
trans-lation along that axis. In other words: the lines of preparation
serve no other purpose than that of expressing, by a series of
repetitions, the relation between the two corresponding projections of the describing line in its successive positions.
But the surface of the helicoidal blade may be defined too,
as the locus of a system of co-axial helical lines, all of the same
given pitch, and each of them fully determined by its radius:
the radius of its "own" cylinder; the duty of the describing line
being now reduced to the simple "telescopilig" of these cylindricál
tubes unto their, desired relative axial positions.
By this definition, a point on the surface of the blade, whether
in its outline or not, is a point of a helix. If the point is given
in the end-view projection, that helix is completely known, and
so is too the angular distance of the given point from the
zero-point of its own helix. Then the axial, or fore-and-aft, position
of the point, relatively to the zero-plane at right angles to the
axis, is fully determined, by the amount d of the axial advance, corresponding to the angle , and by the amount of the rake, forward or backward, of the whole cylinder tò which the helix
and the point belong; this rake being known by the describing line.
'Hence, if a simple method could be devised to' find d,
corres-ponding to any given value of , the preparatory radial lines
and perpendiculars to the axis might be dispensed with.
Such a method however, does really
exist, and the third
It has already been shown, that d may be expressed by the
simple formula:
d=p.cp.
(VII)This may be read as: the axial advance d, corresponding to añ angular motion Q, of a screw of pitch P, is equal to the rectified
length of a circular arc, whose radius is equal to ft = P: 2 r,
and whose central angle is Q.
By this theorem, the chief difficulty of the original problem is transformed for its solution, into the other problem: that of the rectification of circular arcs; and it seems at irst sight, that, in endeavouring to discover a simple, and at the same time
accu-rate, method of delineating a screw propeller, such a transformation
should rather be rejected at once, than adopted for further use; seeing that the exact rectification of circular arcs by graphical
means is an impossibility. But it will be found that the calculated
mathematical errors of the cohstructions, proposed hereafter, are well within the unavoidable errors of execution; in fact, unless the drawing is very large, they are imperceptible.
The construction alluded to is shown by: fig. a, wherein O is the centre of a circle of radius., the central angle. A-OB=Q; hence the arc A B =.p. Q the curve to be rectified to the length d,
set off on the tangent to the arc in the point A, as a distance A C
The radius A O is produced, and on its prolongation a fixed point D is taken, at a distance from O, equal to O D
= i.
fr;Then, the straight line, joining the extreme points B and C of
the arc A B and its tangent A C of equal length, will pass through
the Lfixed point" D, very nearly. Hence, by drawing DB to the point of intersection with the tangent, a segment A C = d is cut off, equal to the given arc A B = . Q, with a very close
