DOI Page / Citation link:
https://doi.org/10.11588/diglit.25888#0076
60
ENGINEER AND MACHINIST’S DRAWING-BOOK.
necessary to make in this place, as, in the ensuing illus-
trations of toothed gearing, which we have to present to
our readers, certain differences may be observed between
the forms and proportions employed, and those we have
already exhibited and announced, or may have occasion
to do in subsequent parts of this work, as governing the
practice of this branch of constructive mechanics. In cases
which we have to bring under notice in the theoretical
part of the investigation, the flanks, or those parts of the
teeth which lie within the pitch circles, are invariably
represented as portions of radii, tangents to the curves of
the faces, or parts beyond these circles, while in the prac-
tical illustrations they are replaced by circular arcs. This
difference arises from the circumstance that, as a compa-
ratively small portion only of the flanks of the teeth come
into actual operation in the communication of the motion,
the form of these parts may be modified, so as to admit of
a greater amount of strength and symmetry of appearance,
than would be compatible with true theoretical form,
which however should be strictly adhered to in the con-
struction of the faces of the teeth. Let it be understood
that the examples now to be brought forward, have been
drawn rather with the view of exhibiting most clearly the
theoretical forms demanded by the various circumstances
which come under review, than those which would be
strictly adopted in actual practice, the modifications ad-
missible into which shall either be explained as we go
along, or will naturally suggest themselves.
Delineation of Spue, and Teundle Geaeing.
System composed of a Pinion Deiving a Rack.
Fig. 1, Plate XXI.—The pitch line M N of the rack,
and the primitive circle A B D of the pinion being laid
down touching one another, divide the latter into twice
the number of equal parts that it is to have of teeth, and
set off the common distance of these parts upon the line
M N, as many times as may be required ; this marks the
thickness of the teeth and width of the spaces in the rack.
Perpendiculars drawn through all these points to the solid
part of the rack, will represent the flanks of the teeth
upon which those of the pinion are to be developed in
succession. The curvature of these latter should, as we
have already remarked, be an involute A c of the circle
A B D. The teeth might be cut off at the point of con-
tact d upon the line M N, for at this position the tooth
A begins its action upon that of the rack E ; but it is
better to allow a little more length; in other words to
describe the circle bounding the points of the teeth with
a radius somewhat greater than C d.
With regard to the form of the spaces in the rack, all
that is required is to set off from M N, as at the point e, a
distance slightly greater than the difference A a of the
radius of the pitch circle, and that of the circle limiting
the points of the teeth, and through this point to draw a
straight line F G parallel to M N. From this line the
flanks of all the teeth of the rack spring, and their points
are terminated by a portion of a cycloid A b, which, how-
ever, may in most instances be replaced by an arc of a
circle. The depth of the spaces in the pinion obviously
depends upon the height of this curved portion of the
teeth ; their outline is formed by a circle drawn from the
centre C, with a radius a little less than the distance from
this point to the straight line, bounding the upper surface
of the teeth of the rack.
System composed of a Rack Deiving a Pinion.
Fig. 2.—In this case the construction is in all respects
identical with that of the preceding example, with this
exception, that the cycloid A b generated by a point A
in the circumference of the circle A E C, in rolling on the
line M N, is the form proper to be given to the teeth of
the rack. The curvature of the teeth of the pinion is an
involute A c as before.
System composed of a Wheel and Tangent, oe
Endless Sceew.
Fig. 3.—In the construction of this variety of gearing,
we must first fix upon the number of teeth in the wheel,
and the distance of its centre from the axis of the screw.
Then conceive a plane passing through the axis E F of the
screw, parallel to the face of the wheel, and let C be the
centre of its primitive circle. If now, a perpendicular
C G be drawn from C upon E F, and C A be taken as the
radius of the pitch circle B A D of the wheel, the differ-
ence A G will represent the radius of a cylinder, which
may be termed the 'primitive cylinder of the screw; and
a line M N drawn through A, parallel to E F, will be a
generatrix of that cylinder, which will serve the purpose
of determining the form of the teeth.
The section having been made through the axis, the
question obviously resolves itself into the case of a rack
driving a pinion ; consequently, the curve of the teeth, or
rather thread, of the screw should be simply a cycloid
generated by a point in the circle A I C, described upon
A C as a diameter, and rolling upon the straight line M N.
It is to be remarked, further, that the outlines of the teeth
are helical surfaces described about the cylinder forming
the screw, with the pitch A b, equal to the distance, mea-
sured upon the primitive circle, between the correspond-
ing points of two contiguous teeth. These curves have
been drawn on our figure, but being for the most part
concealed, they are expressed by dotted lines. The teeth
of the wheel are not, as in ordinary kinds of gearing, set
perpendicularly to the plane of its face, but at an angle,
and with surfaces corresponding to the inclination and
helical form of the thread of the screw. In some instances,
the points of the teeth and bottoms of the spaces are
formed of a concave outline adapted to the convexity of
the screw, in order to present as much bearing surface as
possible to its action. In this kind of gearing, for obvious
reasons, it is invariably the screw that imparts the motion.
Figs. 4, 5, and 6, represent a face and edge elevation,
and a vertical section, of a wheel suitable for being im-
pelled by a tangent screw or worm, as it is sometimes
called.
ENGINEER AND MACHINIST’S DRAWING-BOOK.
necessary to make in this place, as, in the ensuing illus-
trations of toothed gearing, which we have to present to
our readers, certain differences may be observed between
the forms and proportions employed, and those we have
already exhibited and announced, or may have occasion
to do in subsequent parts of this work, as governing the
practice of this branch of constructive mechanics. In cases
which we have to bring under notice in the theoretical
part of the investigation, the flanks, or those parts of the
teeth which lie within the pitch circles, are invariably
represented as portions of radii, tangents to the curves of
the faces, or parts beyond these circles, while in the prac-
tical illustrations they are replaced by circular arcs. This
difference arises from the circumstance that, as a compa-
ratively small portion only of the flanks of the teeth come
into actual operation in the communication of the motion,
the form of these parts may be modified, so as to admit of
a greater amount of strength and symmetry of appearance,
than would be compatible with true theoretical form,
which however should be strictly adhered to in the con-
struction of the faces of the teeth. Let it be understood
that the examples now to be brought forward, have been
drawn rather with the view of exhibiting most clearly the
theoretical forms demanded by the various circumstances
which come under review, than those which would be
strictly adopted in actual practice, the modifications ad-
missible into which shall either be explained as we go
along, or will naturally suggest themselves.
Delineation of Spue, and Teundle Geaeing.
System composed of a Pinion Deiving a Rack.
Fig. 1, Plate XXI.—The pitch line M N of the rack,
and the primitive circle A B D of the pinion being laid
down touching one another, divide the latter into twice
the number of equal parts that it is to have of teeth, and
set off the common distance of these parts upon the line
M N, as many times as may be required ; this marks the
thickness of the teeth and width of the spaces in the rack.
Perpendiculars drawn through all these points to the solid
part of the rack, will represent the flanks of the teeth
upon which those of the pinion are to be developed in
succession. The curvature of these latter should, as we
have already remarked, be an involute A c of the circle
A B D. The teeth might be cut off at the point of con-
tact d upon the line M N, for at this position the tooth
A begins its action upon that of the rack E ; but it is
better to allow a little more length; in other words to
describe the circle bounding the points of the teeth with
a radius somewhat greater than C d.
With regard to the form of the spaces in the rack, all
that is required is to set off from M N, as at the point e, a
distance slightly greater than the difference A a of the
radius of the pitch circle, and that of the circle limiting
the points of the teeth, and through this point to draw a
straight line F G parallel to M N. From this line the
flanks of all the teeth of the rack spring, and their points
are terminated by a portion of a cycloid A b, which, how-
ever, may in most instances be replaced by an arc of a
circle. The depth of the spaces in the pinion obviously
depends upon the height of this curved portion of the
teeth ; their outline is formed by a circle drawn from the
centre C, with a radius a little less than the distance from
this point to the straight line, bounding the upper surface
of the teeth of the rack.
System composed of a Rack Deiving a Pinion.
Fig. 2.—In this case the construction is in all respects
identical with that of the preceding example, with this
exception, that the cycloid A b generated by a point A
in the circumference of the circle A E C, in rolling on the
line M N, is the form proper to be given to the teeth of
the rack. The curvature of the teeth of the pinion is an
involute A c as before.
System composed of a Wheel and Tangent, oe
Endless Sceew.
Fig. 3.—In the construction of this variety of gearing,
we must first fix upon the number of teeth in the wheel,
and the distance of its centre from the axis of the screw.
Then conceive a plane passing through the axis E F of the
screw, parallel to the face of the wheel, and let C be the
centre of its primitive circle. If now, a perpendicular
C G be drawn from C upon E F, and C A be taken as the
radius of the pitch circle B A D of the wheel, the differ-
ence A G will represent the radius of a cylinder, which
may be termed the 'primitive cylinder of the screw; and
a line M N drawn through A, parallel to E F, will be a
generatrix of that cylinder, which will serve the purpose
of determining the form of the teeth.
The section having been made through the axis, the
question obviously resolves itself into the case of a rack
driving a pinion ; consequently, the curve of the teeth, or
rather thread, of the screw should be simply a cycloid
generated by a point in the circle A I C, described upon
A C as a diameter, and rolling upon the straight line M N.
It is to be remarked, further, that the outlines of the teeth
are helical surfaces described about the cylinder forming
the screw, with the pitch A b, equal to the distance, mea-
sured upon the primitive circle, between the correspond-
ing points of two contiguous teeth. These curves have
been drawn on our figure, but being for the most part
concealed, they are expressed by dotted lines. The teeth
of the wheel are not, as in ordinary kinds of gearing, set
perpendicularly to the plane of its face, but at an angle,
and with surfaces corresponding to the inclination and
helical form of the thread of the screw. In some instances,
the points of the teeth and bottoms of the spaces are
formed of a concave outline adapted to the convexity of
the screw, in order to present as much bearing surface as
possible to its action. In this kind of gearing, for obvious
reasons, it is invariably the screw that imparts the motion.
Figs. 4, 5, and 6, represent a face and edge elevation,
and a vertical section, of a wheel suitable for being im-
pelled by a tangent screw or worm, as it is sometimes
called.