Chapter XXII: Part II: Theory of Machines (2)

Consequently, one of the forms suitable for the teeth of wheels is the
involute of a circle; and the obliquity of the action of such teeth is
the angle whose cosine is the ratio of the radius of their base-circle
to that of the pitch-circle of the wheel.

All involute teeth of the same pitch work smoothly together.

To find the length of the path of contact on either side of the
pitch-point I, it is to be observed that the distance between the
fronts of two successive teeth, as measured along P1IP2, is less than
the pitch in the ratio of cos obliquity : I; and consequently that, if
distances equal to the pitch be marked off either way from I towards
P1 and P2 respectively, as the extremities of the path of contact, and
if, according to Principle IV. of § 45, the addendum-circles be
described through the points so found, there will always be at least
two pairs of teeth in action at once. In practice it is usual to make
the path of contact somewhat longer, viz. about 2.4 times the pitch;
and with this length of path, and the obliquity already mentioned of
14½°, the addendum is about 3.1 of the pitch.

The teeth of a _rack_, to work correctly with wheels having involute
teeth, should have plane surfaces perpendicular to the line of
connexion, and consequently making with the direction of motion of the
rack angles equal to the complement of the obliquity of action.

§ 47. _Teeth for a given Path of Contact: Sang's Method._--In the
preceding section the form of the teeth is found by assuming a figure
for the path of contact, viz. the straight line. Any other convenient
figure may be assumed for the path of contact, and the corresponding
forms of the teeth found by determining what curves a point T, moving
along the assumed path of contact, will trace on two disks rotating
round the centres of the wheels with angular velocities bearing that
relation to the component velocity of T along TI, which is given by
Principle II. of § 45, and by equation (25). This method of finding
the forms of the teeth of wheels forms the subject of an elaborate and
most interesting treatise by Edward Sang.

All wheels having teeth of the same pitch, traced from the same path
of contact, work correctly together, and are said to belong to the
same set.

§ 48. _Teeth traced by Rolling Curves._--If any curve R (fig. 103) be
rolled on the inside of the pitch-circle BB of a wheel, it appears,
from § 30, that the instantaneous axis of the rolling curve at any
instant will be at the point I, where it touches the pitch-circle for
the moment, and that consequently the line AT, traced by a
tracing-point T, fixed to the rolling curve upon the plane of the
wheel, will be everywhere perpendicular to the straight line TI; so
that the traced curve AT will be suitable for the flank of a tooth, in
which T is the point of contact corresponding to the position I of the
pitch-point. If the same rolling curve R, with the same tracing-point
T, be rolled on the _outside_ of any other pitch-circle, it will have
the _face_ of a tooth suitable to work with the _flank_ AT.

In like manner, if either the same or any other rolling curve R´ be
rolled the opposite way, on the _outside_ of the pitch-circle BB, so
that the tracing point T´ shall start from A, it will trace the face
AT´ of a tooth suitable to work with a _flank_ traced by rolling the
same curve R´ with the same tracing-point T´ _inside_ any other
pitch-circle.

The figure of the _path of contact_ is that traced on a fixed plane by
the tracing-point, when the rolling curve is rotated in such a manner
as always to touch a fixed straight line EIE (or E´I´E´, as the case
may be) at a fixed point I (or I´).

If the same rolling curve and tracing-point be used to trace both the
faces and the flanks of the teeth of a number of wheels of different
sizes but of the same pitch, all those wheels will work correctly
together, and will form a _set_. The teeth of a _rack_, of the same
set, are traced by rolling the rolling curve on both sides of a
straight line.

The teeth of wheels of any figure, as well as of circular wheels, may
be traced by rolling curves on their pitch-surfaces; and all teeth of
the same pitch, traced by the same rolling curve with the same
tracing-point, will work together correctly if their pitch-surfaces
are in rolling contact.

§ 49. _Epicycloidal Teeth._--The most convenient rolling curve is the
circle. The path of contact which it traces is identical with itself;
and the flanks of the teeth are internal and their faces external
epicycloids for wheels, and both flanks and faces are cycloids for a
rack.

For a pitch-circle of twice the radius of the rolling or _describing_
circle (as it is called) the internal epicycloid is a straight line,
being, in fact, a diameter of the pitch-circle, so that the flanks of
the teeth for such a pitch-circle are planes radiating from the axis.
For a smaller pitch-circle the flanks would be convex and _in-curved_
or _under-cut_, which would be inconvenient; therefore the smallest
wheel of a set should have its pitch-circle of twice the radius of the
describing circle, so that the flanks may be either straight or
concave.

In fig. 104 let BB´ be part of the pitch-circle of a wheel with
epicycloidal teeth; CIC´ the line of centres; I the pitch-point; EIE´
a straight tangent to the pitch-circle at that point; R the internal
and R´ the equal external describing circles, so placed as to touch
the pitch-circle and each other at I. Let DID´ be the path of contact,
consisting of the arc of approach DI and the arc of recess ID´. In
order that there may always be at least two pairs of teeth in action,
each of those arcs should be equal to the pitch.

The obliquity of the action in passing the line of centres is nothing;
the maximum obliquity is the angle EID = E´ID; and the mean obliquity
is one-half of that angle.

It appears from experience that the mean obliquity should not exceed
15°; therefore the maximum obliquity should be about 30°; therefore
the equal arcs DI and ID´ should each be one-sixth of a circumference;
therefore the circumference of the describing circle should be _six
times the pitch_.

It follows that the smallest pinion of a set in which pinion the
flanks are straight should have twelve teeth.

§ 50. _Nearly Epicycloidal Teeth: Willis's Method._--To facilitate the
drawing of epicycloidal teeth in practice, Willis showed how to
approximate to their figure by means of two circular arcs--one
concave, for the flank, and the other convex, for the face--and each
having for its radius the _mean_ radius of curvature of the
epicycloidal arc. Willis's formulae are founded on the following
properties of epicycloids:--

Let R be the radius of the pitch-circle; r that of the describing
circle; [theta] the angle made by the normal TI to the epicycloid at a
given point T, with a tangent to the circle at I--that is, the
obliquity of the action at T.

Then the radius of curvature of the epicycloid at T is--

R - r \
For an internal epicycloid, [rho] = 4r sin [theta]------ |
R - 2r |
> (28)
R + r |
For an external epicycloid, [rho]´ = 4r sin [theta]------ |
R + 2r /

Also, to find the position of the centres of curvature relatively to
the pitch-circle, we have, denoting the chord of the describing circle
TI by c, c = 2r sin [theta]; and therefore

R \
For the flank, [rho] - c = 2r sin [theta]------ |
R - 2r |
> (29)
R |
For the face, [rho]´ - c = 2r sin [theta]------ |
R + 2r /

For the proportions approved of by Willis, sin [theta] = ¼ nearly; r =
p (the pitch) nearly; c = ½p nearly; and, if N be the number of teeth
in the wheel, r/R = 6/N nearly; therefore, approximately,

[rho] - c = p/2 · N/N - 12 \ (30)
[rho]´ - c = p/2 · N/N + 12 /

Hence the following construction (fig. 105). Let BB be part of the
pitch-circle, and a the point where a tooth is to cross it. Set off ab
= ac - ½p. Draw radii bd, ce; draw fb, cg, making angles of 75½° with
those radii. Make bf = p´ - c, cg = p - c. From f, with the radius fa,
draw the circular arc ah; from g, with the radius ga, draw the
circular arc ak. Then ah is the face and ak the flank of the tooth
required.

To facilitate the application of this rule, Willis published tables of
[rho] - c and [rho]´ - c, and invented an instrument called the
"odontograph."

§ 51. _Trundles and Pin-Wheels._--If a wheel or trundle have
cylindrical pins or staves for teeth, the faces of the teeth of a
wheel suitable for driving it are described by first tracing external
epicycloids, by rolling the pitch-circle of the pin-wheel or trundle
on the pitch-circle of the driving-wheel, with the centre of a stave
for a tracing-point, and then drawing curves parallel to, and within
the epicycloids, at a distance from them equal to the radius of a
stave. Trundles having only six staves will work with large wheels.

§ 52. _Backs of Teeth and Spaces._--Toothed wheels being in general
intended to rotate either way, the _backs_ of the teeth are made
similar to the fronts. The _space_ between two teeth, measured on the
pitch-circle, is made about (1/6)th part wider than the thickness of
the tooth on the pitch-circle--that is to say,

Thickness of tooth = 5/11 pitch;
Width of space = 6/11 pitch.

The difference of 1/11 of the pitch is called the _back-lash_. The
clearance allowed between the points of teeth and the bottoms of the
spaces between the teeth of the other wheel is about one-tenth of the
pitch.

§ 53. _Stepped and Helical Teeth._--R. J. Hooke invented the making of
the fronts of teeth in a series of steps with a view to increase the
smoothness of action. A wheel thus formed resembles in shape a series
of equal and similar toothed disks placed side by side, with the teeth
of each a little behind those of the preceding disk. He also invented,
with the same object, teeth whose fronts, instead of being parallel to
the line of contact of the pitch-circles, cross it obliquely, so as to
be of a screw-like or helical form. In wheel-work of this kind the
contact of each pair of teeth commences at the foremost end of the
helical front, and terminates at the aftermost end; and the helix is
of such a pitch that the contact of one pair of teeth shall not
terminate until that of the next pair has commenced.

Stepped and helical teeth have the desired effect of increasing the
smoothness of motion, but they require more difficult and expensive
workmanship than common teeth; and helical teeth are, besides, open to
the objection that they exert a laterally oblique pressure, which
tends to increase resistance, and unduly strain the machinery.

§ 54. _Teeth of Bevel-Wheels._--The acting surfaces of the teeth of
bevel-wheels are of the conical kind, generated by the motion of a
line passing through the common apex of the pitch-cones, while its
extremity is carried round the outlines of the cross section of the
teeth made by a sphere described about that apex.

The operations of describing the exact figures of the teeth of
bevel-wheels, whether by involutes or by rolling curves, are in every
respect analogous to those for describing the figures of the teeth of
spur-wheels, except that in the case of bevel-wheels all those
operations are to be performed on the surface of a sphere described
about the apex instead of on a plane, substituting _poles_ for
_centres_, and _great circles_ for _straight lines_.

In consideration of the practical difficulty, especially in the case
of large wheels, of obtaining an accurate spherical surface, and of
drawing upon it when obtained, the following approximate method,
proposed originally by Tredgold, is generally used:--

Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I,
OB2I their pitch cones; OC1, OC2 their axes; OI their line of contact.
Perpendicular to OI draw A1IA2, cutting the axes in A1, A2; make the
outer rims of the patterns and of the wheels portions of the cones
A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be
sufficiently near to a spherical surface described about O for
practical purposes. To find the figures of the teeth, draw on a flat
surface circular arcs ID1, ID2, with the radii A1I, A2I; those arcs
will be the _developments_ of arcs of the pitch-circles B1I, B2I, when
the conical surfaces A1B1I, A2B2I are spread out flat. Describe the
figures of teeth for the developed arcs as for a pair of spur-wheels;
then wrap the developed arcs on the cones, so as to make them coincide
with the pitch-circles, and trace the teeth on the conical surfaces.

§ 55. _Teeth of Skew-Bevel Wheels._--The crests of the teeth of a
skew-bevel wheel are parallel to the generating straight line of the
hyperboloidal pitch-surface; and the transverse sections of the teeth
at a given pitch-circle are similar to those of the teeth of a
bevel-wheel whose pitch surface is a cone touching the hyperboloidal
surface at the given circle.

§ 56. _Cams._--A _cam_ is a single tooth, either rotating continuously
or oscillating, and driving a sliding or turning piece either
constantly or at intervals. All the principles which have been stated
in § 45 as being applicable to teeth are applicable to cams; but in
designing cams it is not usual to determine or take into consideration
the form of the ideal pitch-surface, which would give the same
comparative motion by rolling contact that the cam gives by sliding
contact.

§ 57. _Screws._--The figure of a screw is that of a convex or concave
cylinder, with one or more helical projections, called _threads_,
winding round it. Convex and concave screws are distinguished
technically by the respective names of _male_ and _female_; a short
concave screw is called a _nut_; and when a _screw_ is spoken of
without qualification a _convex_ screw is usually understood.

The relation between the _advance_ and the _rotation_, which compose
the motion of a screw working in contact with a fixed screw or helical
guide, has already been demonstrated in § 32; and the same relation
exists between the magnitudes of the rotation of a screw about a fixed
axis and the advance of a shifting nut in which it rotates. The
advance of the nut takes place in the opposite direction to that of
the advance of the screw in the case in which the nut is fixed. The
_pitch_ or _axial pitch_ of a screw has the meaning assigned to it in
that section, viz. the distance, measured parallel to the axis,
between the corresponding points in two successive turns of the _same
thread_. If, therefore, the screw has several equidistant threads, the
true pitch is equal to the _divided axial pitch_, as measured between
two adjacent threads, multiplied by the number of threads.

If a helix be described round the screw, crossing each turn of the
thread at right angles, the distance between two corresponding points
on two successive turns of the same thread, measured along this
_normal helix_, may be called the _normal pitch_; and when the screw
has more than one thread the normal pitch from thread to thread may be
called the _normal divided pitch_.

The distance from thread to thread, measured on a circle described
about the axis of the screw, called the pitch-circle, may be called
the _circumferential pitch_; for a screw of one thread it is one
circumference; for a screw of n threads, (one circumference)/n.

Let r denote the radius of the pitch circle;
n the number of threads;
[theta] the obliquity of the threads to the pitch circle, and of the
normal helix to the axis;

P_a \ / pitch
P_a > the axial <
--- = p_a | |
n / \ divided pitch;

P_n \ / pitch
P_n > the normal <
--- = p_n | |
n / \ divided pitch;

P_c the circumferential pitch;

then

2[pi]r \
p_c = p_a cot [theta] = p_n cos [theta] = ------, |
n |
|
2[pi]r tan [theta] |
p_a = p_n sec [theta] = p_c tan [theta] = ------------------, > (31)
n |
|
2[pi]r sin [theta] |
p_n = p_c sin [theta] = p_a cos [theta] = ------------------, |
n /

If a screw rotates, the number of threads which pass a fixed point in
one revolution is the number of threads in the screw.

A pair of convex screws, each rotating about its axis, are used as an
elementary combination to transmit motion by the sliding contact of
their threads. Such screws are commonly called _endless screws_. At
the point of contact of the screws their threads must be parallel; and
their line of connexion is the common perpendicular to the acting
surfaces of the threads at their point of contact. Hence the following
principles:--

I. If the screws are both right-handed or both left-handed, the angle
between the directions of their axes is the sum of their obliquities;
if one is right-handed and the other left-handed, that angle is the
difference of their obliquities.

II. The normal pitch for a screw of one thread, and the normal divided
pitch for a screw of more than one thread, must be the same in each
screw.

III. The angular velocities of the screws are inversely as their
numbers of threads.

Hooke's wheels with oblique or helical teeth are in fact screws of
many threads, and of large diameters as compared with their lengths.

The ordinary position of a pair of endless screws is with their axes
at right angles to each other. When one is of considerably greater
diameter than the other, the larger is commonly called in practice a
_wheel_, the name _screw_ being applied to the smaller only; but they
are nevertheless both screws in fact.

To make the teeth of a pair of endless screws fit correctly and work
smoothly, a hardened steel screw is made of the figure of the smaller
screw, with its thread or threads notched so as to form a cutting
tool; the larger screw, or "wheel," is cast approximately of the
required figure; the larger screw and the steel screw are fitted up in
their proper relative position, and made to rotate in contact with
each other by turning the steel screw, which cuts the threads of the
larger screw to their true figure.

§ 58. _Coupling of Parallel Axes--Oldham's Coupling._--A _coupling_ is
a mode of connecting a pair of shafts so that they shall rotate in the
same direction with the same mean angular velocity. If the axes of the
shafts are in the same straight line, the coupling consists in so
connecting their contiguous ends that they shall rotate as one piece;
but if the axes are not in the same straight line combinations of
mechanism are required. A coupling for parallel shafts which acts by
_sliding contact_ was invented by Oldham, and is represented in fig.
107. C1, C2 are the axes of the two parallel shafts; D1, D2 two disks
facing each other, fixed on the ends of the two shafts respectively;
E1E1 a bar sliding in a diametral groove in the face of D1; E2E2 a bar
sliding in a diametral groove in the face of D2: those bars are fixed
together at A, so as to form a rigid cross. The angular velocities of
the two disks and of the cross are all equal at every instant; the
middle point of the cross, at A, revolves in the dotted circle
described upon the line of centres C1C2 as a diameter twice for each
turn of the disks and cross; the instantaneous axis of rotation of the
cross at any instant is at I, the point in the circle C1C2
diametrically opposite to A.

Oldham's coupling may be used with advantage where the axes of the
shafts are intended to be as nearly in the same straight line as is
possible, but where there is some doubt as to the practibility or
permanency of their exact continuity.

§ 59. _Wrapping Connectors--Belts, Cords and Chains._--Flat belts of
leather or of gutta percha, round cords of catgut, hemp or other
material, and metal chains are used as wrapping connectors to transmit
rotatory motion between pairs of pulleys and drums.

_Belts_ (the most frequently used of all wrapping connectors) require
nearly cylindrical pulleys. A belt tends to move towards that part of
a pulley whose radius is greatest; pulleys for belts, therefore, are
slightly swelled in the middle, in order that the belt may remain on
the pulley, unless forcibly shifted. A belt when in motion is shifted
off a pulley, or from one pulley on to another of equal size alongside
of it, by pressing against that part of the belt which is moving
_towards_ the pulley.

_Cords_ require either cylindrical drums with ledges or grooved
pulleys.

_Chains_ require pulleys or drums, grooved, notched and toothed, so as
to fit the links of the chain.

Wrapping connectors for communicating continuous motion are endless.

Wrapping connectors for communicating reciprocating motion have
usually their ends made fast to the pulleys or drums which they
connect, and which in this case may be sectors.

The line of connexion of two pieces connected by a wrapping connector
is the centre line of the belt, cord or chain; and the comparative
motions of the pieces are determined by the principles of § 36 if both
pieces turn, and of § 37 if one turns and the other shifts, in which
latter case the motion must be reciprocating.

The _pitch-line_ of a pulley or drum is a curve to which the line of
connexion is always a tangent--that is to say, it is a curve parallel
to the acting surface of the pulley or drum, and distant from it by
half the thickness of the wrapping connector.

Pulleys and drums for communicating a constant velocity ratio are
circular. The _effective radius_, or radius of the pitch-circle of a
circular pulley or drum, is equal to the real radius added to half the
thickness of the connector. The angular velocities of a pair of
connected circular pulleys or drums are inversely as the effective
radii.

A _crossed_ belt, as in fig. 108, A, reverses the direction of the
rotation communicated; an _uncrossed_ belt, as in fig. 108, B,
preserves that direction.

The _length_ L of an endless belt connecting a pair of pulleys whose
effective radii are r1, r2, with parallel axes whose distance apart is
c, is given by the following formulae, in each of which the first
term, containing the radical, expresses the length of the straight
parts of the belt, and the remainder of the formula the length of the
curved parts.

For a crossed belt:--

/ r1 + r2 \
L = 2[root][c² - (r1 + r2)²] + (r1 + r2)( [pi] - 2 sin^-1 ------- ); (32 A)
\ c /
and for an uncrossed belt:--

r1 - r2
L = 2[root][c² - (r1 - r2)²] + [pi](r1 + r2 + 2(r1 - r2) sin^-1 -------; (32 B)
c
in which r1 is the greater radius, and r2 the less.

When the axes of a pair of pulleys are not parallel, the pulleys
should be so placed that the part of the belt which is _approaching_
each pulley shall be in the plane of the pulley.

§ 60. _Speed-Cones._--A pair of speed-cones (fig. 109) is a
contrivance for varying and adjusting the velocity ratio communicated
between a pair of parallel shafts by means of a belt. The speed-cones
are either continuous cones or conoids, as A, B, whose velocity ratio
can be varied gradually while they are in motion by shifting the belt,
or sets of pulleys whose radii vary by steps, as C, D, in which case
the velocity ratio can be changed by shifting the belt from one pair
of pulleys to another.

In order that the belt may fit accurately in every possible position
on a pair of speed-cones, the quantity L must be constant, in
equations (32 A) or (32 B), according as the belt is crossed or
uncrossed.

For a _crossed_ belt, as in A and C, fig. 109, L depends solely on c
and on r1 + r2. Now c is constant because the axes are parallel;
therefore the _sum of the radii_ of the pitch-circles connected in
every position of the belt is to be constant. That condition is
fulfilled by a pair of continuous cones generated by the revolution of
two straight lines inclined opposite ways to their respective axes at
equal angles.

For an uncrossed belt, the quantity L in equation (32 B) is to be made
constant. The exact fulfilment of this condition requires the solution
of a transcendental equation; but it may be fulfilled with accuracy
sufficient for practical purposes by using, instead of (32 B) the
following _approximate_ equation:--

L nearly = 2c + [pi](r1 + r2) + (r1 - r2)²/c. (33)

The following is the most convenient practical rule for the
application of this equation:--

Let the speed-cones be equal and similar conoids, as in B, fig. 109,
but with their large and small ends turned opposite ways. Let r1 be
the radius of the large end of each, r2 that of the small end, r0 that
of the middle; and let v be the _sagitta_, measured perpendicular to
the axes, of the arc by whose revolution each of the conoids is
generated, or, in other words, the _bulging_ of the conoids in the
middle of their length. Then

v = r0 - (r1 + r2)/2 = (r1 - r2)²/2[pi]c. (34)

2[pi] = 6.2832; but 6 may be used in most practical cases without
sensible error.

The radii at the middle and end being thus determined, make the
generating curve an arc either of a circle or of a parabola.

§ 61. _Linkwork in General._--The pieces which are connected by
linkwork, if they rotate or oscillate, are usually called _cranks_,
_beams_ and levers. The _link_ by which they are connected is a rigid
rod or bar, which may be straight or of any other figure; the straight
figure being the most favourable to strength, is always used when
there is no special reason to the contrary. The link is known by
various names in various circumstances, such as _coupling-rod_,
_connecting-rod_, _crank-rod_, _eccentric-rod_, &c. It is attached to
the pieces which it connects by two pins, about which it is free to
turn. The effect of the link is to maintain the distance between the
axes of those pins invariable; hence the common perpendicular of the
axes of the pins is _the line of connexion_, and its extremities may
be called the _connected points_. In a turning piece, the
perpendicular let fall from its connected point upon its axis of
rotation is the _arm_ or _crank-arm_.

The axes of rotation of a pair of turning pieces connected by a link
are almost always parallel, and perpendicular to the line of connexion
in which case the angular velocity ratio at any instant is the
reciprocal of the ratio of the common perpendiculars let fall from the
line of connexion upon the respective axes of rotation.

If at any instant the direction of one of the crank-arms coincides
with the line of connexion, the common perpendicular of the line of
connexion and the axis of that crank-arm vanishes, and the directional
relation of the motions becomes indeterminate. The position of the
connected point of the crank-arm in question at such an instant is
called a _dead-point_. The velocity of the other connected point at
such an instant is null, unless it also reaches a dead-point at the
same instant, so that the line of connexion is in the plane of the two
axes of rotation, in which case the velocity ratio is indeterminate.
Examples of dead-points, and of the means of preventing the
inconvenience which they tend to occasion, will appear in the sequel.

§ 62. _Coupling of Parallel Axes._--Two or more parallel shafts (such
as those of a locomotive engine, with two or more pairs of driving
wheels) are made to rotate with constantly equal angular velocities by
having equal cranks, which are maintained parallel by a coupling-rod
of such a length that the line of connexion is equal to the distance
between the axes. The cranks pass their dead-points simultaneously. To
obviate the unsteadiness of motion which this tends to cause, the
shafts are provided with a second set of cranks at right angles to the
first, connected by means of a similar coupling-rod, so that one set
of cranks pass their dead points at the instant when the other set are
farthest from theirs.

§ 63. _Comparative Motion of Connected Points._--As the link is a
rigid body, it is obvious that its action in communicating motion may
be determined by finding the comparative motion of the connected
points, and this is often the most convenient method of proceeding.

If a connected point belongs to a turning piece, the direction of its
motion at a given instant is perpendicular to the plane containing the
axis and crank-arm of the piece. If a connected point belongs to a
shifting piece, the direction of its motion at any instant is given,
and a plane can be drawn perpendicular to that direction.

The line of intersection of the planes perpendicular to the paths of
the two connected points at a given instant is the _instantaneous axis
of the link_ at that instant; and the _velocities of the connected
points are directly as their distances from that axis_.

In drawing on a plane surface, the two planes perpendicular to the
paths of the connected points are represented by two lines (being
their sections by a plane normal to them), and the instantaneous axis
by a point (fig. 110); and, should the length of the two lines render
it impracticable to produce them until they actually intersect, the
velocity ratio of the connected points may be found by the principle
that it is equal to the ratio of the segments which a line parallel to
the line of connexion cuts off from any two lines drawn from a given
point, perpendicular respectively to the paths of the connected
points.

To illustrate this by one example. Let C1 be the axis, and T1 the
connected point of the beam of a steam-engine; T1T2 the connecting or
crank-rod; T2 the other connected point, and the centre of the
crank-pin; C2 the axis of the crank and its shaft. Let v1 denote the
velocity of T1 at any given instant; v2 that of T2. To find the ratio
of these velocities, produce C1T1, C2T2 till they intersect in K; K is
the instantaneous axis of the connecting rod, and the velocity ratio
is

v1 : v2 :: KT1 : KT2. (35)

Should K be inconveniently far off, draw any triangle with its sides
respectively parallel to C1T1, C2T2 and T1T2; the ratio of the two
sides first mentioned will be the velocity ratio required. For
example, draw C2A parallel to C1T1, cutting T1T2 in A; then

v1 : v2 :: C2A : C2T2. (36)

§ 64. _Eccentric._--An eccentric circular disk fixed on a shaft, and
used to give a reciprocating motion to a rod, is in effect a crank-pin
of sufficiently large diameter to surround the shaft, and so to avoid
the weakening of the shaft which would arise from bending it so as to
form an ordinary crank. The centre of the eccentric is its connected
point; and its eccentricity, or the distance from that centre to the
axis of the shaft, is its crank-arm.

An eccentric may be made capable of having its eccentricity altered by
means of an adjusting screw, so as to vary the extent of the
reciprocating motion which it communicates.

§ 65. _Reciprocating Pieces--Stroke--Dead-Points._--The distance
between the extremities of the path of the connected point in a
reciprocating piece (such as the piston of a steam-engine) is called
the _stroke_ or _length of stroke_ of that piece. When it is connected
with a continuously turning piece (such as the crank of a
steam-engine) the ends of the stroke of the reciprocating piece
correspond to the _dead-points_ of the path of the connected point of
the turning piece, where the line of connexion is continuous with or
coincides with the crank-arm.

Let S be the length of stroke of the reciprocating piece, L the length
of the line of connexion, and R the crank-arm of the continuously
turning piece. Then, if the two ends of the stroke be in one straight
line with the axis of the crank,

S = 2R; (37)

and if these ends be not in one straight line with that axis, then S,
L - R, and L + R, are the three sides of a triangle, having the angle
opposite S at that axis; so that, if [theta] be the supplement of the
arc between the dead-points,

S² = 2(L² + R²) - 2(L² - R²) cos [theta], \
|
2L² + 2R² - S² > (38)
cos [theta] = -------------- |
2(L² - R²) /

§ 66. _Coupling of Intersecting Axes--Hooke's Universal
Joint._--Intersecting axes are coupled by a contrivance of Hooke's,
known as the "universal joint," which belongs to the class of linkwork
(see fig. 111). Let O be the point of intersection of the axes OC1,
OC2, and [theta] their angle of inclination to each other. The pair of
shafts C1, C2 terminate in a pair of forks F1, F2 in bearings at the
extremities of which turn the gudgeons at the ends of the arms of a
rectangular cross, having its centre at O. This cross is the link; the
connected points are the centres of the bearings F1, F2. At each
instant each of those points moves at right angles to the central
plane of its shaft and fork, therefore the line of intersection of the
central planes of the two forks at any instant is the instantaneous
axis of the cross, and the _velocity ratio_ of the points F1, F2
(which, as the forks are equal, is also the _angular velocity ratio_
of the shafts) is equal to the ratio of the distances of those points
from that instantaneous axis. The _mean_ value of that velocity ratio
is that of equality, for each successive _quarter-turn_ is made by
both shafts in the same time; but its actual value fluctuates between
the limits:--

[alpha]2 1 \
-------- = ----------- when F1 is the plane of OC1C2 |
[alpha]1 cos [theta] |
> (39)
[alpha]2 |
and -------- = cos [theta] when F2 is in that plane. |
[alpha]1 /

Its value at intermediate instants is given by the following
equations: let [phi]1, [phi]2 be the angles respectively made by the
central planes of the forks and shafts with the plane OC1C2 at a given
instant; then

cos [theta] = tan [phi]1 tan [phi]2, \
|
[alpha]2 d[phi]2 tan [phi]1 + cot [phi]1 > (40)
--------- = - ------- = -----------------------. |
[alpha]1 d[phi]1 tan [phi]2 + cot [phi]2 /

§ 67. _Intermittent Linkwork--Click and Ratchet._--A click acting upon
a ratchet-wheel or rack, which it pushes or pulls through a certain
arc at each forward stroke and leaves at rest at each backward stroke,
is an example of intermittent linkwork. During the forward stroke the
action of the click is governed by the principles of linkwork; during
the backward stroke that action ceases. A _catch_ or _pall_, turning
on a fixed axis, prevents the ratchet-wheel or rack from reversing its
motion.

_Division 5.--Trains of Mechanism._

§ 68. _General Principles.--A train of mechanism_ consists of a series
of pieces each of which is follower to that which drives it and driver
to that which follows it.

The comparative motion of the first driver and last follower is
obtained by combining the proportions expressing by their terms the
velocity ratios and by their signs the directional relations of the
several elementary combinations of which the train consists.

§ 69. _Trains of Wheelwork._--Let A1, A2, A3, &c., A_(m-1), A_m denote
a series of axes, and [alpha]1, [alpha]2, [alpha]3, &c.,
[alpha]_(m-1), [alpha]_m their angular velocities. Let the axis A1
carry a wheel of N1 teeth, driving a wheel of n2 teeth on the axis A2,
which carries also a wheel of N2 teeth, driving a wheel of n3 teeth on
the axis A3, and so on; the numbers of teeth in drivers being denoted
by N´s, and in followers by n's, and the axes to which the wheels are
fixed being denoted by numbers. Then the resulting velocity ratio is
denoted by

[alpha]_m [alpha]2 [alpha]3 [alpha]_m N1 · N2 ... &c. ... N_(m-1)
--------- = -------- · -------- · &c. ... ------------- = ---------------------------; (41)
[alpha]1 [alpha]1 [alpha]2 [alpha]_(m-1) n2 · n3 ... &c. ... n_m

that is to say, the velocity ratio of the last and first axes is the
ratio of the product of the numbers of teeth in the drivers to the
product of the numbers of teeth in the followers.

Supposing all the wheels to be in outside gearing, then, as each
elementary combination reverses the direction of rotation, and as the
number of elementary combinations m - 1 is one less than the number
of axes m, it is evident that if m is odd the direction of rotation is
preserved, and if even reversed.

It is often a question of importance to determine the number of teeth
in a train of wheels best suited for giving a determinate velocity
ratio to two axes. It was shown by Young that, to do this with the
_least total number of teeth_, the velocity ratio of each elementary
combination should approximate as nearly as possible to 3.59. This
would in many cases give too many axes; and, as a useful practical
rule, it may be laid down that from 3 to 6 ought to be the limit of
the velocity ratio of an elementary combination in wheel-work. The
smallest number of teeth in a pinion for epicycloidal teeth ought to
be _twelve_ (see § 49)--but it is better, for smoothness of motion,
not to go below _fifteen_; and for involute teeth the smallest number
is about _twenty-four_.

Let B/C be the velocity ratio required, reduced to its least terms,
and let B be greater than C. If B/C is not greater than 6, and C lies
between the prescribed minimum number of teeth (which may be called t)
and its double 2t, then one pair of wheels will answer the purpose,
and B and C will themselves be the numbers required. Should B and C be
inconveniently large, they are, if possible, to be resolved into
factors, and those factors (or if they are too small, multiples of
them) used for the number of teeth. Should B or C, or both, be at once
inconveniently large and prime, then, instead of the exact ratio B/C
some ratio approximating to that ratio, and capable of resolution into
convenient factors, is to be found by the method of continued
fractions.

Should B/C be greater than 6, the best number of elementary
combinations m - 1 will lie between

(log B - log C) log B - log C
--------------- and -------------.
log 6 log 3

Then, if possible, B and C themselves are to be resolved each into m -
1 factors (counting 1 as a factor), which factors, or multiples of
them, shall be not less than t nor greater than 6t; or if B and C
contain inconveniently large prime factors, an approximate velocity
ratio, found by the method of continued fractions, is to be
substituted for B/C as before.

So far as the resultant velocity ratio is concerned, the _order_ of
the drivers N and of the followers n is immaterial: but to secure
equable wear of the teeth, as explained in § 44, the wheels ought to
be so arranged that, for each elementary combination, the greatest
common divisor of N and n shall be either 1, or as small as possible.

§ 70. _Double Hooke's Coupling._--It has been shown in § 66 that the
velocity ratio of a pair of shafts coupled by a universal joint
fluctuates between the limits cos [theta] and 1/cos [theta]. Hence one
or both of the shafts must have a vibratory and unsteady motion,
injurious to the mechanism and framework. To obviate this evil a short
intermediate shaft is introduced, making equal angles with the first
and last shaft, coupled with each of them by a Hooke's joint, and
having its own two forks in the same plane. Let [alpha]1, [alpha]2,
[alpha]3 be the angular velocities of the first, intermediate, and
last shaft in this _train of two Hooke's couplings_. Then, from the
principles of § 60 it is evident that at each instant
[alpha]2/[alpha]1 = [alpha]2/[alpha]3, and consequently that [alpha]3
= [alpha]1; so that the fluctuations of angular velocity ratio caused
by the first coupling are exactly neutralized by the second, and the
first and last shafts have equal angular velocities at each instant.

§ 71. _Converging and Diverging Trains of Mechanism._--Two or more
trains of mechanism may converge into one--as when the two pistons of
a pair of steam-engines, each through its own connecting-rod, act upon
one crank-shaft. One train of mechanism may _diverge_ into two or
more--as when a single shaft, driven by a prime mover, carries several
pulleys, each of which drives a different machine. The principles of
comparative motion in such converging and diverging trains are the
same as in simple trains.

_Division 6.--Aggregate Combinations._

§ 72. _General Principles._--Willis designated as "aggregate
combinations" those assemblages of pieces of mechanism in which the
motion of one follower is the _resultant_ of component motions
impressed on it by more than one driver. Two classes of aggregate
combinations may be distinguished which, though not different in their
actual nature, differ in the _data_ which they present to the
designer, and in the method of solution to be followed in questions
respecting them.

Class I. comprises those cases in which a piece A is not carried
directly by the frame C, but by another piece B, _relatively_ to which
the motion of A is given--the motion of the piece B relatively to the
frame C being also given. Then the motion of A relatively to the frame
C is the _resultant_ of the motion of A relatively to B and of B
relatively to C; and that resultant is to be found by the principles
already explained in Division 3 of this Chapter §§ 27-32.

Class II. comprises those cases in which the motions of three points
in one follower are determined by their connexions with two or with
three different drivers.

This classification is founded on the kinds of problems arising from
the combinations. Willis adopts another classification founded on the
_objects_ of the combinations, which objects he divides into two
classes, viz. (1) to produce _aggregate velocity_, or a velocity which
is the resultant of two or more components in the same path, and (2)
to produce _an aggregate path_--that is, to make a given point in a
rigid body move in an assigned path by communicating certain motions
to other points in that body.

It is seldom that one of these effects is produced without at the same
time producing the other; but the classification of Willis depends
upon which of those two effects, even supposing them to occur
together, is the practical object of the mechanism.

§ 73. _Differential Windlass._--The axis C (fig. 112) carries a larger
barrel AE and a smaller barrel DB, rotating as one piece with the
angular velocity [alpha]1 in the direction AE. The pulley or _sheave_
FG has a weight W hung to its centre. A cord has one end made fast to
and wrapped round the barrel AE; it passes from A under the sheave FG,
and has the other end wrapped round and made fast to the barrel BD.
Required the relation between the velocity of translation v2 of W and
the angular velocity [alpha]1 of the _differential barrel_.

In this case v2 is an _aggregate velocity_, produced by the joint
action of the two drivers AE and BD, transmitted by wrapping
connectors to FG, and combined by that sheave so as to act on the
follower W, whose motion is the same with that of the centre of FG.

The velocity of the point F is [alpha]1·AC, _upward_ motion being
considered positive. The velocity of the point G is -[alpha]1·CB,
_downward_ motion being negative. Hence the instantaneous axis of the
sheave FG is in the diameter FG, at the distance

FG AC - BC
--- · -------
2 AC + BC

from the centre towards G; the angular velocity of the sheave is

AC + BC
[alpha]2 = [alpha]1 · -------;
FG

and, consequently, the velocity of its centre is

FG AC - BC [alpha]1(AC - BC)
v2 = [alpha]2 · --- · ------- = -----------------, (42)
2 AC + BC 2

or the _mean between the velocities of the two vertical parts of the
cord_.

If the cord be fixed to the framework at the point B, instead of being
wound on a barrel, the velocity of W is half that of AF.

A case containing several sheaves is called a _block_. A _fall-block_
is attached to a fixed point; a _running-block_ is movable to and from
a fall-block, with which it is connected by two or more plies of a
rope. The whole combination constitutes a _tackle_ or _purchase_. (See
PULLEYS for practical applications of these principles.)

§ 74. _Differential Screw._--On the same axis let there be two screws
of the respective pitches p1 and p2, made in one piece, and rotating
with the angular velocity [alpha]. Let this piece be called B. Let the
first screw turn in a fixed nut C, and the second in a sliding nut A.
The velocity of advance of B relatively to C is (according to § 32)
[alpha]p1, and of A relatively to B (according to § 57) -[alpha]p2;
hence the velocity of A relatively to C is

[alpha](p1 - p2), (46)

being the same with the velocity of advance of a screw of the pitch p1
- p2. This combination, called _Hunter's_ or the _differential screw_,
combines the strength of a large thread with the slowness of motion
due to a small one.

§ 75. _Epicyclic Trains._--The term _epicyclic train_ is used by
Willis to denote a train of wheels carried by an arm, and having
certain rotations relatively to that arm, which itself rotates. The
arm may either be driven by the wheels or assist in driving them. The
comparative motions of the wheels and of the arm, and the _aggregate
paths_ traced by points in the wheels, are determined by the
principles of the composition of rotations, and of the description of
rolling curves, explained in §§ 30, 31.

§ 76. _Link Motion._--A slide valve operated by a link motion receives
an aggregate motion from the mechanism driving it. (See STEAM-ENGINE
for a description of this and other types of mechanism of this class.)

§ 77. _Parallel Motions._--A _parallel motion_ is a combination of
turning pieces in mechanism designed to guide the motion of a
reciprocating piece either exactly or approximately in a straight
line, so as to avoid the friction which arises from the use of
straight guides for that purpose.

Fig. 113 represents an exact parallel motion, first proposed, it is
believed, by Scott Russell. The arm CD turns on the axis C, and is
jointed at D to the middle of the bar ADB, whose length is double of
that of CD, and one of whose ends B is jointed to a slider, sliding in
straight guides along the line CB. Draw BE perpendicular to CB,
cutting CD produced in E, then E is the instantaneous axis of the bar
ADB; and the direction of motion of A is at every instant
perpendicular to EA--that is, along the straight line ACa. While the
stroke of A is ACa, extending to equal distances on either side of C,
and equal to twice the chord of the arc Dd, the stroke of B is only
equal to twice the sagitta; and thus A is guided through a
comparatively long stroke by the sliding of B through a comparatively
short stroke, and by rotatory motions at the joints C, D, B.

§ 78.* An example of an approximate straight-line motion composed of
three bars fixed to a frame is shown in fig. 114. It is due to P. L.
Tchebichev of St Petersburg. The links AB and CD are equal in length
and are centred respectively at A and C. The ends D and B are joined
by a link DB. If the respective lengths are made in the proportions AC
: CD : DB = 1 : 1.3 : 0.4 the middle point P of DB will describe an
approximately straight line parallel to AC within limits of length
about equal to AC. C. N. Peaucellier, a French engineer officer, was
the first, in 1864, to invent a linkwork with which an exact straight
line could be drawn. The linkwork is shown in fig. 115, from which it
will be seen that it consists of a rhombus of four equal bars ABCD,
jointed at opposite corners with two equal bars BE and DE. The seventh
link AF is equal in length to halt the distance EA when the mechanism
is in its central position. The points E and F are fixed. It can be
proved that the point C always moves in a straight line at right
angles to the line EF. The more general property of the mechanism
corresponding to proportions between the lengths FA and EF other than
that of equality is that the curve described by the point C is the
inverse of the curve described by A. There are other arrangements of
bars giving straight-line motions, and these arrangements together
with the general properties of mechanisms of this kind are discussed
in _How to Draw a Straight Line_ by A. B. Kempe (London, 1877).

§ 79.* _The Pantograph._--If a parallelogram of links (fig. 116), be
fixed at any one point a in any one of the links produced in either
direction, and if any straight line be drawn from this point to cut
the links in the points b and c, then the points a, b, c will be in a
straight line for all positions of the mechanism, and if the point b
be guided in any curve whatever, the point c will trace a similar
curve to a scale enlarged in the ratio ab : ac. This property of the
parallelogram is utilized in the construction of the pantograph, an
instrument used for obtaining a copy of a map or drawing on a
different scale. Professor J. J. Sylvester discovered that this
property of the parallelogram is not confined to points lying in one
line with the fixed point. Thus if b (fig. 117) be any point on the
link CD, and if a point c be taken on the link DE such that the
triangles CbD and DcE are similar and similarly situated with regard
to their respective links, then the ratio of the distances ab and ac
is constant, and the angle bac is constant for all positions of the
mechanism; so that, if b is guided in any curve, the point c will
describe a similar curve turned through an angle bac, the scales of
the curves being in the ratio ab to ac. Sylvester called an instrument
based on this property a plagiograph or a skew pantograph.

The combination of the parallelogram with a straight-line motion, for
guiding one of the points in a straight line, is illustrated in Watt's
parallel motion for steam-engines. (See STEAM-ENGINE.)

§ 80.* _The Reuleaux System of Analysis._--If two pieces, A and B,
(fig. 118) are jointed together by a pin, the pin being fixed, say, to
A, the only relative motion possible between the pieces is one of
turning about the axis of the pin. Whatever motion the pair of pieces
may have as a whole each separate piece shares in common, and this
common motion in no way affects the relative motion of A and B. The
motion of one piece is said to be completely constrained relatively to
the other piece. Again, the pieces A and B (fig. 119) are paired
together as a slide, and the only relative motion possible between
them now is that of sliding, and therefore the motion of one
relatively to the other is completely constrained. The pieces may be
paired together as a screw and nut, in which case the relative motion
is compounded of turning with sliding.