_Pitch Surfaces of Spur Wheels._--Let two smooth rollers be placed in
contact with their axes parallel, and let one of them rotate about its
axis; then if there is no slipping the other roller will rotate in the
opposite direction with the same surface velocity; and if D_{1}, D_{2}
be the diameters of the rollers, and N_{1}, N_{2} their speeds in
revolutions per minute, it follows as in belt gearing that--
N_{1} D_{2}
----- = -----.
N_{2} D_{1}
If there be considerable resistance to the motion of the follower
slipping may take place, and it may stop. To prevent this the rollers
may be provided with teeth; then they become _spur wheels_; and if the
teeth be so shaped that the ratio of the speeds of the toothed rollers
at any instant is the same as that of the smooth rollers, the surfaces
of the latter are called the _pitch surfaces_ of the former.
_Pitch Circle._--A section of the pitch surface of a toothed wheel by a
plane perpendicular to its axis is a circle, and is called a _pitch
circle_. We may also say that the pitch circle is the edge of the pitch
surface. The pitch circle is generally traced on the side of a toothed
wheel, and is rather nearer the points of the teeth than the roots.
_Pitch of Teeth._--The distance from the centre of one tooth to the
centre of the next, or from the front of one to the front of the next,
_measured at the pitch circle_, is called the _pitch of the teeth_. If D
be the diameter of the pitch circle of a wheel, _n_ the number of teeth,
and _p_ the pitch of the teeth, then D x 3.1416 = _n_ x _p_.
[Illustration: FIG. 34.]
By the diameter of a wheel is meant the diameter of its pitch circle.
_Form and Proportions of Teeth._--The ordinary form of wheel teeth is
shown in fig. 34. The curves of the teeth should be cycloidal curves,
although they are generally drawn in as arcs of circles. It does not
fall within the scope of this work to discuss the correct forms of wheel
teeth. The student will find the theory of the teeth of wheels clearly
and fully explained in Goodeve's 'Elements of Mechanism,' and in Unwin's
'Machine Design.'
The following proportions for the teeth of ordinary toothed wheels may
be taken as representing average practice:--
Pitch of teeth = _p_ = arc _a b c_ (fig. 34).
Thickness of tooth = _b c_ = .48_p_.
Width of space = _a b_ = .52_p_.
Total height of tooth = _h_ = .7_p_.
Height of tooth above pitch line = _k_ = .3_p_.
Depth of tooth below pitch line = _l_ = .4_p_.
Width of tooth = 2_p_ to 3_p_.
EXERCISE 34: _Spur Wheel._--Fig. 35 shows the elevation and
sectional plan of a portion of a cast-iron spur wheel. The diameter
of the pitch circle is 23-7/8 inches, and the pitch of the teeth is
1-1/2 inches, so that there will be 50 teeth in the wheel. The
wheel has six arms. Draw a complete elevation of the wheel and a
half sectional plan, also a half-plan without any section. Draw
also a cross section of one arm. Scale 4 inches to a foot.
[Illustration: FIG. 35.]
_Mortise Wheels._--When two wheels gearing together run at a high speed
the teeth of one are made of wood. These teeth, or cogs, as they are
generally called, have tenons formed on them, which fit into mortises in
the rim of the wheel. This wheel with the wooden teeth is called a
_mortise wheel_. An example of a mortise wheel is shown in fig. 36.
[Illustration: FIG. 36.]
_Bevil Wheels._--In bevil wheels the pitch surfaces are parts of cones.
Bevil wheels are used to connect shafts which are inclined to one
another, whereas spur wheels are used to connect parallel shafts. In
fig. 36 is shown a pair of bevil wheels in gear, one of them being a
mortise wheel. At (_a_) is a separate drawing, to a smaller scale, of
the pitch cones. The pitch cones are shown on the drawing of the
complete wheels by dotted lines.
The diameters of bevil wheels are the diameters of the bases of their
pitch cones.
EXERCISE 35: _Pair of Bevil Wheels._--Draw the sectional elevation
of the bevil wheels shown in gear in fig. 36. Commence by drawing
the centre lines of the shafts, which in this example are at right
angles to one another; then draw the pitch cones shown by dotted
lines. Next put in the teeth which come into the plane of the
section, then complete the sections of the wheels. The pinion or
smaller wheel has 25 teeth, and the wheel has 50 teeth, which makes
the pitch a little over 3 inches. Each tooth of the mortise wheel
is secured as shown by an iron pin 5/16 inch diameter. Scale 3
inches to a foot.