0.5

1 cm

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.