Chapter XXI: Part II: Theory of Machines (1)

§ 20. _Parts of a Machine: Frame and Mechanism._--The parts of a
machine may be distinguished into two principal divisions,--the frame,
or fixed parts, and the _mechanism_, or moving parts. The frame is a
structure which supports the pieces of the mechanism, and to a certain
extent determines the nature of their motions.

The form and arrangement of the pieces of the frame depend upon the
arrangement and the motions of the mechanism; the dimensions of the
pieces of the frame required in order to give it stability and
strength are determined from the pressures applied to it by means of
the mechanism. It appears therefore that in general the mechanism is
to be designed first and the frame afterwards, and that the designing
of the frame is regulated by the principles of the stability of
structures and of the strength and stiffness of materials,--care being
taken to adapt the frame to the most severe load which can be thrown
upon it at any period of the action of the mechanism.

Each independent piece of the mechanism also is a structure, and its
dimensions are to be adapted, according to the principles of the
strength and stiffness of materials, to the most severe load to which
it can be subjected during the action of the machine.

§ 21. _Definition and Division of the Theory of Machines._--From what
has been said in the last section it appears that the department of
the art of designing machines which has reference to the stability of
the frame and to the stiffness and strength of the frame and mechanism
is a branch of the art of construction. It is therefore to be
separated from the _theory of machines_, properly speaking, which has
reference to the action of machines considered as moving. In the
action of a machine the following three things take place:--

_Firstly_, Some natural source of energy communicates motion and force
to a piece or pieces of the mechanism, called the _receiver of power_
or _prime mover_.

_Secondly_, The motion and force are transmitted from the prime mover
through the _train of mechanism_ to the _working piece_ or _pieces_,
and during that transmission the motion and force are modified in
amount and direction, so as to be rendered suitable for the purpose to
which they are to be applied.

_Thirdly_, The working piece or pieces by their motion, or by their
motion and force combined, produce some useful effect.

Such are the phenomena of the action of a machine, arranged in the
order of _causation_. But in studying or treating of the theory of
machines, the order of _simplicity_ is the best; and in this order the
first branch of the subject is the modification of motion and force by
the train of mechanism; the next is the effect or purpose of the
machine; and the last, or most complex, is the action of the prime
mover.

The modification of motion and the modification of force take place
together, and are connected by certain laws; but in the study of the
theory of machines, as well as in that of pure mechanics, much
advantage has been gained in point of clearness and simplicity by
first considering alone the principles of the modification of motion,
which are founded upon what is now known as Kinematics, and afterwards
considering the principles of the combined modification of motion and
force, which are founded both on geometry and on the laws of dynamics.
The separation of kinematics from dynamics is due mainly to G. Monge,
Ampère and R. Willis.

The theory of machines in the present article will be considered under
the following heads:--

I. PURE MECHANISM, or APPLIED KINEMATICS; being the theory of machines
considered simply as modifying motion.

II. APPLIED DYNAMICS; being the theory of machines considered as
modifying both motion and force.

CHAP. I. ON PURE MECHANISM

§ 22. _Division of the Subject._--Proceeding in the order of
simplicity, the subject of Pure Mechanism, or Applied Kinematics, may
be thus divided:--

_Division 1._--Motion of a point.

_Division 2._--Motion of the surface of a fluid.

_Division 3._--Motion of a rigid solid.

_Division 4._--Motions of a pair of connected pieces, or of an
"elementary combination" in mechanism.

_Division 5._--Motions of trains of pieces of mechanism.

_Division 6._--Motions of sets of more than two connected pieces, or of
"aggregate combinations."

A point is the boundary of a line, which is the boundary of a surface,
which is the boundary of a volume. Points, lines and surfaces have no
independent existence, and consequently those divisions of this
chapter which relate to their motions are only preliminary to the
subsequent divisions, which relate to the motions of bodies.

_Division 1. Motion of a Point._

§ 23. _Comparative Motion._--The comparative motion of two points is
the relation which exists between their motions, without having regard
to their absolute amounts. It consists of two elements,--the _velocity
ratio_, which is the ratio of any two magnitudes bearing to each other
the proportions of the respective velocities of the two points at a
given instant, and the _directional relation_, which is the relation
borne to each other by the respective directions of the motions of the
two points at the same given instant.

It is obvious that the motions of a pair of points may be varied in
any manner, whether by direct or by lateral deviation, and yet that
their _comparative motion_ may remain constant, in consequence of the
deviations taking place in the same proportions, in the same
directions and at the same instants for both points.

Robert Willis (1800-1875) has the merit of having been the first to
simplify considerably the theory of pure mechanism, by pointing out
that that branch of mechanics relates wholly to comparative motions.

The comparative motion of two points at a given instant is capable of
being completely expressed by one of Sir William Hamilton's
Quaternions,--the "tensor" expressing the velocity ratio, and the
"versor" the directional relation.

Graphical methods of analysis founded on this way of representing
velocity and acceleration were developed by R. H. Smith in a paper
communicated to the Royal Society of Edinburgh in 1885, and
illustrations of the method will be found below.

_Division 2. Motion of the Surface of a Fluid Mass._

§ 24. _General Principle._--A mass of fluid is used in mechanism to
transmit motion and force between two or more movable portions (called
_pistons_ or _plungers_) of the solid envelope or vessel in which the
fluid is contained; and, when such transmission is the sole action, or
the only appreciable action of the fluid mass, its volume is either
absolutely constant, by reason of its temperature and pressure being
maintained constant, or not sensibly varied.

Let a represent the area of the section of a piston made by a plane
perpendicular to its direction of motion, and v its velocity, which is
to be considered as positive when outward, and negative when inward.
Then the variation of the cubic contents of the vessel in a unit of
time by reason of the motion of one piston is va. The condition that
the volume of the fluid mass shall remain unchanged requires that
there shall be more than one piston, and that the velocities and areas
of the pistons shall be connected by the equation--

[Sigma]·va = 0. (1)

§ 25. _Comparative Motion of Two Pistons._--If there be but two
pistons, whose areas are a1 and a2, and their velocities v1 and v2,
their comparative motion is expressed by the equation--

v2/v1 = -a1/a2; (2)

that is to say, their velocities are opposite as to inwardness and
outwardness and inversely proportional to their areas.

§ 26. _Applications: Hydraulic Press: Pneumatic
Power-Transmitter._--In the hydraulic press the vessel consists of two
cylinders, viz. the pump-barrel and the press-barrel, each having its
piston, and of a passage connecting them having a valve opening
towards the press-barrel. The action of the enclosed water in
transmitting motion takes place during the inward stroke of the
pump-plunger, when the above-mentioned valve is open; and at that time
the press-plunger moves outwards with a velocity which is less than
the inward velocity of the pump-plunger, in the same ratio that the
area of the pump-plunger is less than the area of the press-plunger.
(See HYDRAULICS.)

In the pneumatic power-transmitter the motion of one piston is
transmitted to another at a distance by means of a mass of air
contained in two cylinders and an intervening tube. When the pressure
and temperature of the air can be maintained constant, this machine
fulfils equation (2), like the hydraulic press. The amount and effect
of the variations of pressure and temperature undergone by the air
depend on the principles of the mechanical action of heat, or
THERMODYNAMICS (q.v.), and are foreign to the subject of pure
mechanism.

_Division 3. Motion of a Rigid Solid._

§ 27. _Motions Classed._--In problems of mechanism, each solid piece
of the machine is supposed to be so stiff and strong as not to undergo
any sensible change of figure or dimensions by the forces applied to
it--a supposition which is realized in practice if the machine is
skilfully designed.

This being the case, the various possible motions of a rigid solid
body may all be classed under the following heads: (1) _Shifting or
Translation_; (2) _Turning or Rotation_; (3) _Motions compounded of
Shifting and Turning_.

The most common forms for the paths of the points of a piece of
mechanism, whose motion is simple shifting, are the straight line and
the circle.

Shifting in a straight line is regulated either by straight fixed
guides, in contact with which the moving piece slides, or by
combinations of link-work, called _parallel motions_, which will be
described in the sequel. Shifting in a straight line is usually
_reciprocating_; that is to say, the piece, after shifting through a
certain distance, returns to its original position by reversing its
motion.

Circular shifting is regulated by attaching two or more points of the
shifting piece to ends of equal and parallel rotating cranks, or by
combinations of wheel-work to be afterwards described. As an example
of circular shifting may be cited the motion of the coupling rod, by
which the parallel and equal cranks upon two or more axles of a
locomotive engine are connected and made to rotate simultaneously. The
coupling rod remains always parallel to itself, and all its points
describe equal and similar circles relatively to the frame of the
engine, and move in parallel directions with equal velocities at the
same instant.

§ 28. _Rotation about a Fixed Axis: Lever, Wheel and Axle._--The fixed
axis of a turning body is a line fixed relatively to the body and
relatively to the fixed space in which the body turns. In mechanism it
is usually the central line either of a rotating shaft or axle having
journals, gudgeons, or pivots turning in fixed bearings, or of a fixed
spindle or dead centre round which a rotating bush turns; but it may
sometimes be entirely beyond the limits of the turning body. For
example, if a sliding piece moves in circular fixed guides, that piece
rotates about an ideal fixed axis traversing the centre of those
guides.

Let the angular velocity of the rotation be denoted by [alpha] =
d[theta]/dt, then the linear velocity of any point A at the distance r
from the axis is [alpha]r; and the path of that point is a circle of
the radius r described about the axis.

This is the principle of the modification of motion by the lever,
which consists of a rigid body turning about a fixed axis called a
fulcrum, and having two points at the same or different distances from
that axis, and in the same or different directions, one of which
receives motion and the other transmits motion, modified in direction
and velocity according to the above law.

In the wheel and axle, motion is received and transmitted by two
cylindrical surfaces of different radii described about their common
fixed axis of turning, their velocity-ratio being that of their radii.

§ 29. _Velocity Ratio of Components of Motion._--As the distance
between any two points in a rigid body is invariable, the projections
of their velocities upon the line joining them must be equal. Hence it
follows that, if A in fig. 90 be a point in a rigid body CD, rotating
round the fixed axis F, the component of the velocity of A in any
direction AP parallel to the plane of rotation is equal to the total
velocity of the point m, found by letting fall Fm perpendicular to AP;
that is to say, is equal to

[alpha]·Fm.

Hence also the ratio of the components of the velocities of two points
A and B in the directions AP and BW respectively, both in the plane of
rotation, is equal to the ratio of the perpendiculars Fm and Fn.

§ 30. _Instantaneous Axis of a Cylinder rolling on a Cylinder._--Let a
cylinder bbb, whose axis of figure is B and angular velocity [gamma],
roll on a fixed cylinder [alpha][alpha][alpha], whose axis of figure
is A, either outside (as in fig. 91), when the rolling will be towards
the same hand as the rotation, or inside (as in fig. 92), when the
rolling will be towards the opposite hand; and at a given instant let
T be the line of contact of the two cylindrical surfaces, which is at
their common intersection with the plane AB traversing the two axes of
figure.

The line T on the surface bbb has for the instant no velocity in a
direction perpendicular to AB; because for the instant it touches,
without sliding, the line T on the fixed surface aaa.

The line T on the surface bbb has also for the instant no velocity in
the plane AB; for it has just ceased to move towards the fixed surface
aaa, and is just about to begin to move away from that surface.

The line of contact T, therefore, on the surface of the cylinder bbb,
is _for the instant_ at rest, and is the "instantaneous axis" about
which the cylinder bbb turns, together with any body rigidly attached
to that cylinder.

To find, then, the direction and velocity at the given instant of any
point P, either in or rigidly attached to the rolling cylinder T, draw
the plane PT; the direction of motion of P will be perpendicular to
that plane, and towards the right or left hand according to the
direction of the rotation of bbb; and the velocity of P will be

v_P = [gamma]·PT, (3)

PT denoting the perpendicular distance of P from T. The path of P is a
curve of the kind called _epitrochoids_. If P is in the circumference
of bbb, that path becomes an _epicycloid_.

The velocity of any point in the axis of figure B is

v_B = [gamma]·TB; (4)

and the path of such a point is a circle described about A with the
radius AB, being for outside rolling the sum, and for inside rolling
the difference, of the radii of the cylinders.

Let [alpha] denote the angular velocity with which the _plane of axes_
AB rotates about the fixed axis A. Then it is evident that

v_B = [alpha]·AB, (5)

and consequently that

[alpha] = [gamma]·TB/AB. (6)

For internal rolling, as in fig. 92, AB is to be treated as negative,
which will give a negative value to [alpha], indicating that in this
case the rotation of AB round A is contrary to that of the cylinder
bbb.

The angular velocity of the rolling cylinder, _relatively to the plane
of axes_ AB, is obviously given by the equation--

[beta] = [gamma] - [alpha] \
>, (7)
whence [beta] = [gamma]·TA/AB /

care being taken to attend to the sign of [alpha], so that when that
is negative the arithmetical values of [gamma] and [alpha] are to be
added in order to give that of [beta].

The whole of the foregoing reasonings are applicable, not merely when
aaa and bbb are actual cylinders, but also when they are the
osculating cylinders of a pair of cylindroidal surfaces of varying
curvature, A and B being the axes of curvature of the parts of those
surfaces which are in contact for the instant under consideration.

§ 31. _Instantaneous Axis of a Cone rolling on a Cone._--Let Oaa (fig.
93) be a fixed cone, OA its axis, Obb a cone rolling on it, OB the
axis of the rolling cone, OT the line of contact of the two cones at
the instant under consideration. By reasoning similar to that of § 30,
it appears that OT is the instantaneous axis of rotation of the
rolling cone.

Let [gamma] denote the total angular velocity of the rotation of the
cone B about the instantaneous axis, [beta] its angular velocity about
the axis OB _relatively_ to the plane AOB, and [alpha] the angular
velocity with which the plane AOB turns round the axis OA. It is
required to find the ratios of those angular velocities.

_Solution._--In OT take any point E, from which draw EC parallel to
OA, and ED parallel to OB, so as to construct the parallelogram OCED.
Then

OD : OC : OE :: [alpha] : [beta] : [gamma]. (8)

Or because of the proportionality of the sides of triangles to the
sines of the opposite angles,

sin TOB : sin TOA : sin AOB :: [alpha] : [beta] : [gamma], (8 A)

that is to say, the angular velocity about each axis is proportional
to the sine of the angle between the other two.

_Demonstration._--From C draw CF perpendicular to OA, and CG
perpendicular to OE

area ECO
Then CF = 2 × --------,
CE

area ECO
and CG = 2 × --------;
OE

:. CG : CF :: CE = OD : OE.

Let v_c denote the linear velocity of the point C. Then

v_c = [alpha] · CF = [gamma]·CG
:. [gamma] : [alpha] :: CF : CG :: OE : OD,

which is one part of the solution above stated. From E draw EH
perpendicular to OB, and EK to OA. Then it can be shown as before that

EK : EH :: OC : OD.

Let v_E be the linear velocity of the point E _fixed in the plane of
axes_ AOB. Then

v_K = [alpha] · EK.

Now, as the line of contact OT is for the instant at rest on the
rolling cone as well as on the fixed cone, the linear velocity of the
point E fixed to the plane AOB relatively to the rolling cone is the
same with its velocity relatively to the fixed cone. That is to say,

[beta]·EH = v_E = [alpha]·EK;

therefore

[alpha] : [beta] :: EH : EK :: OD : OC,

which is the remainder of the solution.

The path of a point P in or attached to the rolling cone is a
spherical epitrochoid traced on the surface of a sphere of the radius
OP. From P draw PQ perpendicular to the instantaneous axis. Then the
motion of P is perpendicular to the plane OPQ, and its velocity is

v_P = [gamma]·PQ. (9)

The whole of the foregoing reasonings are applicable, not merely when
A and B are actual regular cones, but also when they are the
osculating regular cones of a pair of irregular conical surfaces,
having a common apex at O.

§ 32. _Screw-like or Helical Motion._--Since any displacement in a
plane can be represented in general by a rotation, it follows that the
only combination of translation and rotation, in which a complex
movement which is not a mere rotation is produced, occurs when there
is a translation _perpendicular to the plane and parallel to the axis_
of rotation.

Such a complex motion is called _screw-like_ or _helical_ motion; for
each point in the body describes a _helix_ or _screw_ round the axis
of rotation, fixed or instantaneous as the case may be. To cause a
body to move in this manner it is usually made of a helical or
screw-like figure, and moves in a guide of a corresponding figure.
Helical motion and screws adapted to it are said to be right- or
left-handed according to the appearance presented by the rotation to
an observer looking towards the direction of the translation. Thus the
screw G in fig. 94 is right-handed.

The translation of a body in helical motion is called its _advance_.
Let v_x denote the velocity of advance at a given instant, which of
course is common to all the particles of the body; [alpha] the angular
velocity of the rotation at the same instant; 2[pi] = 6.2832 nearly,
the circumference of a circle of the radius unity. Then

T = 2[pi]/[alpha] (10)

is the time of one turn at the rate [alpha]; and

p = v_x T = 2[pi]v_x/[alpha] (11)

is the _pitch_ or _advance per turn_--a length which expresses the
_comparative motion_ of the translation and the rotation.

The pitch of a screw is the distance, measured parallel to its axis,
between two successive turns of the same _thread_ or helical
projection.

Let r denote the perpendicular distance of a point in a body moving
helically from the axis. Then

v_r = [alpha]r (12)

is the component of the velocity of that point in a plane
perpendicular to the axis, and its total velocity is

v = [root](v_x² + v_r²). (13)

The ratio of the two components of that velocity is

v_x/v_r = p/2[pi]r = tan [theta]. (14)

where [theta] denotes the angle made by the helical path of the point
with a plane perpendicular to the axis.

_Division 4. Elementary Combinations in Mechanism_

§ 33. _Definitions._--An _elementary combination_ in mechanism
consists of two pieces whose kinds of motion are determined by their
connexion with the frame, and their comparative motion by their
connexion with each other--that connexion being effected either by
direct contact of the pieces, or by a connecting piece, which is not
connected with the frame, and whose motion depends entirely on the
motions of the pieces which it connects.

The piece whose motion is the cause is called the _driver_; the piece
whose motion is the effect, the _follower_.

The connexion of each of those two pieces with the frame is in general
such as to determine the path of every point in it. In the
investigation, therefore, of the comparative motion of the driver and
follower, in an elementary combination, it is unnecessary to consider
relations of angular direction, which are already fixed by the
connexion of each piece with the frame; so that the inquiry is
confined to the determination of the velocity ratio, and of the
directional relation, so far only as it expresses the connexion
between _forward_ and _backward_ movements of the driver and follower.
When a continuous motion of the driver produces a continuous motion of
the follower, forward or backward, and a reciprocating motion a motion
reciprocating at the same instant, the directional relation is said to
be _constant_. When a continuous motion produces a reciprocating
motion, or vice versa, or when a reciprocating motion produces a
motion not reciprocating at the same instant, the directional relation
is said to be _variable_.

The _line of action_ or _of connexion_ of the driver and follower is a
line traversing a pair of points in the driver and follower
respectively, which are so connected that the component of their
velocity relatively to each other, resolved along the line of
connexion, is null. There may be several or an indefinite number of
lines of connexion, or there may be but one; and a line of connexion
may connect either the same pair of points or a succession of
different pairs.

§ 34. _General Principle._--From the definition of a line of connexion
it follows that _the components of the velocities of a pair of
connected points along their line of connexion are equal_. And from
this, and from the property of a rigid body, already stated in § 29,
it follows, that _the components along a line of connexion of all the
points traversed by that line, whether in the driver or in the
follower, are equal_; and consequently, _that the velocities of any
pair of points traversed by a line of connexion are to each other
inversely as the cosines, or directly as the secants, of the angles
made by the paths of those points with the line of connexion_.

The general principle stated above in different forms serves to solve
every problem in which--the mode of connexion of a pair of pieces
being given--it is required to find their comparative motion at a
given instant, or vice versa.

§ 35. _Application to a Pair of Shifting Pieces._--In fig. 95, let
P1P2 be the line of connexion of a pair of pieces, each of which has a
motion of translation or shifting. Through any point T in that line
draw TV1, TV2, respectively parallel to the simultaneous direction of
motion of the pieces; through any other point A in the line of
connexion draw a plane perpendicular to that line, cutting TV1, TV2 in
V1, V2; then, velocity of piece 1 : velocity of piece 2 :: TV1 : TV2.
Also TA represents the equal components of the velocities of the
pieces parallel to their line of connexion, and the line V1V2
represents their velocity relatively to each other.

§ 36. _Application to a Pair of Turning Pieces._--Let [alpha]1,
[alpha]2 be the angular velocities of a pair of turning pieces;
[theta]1, [theta]2 the angles which their line of connexion makes with
their respective planes of rotation; r1, r2 the common perpendiculars
let fall from the line of connexion upon the respective axes of
rotation of the pieces. Then the equal components, along the line of
connexion, of the velocities of the points where those perpendiculars
meet that line are--

[alpha]1r1 cos [theta]1 = [alpha]2r2 cos [theta]2;

consequently, the comparative motion of the pieces is given by the
equation

[alpha]2 r1 cos [theta]1
-------- = ---------------. (15)
[alpha]1 r2 cos [theta]2

§ 37. _Application to a Shifting Piece and a Turning Piece._--Let a
shifting piece be connected with a turning piece, and at a given
instant let [alpha]1 be the angular velocity of the turning piece, r1
the common perpendicular of its axis of rotation and the line of
connexion, [theta]1 the angle made by the line of connexion with the
plane of rotation, [theta]2 the angle made by the line of connexion
with the direction of motion of the shifting piece, v2 the linear
velocity of that piece. Then

[alpha]1r1 cos [theta]1 = v2 cos [theta]2; (16)

which equation expresses the comparative motion of the two pieces.

§ 38. _Classification of Elementary Combinations in Mechanism._--The
first systematic classification of elementary combinations in
mechanism was that founded by Monge, and fully developed by Lanz and
Bétancourt, which has been generally received, and has been adopted in
most treatises on applied mechanics. But that classification is
founded on the absolute instead of the comparative motions of the
pieces, and is, for that reason, defective, as Willis pointed out in
his admirable treatise _On the Principles of Mechanism_.

Willis's classification is founded, in the first place, on comparative
motion, as expressed by velocity ratio and directional relation, and
in the second place, on the mode of connexion of the driver and
follower. He divides the elementary combinations in mechanism into
three classes, of which the characters are as follows:--

Class A: Directional relation constant; velocity ratio constant.

Class B: Directional relation constant; velocity ratio varying.

Class C: Directional relation changing periodically; velocity ratio
constant or varying.

Each of those classes is subdivided by Willis into five divisions, of
which the characters are as follows:--

Division A: Connexion by rolling contact.
" B: " " sliding contact.
" C: " " wrapping connectors.
" D: " " link-work.
" E: " " reduplication.

In the Reuleaux system of analysis of mechanisms the principle of
comparative motion is generalized, and mechanisms apparently very
diverse in character are shown to be founded on the same sequence of
elementary combinations forming a kinematic chain. A short description
of this system is given in § 80, but in the present article the
principle of Willis's classification is followed mainly. The
arrangement is, however, modified by taking the _mode of connexion_ as
the basis of the primary classification, and by removing the subject
of connexion by reduplication to the section of aggregate
combinations. This modified arrangement is adopted as being better
suited than the original arrangement to the limits of an article in an
encyclopaedia; but it is not disputed that the original arrangement
may be the best for a separate treatise.

§ 39. _Rolling Contact: Smooth Wheels and Racks._--In order that two
pieces may move in rolling contact, it is necessary that each pair of
points in the two pieces which touch each other should at the instant
of contact be moving in the same direction with the same velocity. In
the case of two _shifting_ pieces this would involve equal and
parallel velocities for all the points of each piece, so that there
could be no rolling, and, in fact, the two pieces would move like one;
hence, in the case of rolling contact, either one or both of the
pieces must rotate.

The direction of motion of a point in a turning piece being
perpendicular to a plane passing through its axis, the condition that
each pair of points in contact with each other must move in the same
direction leads to the following consequences:--

I. That, when both pieces rotate, their axes, and all their points of
contact, lie in the same plane.

II. That, when one piece rotates, and the other shifts, the axis of
the rotating piece, and all the points of contact, lie in a plane
perpendicular to the direction of motion of the shifting piece.

The condition that the velocity of each pair of points of contact must
be equal leads to the following consequences:--

III. That the angular velocities of a pair of turning pieces in
rolling contact must be inversely as the perpendicular distances of
any pair of points of contact from the respective axes.

IV. That the linear velocity of a shifting piece in rolling contact
with a turning piece is equal to the product of the angular velocity
of the turning piece by the perpendicular distance from its axis to a
pair of points of contact.

The _line of contact_ is that line in which the points of contact are
all situated. Respecting this line, the above Principles III. and IV.
lead to the following conclusions:--

V. That for a pair of turning pieces with parallel axes, and for a
turning piece and a shifting piece, the line of contact is straight,
and parallel to the axes or axis; and hence that the rolling surfaces
are either plane or cylindrical (the term "cylindrical" including all
surfaces generated by the motion of a straight line parallel to
itself).

VI. That for a pair of turning pieces with intersecting axes the line
of contact is also straight, and traverses the point of intersection
of the axes; and hence that the rolling surfaces are conical, with a
common apex (the term "conical" including all surfaces generated by
the motion of a straight line which traverses a fixed point).

Turning pieces in rolling contact are called _smooth_ or _toothless
wheels_. Shifting pieces in rolling contact with turning pieces may be
called _smooth_ or _toothless racks_.

VII. In a pair of pieces in rolling contact every straight line
traversing the line of contact is a line of connexion.

§ 40. _Cylindrical Wheels and Smooth Racks._--In designing cylindrical
wheels and smooth racks, and determining their comparative motion, it
is sufficient to consider a section of the pair of pieces made by a
plane perpendicular to the axis or axes.

The points where axes intersect the plane of section are called
_centres_; the point where the line of contact intersects it, the
_point of contact_, or _pitch-point_; and the wheels are described as
_circular_, _elliptical_, &c., according to the forms of their
sections made by that plane.

When the point of contact of two wheels lies between their centres,
they are said to be in _outside gearing_; when beyond their centres,
in _inside gearing_, because the rolling surface of the larger wheel
must in this case be turned inward or towards its centre.

From Principle III. of § 39 it appears that the angular velocity-ratio
of a pair of wheels is the inverse ratio of the distances of the point
of contact from the centres respectively.

For outside gearing that ratio is _negative_, because the wheels turn
contrary ways; for inside gearing it is _positive_, because they turn
the same way.

If the velocity ratio is to be constant, as in Willis's Class A, the
wheels must be circular; and this is the most common form for wheels.

If the velocity ratio is to be variable, as in Willis's Class B, the
figures of the wheels are a pair of _rolling curves_, subject to the
condition that the distance between their _poles_ (which are the
centres of rotation) shall be constant.

The following is the geometrical relation which must exist between
such a pair of curves:--

Let C1, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2
any pair of points of contact; U1, U2 any other pair of points of
contact. Then, for every possible pair of points of contact, the two
following equations must be simultaneously fulfilled:--

Sum of radii, C1U1 + C2U2 = C1T1 + C2T2 = constant;
arc, T2U2 = T1U1. (17)

A condition equivalent to the above, and necessarily connected with
it, is, that at each pair of points of contact the inclinations of the
curves to their radii-vectores shall be equal and contrary; or,
denoting by r1, r2 the radii-vectores at any given pair of points of
contact, and s the length of the equal arcs measured from a certain
fixed pair of points of contact--

dr2/ds = -dr1/ds; (18)

which is the differential equation of a pair of rolling curves whose
poles are at a constant distance apart.

For full details as to rolling curves, see Willis's work, already
mentioned, and Clerk Maxwell's paper on Rolling Curves, _Trans. Roy.
Soc. Edin._, 1849.

A rack, to work with a circular wheel, must be straight. To work with
a wheel of any other figure, its section must be a rolling curve,
subject to the condition that the perpendicular distance from the pole
or centre of the wheel to a straight line parallel to the direction of
the motion of the rack shall be constant. Let r1 be the radius-vector
of a point of contact on the wheel, x2 the ordinate from the straight
line before mentioned to the corresponding point of contact on the
rack. Then

dx2/ds = -dr1/ds (19)

is the differential equation of the pair of rolling curves.

To illustrate this subject, it may be mentioned that an ellipse
rotating about one focus rolls completely round in outside gearing
with an equal and similar ellipse also rotating about one focus, the
distance between the axes of rotation being equal to the major axis of
the ellipses, and the velocity ratio varying from (1 +
eccentricity)/(1 - eccentricity) to (1 - eccentricity)/(1 +
eccentricity); an hyperbola rotating about its further focus rolls in
inside gearing, through a limited arc, with an equal and similar
hyperbola rotating about its nearer focus, the distance between the
axes of rotation being equal to the axis of the hyperbolas, and the
velocity ratio varying between (eccentricity + 1)/(eccentricity - 1)
and unity; and a parabola rotating about its focus rolls with an equal
and similar parabola, shifting parallel to its directrix.

§ 41. _Conical or Bevel and Disk Wheels._--From Principles III. and
VI. of § 39 it appears that the angular velocities of a pair of wheels
whose axes meet in a point are to each other inversely as the sines of
the angles which the axes of the wheels make with the line of contact.
Hence we have the following construction (figs. 97 and 98).--Let O be
the apex or point of intersection of the two axes OC1, OC2. The
angular velocity ratio being given, it is required to find the line of
contact. On OC1, OC2 take lengths OA1, OA2, respectively proportional
to the angular velocities of the pieces on whose axes they are taken.
Complete the parallelogram OA1EA2; the diagonal OET will be the line
of contact required.

When the velocity ratio is variable, the line of contact will shift
its position in the plane C1OC2, and the wheels will be cones, with
eccentric or irregular bases. In every case which occurs in practice,
however, the velocity ratio is constant; the line of contact is
constant in position, and the rolling surfaces of the wheels are
regular circular cones (when they are called _bevel wheels_); or one
of a pair of wheels may have a flat disk for its rolling surface, as
W2 in fig. 98, in which case it is a _disk wheel_. The rolling
surfaces of actual wheels consist of frusta or zones of the complete
cones or disks, as shown by W1, W2 in figs. 97 and 98.

§ 42. _Sliding Contact (lateral): Skew-Bevel Wheels._--An hyperboloid
of revolution is a surface resembling a sheaf or a dice box, generated
by the rotation of a straight line round an axis from which it is at a
constant distance, and to which it is inclined at a constant angle. If
two such hyperboloids E, F, equal or unequal, be placed in the closest
possible contact, as in fig. 99, they will touch each other along one
of the generating straight lines of each, which will form their line
of contact, and will be inclined to the axes AG, BH in opposite
directions. The axes will not be parallel, nor will they intersect
each other.

The motion of two such hyperboloids, turning in contact with each
other, has hitherto been classed amongst cases of rolling contact; but
that classification is not strictly correct, for, although the
component velocities of a pair of points of contact in a direction at
right angles to the line of contact are equal, still, as the axes are
parallel neither to each other nor to the line of contact, the
velocities of a pair of points of contact have components along the
line of contact which are unequal, and their difference constitutes a
_lateral sliding_.

The directions and positions of the axes being given, and the required
angular velocity ratio, the following construction serves to determine
the line of contact, by whose rotation round the two axes respectively
the hyperboloids are generated:--

In fig. 100, let B1C1, B2C2 be the two axes; B1B2 their common
perpendicular. Through any point O in this common perpendicular draw
OA1 parallel to B1C1 and OA2 parallel to B2C2; make those lines
proportional to the angular velocities about the axes to which they
are respectively parallel; complete the parallelogram OA1EA2, and draw
the diagonal OE; divide B1B2 in D into two parts, _inversely_
proportional to the angular velocities about the axes which they
respectively adjoin; through D parallel to OE draw DT. This will be
the line of contact.

A pair of thin frusta of a pair of hyperboloids are used in practice
to communicate motion between a pair of axes neither parallel nor
intersecting, and are called _skew-bevel wheels_.

In skew-bevel wheels the properties of a line of connexion are not
possessed by every line traversing the line of contact, but only by
every line traversing the line of contact at right angles.

If the velocity ratio to be communicated were variable, the point D
would alter its position, and the line DT its direction, at different
periods of the motion, and the wheels would be hyperboloids of an
eccentric or irregular cross-section; but forms of this kind are not
used in practice.

§ 43. _Sliding Contact (circular): Grooved Wheels._--As the adhesion
or friction between a pair of smooth wheels is seldom sufficient to
prevent their slipping on each other, contrivances are used to
increase their mutual hold. One of those consists in forming the rim
of each wheel into a series of alternate ridges and grooves parallel
to the plane of rotation; it is applicable to cylindrical and bevel
wheels, but not to skew-bevel wheels. The comparative motion of a pair
of wheels so ridged and grooved is the same as that of a pair of
smooth wheels in rolling contact, whose cylindrical or conical
surfaces lie midway between the tops of the ridges and bottoms of the
grooves, and those ideal smooth surfaces are called the _pitch
surfaces_ of the wheels.

The relative motion of the faces of contact of the ridges and grooves
is a _rotatory sliding_ or _grinding_ motion, about the line of
contact of the pitch-surfaces as an instantaneous axis.

Grooved wheels have hitherto been but little used.

§ 44. _Sliding Contact (direct): Teeth of Wheels, their Number and
Pitch._--The ordinary method of connecting a pair of wheels, or a
wheel and a rack, and the only method which ensures the exact
maintenance of a given numerical velocity ratio, is by means of a
series of alternate ridges and hollows parallel or nearly parallel to
the successive lines of contact of the ideal smooth wheels whose
velocity ratio would be the same with that of the toothed wheels. The
ridges are called _teeth_; the hollows, _spaces_. The teeth of the
driver push those of the follower before them, and in so doing
sliding takes place between them in a direction across their lines of
contact.

The _pitch-surfaces_ of a pair of toothed wheels are the ideal smooth
surfaces which would have the same comparative motion by rolling
contact that the actual wheels have by the sliding contact of their
teeth. The _pitch-circles_ of a pair of circular toothed wheels are
sections of their pitch-surfaces, made for _spur-wheels_ (that is, for
wheels whose axes are parallel) by a plane at right angles to the
axes, and for bevel wheels by a sphere described about the common
apex. For a pair of skew-bevel wheels the pitch-circles are a pair of
contiguous rectangular sections of the pitch-surfaces. The
_pitch-point_ is the point of contact of the pitch-circles.

The pitch-surface of a wheel lies intermediate between the points of
the teeth and the bottoms of the hollows between them. That part of
the acting surface of a tooth which projects beyond the pitch-surface
is called the _face_; that part which lies within the pitch-surface,
the _flank_.

Teeth, when not otherwise specified, are understood to be made in one
piece with the wheel, the material being generally cast-iron, brass or
bronze. Separate teeth, fixed into mortises in the rim of the wheel,
are called _cogs_. A _pinion_ is a small toothed wheel; a _trundle_ is
a pinion with cylindrical _staves_ for teeth.

The radius of the pitch-circle of a wheel is called the _geometrical
radius_; a circle touching the ends of the teeth is called the
_addendum circle_, and its radius the _real radius_; the difference
between these radii, being the projection of the teeth beyond the
pitch-surface, is called the _addendum_.

The distance, measured along the pitch-circle, from the face of one
tooth to the face of the next, is called the _pitch_. The pitch and
the number of teeth in wheels are regulated by the following
principles:--

I. In wheels which rotate continuously for one revolution or more, it
is obviously necessary _that the pitch should be an aliquot part of
the circumference_.

In wheels which reciprocate without performing a complete revolution
this condition is not necessary. Such wheels are called _sectors_.

II. In order that a pair of wheels, or a wheel and a rack, may work
correctly together, it is in all cases essential _that the pitch
should be the same in each_.

III. Hence, in any pair of circular wheels which work together, the
numbers of teeth in a complete circumference are directly as the radii
and inversely as the angular velocities.

IV. Hence also, in any pair of circular wheels which rotate
continuously for one revolution or more, the ratio of the numbers of
teeth and its reciprocal the angular velocity ratio must be
expressible in whole numbers.

From this principle arise problems of a kind which will be referred to
in treating of _Trains of Mechanism_.

V. Let n, N be the respective numbers of teeth in a pair of wheels, N
being the greater. Let t, T be a pair of teeth in the smaller and
larger wheel respectively, which at a particular instant work
together. It is required to find, first, how many pairs of teeth must
pass the line of contact of the pitch-surfaces before t and T work
together again (let this number be called a); and, secondly, with how
many different teeth of the larger wheel the tooth t will work at
different times (let this number be called b); thirdly, with how many
different teeth of the smaller wheel the tooth T will work at
different times (let this be called c).

CASE 1. If n is a divisor of N,

a = N; b = N/n; c = 1. (20)

CASE 2. If the greatest common divisor of N and n be d, a number less
than n, so that n = md, N = Md; then

a = mN = Mn = Mmd; b = M; c = m. (21)

CASE 3. If N and n be prime to each other,

a = nN; b = N; c = n. (22)

It is considered desirable by millwrights, with a view to the
preservation of the uniformity of shape of the teeth of a pair of
wheels, that each given tooth in one wheel should work with as many
different teeth in the other wheel as possible. They therefore study
that the numbers of teeth in each pair of wheels which work together
shall either be prime to each other, or shall have their greatest
common divisor as small as is consistent with a velocity ratio suited
for the purposes of the machine.

§ 45. _Sliding Contact: Forms of the Teeth of Spur-wheels and
Racks._--A line of connexion of two pieces in sliding contact is a
line perpendicular to their surfaces at a point where they touch.
Bearing this in mind, the principle of the comparative motion of a
pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel
and a rack, is found by applying the principles stated generally in §§
36 and 37 to the case of parallel axes for a pair of spur-wheels, and
to the case of an axis perpendicular to the direction of shifting for
a wheel and a rack.

In fig. 101, let C1, C2 be the centres of a pair of spur-wheels;
B1IB1´, B2IB2´ portions of their pitch-circles, touching at I, the
pitch-point. Let the wheel 1 be the driver, and the wheel 2 the
follower.

Let D1TB1A1, D2TB2A2 be the positions, at a given instant, of the
acting surfaces of a pair of teeth in the driver and follower
respectively, touching each other at T; the line of connexion of those
teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2 be
perpendiculars let fall from the centres of the wheels on the line of
contact. Then, by § 36, the angular velocity-ratio is

[alpha]2/[alpha]1 = C1P1/C2P2. (23)

The following principles regulate the forms of the teeth and their
relative motions:--

I. The angular velocity ratio due to the sliding contact of the teeth
will be the same with that due to the rolling contact of the
pitch-circles, if the line of connexion of the teeth cuts the line of
centres at the pitch-point.

For, let P1P2 cut the line of centres at I; then, by similar
triangles,

[alpha]1 : [alpha]2 :: C2P2 : C1P1 :: IC2 :: IC1; (24)

which is also the angular velocity ratio due to the rolling contact of
the circles B1IB1´, B2IB2´.

This principle determines the _forms_ of all teeth of spur-wheels. It
also determines the forms of the teeth of straight racks, if one of
the centres be removed, and a straight line EIE´, parallel to the
direction of motion of the rack, and perpendicular to C1IC2, be
substituted for a pitch-circle.

II. The component of the velocity of the point of contact of the teeth
T along the line of connexion is

[alpha]1·C1P1 = [alpha]2·C2P2. (25)

III. The relative velocity perpendicular to P1P2 of the teeth at their
point of contact--that is, their _velocity of sliding_ on each
other--is found by supposing one of the wheels, such as 1, to be
fixed, the line of centres C1C2 to rotate backwards round C1 with the
angular velocity [alpha]1, and the wheel 2 to rotate round C2 as
before, with the angular velocity [alpha]2 relatively to the line of
centres C1C2, so as to have the same motion as if its pitch-circle
_rolled_ on the pitch-circle of the first wheel. Thus the _relative_
motion of the wheels is unchanged; but 1 is considered as fixed, and 2
has the total motion, that is, a rotation about the instantaneous axis
I, with the angular velocity [alpha]1 + [alpha]2. Hence the _velocity
of sliding_ is that due to this rotation about I, with the radius IT;
that is to say, its value is

([alpha]1 + [alpha]2)·IT; (26)

so that it is greater the farther the point of contact is from the
line of centres; and at the instant when that point passes the line of
centres, and coincides with the _pitch-point_, the velocity of sliding
is null, and the action of the teeth is, for the instant, that of
rolling contact.

IV. The _path of contact_ is the line traversing the various positions
of the point T. If the line of connexion preserves always the same
position, the path of contact coincides with it, and is straight; in
other cases the path of contact is curved.

It is divided by the pitch-point I into two parts--the _arc_ or _line
of approach_ described by T in approaching the line of centres, and
the _arc_ or _line of recess_ described by T after having passed the
line of centres.

During the _approach_, the _flank_ D1B1 of the driving tooth drives
the face D2B2 of the following tooth, and the teeth are sliding
_towards_ each other. During the _recess_ (in which the position of
the teeth is exemplified in the figure by curves marked with accented
letters), the _face_ B1´A1´ of the driving tooth drives the _flank_
B2´A2´ of the following tooth, and the teeth are sliding _from_ each
other.

The path of contact is bounded where the approach commences by the
addendum-circle of the follower, and where the recess terminates by
the addendum-circle of the driver. The length of the path of contact
should be such that there shall always be at least one pair of teeth
in contact; and it is better still to make it so long that there shall
always be at least two pairs of teeth in contact.

V. The _obliquity_ of the action of the teeth is the angle EIT = IC1,
P1 = IC2P2.

In practice it is found desirable that the mean value of the obliquity
of action during the contact of teeth should not exceed 15°, nor the
maximum value 30°.

It is unnecessary to give separate figures and demonstrations for
inside gearing. The only modification required in the formulae is,
that in equation (26) the _difference_ of the angular velocities
should be substituted for their sum.

§ 46. _Involute Teeth._--The simplest form of tooth which fulfils the
conditions of § 45 is obtained in the following manner (see fig. 102).
Let C1, C2 be the centres of two wheels, B1IB1´, B2IB2´ their
pitch-circles, I the pitch-point; let the obliquity of action of the
teeth be constant, so that the same straight line P1IP2 shall
represent at once the constant line of connexion of teeth and the path
of contact. Draw C1P1, C2P2 perpendicular to P1IP2, and with those
lines as radii describe about the centres of the wheels the circles
D1D1´, D2D2´, called _base-circles_. It is evident that the radii of
the base-circles bear to each other the same proportions as the radii
of the pitch-circles, and also that

C1P1 = IC1 · cos obliquity \ (27)
C2P2 = IC2 · cos obliquity /

(The obliquity which is found to answer best in practice is about
14½°; its cosine is about 31/22, and its sine about ¼. These values
though not absolutely exact, are near enough to the truth for
practical purposes.)

Suppose the base-circles to be a pair of circular pulleys connected by
means of a cord whose course from pulley to pulley is P1IP2. As the
line of connexion of those pulleys is the same as that of the proposed
teeth, they will rotate with the required velocity ratio. Now, suppose
a tracing point T to be fixed to the cord, so as to be carried along
the path of contact P1IP2, that point will trace on a plane rotating
along with the wheel 1 part of the involute of the base-circle D1D1´,
and on a plane rotating along with the wheel 2 part of the involute of
the base-circle D2D2´; and the two curves so traced will always touch
each other in the required point of contact T, and will therefore
fulfil the condition required by Principle I. of § 45.