DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0075
DRAWING OF MACHINERY.
59
tances. It is therefore necessary that those divisions of
the circumference of the one, which indicate the thickness
of the teeth and width of the spaces, should be contained
an exact number of times in the circumference of the
other. Hence it follows that the numbers of the teeth
in the two wheels are proportional to their respective radii
or diameters ; and, on the other hand, the lengths of these
diameters depend on the relative velocities at which the
two wheels are intended to revolve. These are the points
which first demand our attention, and which wre shall
proceed shortly to elucidate.
Prop. I.—If the distance between the parallel axes of
two wheels, and their relative velocities be known, the
diameter of each wheel may be found.
The velocities of wheels gearing together, are in the
inverse ratio of their diameters; or, in other words, the
one wheel revolves with so much the less velocity as its
diameter exceeds that of the other, and vice versa.
Thus, the velocity is doubled, by causing a wheel of a
certain diameter to work into another of half that dia-
meter; and conversely to reduce the velocity by one-half,
it is necessary to employ a wheel of twice the diameter.
This being premised, let us now call A and B the two
axes; suppose them to be situated at the distance of three
feet apart, and let it be required to make the axis A
revolve twice during one revolution of the axis B.
Divide the distance between the axes into three equal
parts ; from the point A, as a centre, with a radius equal
to one of these parts, describe a circle; then, from the
centre B, with a radius equal to the sum of the two
remaining parts, describe another circle, which will touch
the former in a point situated upon the line joining their
centres. These circles indicate the true diameters of the
wheels; they are called the primitive or proportional
circles, or, more frequently, in reference to the teeth, the
pitch circles, of the wheels.
Hence it appears that, in order to the solution of the
general problem, it is only necessary to divide the dis-
tance between the axes into as many equal parts as there
are units in the sum of the numbers expressing the velo-
cities of each wheel; then to take for the radius of the
smaller wheel a number of parts corresponding with
the number which denotes the lower velocity; and con-
versely.
Now, since the circumferences of circles are directly
as their diameters, it is obvious that, in any pair of
wheels gearing together, the number of teeth in each is
also directly proportional to the respective diameters ;
so that, when the number of teeth in the one is known,
that of the other may be found by the following simple
proportion :—
Prop. II.—The number of teeth in the one wheel is to
the number of teeth in the other as the diameter of the
first is to that of the second.
Supposing the number of teeth in both wheels, and the
distance of their centres to be known, their diameters
may be found thus :—
Prop. III.—The sum of the numbers of teeth in the
two wheels is to the distance between their centres as the
number of teeth in either wheel is to its radius.
In the case of a rack gearing with a wheel, the primi-
tive circle becomes a straight line, which, in determining
the form of the teeth, must be drawn touching the pitch-
circle of the wheel. In a trundle, the pitch-circle passes
through the centres of the staves. In the examples of
trundle gearing, which form the subject of Plate XXII., it
will be observed that the diameter of the staves is equal
to the thickness of the teeth of the wheel or rack which
imparts the motion; this circumstance, however, which
is by no means essential, does not frustrate the application
of the principles we shall establish, to other cases.
In a pair of toothed wheels working together, the first
and most important objects to be attained are the reduc-
tion of the friction to a minimum, and the equable
distribution of the strain over the teeth as they come
successively into action. To satisfy these conditions, it
is necessary that the form of the teeth should be such
that the working surface of a tooth of the driving
wheel should roll over, or be developed upon, that of the
driven wheel, without any sliding or rubbing action.
Now, taking as an example the pair of spur-wheels repre-
sented at Fig. 1, Plate XXIII., and supposing the smaller
to be the driver; if a circle, 0 G C, be described with
the radius 0 C of the driven wheel as a diameter, and
if this circle be conceived to roll in the interior of the
primitive circle ABC, the line in which the point C
will travel, will coincide, as we have already demon-
strated, with the radius 0 C. And supposing the circle
0 G C, to roll on the exterior of the primitive circle
C D E of the driving wheel, the same point 0 will de-
scribe an epicycloid, which will, in every possible position,
be a tangent to the straight line 0 C, upon which it will
be developed, or, in other words, will roll without sliding,
since the curve and straight line are generated by one
and the same point in the circle 0 G C. Hence it follows
that the point of contact of any two teeth must always
be in the circle 0 G C.
In the case of a pinion driven by an internal spur
wheel, Fig. 3, Plate XXII., it is obvious that the
same construction which we have above described is
applicable. When a rack is substituted for the driving
wheel, Fig. 2, Plate XXI., the circle A E C, drawn
with a diameter equal to the radius A C of the primitive
circle of the pinion, generates a cycloid, which fulfils, in
this case, the required conditions. But if, on the other
hand, the pinion is the driver, as in Fig. I, Plate XXI.,
the generating circle then becomes a straight line, M N,
which, by moving round the circle BAD, would cause
any point, such as A, to describe an involute A c; this
curve, therefore, will act in the required manner upon
the flank of a tooth of the rack, provided the latter be
perpendicular to the pitch line M N.
These illustrations involve the statement of the theore-
tical principle which determines the form of the teeth of
all wheels working in gear ; but in practice certain modi-
fications are admissible. This observation we conceive it
59
tances. It is therefore necessary that those divisions of
the circumference of the one, which indicate the thickness
of the teeth and width of the spaces, should be contained
an exact number of times in the circumference of the
other. Hence it follows that the numbers of the teeth
in the two wheels are proportional to their respective radii
or diameters ; and, on the other hand, the lengths of these
diameters depend on the relative velocities at which the
two wheels are intended to revolve. These are the points
which first demand our attention, and which wre shall
proceed shortly to elucidate.
Prop. I.—If the distance between the parallel axes of
two wheels, and their relative velocities be known, the
diameter of each wheel may be found.
The velocities of wheels gearing together, are in the
inverse ratio of their diameters; or, in other words, the
one wheel revolves with so much the less velocity as its
diameter exceeds that of the other, and vice versa.
Thus, the velocity is doubled, by causing a wheel of a
certain diameter to work into another of half that dia-
meter; and conversely to reduce the velocity by one-half,
it is necessary to employ a wheel of twice the diameter.
This being premised, let us now call A and B the two
axes; suppose them to be situated at the distance of three
feet apart, and let it be required to make the axis A
revolve twice during one revolution of the axis B.
Divide the distance between the axes into three equal
parts ; from the point A, as a centre, with a radius equal
to one of these parts, describe a circle; then, from the
centre B, with a radius equal to the sum of the two
remaining parts, describe another circle, which will touch
the former in a point situated upon the line joining their
centres. These circles indicate the true diameters of the
wheels; they are called the primitive or proportional
circles, or, more frequently, in reference to the teeth, the
pitch circles, of the wheels.
Hence it appears that, in order to the solution of the
general problem, it is only necessary to divide the dis-
tance between the axes into as many equal parts as there
are units in the sum of the numbers expressing the velo-
cities of each wheel; then to take for the radius of the
smaller wheel a number of parts corresponding with
the number which denotes the lower velocity; and con-
versely.
Now, since the circumferences of circles are directly
as their diameters, it is obvious that, in any pair of
wheels gearing together, the number of teeth in each is
also directly proportional to the respective diameters ;
so that, when the number of teeth in the one is known,
that of the other may be found by the following simple
proportion :—
Prop. II.—The number of teeth in the one wheel is to
the number of teeth in the other as the diameter of the
first is to that of the second.
Supposing the number of teeth in both wheels, and the
distance of their centres to be known, their diameters
may be found thus :—
Prop. III.—The sum of the numbers of teeth in the
two wheels is to the distance between their centres as the
number of teeth in either wheel is to its radius.
In the case of a rack gearing with a wheel, the primi-
tive circle becomes a straight line, which, in determining
the form of the teeth, must be drawn touching the pitch-
circle of the wheel. In a trundle, the pitch-circle passes
through the centres of the staves. In the examples of
trundle gearing, which form the subject of Plate XXII., it
will be observed that the diameter of the staves is equal
to the thickness of the teeth of the wheel or rack which
imparts the motion; this circumstance, however, which
is by no means essential, does not frustrate the application
of the principles we shall establish, to other cases.
In a pair of toothed wheels working together, the first
and most important objects to be attained are the reduc-
tion of the friction to a minimum, and the equable
distribution of the strain over the teeth as they come
successively into action. To satisfy these conditions, it
is necessary that the form of the teeth should be such
that the working surface of a tooth of the driving
wheel should roll over, or be developed upon, that of the
driven wheel, without any sliding or rubbing action.
Now, taking as an example the pair of spur-wheels repre-
sented at Fig. 1, Plate XXIII., and supposing the smaller
to be the driver; if a circle, 0 G C, be described with
the radius 0 C of the driven wheel as a diameter, and
if this circle be conceived to roll in the interior of the
primitive circle ABC, the line in which the point C
will travel, will coincide, as we have already demon-
strated, with the radius 0 C. And supposing the circle
0 G C, to roll on the exterior of the primitive circle
C D E of the driving wheel, the same point 0 will de-
scribe an epicycloid, which will, in every possible position,
be a tangent to the straight line 0 C, upon which it will
be developed, or, in other words, will roll without sliding,
since the curve and straight line are generated by one
and the same point in the circle 0 G C. Hence it follows
that the point of contact of any two teeth must always
be in the circle 0 G C.
In the case of a pinion driven by an internal spur
wheel, Fig. 3, Plate XXII., it is obvious that the
same construction which we have above described is
applicable. When a rack is substituted for the driving
wheel, Fig. 2, Plate XXI., the circle A E C, drawn
with a diameter equal to the radius A C of the primitive
circle of the pinion, generates a cycloid, which fulfils, in
this case, the required conditions. But if, on the other
hand, the pinion is the driver, as in Fig. I, Plate XXI.,
the generating circle then becomes a straight line, M N,
which, by moving round the circle BAD, would cause
any point, such as A, to describe an involute A c; this
curve, therefore, will act in the required manner upon
the flank of a tooth of the rack, provided the latter be
perpendicular to the pitch line M N.
These illustrations involve the statement of the theore-
tical principle which determines the form of the teeth of
all wheels working in gear ; but in practice certain modi-
fications are admissible. This observation we conceive it