Improvements in Gear Wheels.
Year: 1895

page_text: PROVISIONAL SPECIFICATION.
Improvements in Gear Wheels.
I, ARCHIBALD SHARP, Engineer, 13, Fairlawn Grove, Chiswick, in the County
of Middlesex, do hereby declare the nature of this invention to be as follows:-
My invention relates to the teeth of gear wheels. The form of the teeth.is
usually defined so that the velocity-ratio of the two wheels in gear will be constant:
the well known involute or cycloidal teeth satisfy this condition. Since however,
it is very difficult to form these curves correctly, it is usual to approximate to
the exact involute or cycloidal arc by a circular arc which will agree with the
correct curve as closely as possible. I have investigated the forms of circular
teeth which for a given are of contact will give the least possible variation from
constant velocity-ratio, and I find the following results :-
=
Let r and r, be the radii of the pitch circles of the smaller and larger wheels
respectively of a pair in gear, h₁ and h, the radii of curvature of the circular
teeth, and, and 4, the distances of the centres of curvature of the teeth from the
centres of the wheels respectively; then during contact of a pair of circular teeth,
the motion of the wheels is the same as that of two levers of length ₁ and l, (whose
fulcra are situated at the wheel centres) connected by a link of length / h₁ x hq.
(This link is referred to hereinafter as the equivalent link.)
With circular teeth the velocity-ratio is not constant, but by suitably choosing
the values of 1, 2, and 1, the variation of the velocity-ratio may be made
extremely small. During contact of the same pair of teeth, the actual velocity-
ratio may be three times made exactly equal to the assigned velocity-ratio 1. The
Τι
variation will be a minimum when these three positions are at equal angular
intervals. When the wheels (or the equivalent mechanism) are in these three
positions, let the distances from the pitch point (the point of contact of the
two pitch circles) of that end of the equivalent link which is jointed to the lever
of length be p+q, p, and p−g respectively; the length of the arc of contact of
"1
}
a pair of teeth is 29. Writing R == 3, p = mr₁, and q = n r₁, I find that the
circular teeth which give the least variation from the assigned velocity-ratio R are
determined by the following equations

page_text: Also, the percentage variation of the velocity-ratio is given by the equation
Equations (1) to (4) apply to internal gearing by writing R negative, and to the
case of a rack and pinion by writing R = o. If the arc of contact be equal to
the pitch of the teeth, N, the number of teeth in the small wheel, will be
approximately equal to .
n
If the length of the equivalent link be assumed, then writing L
substituting in (2) and (3)
h
and
we get
(六)
(K + 2)
27 R (R+1)
L' +
R+2
3 R
n³
(5)
2
th
(2 R +1)
=
L' +
(2 R+1)
n³
(6)
27 R' (R + 1)'
3 R
Equations (5) and (6) are useful for the design of wheels, the teeth of which have
to be shaped by milling cutters. With a set of a small number of milling cutters
formed to cut circular arcs of different radii, it will be possible to cut the teeth
of any pair of wheels so that they gear together with practically constant velocity-
ratio. The same milling cutter may be used in cutting wheels of quite different
pitches instead of requiring, as in the present system of cutting involute or
cycloidal wheel teeth, a separate set of cutters for each pitch. Milling cutters of
radii h, and h, having been chosen, the cutter of radius h₁ is adjusted in the wheel
cutting machine so that the centre of curvature of the cutter is at a distance
from the wheel centre; 4, being determined by Equation (5). Similarly for ...
In toothed wheels the contact between a pair of teeth takes place along a line.
But from the fact that the curvature of circular teeth is the same at all points, a
pair of wheels may be made in which the teeth of one wheel are circular arcs
and those of the other are replaced by circular pins on which rotate slide blocks
which press against the circular teeth of the other wheel, and have their rubbing
surfaces made to the same radius as that of the circular teeth. The contact
between the various parts being now distributed over a surface the wear and tear
will be less than with ordinary spur wheels.

page_text: may assume L=
h
T1
in (5) and (6), we get
-
πT
πT
and n =
Nk
Substituting these values of L and N
0
Equations (7), (8), and (9) are applicable to internal and external gearing, and
to the case of a rack and pinion.
ROLLER-AND-PIN BEVIL WHEELS.
Equations (7) and (8) may be applied to the design of the development of a
pair of bevil wheels. The teeth of the bevil wheels will be replaced by conical
rollers which may turn on cylindrical pins fixed to the rims of the wheels, and
having their common vertex coincident with that of the two pitch cones.
Dated this Eleventh day of April 1895.
Improvements in Gear wheels.
I, ARCHIBALD SHARP, of 13, Fairlawn Grove, Chiswick, in the County of
Middlesex, Engineer, do hereby declare the nature of this invention and in what
manner the same is to be performed to be particularly described and ascertained
in and by the following statement :-
.
My invention relates to the teeth of gear-wheels. The shape of the teeth of
gear-wheels is usually defined so that the velocity ratio of two wheels in gear
will be constant; the well-known involute and cycloidal teeth satisfying this
condition. These curves, however, are more difficult to form correctly than circles,
and in many cases it is usual to approximate to the true shape of the cycloidal or
involute teeth by circular arcs. However, if the outlines of the teeth of a pair of
wheels in gear be circular arcs the velocity ratio will not be constant during the
time of contact of a pair of teeth. I have investigated the forms of circular
teeth which will give the least possible variation from constant velocity ratio, and
I find the following results :-
During the contact of the same pair of circular teeth, the velocity ratio though
continually varying, may be three times and only three times exactly equal to an
assigned value.
If A and B (Fig. 1) be the centres of the two wheels the teeth of which are
circular arcs having the centres M and N respectively, while a pair of teeth are
in contact the angular velocity-ratio of the wheels is, as is well-known, the same
as that of the links A M and BN connected by the link M N. Let r₁ and r, be

page_text: the radii of the pitch-circles, O the pitch-point, and the lengths of the
Inks A Mand BN respectively, and the length of the connecting link M N.
In any position of the four-link chain A B M 'N, let M N cut A B in the point e.
Let Me, and Oe=y. Expressing the cosine of the angle Be N in terms
of the three sides of the triangle Be N, and the cosine of the angle A e M in
terms of the sides of the triangle A e M,
+ n ( li + li*) + rg (ri – li) } * −rgh (ri – I) = 0.
In order that the variation of the velocity-ratio of the wheels, during contact of
a pair of teeth, may be as small as possible, it seems a fair assumption that the
three positions of the crank A M, in which the actual velocity-ratio is exactly 2
equal to r, shall be at equal angular intervals. That the three roots of (3)
should be of the form p-q, p, and p+9, approximately expresses this condition.
The equation having these roots is

page_text: Equations 11-14 apply to internal gearing by writing R negative; and to the
case of a rack and pinion by writing R = ∞.
With circular teeth designed as above the distance oe (Fig. 7) and therefore also
the variation of the velocity-ratio is much less than with circular teeth designed
by the already known methods; viz., by fitting as closely as possible a circular arc
to the arc of an involute or cycloidal curve.
If M N, the sum of the radii of curvature of the circular teeth, be arbitrarily
chosen then Equations (11), (12) and (13) may be more convenient written
thue

page_text: From the fact that a circle is a closed curve, the teeth may be made of complete
circles (Fig. 8). These complete circular teeth may be fastened rigidly to their
respective wheels, or each tooth may be made of a roller and pin. Thus, in
Figure 8 the rollers w are free to revolve on pins v, which are driven tight into
holes in the projections u from the rims t of the wheels. In this case the arc of
contact, which is approximately 2 q, must be equal to or a little greater than 2 h,
so that one pair of teeth may always be in gear... A roller and pin wheel may be
made in steps, so that the arc during which the same pair of rollers remain in
contact may be reduced. Each roller and pin may have a plain cylindrical or
conical bearing between each roller and pin. Figure 9 is an elevation, and
Figure 10 a view, partly in section, of a pair of roller and pin wheels with two steps
the reference letters being the same as in Figure 8. In each wheel the centres of
the rollers of one step are midway between the centres of the rollers of the other
step. In N be the number of teeth in one step of the smaller wheel, and k the.
b
T
+
=1924
N'k
(20)
Roller-and-pin wheels may be made for internal gearing. Figure 11 shows a
pair, in which R= 1.5, N = 4, k = 1, and V1.3 per cent.
In toothed wheels of the ordinary type the contact between a pair of teeth takes
place along a line. But from the fact that the curvature of circular teeth is the
same at all points, a pair of wheels may be made as in Figure 15, in which the
teeth of one of the wheels are circular arcs, but those of the other are replaced by
circular pins v on which are centred solid blocks w which press against the circular
teeth and have their rubbing surfaces made to the same radius as that of the
circular teeth. As the wheels revolve, the slide block in contact at the moment
will rotate on its pin; thus provision must be made to allow of a certain amount of
rotation. Further rotation of the slide blocks w may be prevented by pins s fixed
to the wheel and projecting into slots in the slide blocks.
'Circular pins have been used instead of teeth, when the wheels are equal and
turn in the same direction, i.e., R= 1; also in one wheel of a pair, the outline
of the teeth of the other wheel being a curve parallel to an epicycloid; but with
these exceptions circular pins or rollers have never before, to my knowledge, been
proposed for the teeth of spur-wheels. Such wheels are advantageous from a
mechanical point of view, in so far as the relative sliding of the bigher pair in
spur-wheel teeth, is here replaced by the relative turning of the two lower pairs
(the rollers and pins); the relative motion of the surfaces in rubbing contact being
less in the latter than in the former case. The roller and pin, forming a cylindrical
journal, can be much better lubricated than the teeth of spur-wheels; and since
they have surface contact, as against line contact of spur-wheels, the frictional
resistance and the wear is greatly reduced. Even assuming that the lubrication of
the rollers is so imperfect that they stick on their pins, and that, therefore, the
rollers of one wheel rub on those of the other, they will at least be moved slowly
round on their pins; and the wear will be distributed round their entire outer

page_text: surfaces, instead of being confined, as in fixed teeth, to small parts of the surfaces
in the neighbourhood of the pitch lines.
+
:
If the rollers are made of hard steel, it seems that the power that may safely be
transmitted, may be very much higher than would be possible with a pair of
ordinary spur-wheels of the same size. Again, with pins of wrought-iron or mild
steel on the small wheel large wooden rollers might be used on the large wheels ;
this would give the easy, noiseless running of mortise wheels with a much stronger,
though perhaps more costly, construction..
The teeth of bevel wheels can be designed on the same general principles; and
bevel wheels can be made with roller-and-pin teeth. Figure 12. is a view, partly in
section Figure 13 a part front and part back view, and Figure 14 a development
of a pair of roller-and-pin bevel wheels. With bevel wheels the rollers w will be
conical and will have a common vertex with the two pitch cones: they may revolve
on cylindrical pins v driven tight into holes in the parts u projecting from the
rim t. The pins may be made slightly taper, so as to give a small amount of
adjustment to each roller. Bevel wheels may of course be made with steps; also
with external or internal gear; and with axes inclined at any angle.
Equations (16) and (17) are useful for the design of wheels the teeth of which
have to be shaped by milling cutters. With a set of a small number of milling
cutters formed to cut circular arcs of different radii it will be possible to cut any
pair of wheels with circular teeth to gear together with practically constant
velocity-ratio. The same milling cutter may be used in cutting wheels of quite
different pitches, instead of requiring, as in the present systems of cutting wheel
teeth, a separate series of cutters for each pitch. The part M of the sliding
machine (Fig. 2) carrying the cutter would be adjusted relative to the table on
which the wheel blank W rests, until the distance of C. the centre of the arc of
outline of the cutter, from the centre of rotation of the table of the milling machine,
is exactly equal to ₁ or .
Figures 3-7 show a few cases of wheels with circular teeth designed as above;
the pitch point O, and the three positions m, n, m, n, and m, n, of the equivalent
link M N in which the velocity-ratio is exactly equal to the assigned value R being
shown in each figure. In Figure 3 R = 1, m = 4.40, n = 0.35 N the number of
teeth in the smaller wheel of the pair
==== 9,
h
= 0.80 4 ↳
T1
T1 T
0.931, and V
= 4.1 per
Figure 5 shows a rack and
Figures 6 and 7 are cases of
0'4, n = 03; In Figure 7, 9 =
cent. In Figure 4, R = 10, m = 0.60), n = 0:55.
pinion, that is R 8, m = 0-020, n = 0.45, N = 7.
internal gearing; In Figure 6 R = 1.5, m =
m = 04, n = 0.8 aud N = 4.
-
3,
In Figures 3-7 the working circular part of each tooth is shown by a thick
line, the remainder near the root is designed in the usual way, so that points of the
teeth of the other wheel may clear. The height of the working portions of the
teeth may also be determined in the usual manner.
I wish to state distinctly that I do not claim exclusively the making of gear
wheels with circular teeth; also that I am aware that pin gearing has already
been extensively used, the teeth of one wheel being circular pins, those of the
other being curves parallel to epicycloids; and also, that in a pair of equal wheels
turning in the same direction, circular pins have been used on both wheels, the
sum of the radii of the two sets of pins being equal to the distance between the
wheel centres.
Having now particularly described and ascertained the nature of my said
invention and in what manner the same is to be performed, I declare that what I
claim is :-
(1.) Constructing wheels with circular teeth, the radii of curvature of the teeth
and the positions of the centres of curvature being determined by Equations 11, 12
and 12

page_text: (2.) The shaping of circular wheel teeth of any size and pitch by one of a
small series of milling cutters, the position of the centre of the arc of outline of
the cutter being calculated from the sum of the chosen radii of curvature of the
teeth as herein before described. (Equations 16 or 17).
(3.) Gear wheels (spur wheels and bevel wheels) in which teeth are replaced by
complete cylindrical or conical rollers free to revolve on pins fixed to the rim of
the wheel, substantially as described and illustrated in Figures 8-14; the rollers
for the least possible variation of velocity ratio being designed from Equations 18
and 19.
"I
(4.) A pair of gear wheels (spur wheels or bevel wheels) one of which has
circular teeth, the other circular pins on which rotate slide blocks which rub on the
circular teeth and have their rubbing surfaces made to the same radius as the
said circular teeth, substantially as described in Figure 15.
Dated this 2nd day of November 1895:
ROBERT E. PHILLIPS, A880C. M. Inst. C.E., M.I. Mech. E.,
Consulting Engineer and Patent Agent,
70, Chancery Lane, London, W.C., Agent for the Applicant.