§ 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.
[Illustration: FIG. 90.]
§ 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.
[Illustration: FIG. 91.]
[Illustration: FIG. 92.]
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.
[Illustration: FIG. 93.]
§ 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.
[Illustration: FIG. 94.]
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.
[Illustration: FIG. 95.]
§ 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.
[Illustration: FIG. 96.]
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.
[Illustration: FIG. 97.]
§ 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.
[Illustration: FIG. 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.
[Illustration: FIG. 99.]
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:--
[Illustration: FIG. 100.]
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.
[Illustration: FIG. 101.]
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.)
[Illustration: FIG. 102.]
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.
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.
[Illustration: FIG. 103.]
§ 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.
[Illustration: FIG. 104.]
§ 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 /
[Illustration: FIG. 105.]
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.
[Illustration: FIG. 106.]
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.
[Illustration: FIG. 107.]
§ 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.
[Illustration: FIG. 108.]
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.
[Illustration: FIG. 109.]
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_.
[Illustration: FIG. 110.]
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) /
[Illustration: FIG. 111.]
§ 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.
[Illustration: FIG. 112.]
§ 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.)
[Illustration: FIG. 113.]
§ 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.
[Illustration: FIG. 114.]
[Illustration: FIG. 115.]
§ 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).
[Illustration: FIG. 116.]
[Illustration: FIG. 117.]
§ 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.
[Illustration: FIG. 118.]
[Illustration: FIG. 119.]
These combinations of pieces are known individually as _kinematic
pairs of elements_, or briefly _kinematic pairs_. The three pairs
mentioned above have each the peculiarity that contact between the two
pieces forming the pair is distributed over a surface. Kinematic pairs
which have surface contact are classified as _lower pairs_. Kinematic
pairs in which contact takes place along a line only are classified as
_higher pairs_. A pair of spur wheels in gear is an example of a
higher pair, because the wheels have contact between their teeth along
lines only.
A _kinematic link_ of the simplest form is made by joining up the
halves of two kinematic pairs by means of a rigid link. Thus if A1B1
represent a turning pair, and A2B2 a second turning pair, the rigid
link formed by joining B1 to B2 is a kinematic link. Four links of
this kind are shown in fig. 120 joined up to form a _closed kinematic
chain_.
[Illustration: FIG. 120.]
In order that a kinematic chain may be made the basis of a mechanism,
every point in any link of it must be completely constrained with
regard to every other link. Thus in fig. 120 the motion of a point a
in the link A1A2 is completely constrained with regard to the link
B1B4 by the turning pair A1B1, and it can be proved that the motion of
a relatively to the non-adjacent link A3A4 is completely constrained,
and therefore the four-bar chain, as it is called, can be and is used
as the basis of many mechanisms. Another way of considering the
question of constraint is to imagine any one link of the chain fixed;
then, however the chain be moved, the path of a point, as a, will
always remain the same. In a five-bar chain, if a is a point in a link
non-adjacent to a fixed link, its path is indeterminate. Still another
way of stating the matter is to say that, if any one link in the chain
be fixed, any point in the chain must have only one degree of freedom.
In a five-bar chain a point, as a, in a link non-adjacent to the fixed
link has two degrees of freedom and the chain cannot therefore be used
for a mechanism. These principles may be applied to examine any
possible combination of links forming a kinematic chain in order to
test its suitability for use as a mechanism. Compound chains are
formed by the superposition of two or more simple chains, and in these
more complex chains links will be found carrying three, or even more,
halves of kinematic pairs. The Joy valve gear mechanism is a good
example of a compound kinematic chain.
[Illustration: FIG. 121.]
A chain built up of three turning pairs and one sliding pair, and
known as the _slider crank chain_, is shown in fig. 121. It will be
seen that the piece A1 can only slide relatively to the piece B1, and
these two pieces therefore form the sliding pair. The piece A1 carries
the pin B4, which is one half of the turning pair A4 B4. The piece A1
together with the pin B4 therefore form a kinematic link A1B4. The
other links of the chain are, B1A2, B2B3, A3A4. In order to convert a
chain into a mechanism it is necessary to fix one link in it. Any one
of the links may be fixed. It follows therefore that there are as many
possible mechanisms as there are links in the chain. For example,
there is a well-known mechanism corresponding to the fixing of three
of the four links of the slider crank chain (fig. 121). If the link d
is fixed the chain at once becomes the mechanism of the ordinary steam
engine; if the link e is fixed the mechanism obtained is that of the
oscillating cylinder steam engine; if the link c is fixed the
mechanism becomes either the Whitworth quick-return motion or the
slot-bar motion, depending upon the proportion between the lengths of
the links c and e. These different mechanisms are called _inversions_
of the slider crank chain. What was the fixed framework of the
mechanism in one case becomes a moving link in an inversion.
The Reuleaux system, therefore, consists essentially of the analysis
of every mechanism into a kinematic chain, and since each link of the
chain may be the fixed frame of a mechanism quite diverse mechanisms
are found to be merely inversions of the same kinematic chain. Franz
Reuleaux's _Kinematics of Machinery_, translated by Sir A. B. W.
Kennedy (London, 1876), is the book in which the system is set forth
in all its completeness. In _Mechanics of Machinery_, by Sir A. B. W.
Kennedy (London, 1886), the system was used for the first time in an
English textbook, and now it has found its way into most modern
textbooks relating to the subject of mechanism.
§ 81.* _Centrodes, Instantaneous Centres, Velocity Image, Velocity
Diagram._--Problems concerning the relative motion of the several
parts of a kinematic chain may be considered in two ways, in addition
to the way hitherto used in this article and based on the principle of
§ 34. The first is by the method of instantaneous centres, already
exemplified in § 63, and rolling centroids, developed by Reuleaux in
connexion with his method of analysis. The second is by means of
Professor R. H. Smith's method already referred to in § 23.
_Method 1._--By reference to § 30 it will be seen that the motion of a
cylinder rolling on a fixed cylinder is one of rotation about an
instantaneous axis T, and that the velocity both as regards direction
and magnitude is the same as if the rolling piece B were for the
instant turning about a fixed axis coincident with the instantaneous
axis. If the rolling cylinder B and its path A now be assumed to
receive a common plane motion, what was before the velocity of the
point P becomes the velocity of P relatively to the cylinder A, since
the motion of B relatively to A still takes place about the
instantaneous axis T. If B stops rolling, then the two cylinders
continue to move as though they were parts of a rigid body. Notice
that the shape of either rolling curve (fig. 91 or 92) may be found by
considering each fixed in turn and then tracing out the locus of the
instantaneous axis. These rolling cylinders are sometimes called
axodes, and a section of an axode in a plane parallel to the plane of
motion is called a centrode. The axode is hence the locus of the
instantaneous axis, whilst the centrode is the locus of the
instantaneous centre in any plane parallel to the plane of motion.
There is no restriction on the shape of these rolling axodes; they may
have any shape consistent with rolling (that is, no slipping is
permitted), and the relative velocity of a point P is still found by
considering it with regard to the instantaneous centre.
Reuleaux has shown that the relative motion of any pair of
non-adjacent links of a kinematic chain is determined by the rolling
together of two ideal cylindrical surfaces (cylindrical being used
here in the general sense), each of which may be assumed to be formed
by the extension of the material of the link to which it corresponds.
These surfaces have contact at the instantaneous axis, which is now
called the instantaneous axis of the two links concerned. To find the
form of these surfaces corresponding to a particular pair of
non-adjacent links, consider each link of the pair fixed in turn, then
the locus of the instantaneous axis is the axode corresponding to the
fixed link, or, considering a plane of motion only, the locus of the
instantaneous centre is the centrode corresponding to the fixed link.
To find the instantaneous centre for a particular link corresponding
to any given configuration of the kinematic chain, it is only
necessary to know the direction of motion of any two points in the
link, since lines through these points respectively at right angles to
their directions of motion intersect in the instantaneous centre.
[Illustration: FIG. 122.]
To illustrate this principle, consider the four-bar chain shown in
fig. 122 made up of the four links, a, b, c, d. Let a be the fixed
link, and consider the link c. Its extremities are moving respectively
in directions at right angles to the links b and d; hence produce the
links b and d to meet in the point O_(ac). This point is the
instantaneous centre of the motion of the link c relatively to the
fixed link a, a fact indicated by the suffix ac placed after the
letter O. The process being repeated for different values of the angle
[theta] the curve through the several points Oac is the centroid which
may be imagined as formed by an extension of the material of the link
a. To find the corresponding centroid for the link c, fix c and repeat
the process. Again, imagine d fixed, then the instantaneous centre
O_(bd) of b with regard to d is found by producing the links c and a
to intersect in O_(bd), and the shapes of the centroids belonging
respectively to the links b and d can be found as before. The axis
about which a pair of adjacent links turn is a permanent axis, and is
of course the axis of the pin which forms the point. Adding the
centres corresponding to these several axes to the figure, it will be
seen that there are six centres in connexion with the four-bar chain
of which four are permanent and two are instantaneous or virtual
centres; and, further, that whatever be the configuration of the chain
these centres group themselves into three sets of three, each set
lying on a straight line. This peculiarity is not an accident or a
special property of the four-bar chain, but is an illustration of a
general law regarding the subject discovered by Aronhold and Sir A. B.
W. Kennedy independently, which may be thus stated: If any three
bodies, a, b, c, have plane motion their three virtual centres,
O_(ab), O_(bc), O_(ac), are three points on one straight line. A proof
of this will be found in _The Mechanics of Machinery_ quoted above.
Having obtained the set of instantaneous centres for a chain, suppose
a is the fixed link of the chain and c any other link; then O_(ac) is
the instantaneous centre of the two links and may be considered for
the instant as the trace of an axis fixed to an extension of the link
a about which c is turning, and thus problems of instantaneous
velocity concerning the link c are solved as though the link c were
merely rotating for the instant about a fixed axis coincident with the
instantaneous axis.
[Illustration: FIG. 123.]
[Illustration: FIG. 124.]
_Method 2._--The second method is based upon the vector representation
of velocity, and may be illustrated by applying it to the four-bar
chain. Let AD (fig. 123) be the fixed link. Consider the link BC, and
let it be required to find the velocity of the point B having given
the velocity of the point C. The principle upon which the solution is
based is that the only motion which B can have relatively to an axis
through C fixed to the link CD is one of turning about C. Choose any
pole O (fig. 124). From this pole set out Oc to represent the velocity
of the point C. The direction of this must be at right angles to the
line CD, because this is the only direction possible to the point C.
If the link BC moves without turning, Oc will also represent the
velocity of the point B; but, if the link is turning, B can only move
about the axis C, and its direction of motion is therefore at right
angles to the line CB. Hence set out the possible direction of B´s
motion in the velocity diagram, namely cb1, at right angles to CB. But
the point B must also move at right angles to AB in the case under
consideration. Hence draw a line through O in the velocity diagram at
right angles to AB to cut cb1 in b. Then Ob is the velocity of the
point b in magnitude and direction, and cb is the tangential velocity
of B relatively to C. Moreover, whatever be the actual magnitudes of
the velocities, the instantaneous velocity ratio of the points C and B
is given by the ratio Oc/Ob.
A most important property of the diagram (figs. 123 and 124) is the
following: If points X and x are taken dividing the link BC and the
tangential velocity cb, so that cx:xb = CX:XB, then Ox represents the
velocity of the point X in magnitude and direction. The line cb has
been called the _velocity image_ of the rod, since it may be looked
upon as a scale drawing of the rod turned through 90° from the actual
rod. Or, put in another way, if the link CB is drawn to scale on the
new length cb in the velocity diagram (fig. 124), then a vector drawn
from O to any point on the new drawing of the rod will represent the
velocity of that point of the actual rod in magnitude and direction.
It will be understood that there is a new velocity diagram for every
new configuration of the mechanism, and that in each new diagram the
image of the rod will be different in scale. Following the method
indicated above for a kinematic chain in general, there will be
obtained a velocity diagram similar to that of fig. 124 for each
configuration of the mechanism, a diagram in which the velocity of the
several points in the chain utilized for drawing the diagram will
appear to the same scale, all radiating from the pole O. The lines
joining the ends of these several velocities are the several
tangential velocities, each being the velocity image of a link in the
chain. These several images are not to the same scale, so that
although the images may be considered to form collectively an image of
the chain itself, the several members of this chain-image are to
different scales in any one velocity diagram, and thus the chain-image
is distorted from the actual proportions of the mechanism which it
represents.
[Illustration: FIG. 125.]
§ 82.* _Acceleration Diagram. Acceleration Image._--Although it is
possible to obtain the acceleration of points in a kinematic chain
with one link fixed by methods which utilize the instantaneous centres
of the chain, the vector method more readily lends itself to this
purpose. It should be understood that the instantaneous centre
considered in the preceding paragraphs is available only for
estimating relative velocities; it cannot be used in a similar manner
for questions regarding acceleration. That is to say, although the
instantaneous centre is a centre of no velocity for the instant, it is
not a centre of no acceleration, and in fact the centre of no
acceleration is in general a quite different point. The general
principle on which the method of drawing an acceleration diagram
depends is that if a link CB (fig. 125) have plane motion and the
acceleration of any point C be given in magnitude and direction, the
acceleration of any other point B is the vector sum of the
acceleration of C, the radial acceleration of B about C and the
tangential acceleration of B about C. Let A be any origin, and let Ac
represent the acceleration of the point C, ct the radial acceleration
of B about C which must be in a direction parallel to BC, and tb the
tangential acceleration of B about C, which must of course be at right
angles to ct; then the vector sum of these three magnitudes is Ab, and
this vector represents the acceleration of the point B. The directions
of the radial and tangential accelerations of the point B are always
known when the position of the link is assigned, since these are to be
drawn respectively parallel to and at right angles to the link itself.
The magnitude of the radial acceleration is given by the expression
v²/BC, v being the velocity of the point B about the point C. This
velocity can always be found from the velocity diagram of the chain of
which the link forms a part. If dw/dt is the angular acceleration of
the link, dw/dt × CB is the tangential acceleration of the point B
about the point C. Generally this tangential acceleration is unknown
in magnitude, and it becomes part of the problem to find it. An
important property of the diagram is that if points X and x are taken
dividing the link CB and the whole acceleration of B about C, namely,
cb in the same ratio, then Ax represents the acceleration of the point
X in magnitude and direction; cb is called the acceleration image of
the rod. In applying this principle to the drawing of an acceleration
diagram for a mechanism, the velocity diagram of the mechanism must be
first drawn in order to afford the means of calculating the several
radial accelerations of the links. Then assuming that the acceleration
of one point of a particular link of the mechanism is known together
with the corresponding configuration of the mechanism, the two vectors
Ac and ct can be drawn. The direction of tb, the third vector in the
diagram, is also known, so that the problem is reduced to the
condition that b is somewhere on the line tb. Then other conditions
consequent upon the fact that the link forms part of a kinematic chain
operate to enable b to be fixed. These methods are set forth and
exemplified in _Graphics_, by R. H. Smith (London, 1889). Examples,
completely worked out, of velocity and acceleration diagrams for the
slider crank chain, the four-bar chain, and the mechanism of the Joy
valve gear will be found in ch. ix. of _Valves and Valve Gear
Mechanism_, by W. E. Dalby (London, 1906).